# Linear Algebra

In addition to (and as part of) its support for multi-dimensional arrays, Julia provides native implementations of many common and useful linear algebra operations which can be loaded with using LinearAlgebra. Basic operations, such as tr, det, and inv are all supported:

julia> A = [1 2 3; 4 1 6; 7 8 1]
3×3 Matrix{Int64}:
1  2  3
4  1  6
7  8  1

julia> tr(A)
3

julia> det(A)
104.0

julia> inv(A)
3×3 Matrix{Float64}:
-0.451923   0.211538    0.0865385
0.365385  -0.192308    0.0576923
0.240385   0.0576923  -0.0673077

As well as other useful operations, such as finding eigenvalues or eigenvectors:

julia> A = [-4. -17.; 2. 2.]
2×2 Matrix{Float64}:
-4.0  -17.0
2.0    2.0

julia> eigvals(A)
2-element Vector{ComplexF64}:
-1.0 - 5.0im
-1.0 + 5.0im

julia> eigvecs(A)
2×2 Matrix{ComplexF64}:
0.945905-0.0im        0.945905+0.0im
-0.166924+0.278207im  -0.166924-0.278207im

In addition, Julia provides many factorizations which can be used to speed up problems such as linear solve or matrix exponentiation by pre-factorizing a matrix into a form more amenable (for performance or memory reasons) to the problem. See the documentation on factorize for more information. As an example:

julia> A = [1.5 2 -4; 3 -1 -6; -10 2.3 4]
3×3 Matrix{Float64}:
1.5   2.0  -4.0
3.0  -1.0  -6.0
-10.0   2.3   4.0

julia> factorize(A)
LU{Float64, Matrix{Float64}, Vector{Int64}}
L factor:
3×3 Matrix{Float64}:
1.0    0.0       0.0
-0.15   1.0       0.0
-0.3   -0.132196  1.0
U factor:
3×3 Matrix{Float64}:
-10.0  2.3     4.0
0.0  2.345  -3.4
0.0  0.0    -5.24947

Since A is not Hermitian, symmetric, triangular, tridiagonal, or bidiagonal, an LU factorization may be the best we can do. Compare with:

julia> B = [1.5 2 -4; 2 -1 -3; -4 -3 5]
3×3 Matrix{Float64}:
1.5   2.0  -4.0
2.0  -1.0  -3.0
-4.0  -3.0   5.0

julia> factorize(B)
BunchKaufman{Float64, Matrix{Float64}, Vector{Int64}}
D factor:
3×3 Tridiagonal{Float64, Vector{Float64}}:
-1.64286   0.0   ⋅
0.0      -2.8  0.0
⋅        0.0  5.0
U factor:
3×3 UnitUpperTriangular{Float64, Matrix{Float64}}:
1.0  0.142857  -0.8
⋅   1.0       -0.6
⋅    ⋅         1.0
permutation:
3-element Vector{Int64}:
1
2
3

Here, Julia was able to detect that B is in fact symmetric, and used a more appropriate factorization. Often it's possible to write more efficient code for a matrix that is known to have certain properties e.g. it is symmetric, or tridiagonal. Julia provides some special types so that you can "tag" matrices as having these properties. For instance:

julia> B = [1.5 2 -4; 2 -1 -3; -4 -3 5]
3×3 Matrix{Float64}:
1.5   2.0  -4.0
2.0  -1.0  -3.0
-4.0  -3.0   5.0

julia> sB = Symmetric(B)
3×3 Symmetric{Float64, Matrix{Float64}}:
1.5   2.0  -4.0
2.0  -1.0  -3.0
-4.0  -3.0   5.0

sB has been tagged as a matrix that's (real) symmetric, so for later operations we might perform on it, such as eigenfactorization or computing matrix-vector products, efficiencies can be found by only referencing half of it. For example:

julia> B = [1.5 2 -4; 2 -1 -3; -4 -3 5]
3×3 Matrix{Float64}:
1.5   2.0  -4.0
2.0  -1.0  -3.0
-4.0  -3.0   5.0

julia> sB = Symmetric(B)
3×3 Symmetric{Float64, Matrix{Float64}}:
1.5   2.0  -4.0
2.0  -1.0  -3.0
-4.0  -3.0   5.0

julia> x = [1; 2; 3]
3-element Vector{Int64}:
1
2
3

julia> sB\x
3-element Vector{Float64}:
-1.7391304347826084
-1.1086956521739126
-1.4565217391304346

The \ operation here performs the linear solution. The left-division operator is pretty powerful and it's easy to write compact, readable code that is flexible enough to solve all sorts of systems of linear equations.

## Special matrices

Matrices with special symmetries and structures arise often in linear algebra and are frequently associated with various matrix factorizations. Julia features a rich collection of special matrix types, which allow for fast computation with specialized routines that are specially developed for particular matrix types.

The following tables summarize the types of special matrices that have been implemented in Julia, as well as whether hooks to various optimized methods for them in LAPACK are available.

TypeDescription
SymmetricSymmetric matrix
HermitianHermitian matrix
UpperTriangularUpper triangular matrix
UnitUpperTriangularUpper triangular matrix with unit diagonal
LowerTriangularLower triangular matrix
UnitLowerTriangularLower triangular matrix with unit diagonal
UpperHessenbergUpper Hessenberg matrix
TridiagonalTridiagonal matrix
SymTridiagonalSymmetric tridiagonal matrix
BidiagonalUpper/lower bidiagonal matrix
DiagonalDiagonal matrix
UniformScalingUniform scaling operator

### Elementary operations

Matrix type+-*\Other functions with optimized methods
SymmetricMVinv, sqrt, exp
HermitianMVinv, sqrt, exp
UpperTriangularMVMVinv, det, logdet
UnitUpperTriangularMVMVinv, det, logdet
LowerTriangularMVMVinv, det, logdet
UnitLowerTriangularMVMVinv, det, logdet
UpperHessenbergMMinv, det
SymTridiagonalMMMSMVeigmax, eigmin
TridiagonalMMMSMV
BidiagonalMMMSMV
DiagonalMMMVMVinv, det, logdet, /
UniformScalingMMMVSMVS/

Legend:

KeyDescription
M (matrix)An optimized method for matrix-matrix operations is available
V (vector)An optimized method for matrix-vector operations is available
S (scalar)An optimized method for matrix-scalar operations is available

### Matrix factorizations

Matrix typeLAPACKeigeneigvalseigvecssvdsvdvals
SymmetricSYARI
HermitianHEARI
UpperTriangularTRAAA
UnitUpperTriangularTRAAA
LowerTriangularTRAAA
UnitLowerTriangularTRAAA
SymTridiagonalSTAARIAV
TridiagonalGT
BidiagonalBDAA
DiagonalDIA

Legend:

KeyDescriptionExample
A (all)An optimized method to find all the characteristic values and/or vectors is availablee.g. eigvals(M)
R (range)An optimized method to find the ilth through the ihth characteristic values are availableeigvals(M, il, ih)
I (interval)An optimized method to find the characteristic values in the interval [vl, vh] is availableeigvals(M, vl, vh)
V (vectors)An optimized method to find the characteristic vectors corresponding to the characteristic values x=[x1, x2,...] is availableeigvecs(M, x)

### The uniform scaling operator

A UniformScaling operator represents a scalar times the identity operator, λ*I. The identity operator I is defined as a constant and is an instance of UniformScaling. The size of these operators are generic and match the other matrix in the binary operations +, -, * and \. For A+I and A-I this means that A must be square. Multiplication with the identity operator I is a noop (except for checking that the scaling factor is one) and therefore almost without overhead.

To see the UniformScaling operator in action:

julia> U = UniformScaling(2);

julia> a = [1 2; 3 4]
2×2 Matrix{Int64}:
1  2
3  4

julia> a + U
2×2 Matrix{Int64}:
3  2
3  6

julia> a * U
2×2 Matrix{Int64}:
2  4
6  8

julia> [a U]
2×4 Matrix{Int64}:
1  2  2  0
3  4  0  2

julia> b = [1 2 3; 4 5 6]
2×3 Matrix{Int64}:
1  2  3
4  5  6

julia> b - U
ERROR: DimensionMismatch: matrix is not square: dimensions are (2, 3)
Stacktrace:
[...]

If you need to solve many systems of the form (A+μI)x = b for the same A and different μ, it might be beneficial to first compute the Hessenberg factorization F of A via the hessenberg function. Given F, Julia employs an efficient algorithm for (F+μ*I) \ b (equivalent to (A+μ*I)x \ b) and related operations like determinants.

## Matrix factorizations

Matrix factorizations (a.k.a. matrix decompositions) compute the factorization of a matrix into a product of matrices, and are one of the central concepts in linear algebra.

The following table summarizes the types of matrix factorizations that have been implemented in Julia. Details of their associated methods can be found in the Standard functions section of the Linear Algebra documentation.

TypeDescription
BunchKaufmanBunch-Kaufman factorization
CholeskyCholesky factorization
CholeskyPivotedPivoted Cholesky factorization
LDLtLDL(T) factorization
LULU factorization
QRQR factorization
QRCompactWYCompact WY form of the QR factorization
QRPivotedPivoted QR factorization
LQQR factorization of transpose(A)
HessenbergHessenberg decomposition
EigenSpectral decomposition
GeneralizedEigenGeneralized spectral decomposition
SVDSingular value decomposition
GeneralizedSVDGeneralized SVD
SchurSchur decomposition
GeneralizedSchurGeneralized Schur decomposition

## Standard functions

Linear algebra functions in Julia are largely implemented by calling functions from LAPACK. Sparse matrix factorizations call functions from SuiteSparse. Other sparse solvers are available as Julia packages.

Base.:*Method
*(A::AbstractMatrix, B::AbstractMatrix)

Matrix multiplication.

Examples

julia> [1 1; 0 1] * [1 0; 1 1]
2×2 Matrix{Int64}:
2  1
1  1
Base.:\Method
\(A, B)

Matrix division using a polyalgorithm. For input matrices A and B, the result X is such that A*X == B when A is square. The solver that is used depends upon the structure of A. If A is upper or lower triangular (or diagonal), no factorization of A is required and the system is solved with either forward or backward substitution. For non-triangular square matrices, an LU factorization is used.

For rectangular A the result is the minimum-norm least squares solution computed by a pivoted QR factorization of A and a rank estimate of A based on the R factor.

When A is sparse, a similar polyalgorithm is used. For indefinite matrices, the LDLt factorization does not use pivoting during the numerical factorization and therefore the procedure can fail even for invertible matrices.

See also: factorize, pinv.

Examples

julia> A = [1 0; 1 -2]; B = [32; -4];

julia> X = A \ B
2-element Vector{Float64}:
32.0
18.0

julia> A * X == B
true
Base.:/Method
A / B

Matrix right-division: A / B is equivalent to (B' \ A')' where \ is the left-division operator. For square matrices, the result X is such that A == X*B.

See also: rdiv!.

Examples

julia> A = Float64[1 4 5; 3 9 2]; B = Float64[1 4 2; 3 4 2; 8 7 1];

julia> X = A / B
2×3 Matrix{Float64}:
-0.65   3.75  -1.2
3.25  -2.75   1.0

julia> isapprox(A, X*B)
true

julia> isapprox(X, A*pinv(B))
true
LinearAlgebra.SingularExceptionType
SingularException

Exception thrown when the input matrix has one or more zero-valued eigenvalues, and is not invertible. A linear solve involving such a matrix cannot be computed. The info field indicates the location of (one of) the singular value(s).

LinearAlgebra.PosDefExceptionType
PosDefException

Exception thrown when the input matrix was not positive definite. Some linear algebra functions and factorizations are only applicable to positive definite matrices. The info field indicates the location of (one of) the eigenvalue(s) which is (are) less than/equal to 0.

LinearAlgebra.ZeroPivotExceptionType
ZeroPivotException <: Exception

Exception thrown when a matrix factorization/solve encounters a zero in a pivot (diagonal) position and cannot proceed. This may not mean that the matrix is singular: it may be fruitful to switch to a different factorization such as pivoted LU that can re-order variables to eliminate spurious zero pivots. The info field indicates the location of (one of) the zero pivot(s).

LinearAlgebra.dotFunction
dot(x, y)
x ⋅ y

Compute the dot product between two vectors. For complex vectors, the first vector is conjugated.

dot also works on arbitrary iterable objects, including arrays of any dimension, as long as dot is defined on the elements.

dot is semantically equivalent to sum(dot(vx,vy) for (vx,vy) in zip(x, y)), with the added restriction that the arguments must have equal lengths.

x ⋅ y (where ⋅ can be typed by tab-completing \cdot in the REPL) is a synonym for dot(x, y).

Examples

julia> dot([1; 1], [2; 3])
5

julia> dot([im; im], [1; 1])
0 - 2im

julia> dot(1:5, 2:6)
70

julia> x = fill(2., (5,5));

julia> y = fill(3., (5,5));

julia> dot(x, y)
150.0
LinearAlgebra.dotMethod
dot(x, A, y)

Compute the generalized dot product dot(x, A*y) between two vectors x and y, without storing the intermediate result of A*y. As for the two-argument dot(_,_), this acts recursively. Moreover, for complex vectors, the first vector is conjugated.

Julia 1.4

Three-argument dot requires at least Julia 1.4.

Examples

julia> dot([1; 1], [1 2; 3 4], [2; 3])
26

julia> dot(1:5, reshape(1:25, 5, 5), 2:6)
4850

julia> ⋅(1:5, reshape(1:25, 5, 5), 2:6) == dot(1:5, reshape(1:25, 5, 5), 2:6)
true
LinearAlgebra.crossFunction
cross(x, y)
×(x,y)

Compute the cross product of two 3-vectors.

Examples

julia> a = [0;1;0]
3-element Vector{Int64}:
0
1
0

julia> b = [0;0;1]
3-element Vector{Int64}:
0
0
1

julia> cross(a,b)
3-element Vector{Int64}:
1
0
0
LinearAlgebra.axpy!Function
axpy!(α, x::AbstractArray, y::AbstractArray)

Overwrite y with x * α + y and return y. If x and y have the same axes, it's equivalent with y .+= x .* a.

Examples

julia> x = [1; 2; 3];

julia> y = [4; 5; 6];

julia> axpy!(2, x, y)
3-element Vector{Int64}:
6
9
12
LinearAlgebra.axpby!Function
axpby!(α, x::AbstractArray, β, y::AbstractArray)

Overwrite y with x * α + y * β and return y. If x and y have the same axes, it's equivalent with y .= x .* a .+ y .* β.

Examples

julia> x = [1; 2; 3];

julia> y = [4; 5; 6];

julia> axpby!(2, x, 2, y)
3-element Vector{Int64}:
10
14
18
LinearAlgebra.rotate!Function
rotate!(x, y, c, s)

Overwrite x with c*x + s*y and y with -conj(s)*x + c*y. Returns x and y.

Julia 1.5

rotate! requires at least Julia 1.5.

LinearAlgebra.reflect!Function
reflect!(x, y, c, s)

Overwrite x with c*x + s*y and y with conj(s)*x - c*y. Returns x and y.

Julia 1.5

reflect! requires at least Julia 1.5.

LinearAlgebra.factorizeFunction
factorize(A)

Compute a convenient factorization of A, based upon the type of the input matrix. factorize checks A to see if it is symmetric/triangular/etc. if A is passed as a generic matrix. factorize checks every element of A to verify/rule out each property. It will short-circuit as soon as it can rule out symmetry/triangular structure. The return value can be reused for efficient solving of multiple systems. For example: A=factorize(A); x=A\b; y=A\C.

Properties of Atype of factorization
Positive-definiteCholesky (see cholesky)
Dense Symmetric/HermitianBunch-Kaufman (see bunchkaufman)
Sparse Symmetric/HermitianLDLt (see ldlt)
TriangularTriangular
DiagonalDiagonal
BidiagonalBidiagonal
TridiagonalLU (see lu)
Symmetric real tridiagonalLDLt (see ldlt)
General squareLU (see lu)
General non-squareQR (see qr)

If factorize is called on a Hermitian positive-definite matrix, for instance, then factorize will return a Cholesky factorization.

Examples

julia> A = Array(Bidiagonal(fill(1.0, (5, 5)), :U))
5×5 Matrix{Float64}:
1.0  1.0  0.0  0.0  0.0
0.0  1.0  1.0  0.0  0.0
0.0  0.0  1.0  1.0  0.0
0.0  0.0  0.0  1.0  1.0
0.0  0.0  0.0  0.0  1.0

julia> factorize(A) # factorize will check to see that A is already factorized
5×5 Bidiagonal{Float64, Vector{Float64}}:
1.0  1.0   ⋅    ⋅    ⋅
⋅   1.0  1.0   ⋅    ⋅
⋅    ⋅   1.0  1.0   ⋅
⋅    ⋅    ⋅   1.0  1.0
⋅    ⋅    ⋅    ⋅   1.0

This returns a 5×5 Bidiagonal{Float64}, which can now be passed to other linear algebra functions (e.g. eigensolvers) which will use specialized methods for Bidiagonal types.

LinearAlgebra.DiagonalType
Diagonal(V::AbstractVector)

Construct a matrix with V as its diagonal.

See also diag, diagm.

Examples

julia> Diagonal([1, 10, 100])
3×3 Diagonal{Int64, Vector{Int64}}:
1   ⋅    ⋅
⋅  10    ⋅
⋅   ⋅  100

julia> diagm([7, 13])
2×2 Matrix{Int64}:
7   0
0  13
Diagonal(A::AbstractMatrix)

Construct a matrix from the diagonal of A.

Examples

julia> A = permutedims(reshape(1:15, 5, 3))
3×5 Matrix{Int64}:
1   2   3   4   5
6   7   8   9  10
11  12  13  14  15

julia> Diagonal(A)
3×3 Diagonal{Int64, Vector{Int64}}:
1  ⋅   ⋅
⋅  7   ⋅
⋅  ⋅  13

julia> diag(A, 2)
3-element Vector{Int64}:
3
9
15
Diagonal{T}(undef, n)

Construct an uninitialized Diagonal{T} of length n. See undef.

LinearAlgebra.BidiagonalType
Bidiagonal(dv::V, ev::V, uplo::Symbol) where V <: AbstractVector

Constructs an upper (uplo=:U) or lower (uplo=:L) bidiagonal matrix using the given diagonal (dv) and off-diagonal (ev) vectors. The result is of type Bidiagonal and provides efficient specialized linear solvers, but may be converted into a regular matrix with convert(Array, _) (or Array(_) for short). The length of ev must be one less than the length of dv.

Examples

julia> dv = [1, 2, 3, 4]
4-element Vector{Int64}:
1
2
3
4

julia> ev = [7, 8, 9]
3-element Vector{Int64}:
7
8
9

julia> Bu = Bidiagonal(dv, ev, :U) # ev is on the first superdiagonal
4×4 Bidiagonal{Int64, Vector{Int64}}:
1  7  ⋅  ⋅
⋅  2  8  ⋅
⋅  ⋅  3  9
⋅  ⋅  ⋅  4

julia> Bl = Bidiagonal(dv, ev, :L) # ev is on the first subdiagonal
4×4 Bidiagonal{Int64, Vector{Int64}}:
1  ⋅  ⋅  ⋅
7  2  ⋅  ⋅
⋅  8  3  ⋅
⋅  ⋅  9  4
Bidiagonal(A, uplo::Symbol)

Construct a Bidiagonal matrix from the main diagonal of A and its first super- (if uplo=:U) or sub-diagonal (if uplo=:L).

Examples

julia> A = [1 1 1 1; 2 2 2 2; 3 3 3 3; 4 4 4 4]
4×4 Matrix{Int64}:
1  1  1  1
2  2  2  2
3  3  3  3
4  4  4  4

julia> Bidiagonal(A, :U) # contains the main diagonal and first superdiagonal of A
4×4 Bidiagonal{Int64, Vector{Int64}}:
1  1  ⋅  ⋅
⋅  2  2  ⋅
⋅  ⋅  3  3
⋅  ⋅  ⋅  4

julia> Bidiagonal(A, :L) # contains the main diagonal and first subdiagonal of A
4×4 Bidiagonal{Int64, Vector{Int64}}:
1  ⋅  ⋅  ⋅
2  2  ⋅  ⋅
⋅  3  3  ⋅
⋅  ⋅  4  4
LinearAlgebra.SymTridiagonalType
SymTridiagonal(dv::V, ev::V) where V <: AbstractVector

Construct a symmetric tridiagonal matrix from the diagonal (dv) and first sub/super-diagonal (ev), respectively. The result is of type SymTridiagonal and provides efficient specialized eigensolvers, but may be converted into a regular matrix with convert(Array, _) (or Array(_) for short).

For SymTridiagonal block matrices, the elements of dv are symmetrized. The argument ev is interpreted as the superdiagonal. Blocks from the subdiagonal are (materialized) transpose of the corresponding superdiagonal blocks.

Examples

julia> dv = [1, 2, 3, 4]
4-element Vector{Int64}:
1
2
3
4

julia> ev = [7, 8, 9]
3-element Vector{Int64}:
7
8
9

julia> SymTridiagonal(dv, ev)
4×4 SymTridiagonal{Int64, Vector{Int64}}:
1  7  ⋅  ⋅
7  2  8  ⋅
⋅  8  3  9
⋅  ⋅  9  4

julia> A = SymTridiagonal(fill([1 2; 3 4], 3), fill([1 2; 3 4], 2));

julia> A[1,1]
2×2 Symmetric{Int64, Matrix{Int64}}:
1  2
2  4

julia> A[1,2]
2×2 Matrix{Int64}:
1  2
3  4

julia> A[2,1]
2×2 Matrix{Int64}:
1  3
2  4
SymTridiagonal(A::AbstractMatrix)

Construct a symmetric tridiagonal matrix from the diagonal and first superdiagonal of the symmetric matrix A.

Examples

julia> A = [1 2 3; 2 4 5; 3 5 6]
3×3 Matrix{Int64}:
1  2  3
2  4  5
3  5  6

julia> SymTridiagonal(A)
3×3 SymTridiagonal{Int64, Vector{Int64}}:
1  2  ⋅
2  4  5
⋅  5  6

julia> B = reshape([[1 2; 2 3], [1 2; 3 4], [1 3; 2 4], [1 2; 2 3]], 2, 2);

julia> SymTridiagonal(B)
2×2 SymTridiagonal{Matrix{Int64}, Vector{Matrix{Int64}}}:
[1 2; 2 3]  [1 3; 2 4]
[1 2; 3 4]  [1 2; 2 3]
LinearAlgebra.TridiagonalType
Tridiagonal(dl::V, d::V, du::V) where V <: AbstractVector

Construct a tridiagonal matrix from the first subdiagonal, diagonal, and first superdiagonal, respectively. The result is of type Tridiagonal and provides efficient specialized linear solvers, but may be converted into a regular matrix with convert(Array, _) (or Array(_) for short). The lengths of dl and du must be one less than the length of d.

Examples

julia> dl = [1, 2, 3];

julia> du = [4, 5, 6];

julia> d = [7, 8, 9, 0];

julia> Tridiagonal(dl, d, du)
4×4 Tridiagonal{Int64, Vector{Int64}}:
7  4  ⋅  ⋅
1  8  5  ⋅
⋅  2  9  6
⋅  ⋅  3  0
Tridiagonal(A)

Construct a tridiagonal matrix from the first sub-diagonal, diagonal and first super-diagonal of the matrix A.

Examples

julia> A = [1 2 3 4; 1 2 3 4; 1 2 3 4; 1 2 3 4]
4×4 Matrix{Int64}:
1  2  3  4
1  2  3  4
1  2  3  4
1  2  3  4

julia> Tridiagonal(A)
4×4 Tridiagonal{Int64, Vector{Int64}}:
1  2  ⋅  ⋅
1  2  3  ⋅
⋅  2  3  4
⋅  ⋅  3  4
LinearAlgebra.SymmetricType
Symmetric(A, uplo=:U)

Construct a Symmetric view of the upper (if uplo = :U) or lower (if uplo = :L) triangle of the matrix A.

Examples

julia> A = [1 0 2 0 3; 0 4 0 5 0; 6 0 7 0 8; 0 9 0 1 0; 2 0 3 0 4]
5×5 Matrix{Int64}:
1  0  2  0  3
0  4  0  5  0
6  0  7  0  8
0  9  0  1  0
2  0  3  0  4

julia> Supper = Symmetric(A)
5×5 Symmetric{Int64, Matrix{Int64}}:
1  0  2  0  3
0  4  0  5  0
2  0  7  0  8
0  5  0  1  0
3  0  8  0  4

julia> Slower = Symmetric(A, :L)
5×5 Symmetric{Int64, Matrix{Int64}}:
1  0  6  0  2
0  4  0  9  0
6  0  7  0  3
0  9  0  1  0
2  0  3  0  4

Note that Supper will not be equal to Slower unless A is itself symmetric (e.g. if A == transpose(A)).

LinearAlgebra.HermitianType
Hermitian(A, uplo=:U)

Construct a Hermitian view of the upper (if uplo = :U) or lower (if uplo = :L) triangle of the matrix A.

Examples

julia> A = [1 0 2+2im 0 3-3im; 0 4 0 5 0; 6-6im 0 7 0 8+8im; 0 9 0 1 0; 2+2im 0 3-3im 0 4];

julia> Hupper = Hermitian(A)
5×5 Hermitian{Complex{Int64}, Matrix{Complex{Int64}}}:
1+0im  0+0im  2+2im  0+0im  3-3im
0+0im  4+0im  0+0im  5+0im  0+0im
2-2im  0+0im  7+0im  0+0im  8+8im
0+0im  5+0im  0+0im  1+0im  0+0im
3+3im  0+0im  8-8im  0+0im  4+0im

julia> Hlower = Hermitian(A, :L)
5×5 Hermitian{Complex{Int64}, Matrix{Complex{Int64}}}:
1+0im  0+0im  6+6im  0+0im  2-2im
0+0im  4+0im  0+0im  9+0im  0+0im
6-6im  0+0im  7+0im  0+0im  3+3im
0+0im  9+0im  0+0im  1+0im  0+0im
2+2im  0+0im  3-3im  0+0im  4+0im

Note that Hupper will not be equal to Hlower unless A is itself Hermitian (e.g. if A == adjoint(A)).

All non-real parts of the diagonal will be ignored.

Hermitian(fill(complex(1,1), 1, 1)) == fill(1, 1, 1)
LinearAlgebra.LowerTriangularType
LowerTriangular(A::AbstractMatrix)

Construct a LowerTriangular view of the matrix A.

Examples

julia> A = [1.0 2.0 3.0; 4.0 5.0 6.0; 7.0 8.0 9.0]
3×3 Matrix{Float64}:
1.0  2.0  3.0
4.0  5.0  6.0
7.0  8.0  9.0

julia> LowerTriangular(A)
3×3 LowerTriangular{Float64, Matrix{Float64}}:
1.0   ⋅    ⋅
4.0  5.0   ⋅
7.0  8.0  9.0
LinearAlgebra.UpperTriangularType
UpperTriangular(A::AbstractMatrix)

Construct an UpperTriangular view of the matrix A.

Examples

julia> A = [1.0 2.0 3.0; 4.0 5.0 6.0; 7.0 8.0 9.0]
3×3 Matrix{Float64}:
1.0  2.0  3.0
4.0  5.0  6.0
7.0  8.0  9.0

julia> UpperTriangular(A)
3×3 UpperTriangular{Float64, Matrix{Float64}}:
1.0  2.0  3.0
⋅   5.0  6.0
⋅    ⋅   9.0
LinearAlgebra.UnitLowerTriangularType
UnitLowerTriangular(A::AbstractMatrix)

Construct a UnitLowerTriangular view of the matrix A. Such a view has the oneunit of the eltype of A on its diagonal.

Examples

julia> A = [1.0 2.0 3.0; 4.0 5.0 6.0; 7.0 8.0 9.0]
3×3 Matrix{Float64}:
1.0  2.0  3.0
4.0  5.0  6.0
7.0  8.0  9.0

julia> UnitLowerTriangular(A)
3×3 UnitLowerTriangular{Float64, Matrix{Float64}}:
1.0   ⋅    ⋅
4.0  1.0   ⋅
7.0  8.0  1.0
LinearAlgebra.UnitUpperTriangularType
UnitUpperTriangular(A::AbstractMatrix)

Construct an UnitUpperTriangular view of the matrix A. Such a view has the oneunit of the eltype of A on its diagonal.

Examples

julia> A = [1.0 2.0 3.0; 4.0 5.0 6.0; 7.0 8.0 9.0]
3×3 Matrix{Float64}:
1.0  2.0  3.0
4.0  5.0  6.0
7.0  8.0  9.0

julia> UnitUpperTriangular(A)
3×3 UnitUpperTriangular{Float64, Matrix{Float64}}:
1.0  2.0  3.0
⋅   1.0  6.0
⋅    ⋅   1.0
LinearAlgebra.UpperHessenbergType
UpperHessenberg(A::AbstractMatrix)

Construct an UpperHessenberg view of the matrix A. Entries of A below the first subdiagonal are ignored.

Julia 1.3

This type was added in Julia 1.3.

Efficient algorithms are implemented for H \ b, det(H), and similar.

See also the hessenberg function to factor any matrix into a similar upper-Hessenberg matrix.

If F::Hessenberg is the factorization object, the unitary matrix can be accessed with F.Q and the Hessenberg matrix with F.H. When Q is extracted, the resulting type is the HessenbergQ object, and may be converted to a regular matrix with convert(Array, _) (or Array(_) for short).

Iterating the decomposition produces the factors F.Q and F.H.

Examples

julia> A = [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16]
4×4 Matrix{Int64}:
1   2   3   4
5   6   7   8
9  10  11  12
13  14  15  16

julia> UpperHessenberg(A)
4×4 UpperHessenberg{Int64, Matrix{Int64}}:
1   2   3   4
5   6   7   8
⋅  10  11  12
⋅   ⋅  15  16
LinearAlgebra.UniformScalingType
UniformScaling{T<:Number}

Generically sized uniform scaling operator defined as a scalar times the identity operator, λ*I. Although without an explicit size, it acts similarly to a matrix in many cases and includes support for some indexing. See also I.

Julia 1.6

Indexing using ranges is available as of Julia 1.6.

Examples

julia> J = UniformScaling(2.)
UniformScaling{Float64}
2.0*I

julia> A = [1. 2.; 3. 4.]
2×2 Matrix{Float64}:
1.0  2.0
3.0  4.0

julia> J*A
2×2 Matrix{Float64}:
2.0  4.0
6.0  8.0

julia> J[1:2, 1:2]
2×2 Matrix{Float64}:
2.0  0.0
0.0  2.0
LinearAlgebra.IConstant
I

An object of type UniformScaling, representing an identity matrix of any size.

Examples

julia> fill(1, (5,6)) * I == fill(1, (5,6))
true

julia> [1 2im 3; 1im 2 3] * I
2×3 Matrix{Complex{Int64}}:
1+0im  0+2im  3+0im
0+1im  2+0im  3+0im
LinearAlgebra.UniformScalingMethod
(I::UniformScaling)(n::Integer)

Construct a Diagonal matrix from a UniformScaling.

Julia 1.2

This method is available as of Julia 1.2.

Examples

julia> I(3)
3×3 Diagonal{Bool, Vector{Bool}}:
1  ⋅  ⋅
⋅  1  ⋅
⋅  ⋅  1

julia> (0.7*I)(3)
3×3 Diagonal{Float64, Vector{Float64}}:
0.7   ⋅    ⋅
⋅   0.7   ⋅
⋅    ⋅   0.7
LinearAlgebra.LUType
LU <: Factorization

Matrix factorization type of the LU factorization of a square matrix A. This is the return type of lu, the corresponding matrix factorization function.

The individual components of the factorization F::LU can be accessed via getproperty:

ComponentDescription
F.LL (unit lower triangular) part of LU
F.UU (upper triangular) part of LU
F.p(right) permutation Vector
F.P(right) permutation Matrix

Iterating the factorization produces the components F.L, F.U, and F.p.

Examples

julia> A = [4 3; 6 3]
2×2 Matrix{Int64}:
4  3
6  3

julia> F = lu(A)
LU{Float64, Matrix{Float64}, Vector{Int64}}
L factor:
2×2 Matrix{Float64}:
1.0       0.0
0.666667  1.0
U factor:
2×2 Matrix{Float64}:
6.0  3.0
0.0  1.0

julia> F.L * F.U == A[F.p, :]
true

julia> l, u, p = lu(A); # destructuring via iteration

julia> l == F.L && u == F.U && p == F.p
true
LinearAlgebra.luFunction
lu(A::AbstractSparseMatrixCSC; check = true, q = nothing, control = get_umfpack_control()) -> F::UmfpackLU

Compute the LU factorization of a sparse matrix A.

For sparse A with real or complex element type, the return type of F is UmfpackLU{Tv, Ti}, with Tv = Float64 or ComplexF64 respectively and Ti is an integer type (Int32 or Int64).

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

The permutation q can either be a permutation vector or nothing. If no permutation vector is proveded or q is nothing, UMFPACK's default is used. If the permutation is not zero based, a zero based copy is made.

The control vector default to the package's default configs for umfpacks but can be changed passing a vector of length UMFPACK_CONTROL. See the UMFPACK manual for possible configurations. The corresponding variables are named JL_UMFPACK_ since julia uses one based indexing.

The individual components of the factorization F can be accessed by indexing:

ComponentDescription
LL (lower triangular) part of LU
UU (upper triangular) part of LU
pright permutation Vector
qleft permutation Vector
RsVector of scaling factors
:(L,U,p,q,Rs) components

The relation between F and A is

F.L*F.U == (F.Rs .* A)[F.p, F.q]

F further supports the following functions:

See also lu!

Note

lu(A::AbstractSparseMatrixCSC) uses the UMFPACK[ACM832] library that is part of SuiteSparse. As this library only supports sparse matrices with Float64 or ComplexF64 elements, lu converts A into a copy that is of type SparseMatrixCSC{Float64} or SparseMatrixCSC{ComplexF64} as appropriate.

lu(A, pivot = RowMaximum(); check = true) -> F::LU

Compute the LU factorization of A.

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

In most cases, if A is a subtype S of AbstractMatrix{T} with an element type T supporting +, -, * and /, the return type is LU{T,S{T}}.

In general, LU factorization involves a permutation of the rows of the matrix (corresponding to the F.p output described below), known as "pivoting" (because it corresponds to choosing which row contains the "pivot", the diagonal entry of F.U). One of the following pivoting strategies can be selected via the optional pivot argument:

• RowMaximum() (default): the standard pivoting strategy; the pivot corresponds to the element of maximum absolute value among the remaining, to be factorized rows. This pivoting strategy requires the element type to also support abs and <. (This is generally the only numerically stable option for floating-point matrices.)
• RowNonZero(): the pivot corresponds to the first non-zero element among the remaining, to be factorized rows. (This corresponds to the typical choice in hand calculations, and is also useful for more general algebraic number types that support iszero but not abs or <.)
• NoPivot(): pivoting turned off (may fail if a zero entry is encountered).

The individual components of the factorization F can be accessed via getproperty:

ComponentDescription
F.LL (lower triangular) part of LU
F.UU (upper triangular) part of LU
F.p(right) permutation Vector
F.P(right) permutation Matrix

Iterating the factorization produces the components F.L, F.U, and F.p.

The relationship between F and A is

F.L*F.U == A[F.p, :]

F further supports the following functions:

Supported functionLULU{T,Tridiagonal{T}}
/
\
inv
det
logdet
logabsdet
size

Examples

julia> A = [4 3; 6 3]
2×2 Matrix{Int64}:
4  3
6  3

julia> F = lu(A)
LU{Float64, Matrix{Float64}, Vector{Int64}}
L factor:
2×2 Matrix{Float64}:
1.0       0.0
0.666667  1.0
U factor:
2×2 Matrix{Float64}:
6.0  3.0
0.0  1.0

julia> F.L * F.U == A[F.p, :]
true

julia> l, u, p = lu(A); # destructuring via iteration

julia> l == F.L && u == F.U && p == F.p
true
LinearAlgebra.lu!Function
lu!(F::UmfpackLU, A::AbstractSparseMatrixCSC; check=true, reuse_symbolic=true, q=nothing) -> F::UmfpackLU

Compute the LU factorization of a sparse matrix A, reusing the symbolic factorization of an already existing LU factorization stored in F. Unless reuse_symbolic is set to false, the sparse matrix A must have an identical nonzero pattern as the matrix used to create the LU factorization F, otherwise an error is thrown. If the size of A and F differ, all vectors will be resized accordingly.

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

The permutation q can either be a permutation vector or nothing. If no permutation vector is proveded or q is nothing, UMFPACK's default is used. If the permutation is not zero based, a zero based copy is made.

See also lu

Note

lu!(F::UmfpackLU, A::AbstractSparseMatrixCSC) uses the UMFPACK library that is part of SuiteSparse. As this library only supports sparse matrices with Float64 or ComplexF64 elements, lu! will automatically convert the types to those set by the LU factorization or SparseMatrixCSC{ComplexF64} as appropriate.

Julia 1.5

lu! for UmfpackLU requires at least Julia 1.5.

Examples

julia> A = sparse(Float64[1.0 2.0; 0.0 3.0]);

julia> F = lu(A);

julia> B = sparse(Float64[1.0 1.0; 0.0 1.0]);

julia> lu!(F, B);

julia> F \ ones(2)
2-element Vector{Float64}:
0.0
1.0
lu!(A, pivot = RowMaximum(); check = true) -> LU

lu! is the same as lu, but saves space by overwriting the input A, instead of creating a copy. An InexactError exception is thrown if the factorization produces a number not representable by the element type of A, e.g. for integer types.

Examples

julia> A = [4. 3.; 6. 3.]
2×2 Matrix{Float64}:
4.0  3.0
6.0  3.0

julia> F = lu!(A)
LU{Float64, Matrix{Float64}, Vector{Int64}}
L factor:
2×2 Matrix{Float64}:
1.0       0.0
0.666667  1.0
U factor:
2×2 Matrix{Float64}:
6.0  3.0
0.0  1.0

julia> iA = [4 3; 6 3]
2×2 Matrix{Int64}:
4  3
6  3

julia> lu!(iA)
ERROR: InexactError: Int64(0.6666666666666666)
Stacktrace:
[...]
LinearAlgebra.CholeskyType
Cholesky <: Factorization

Matrix factorization type of the Cholesky factorization of a dense symmetric/Hermitian positive definite matrix A. This is the return type of cholesky, the corresponding matrix factorization function.

The triangular Cholesky factor can be obtained from the factorization F::Cholesky via F.L and F.U, where A ≈ F.U' * F.U ≈ F.L * F.L'.

The following functions are available for Cholesky objects: size, \, inv, det, logdet and isposdef.

Iterating the decomposition produces the components L and U.

Examples

julia> A = [4. 12. -16.; 12. 37. -43.; -16. -43. 98.]
3×3 Matrix{Float64}:
4.0   12.0  -16.0
12.0   37.0  -43.0
-16.0  -43.0   98.0

julia> C = cholesky(A)
Cholesky{Float64, Matrix{Float64}}
U factor:
3×3 UpperTriangular{Float64, Matrix{Float64}}:
2.0  6.0  -8.0
⋅   1.0   5.0
⋅    ⋅    3.0

julia> C.U
3×3 UpperTriangular{Float64, Matrix{Float64}}:
2.0  6.0  -8.0
⋅   1.0   5.0
⋅    ⋅    3.0

julia> C.L
3×3 LowerTriangular{Float64, Matrix{Float64}}:
2.0   ⋅    ⋅
6.0  1.0   ⋅
-8.0  5.0  3.0

julia> C.L * C.U == A
true

julia> l, u = C; # destructuring via iteration

julia> l == C.L && u == C.U
true
LinearAlgebra.CholeskyPivotedType
CholeskyPivoted

Matrix factorization type of the pivoted Cholesky factorization of a dense symmetric/Hermitian positive semi-definite matrix A. This is the return type of cholesky(_, ::RowMaximum), the corresponding matrix factorization function.

The triangular Cholesky factor can be obtained from the factorization F::CholeskyPivoted via F.L and F.U, and the permutation via F.p, where A[F.p, F.p] ≈ Ur' * Ur ≈ Lr * Lr' with Ur = F.U[1:F.rank, :] and Lr = F.L[:, 1:F.rank], or alternatively A ≈ Up' * Up ≈ Lp * Lp' with Up = F.U[1:F.rank, invperm(F.p)] and Lp = F.L[invperm(F.p), 1:F.rank].

The following functions are available for CholeskyPivoted objects: size, \, inv, det, and rank.

Iterating the decomposition produces the components L and U.

Examples

julia> X = [1.0, 2.0, 3.0, 4.0];

julia> A = X * X';

julia> C = cholesky(A, RowMaximum(), check = false)
CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}
U factor with rank 1:
4×4 UpperTriangular{Float64, Matrix{Float64}}:
4.0  2.0  3.0  1.0
⋅   0.0  6.0  2.0
⋅    ⋅   9.0  3.0
⋅    ⋅    ⋅   1.0
permutation:
4-element Vector{Int64}:
4
2
3
1

julia> C.U[1:C.rank, :]' * C.U[1:C.rank, :] ≈ A[C.p, C.p]
true

julia> l, u = C; # destructuring via iteration

julia> l == C.L && u == C.U
true
LinearAlgebra.choleskyFunction
cholesky(A::SparseMatrixCSC; shift = 0.0, check = true, perm = nothing) -> CHOLMOD.Factor

Compute the Cholesky factorization of a sparse positive definite matrix A. A must be a SparseMatrixCSC or a Symmetric/Hermitian view of a SparseMatrixCSC. Note that even if A doesn't have the type tag, it must still be symmetric or Hermitian. If perm is not given, a fill-reducing permutation is used. F = cholesky(A) is most frequently used to solve systems of equations with F\b, but also the methods diag, det, and logdet are defined for F. You can also extract individual factors from F, using F.L. However, since pivoting is on by default, the factorization is internally represented as A == P'*L*L'*P with a permutation matrix P; using just L without accounting for P will give incorrect answers. To include the effects of permutation, it's typically preferable to extract "combined" factors like PtL = F.PtL (the equivalent of P'*L) and LtP = F.UP (the equivalent of L'*P).

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

Setting the optional shift keyword argument computes the factorization of A+shift*I instead of A. If the perm argument is provided, it should be a permutation of 1:size(A,1) giving the ordering to use (instead of CHOLMOD's default AMD ordering).

Examples

In the following example, the fill-reducing permutation used is [3, 2, 1]. If perm is set to 1:3 to enforce no permutation, the number of nonzero elements in the factor is 6.

julia> A = [2 1 1; 1 2 0; 1 0 2]
3×3 Matrix{Int64}:
2  1  1
1  2  0
1  0  2

julia> C = cholesky(sparse(A))
SparseArrays.CHOLMOD.Factor{Float64}
type:    LLt
method:  simplicial
maxnnz:  5
nnz:     5
success: true

julia> C.p
3-element Vector{Int64}:
3
2
1

julia> L = sparse(C.L);

julia> Matrix(L)
3×3 Matrix{Float64}:
1.41421   0.0       0.0
0.0       1.41421   0.0
0.707107  0.707107  1.0

julia> L * L' ≈ A[C.p, C.p]
true

julia> P = sparse(1:3, C.p, ones(3))
3×3 SparseMatrixCSC{Float64, Int64} with 3 stored entries:
⋅    ⋅   1.0
⋅   1.0   ⋅
1.0   ⋅    ⋅

julia> P' * L * L' * P ≈ A
true

julia> C = cholesky(sparse(A), perm=1:3)
SparseArrays.CHOLMOD.Factor{Float64}
type:    LLt
method:  simplicial
maxnnz:  6
nnz:     6
success: true

julia> L = sparse(C.L);

julia> Matrix(L)
3×3 Matrix{Float64}:
1.41421    0.0       0.0
0.707107   1.22474   0.0
0.707107  -0.408248  1.1547

julia> L * L' ≈ A
true
Note

This method uses the CHOLMOD[ACM887][DavisHager2009] library from SuiteSparse. CHOLMOD only supports double or complex double element types. Input matrices not of those element types will be converted to SparseMatrixCSC{Float64} or SparseMatrixCSC{ComplexF64} as appropriate.

Many other functions from CHOLMOD are wrapped but not exported from the Base.SparseArrays.CHOLMOD module.

cholesky(A, NoPivot(); check = true) -> Cholesky

Compute the Cholesky factorization of a dense symmetric positive definite matrix A and return a Cholesky factorization. The matrix A can either be a Symmetric or Hermitian AbstractMatrix or a perfectly symmetric or Hermitian AbstractMatrix.

The triangular Cholesky factor can be obtained from the factorization F via F.L and F.U, where A ≈ F.U' * F.U ≈ F.L * F.L'.

The following functions are available for Cholesky objects: size, \, inv, det, logdet and isposdef.

If you have a matrix A that is slightly non-Hermitian due to roundoff errors in its construction, wrap it in Hermitian(A) before passing it to cholesky in order to treat it as perfectly Hermitian.

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

Examples

julia> A = [4. 12. -16.; 12. 37. -43.; -16. -43. 98.]
3×3 Matrix{Float64}:
4.0   12.0  -16.0
12.0   37.0  -43.0
-16.0  -43.0   98.0

julia> C = cholesky(A)
Cholesky{Float64, Matrix{Float64}}
U factor:
3×3 UpperTriangular{Float64, Matrix{Float64}}:
2.0  6.0  -8.0
⋅   1.0   5.0
⋅    ⋅    3.0

julia> C.U
3×3 UpperTriangular{Float64, Matrix{Float64}}:
2.0  6.0  -8.0
⋅   1.0   5.0
⋅    ⋅    3.0

julia> C.L
3×3 LowerTriangular{Float64, Matrix{Float64}}:
2.0   ⋅    ⋅
6.0  1.0   ⋅
-8.0  5.0  3.0

julia> C.L * C.U == A
true
cholesky(A, RowMaximum(); tol = 0.0, check = true) -> CholeskyPivoted

Compute the pivoted Cholesky factorization of a dense symmetric positive semi-definite matrix A and return a CholeskyPivoted factorization. The matrix A can either be a Symmetric or Hermitian AbstractMatrix or a perfectly symmetric or Hermitian AbstractMatrix.

The triangular Cholesky factor can be obtained from the factorization F via F.L and F.U, and the permutation via F.p, where A[F.p, F.p] ≈ Ur' * Ur ≈ Lr * Lr' with Ur = F.U[1:F.rank, :] and Lr = F.L[:, 1:F.rank], or alternatively A ≈ Up' * Up ≈ Lp * Lp' with Up = F.U[1:F.rank, invperm(F.p)] and Lp = F.L[invperm(F.p), 1:F.rank].

The following functions are available for CholeskyPivoted objects: size, \, inv, det, and rank.

The argument tol determines the tolerance for determining the rank. For negative values, the tolerance is the machine precision.

If you have a matrix A that is slightly non-Hermitian due to roundoff errors in its construction, wrap it in Hermitian(A) before passing it to cholesky in order to treat it as perfectly Hermitian.

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

Examples

julia> X = [1.0, 2.0, 3.0, 4.0];

julia> A = X * X';

julia> C = cholesky(A, RowMaximum(), check = false)
CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}
U factor with rank 1:
4×4 UpperTriangular{Float64, Matrix{Float64}}:
4.0  2.0  3.0  1.0
⋅   0.0  6.0  2.0
⋅    ⋅   9.0  3.0
⋅    ⋅    ⋅   1.0
permutation:
4-element Vector{Int64}:
4
2
3
1

julia> C.U[1:C.rank, :]' * C.U[1:C.rank, :] ≈ A[C.p, C.p]
true

julia> l, u = C; # destructuring via iteration

julia> l == C.L && u == C.U
true
LinearAlgebra.cholesky!Function
cholesky!(F::CHOLMOD.Factor, A::SparseMatrixCSC; shift = 0.0, check = true) -> CHOLMOD.Factor

Compute the Cholesky ($LL'$) factorization of A, reusing the symbolic factorization F. A must be a SparseMatrixCSC or a Symmetric/ Hermitian view of a SparseMatrixCSC. Note that even if A doesn't have the type tag, it must still be symmetric or Hermitian.

See also cholesky.

Note

This method uses the CHOLMOD library from SuiteSparse, which only supports doubles or complex doubles. Input matrices not of those element types will be converted to SparseMatrixCSC{Float64} or SparseMatrixCSC{ComplexF64} as appropriate.

cholesky!(A::AbstractMatrix, NoPivot(); check = true) -> Cholesky

The same as cholesky, but saves space by overwriting the input A, instead of creating a copy. An InexactError exception is thrown if the factorization produces a number not representable by the element type of A, e.g. for integer types.

Examples

julia> A = [1 2; 2 50]
2×2 Matrix{Int64}:
1   2
2  50

julia> cholesky!(A)
ERROR: InexactError: Int64(6.782329983125268)
Stacktrace:
[...]
cholesky!(A::AbstractMatrix, RowMaximum(); tol = 0.0, check = true) -> CholeskyPivoted

The same as cholesky, but saves space by overwriting the input A, instead of creating a copy. An InexactError exception is thrown if the factorization produces a number not representable by the element type of A, e.g. for integer types.

LinearAlgebra.lowrankupdateFunction
lowrankupdate(C::Cholesky, v::AbstractVector) -> CC::Cholesky

Update a Cholesky factorization C with the vector v. If A = C.U'C.U then CC = cholesky(C.U'C.U + v*v') but the computation of CC only uses O(n^2) operations.

LinearAlgebra.lowrankdowndateFunction
lowrankdowndate(C::Cholesky, v::AbstractVector) -> CC::Cholesky

Downdate a Cholesky factorization C with the vector v. If A = C.U'C.U then CC = cholesky(C.U'C.U - v*v') but the computation of CC only uses O(n^2) operations.

LinearAlgebra.lowrankupdate!Function
lowrankupdate!(C::Cholesky, v::AbstractVector) -> CC::Cholesky

Update a Cholesky factorization C with the vector v. If A = C.U'C.U then CC = cholesky(C.U'C.U + v*v') but the computation of CC only uses O(n^2) operations. The input factorization C is updated in place such that on exit C == CC. The vector v is destroyed during the computation.

LinearAlgebra.lowrankdowndate!Function
lowrankdowndate!(C::Cholesky, v::AbstractVector) -> CC::Cholesky

Downdate a Cholesky factorization C with the vector v. If A = C.U'C.U then CC = cholesky(C.U'C.U - v*v') but the computation of CC only uses O(n^2) operations. The input factorization C is updated in place such that on exit C == CC. The vector v is destroyed during the computation.

LinearAlgebra.LDLtType
LDLt <: Factorization

Matrix factorization type of the LDLt factorization of a real SymTridiagonal matrix S such that S = L*Diagonal(d)*L', where L is a UnitLowerTriangular matrix and d is a vector. The main use of an LDLt factorization F = ldlt(S) is to solve the linear system of equations Sx = b with F\b. This is the return type of ldlt, the corresponding matrix factorization function.

The individual components of the factorization F::LDLt can be accessed via getproperty:

ComponentDescription
F.LL (unit lower triangular) part of LDLt
F.DD (diagonal) part of LDLt
F.LtLt (unit upper triangular) part of LDLt
F.ddiagonal values of D as a Vector

Examples

julia> S = SymTridiagonal([3., 4., 5.], [1., 2.])
3×3 SymTridiagonal{Float64, Vector{Float64}}:
3.0  1.0   ⋅
1.0  4.0  2.0
⋅   2.0  5.0

julia> F = ldlt(S)
LDLt{Float64, SymTridiagonal{Float64, Vector{Float64}}}
L factor:
3×3 UnitLowerTriangular{Float64, SymTridiagonal{Float64, Vector{Float64}}}:
1.0        ⋅         ⋅
0.333333  1.0        ⋅
0.0       0.545455  1.0
D factor:
3×3 Diagonal{Float64, Vector{Float64}}:
3.0   ⋅        ⋅
⋅   3.66667   ⋅
⋅    ⋅       3.90909
LinearAlgebra.ldltFunction
ldlt(A::SparseMatrixCSC; shift = 0.0, check = true, perm=nothing) -> CHOLMOD.Factor

Compute the $LDL'$ factorization of a sparse matrix A. A must be a SparseMatrixCSC or a Symmetric/Hermitian view of a SparseMatrixCSC. Note that even if A doesn't have the type tag, it must still be symmetric or Hermitian. A fill-reducing permutation is used. F = ldlt(A) is most frequently used to solve systems of equations A*x = b with F\b. The returned factorization object F also supports the methods diag, det, logdet, and inv. You can extract individual factors from F using F.L. However, since pivoting is on by default, the factorization is internally represented as A == P'*L*D*L'*P with a permutation matrix P; using just L without accounting for P will give incorrect answers. To include the effects of permutation, it is typically preferable to extract "combined" factors like PtL = F.PtL (the equivalent of P'*L) and LtP = F.UP (the equivalent of L'*P). The complete list of supported factors is :L, :PtL, :D, :UP, :U, :LD, :DU, :PtLD, :DUP.

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

Setting the optional shift keyword argument computes the factorization of A+shift*I instead of A. If the perm argument is provided, it should be a permutation of 1:size(A,1) giving the ordering to use (instead of CHOLMOD's default AMD ordering).

Note

This method uses the CHOLMOD[ACM887][DavisHager2009] library from SuiteSparse. CHOLMOD only supports double or complex double element types. Input matrices not of those element types will be converted to SparseMatrixCSC{Float64} or SparseMatrixCSC{ComplexF64} as appropriate.

Many other functions from CHOLMOD are wrapped but not exported from the Base.SparseArrays.CHOLMOD module.

ldlt(S::SymTridiagonal) -> LDLt

Compute an LDLt (i.e., $LDL^T$) factorization of the real symmetric tridiagonal matrix S such that S = L*Diagonal(d)*L' where L is a unit lower triangular matrix and d is a vector. The main use of an LDLt factorization F = ldlt(S) is to solve the linear system of equations Sx = b with F\b.

See also bunchkaufman for a similar, but pivoted, factorization of arbitrary symmetric or Hermitian matrices.

Examples

julia> S = SymTridiagonal([3., 4., 5.], [1., 2.])
3×3 SymTridiagonal{Float64, Vector{Float64}}:
3.0  1.0   ⋅
1.0  4.0  2.0
⋅   2.0  5.0

julia> ldltS = ldlt(S);

julia> b = [6., 7., 8.];

julia> ldltS \ b
3-element Vector{Float64}:
1.7906976744186047
0.627906976744186
1.3488372093023255

julia> S \ b
3-element Vector{Float64}:
1.7906976744186047
0.627906976744186
1.3488372093023255
LinearAlgebra.ldlt!Function
ldlt!(F::CHOLMOD.Factor, A::SparseMatrixCSC; shift = 0.0, check = true) -> CHOLMOD.Factor

Compute the $LDL'$ factorization of A, reusing the symbolic factorization F. A must be a SparseMatrixCSC or a Symmetric/Hermitian view of a SparseMatrixCSC. Note that even if A doesn't have the type tag, it must still be symmetric or Hermitian.

See also ldlt.

Note

This method uses the CHOLMOD library from SuiteSparse, which only supports doubles or complex doubles. Input matrices not of those element types will be converted to SparseMatrixCSC{Float64} or SparseMatrixCSC{ComplexF64} as appropriate.

ldlt!(S::SymTridiagonal) -> LDLt

Same as ldlt, but saves space by overwriting the input S, instead of creating a copy.

Examples

julia> S = SymTridiagonal([3., 4., 5.], [1., 2.])
3×3 SymTridiagonal{Float64, Vector{Float64}}:
3.0  1.0   ⋅
1.0  4.0  2.0
⋅   2.0  5.0

julia> ldltS = ldlt!(S);

julia> ldltS === S
false

julia> S
3×3 SymTridiagonal{Float64, Vector{Float64}}:
3.0       0.333333   ⋅
0.333333  3.66667   0.545455
⋅        0.545455  3.90909
LinearAlgebra.QRType
QR <: Factorization

A QR matrix factorization stored in a packed format, typically obtained from qr. If $A$ is an m×n matrix, then

$$$A = Q R$$$

where $Q$ is an orthogonal/unitary matrix and $R$ is upper triangular. The matrix $Q$ is stored as a sequence of Householder reflectors $v_i$ and coefficients $\tau_i$ where:

$$$Q = \prod_{i=1}^{\min(m,n)} (I - \tau_i v_i v_i^T).$$$

Iterating the decomposition produces the components Q and R.

The object has two fields:

• factors is an m×n matrix.

• The upper triangular part contains the elements of $R$, that is R = triu(F.factors) for a QR object F.

• The subdiagonal part contains the reflectors $v_i$ stored in a packed format where $v_i$ is the $i$th column of the matrix V = I + tril(F.factors, -1).

• τ is a vector of length min(m,n) containing the coefficients $au_i$.

LinearAlgebra.QRCompactWYType
QRCompactWY <: Factorization

A QR matrix factorization stored in a compact blocked format, typically obtained from qr. If $A$ is an m×n matrix, then

$$$A = Q R$$$

where $Q$ is an orthogonal/unitary matrix and $R$ is upper triangular. It is similar to the QR format except that the orthogonal/unitary matrix $Q$ is stored in Compact WY format [Schreiber1989]. For the block size $n_b$, it is stored as a m×n lower trapezoidal matrix $V$ and a matrix $T = (T_1 \; T_2 \; ... \; T_{b-1} \; T_b')$ composed of $b = \lceil \min(m,n) / n_b \rceil$ upper triangular matrices $T_j$ of size $n_b$×$n_b$ ($j = 1, ..., b-1$) and an upper trapezoidal $n_b$×$\min(m,n) - (b-1) n_b$ matrix $T_b'$ ($j=b$) whose upper square part denoted with $T_b$ satisfying

$$$Q = \prod_{i=1}^{\min(m,n)} (I - \tau_i v_i v_i^T) = \prod_{j=1}^{b} (I - V_j T_j V_j^T)$$$

such that $v_i$ is the $i$th column of $V$, $\tau_i$ is the $i$th element of [diag(T_1); diag(T_2); …; diag(T_b)], and $(V_1 \; V_2 \; ... \; V_b)$ is the left m×min(m, n) block of $V$. When constructed using qr, the block size is given by $n_b = \min(m, n, 36)$.

Iterating the decomposition produces the components Q and R.

The object has two fields:

• factors, as in the QR type, is an m×n matrix.

• The upper triangular part contains the elements of $R$, that is R = triu(F.factors) for a QR object F.

• The subdiagonal part contains the reflectors $v_i$ stored in a packed format such that V = I + tril(F.factors, -1).

• T is a $n_b$-by-$\min(m,n)$ matrix as described above. The subdiagonal elements for each triangular matrix $T_j$ are ignored.

Note

This format should not to be confused with the older WY representation [Bischof1987].

LinearAlgebra.QRPivotedType
QRPivoted <: Factorization

A QR matrix factorization with column pivoting in a packed format, typically obtained from qr. If $A$ is an m×n matrix, then

$$$A P = Q R$$$

where $P$ is a permutation matrix, $Q$ is an orthogonal/unitary matrix and $R$ is upper triangular. The matrix $Q$ is stored as a sequence of Householder reflectors:

$$$Q = \prod_{i=1}^{\min(m,n)} (I - \tau_i v_i v_i^T).$$$

Iterating the decomposition produces the components Q, R, and p.

The object has three fields:

• factors is an m×n matrix.

• The upper triangular part contains the elements of $R$, that is R = triu(F.factors) for a QR object F.

• The subdiagonal part contains the reflectors $v_i$ stored in a packed format where $v_i$ is the $i$th column of the matrix V = I + tril(F.factors, -1).

• τ is a vector of length min(m,n) containing the coefficients $au_i$.

• jpvt is an integer vector of length n corresponding to the permutation $P$.

LinearAlgebra.qrFunction
qr(A::SparseMatrixCSC; tol=_default_tol(A), ordering=ORDERING_DEFAULT) -> QRSparse

Compute the QR factorization of a sparse matrix A. Fill-reducing row and column permutations are used such that F.R = F.Q'*A[F.prow,F.pcol]. The main application of this type is to solve least squares or underdetermined problems with \. The function calls the C library SPQR[ACM933].

Note

qr(A::SparseMatrixCSC) uses the SPQR library that is part of SuiteSparse. As this library only supports sparse matrices with Float64 or ComplexF64 elements, as of Julia v1.4 qr converts A into a copy that is of type SparseMatrixCSC{Float64} or SparseMatrixCSC{ComplexF64} as appropriate.

Examples

julia> A = sparse([1,2,3,4], [1,1,2,2], [1.0,1.0,1.0,1.0])
4×2 SparseMatrixCSC{Float64, Int64} with 4 stored entries:
1.0   ⋅
1.0   ⋅
⋅   1.0
⋅   1.0

julia> qr(A)
SparseArrays.SPQR.QRSparse{Float64, Int64}
Q factor:
4×4 SparseArrays.SPQR.QRSparseQ{Float64, Int64}
R factor:
2×2 SparseMatrixCSC{Float64, Int64} with 2 stored entries:
-1.41421    ⋅
⋅       -1.41421
Row permutation:
4-element Vector{Int64}:
1
3
4
2
Column permutation:
2-element Vector{Int64}:
1
2
qr(A, pivot = NoPivot(); blocksize) -> F

Compute the QR factorization of the matrix A: an orthogonal (or unitary if A is complex-valued) matrix Q, and an upper triangular matrix R such that

$$$A = Q R$$$

The returned object F stores the factorization in a packed format:

The individual components of the decomposition F can be retrieved via property accessors:

• F.Q: the orthogonal/unitary matrix Q
• F.R: the upper triangular matrix R
• F.p: the permutation vector of the pivot (QRPivoted only)
• F.P: the permutation matrix of the pivot (QRPivoted only)

Iterating the decomposition produces the components Q, R, and if extant p.

The following functions are available for the QR objects: inv, size, and \. When A is rectangular, \ will return a least squares solution and if the solution is not unique, the one with smallest norm is returned. When A is not full rank, factorization with (column) pivoting is required to obtain a minimum norm solution.

Multiplication with respect to either full/square or non-full/square Q is allowed, i.e. both F.Q*F.R and F.Q*A are supported. A Q matrix can be converted into a regular matrix with Matrix. This operation returns the "thin" Q factor, i.e., if A is m×n with m>=n, then Matrix(F.Q) yields an m×n matrix with orthonormal columns. To retrieve the "full" Q factor, an m×m orthogonal matrix, use F.Q*I. If m<=n, then Matrix(F.Q) yields an m×m orthogonal matrix.

The block size for QR decomposition can be specified by keyword argument blocksize :: Integer when pivot == NoPivot() and A isa StridedMatrix{<:BlasFloat}. It is ignored when blocksize > minimum(size(A)). See QRCompactWY.

Julia 1.4

The blocksize keyword argument requires Julia 1.4 or later.

Examples

julia> A = [3.0 -6.0; 4.0 -8.0; 0.0 1.0]
3×2 Matrix{Float64}:
3.0  -6.0
4.0  -8.0
0.0   1.0

julia> F = qr(A)
LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}
Q factor:
3×3 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}:
-0.6   0.0   0.8
-0.8   0.0  -0.6
0.0  -1.0   0.0
R factor:
2×2 Matrix{Float64}:
-5.0  10.0
0.0  -1.0

julia> F.Q * F.R == A
true
Note

qr returns multiple types because LAPACK uses several representations that minimize the memory storage requirements of products of Householder elementary reflectors, so that the Q and R matrices can be stored compactly rather as two separate dense matrices.

LinearAlgebra.qr!Function
qr!(A, pivot = NoPivot(); blocksize)

qr! is the same as qr when A is a subtype of StridedMatrix, but saves space by overwriting the input A, instead of creating a copy. An InexactError exception is thrown if the factorization produces a number not representable by the element type of A, e.g. for integer types.

Julia 1.4

The blocksize keyword argument requires Julia 1.4 or later.

Examples

julia> a = [1. 2.; 3. 4.]
2×2 Matrix{Float64}:
1.0  2.0
3.0  4.0

julia> qr!(a)
LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}
Q factor:
2×2 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}:
-0.316228  -0.948683
-0.948683   0.316228
R factor:
2×2 Matrix{Float64}:
-3.16228  -4.42719
0.0      -0.632456

julia> a = [1 2; 3 4]
2×2 Matrix{Int64}:
1  2
3  4

julia> qr!(a)
ERROR: InexactError: Int64(3.1622776601683795)
Stacktrace:
[...]
LinearAlgebra.LQType
LQ <: Factorization

Matrix factorization type of the LQ factorization of a matrix A. The LQ decomposition is the QR decomposition of transpose(A). This is the return type of lq, the corresponding matrix factorization function.

If S::LQ is the factorization object, the lower triangular component can be obtained via S.L, and the orthogonal/unitary component via S.Q, such that A ≈ S.L*S.Q.

Iterating the decomposition produces the components S.L and S.Q.

Examples

julia> A = [5. 7.; -2. -4.]
2×2 Matrix{Float64}:
5.0   7.0
-2.0  -4.0

julia> S = lq(A)
LQ{Float64, Matrix{Float64}, Vector{Float64}}
L factor:
2×2 Matrix{Float64}:
-8.60233   0.0
4.41741  -0.697486
Q factor:
2×2 LinearAlgebra.LQPackedQ{Float64, Matrix{Float64}, Vector{Float64}}:
-0.581238  -0.813733
-0.813733   0.581238

julia> S.L * S.Q
2×2 Matrix{Float64}:
5.0   7.0
-2.0  -4.0

julia> l, q = S; # destructuring via iteration

julia> l == S.L &&  q == S.Q
true
LinearAlgebra.lqFunction
lq(A) -> S::LQ

Compute the LQ decomposition of A. The decomposition's lower triangular component can be obtained from the LQ object S via S.L, and the orthogonal/unitary component via S.Q, such that A ≈ S.L*S.Q.

Iterating the decomposition produces the components S.L and S.Q.

The LQ decomposition is the QR decomposition of transpose(A), and it is useful in order to compute the minimum-norm solution lq(A) \ b to an underdetermined system of equations (A has more columns than rows, but has full row rank).

Examples

julia> A = [5. 7.; -2. -4.]
2×2 Matrix{Float64}:
5.0   7.0
-2.0  -4.0

julia> S = lq(A)
LQ{Float64, Matrix{Float64}, Vector{Float64}}
L factor:
2×2 Matrix{Float64}:
-8.60233   0.0
4.41741  -0.697486
Q factor:
2×2 LinearAlgebra.LQPackedQ{Float64, Matrix{Float64}, Vector{Float64}}:
-0.581238  -0.813733
-0.813733   0.581238

julia> S.L * S.Q
2×2 Matrix{Float64}:
5.0   7.0
-2.0  -4.0

julia> l, q = S; # destructuring via iteration

julia> l == S.L &&  q == S.Q
true
LinearAlgebra.BunchKaufmanType
BunchKaufman <: Factorization

Matrix factorization type of the Bunch-Kaufman factorization of a symmetric or Hermitian matrix A as P'UDU'P or P'LDL'P, depending on whether the upper (the default) or the lower triangle is stored in A. If A is complex symmetric then U' and L' denote the unconjugated transposes, i.e. transpose(U) and transpose(L), respectively. This is the return type of bunchkaufman, the corresponding matrix factorization function.

If S::BunchKaufman is the factorization object, the components can be obtained via S.D, S.U or S.L as appropriate given S.uplo, and S.p.

Iterating the decomposition produces the components S.D, S.U or S.L as appropriate given S.uplo, and S.p.

Examples

julia> A = [1 2; 2 3]
2×2 Matrix{Int64}:
1  2
2  3

julia> S = bunchkaufman(A) # A gets wrapped internally by Symmetric(A)
BunchKaufman{Float64, Matrix{Float64}, Vector{Int64}}
D factor:
2×2 Tridiagonal{Float64, Vector{Float64}}:
-0.333333  0.0
0.0       3.0
U factor:
2×2 UnitUpperTriangular{Float64, Matrix{Float64}}:
1.0  0.666667
⋅   1.0
permutation:
2-element Vector{Int64}:
1
2

julia> d, u, p = S; # destructuring via iteration

julia> d == S.D && u == S.U && p == S.p
true

julia> S = bunchkaufman(Symmetric(A, :L))
BunchKaufman{Float64, Matrix{Float64}, Vector{Int64}}
D factor:
2×2 Tridiagonal{Float64, Vector{Float64}}:
3.0   0.0
0.0  -0.333333
L factor:
2×2 UnitLowerTriangular{Float64, Matrix{Float64}}:
1.0        ⋅
0.666667  1.0
permutation:
2-element Vector{Int64}:
2
1
LinearAlgebra.bunchkaufmanFunction
bunchkaufman(A, rook::Bool=false; check = true) -> S::BunchKaufman

Compute the Bunch-Kaufman [Bunch1977] factorization of a symmetric or Hermitian matrix A as P'*U*D*U'*P or P'*L*D*L'*P, depending on which triangle is stored in A, and return a BunchKaufman object. Note that if A is complex symmetric then U' and L' denote the unconjugated transposes, i.e. transpose(U) and transpose(L).

Iterating the decomposition produces the components S.D, S.U or S.L as appropriate given S.uplo, and S.p.

If rook is true, rook pivoting is used. If rook is false, rook pivoting is not used.

When check = true, an error is thrown if the decomposition fails. When check = false, responsibility for checking the decomposition's validity (via issuccess) lies with the user.

The following functions are available for BunchKaufman objects: size, \, inv, issymmetric, ishermitian, getindex.

Examples

julia> A = [1 2; 2 3]
2×2 Matrix{Int64}:
1  2
2  3

julia> S = bunchkaufman(A) # A gets wrapped internally by Symmetric(A)
BunchKaufman{Float64, Matrix{Float64}, Vector{Int64}}
D factor:
2×2 Tridiagonal{Float64, Vector{Float64}}:
-0.333333  0.0
0.0       3.0
U factor:
2×2 UnitUpperTriangular{Float64, Matrix{Float64}}:
1.0  0.666667
⋅   1.0
permutation:
2-element Vector{Int64}:
1
2

julia> d, u, p = S; # destructuring via iteration

julia> d == S.D && u == S.U && p == S.p
true

julia> S.U*S.D*S.U' - S.P*A*S.P'
2×2 Matrix{Float64}:
0.0  0.0
0.0  0.0

julia> S = bunchkaufman(Symmetric(A, :L))
BunchKaufman{Float64, Matrix{Float64}, Vector{Int64}}
D factor:
2×2 Tridiagonal{Float64, Vector{Float64}}:
3.0   0.0
0.0  -0.333333
L factor:
2×2 UnitLowerTriangular{Float64, Matrix{Float64}}:
1.0        ⋅
0.666667  1.0
permutation:
2-element Vector{Int64}:
2
1

julia> S.L*S.D*S.L' - A[S.p, S.p]
2×2 Matrix{Float64}:
0.0  0.0
0.0  0.0
LinearAlgebra.EigenType
Eigen <: Factorization

Matrix factorization type of the eigenvalue/spectral decomposition of a square matrix A. This is the return type of eigen, the corresponding matrix factorization function.

If F::Eigen is the factorization object, the eigenvalues can be obtained via F.values and the eigenvectors as the columns of the matrix F.vectors. (The kth eigenvector can be obtained from the slice F.vectors[:, k].)

Iterating the decomposition produces the components F.values and F.vectors.

Examples

julia> F = eigen([1.0 0.0 0.0; 0.0 3.0 0.0; 0.0 0.0 18.0])
Eigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}
values:
3-element Vector{Float64}:
1.0
3.0
18.0
vectors:
3×3 Matrix{Float64}:
1.0  0.0  0.0
0.0  1.0  0.0
0.0  0.0  1.0

julia> F.values
3-element Vector{Float64}:
1.0
3.0
18.0

julia> F.vectors
3×3 Matrix{Float64}:
1.0  0.0  0.0
0.0  1.0  0.0
0.0  0.0  1.0

julia> vals, vecs = F; # destructuring via iteration

julia> vals == F.values && vecs == F.vectors
true
LinearAlgebra.GeneralizedEigenType
GeneralizedEigen <: Factorization

Matrix factorization type of the generalized eigenvalue/spectral decomposition of A and B. This is the return type of eigen, the corresponding matrix factorization function, when called with two matrix arguments.

If F::GeneralizedEigen is the factorization object, the eigenvalues can be obtained via F.values and the eigenvectors as the columns of the matrix F.vectors. (The kth eigenvector can be obtained from the slice F.vectors[:, k].)

Iterating the decomposition produces the components F.values and F.vectors.

Examples

julia> A = [1 0; 0 -1]
2×2 Matrix{Int64}:
1   0
0  -1

julia> B = [0 1; 1 0]
2×2 Matrix{Int64}:
0  1
1  0

julia> F = eigen(A, B)
GeneralizedEigen{ComplexF64, ComplexF64, Matrix{ComplexF64}, Vector{ComplexF64}}
values:
2-element Vector{ComplexF64}:
0.0 - 1.0im
0.0 + 1.0im
vectors:
2×2 Matrix{ComplexF64}:
0.0+1.0im   0.0-1.0im
-1.0+0.0im  -1.0-0.0im

julia> F.values
2-element Vector{ComplexF64}:
0.0 - 1.0im
0.0 + 1.0im

julia> F.vectors
2×2 Matrix{ComplexF64}:
0.0+1.0im   0.0-1.0im
-1.0+0.0im  -1.0-0.0im

julia> vals, vecs = F; # destructuring via iteration

julia> vals == F.values && vecs == F.vectors
true
LinearAlgebra.eigvalsFunction
eigvals(A; permute::Bool=true, scale::Bool=true, sortby) -> values

Return the eigenvalues of A.

For general non-symmetric matrices it is possible to specify how the matrix is balanced before the eigenvalue calculation. The permute, scale, and sortby keywords are the same as for eigen.

Examples

julia> diag_matrix = [1 0; 0 4]
2×2 Matrix{Int64}:
1  0
0  4

julia> eigvals(diag_matrix)
2-element Vector{Float64}:
1.0
4.0

For a scalar input, eigvals will return a scalar.

Example

julia> eigvals(-2)
-2
eigvals(A, B) -> values

Compute the generalized eigenvalues of A and B.

Examples

julia> A = [1 0; 0 -1]
2×2 Matrix{Int64}:
1   0
0  -1

julia> B = [0 1; 1 0]
2×2 Matrix{Int64}:
0  1
1  0

julia> eigvals(A,B)
2-element Vector{ComplexF64}:
0.0 - 1.0im
0.0 + 1.0im
eigvals(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> values

Return the eigenvalues of A. It is possible to calculate only a subset of the eigenvalues by specifying a UnitRange irange covering indices of the sorted eigenvalues, e.g. the 2nd to 8th eigenvalues.

Examples

julia> A = SymTridiagonal([1.; 2.; 1.], [2.; 3.])
3×3 SymTridiagonal{Float64, Vector{Float64}}:
1.0  2.0   ⋅
2.0  2.0  3.0
⋅   3.0  1.0

julia> eigvals(A, 2:2)
1-element Vector{Float64}:
0.9999999999999996

julia> eigvals(A)
3-element Vector{Float64}:
-2.1400549446402604
1.0000000000000002
5.140054944640259
eigvals(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> values

Return the eigenvalues of A. It is possible to calculate only a subset of the eigenvalues by specifying a pair vl and vu for the lower and upper boundaries of the eigenvalues.

Examples

julia> A = SymTridiagonal([1.; 2.; 1.], [2.; 3.])
3×3 SymTridiagonal{Float64, Vector{Float64}}:
1.0  2.0   ⋅
2.0  2.0  3.0
⋅   3.0  1.0

julia> eigvals(A, -1, 2)
1-element Vector{Float64}:
1.0000000000000009

julia> eigvals(A)
3-element Vector{Float64}:
-2.1400549446402604
1.0000000000000002
5.140054944640259
LinearAlgebra.eigvals!Function
eigvals!(A; permute::Bool=true, scale::Bool=true, sortby) -> values

Same as eigvals, but saves space by overwriting the input A, instead of creating a copy. The permute, scale, and sortby keywords are the same as for eigen.

Note

The input matrix A will not contain its eigenvalues after eigvals! is called on it - A is used as a workspace.

Examples

julia> A = [1. 2.; 3. 4.]
2×2 Matrix{Float64}:
1.0  2.0
3.0  4.0

julia> eigvals!(A)
2-element Vector{Float64}:
-0.3722813232690143
5.372281323269014

julia> A
2×2 Matrix{Float64}:
-0.372281  -1.0
0.0        5.37228
eigvals!(A, B; sortby) -> values

Same as eigvals, but saves space by overwriting the input A (and B), instead of creating copies.

Note

The input matrices A and B will not contain their eigenvalues after eigvals! is called. They are used as workspaces.

Examples

julia> A = [1. 0.; 0. -1.]
2×2 Matrix{Float64}:
1.0   0.0
0.0  -1.0

julia> B = [0. 1.; 1. 0.]
2×2 Matrix{Float64}:
0.0  1.0
1.0  0.0

julia> eigvals!(A, B)
2-element Vector{ComplexF64}:
0.0 - 1.0im
0.0 + 1.0im

julia> A
2×2 Matrix{Float64}:
-0.0  -1.0
1.0  -0.0

julia> B
2×2 Matrix{Float64}:
1.0  0.0
0.0  1.0
eigvals!(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> values

Same as eigvals, but saves space by overwriting the input A, instead of creating a copy. irange is a range of eigenvalue indices to search for - for instance, the 2nd to 8th eigenvalues.

eigvals!(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> values

Same as eigvals, but saves space by overwriting the input A, instead of creating a copy. vl is the lower bound of the interval to search for eigenvalues, and vu is the upper bound.

LinearAlgebra.eigmaxFunction
eigmax(A; permute::Bool=true, scale::Bool=true)

Return the largest eigenvalue of A. The option permute=true permutes the matrix to become closer to upper triangular, and scale=true scales the matrix by its diagonal elements to make rows and columns more equal in norm. Note that if the eigenvalues of A are complex, this method will fail, since complex numbers cannot be sorted.

Examples

julia> A = [0 im; -im 0]
2×2 Matrix{Complex{Int64}}:
0+0im  0+1im
0-1im  0+0im

julia> eigmax(A)
1.0

julia> A = [0 im; -1 0]
2×2 Matrix{Complex{Int64}}:
0+0im  0+1im
-1+0im  0+0im

julia> eigmax(A)
ERROR: DomainError with Complex{Int64}[0+0im 0+1im; -1+0im 0+0im]:
A cannot have complex eigenvalues.
Stacktrace:
[...]
LinearAlgebra.eigminFunction
eigmin(A; permute::Bool=true, scale::Bool=true)

Return the smallest eigenvalue of A. The option permute=true permutes the matrix to become closer to upper triangular, and scale=true scales the matrix by its diagonal elements to make rows and columns more equal in norm. Note that if the eigenvalues of A are complex, this method will fail, since complex numbers cannot be sorted.

Examples

julia> A = [0 im; -im 0]
2×2 Matrix{Complex{Int64}}:
0+0im  0+1im
0-1im  0+0im

julia> eigmin(A)
-1.0

julia> A = [0 im; -1 0]
2×2 Matrix{Complex{Int64}}:
0+0im  0+1im
-1+0im  0+0im

julia> eigmin(A)
ERROR: DomainError with Complex{Int64}[0+0im 0+1im; -1+0im 0+0im]:
A cannot have complex eigenvalues.
Stacktrace:
[...]
LinearAlgebra.eigvecsFunction
eigvecs(A::SymTridiagonal[, eigvals]) -> Matrix

Return a matrix M whose columns are the eigenvectors of A. (The kth eigenvector can be obtained from the slice M[:, k].)

If the optional vector of eigenvalues eigvals is specified, eigvecs returns the specific corresponding eigenvectors.

Examples

julia> A = SymTridiagonal([1.; 2.; 1.], [2.; 3.])
3×3 SymTridiagonal{Float64, Vector{Float64}}:
1.0  2.0   ⋅
2.0  2.0  3.0
⋅   3.0  1.0

julia> eigvals(A)
3-element Vector{Float64}:
-2.1400549446402604
1.0000000000000002
5.140054944640259

julia> eigvecs(A)
3×3 Matrix{Float64}:
0.418304  -0.83205      0.364299
-0.656749  -7.39009e-16  0.754109
0.627457   0.5547       0.546448

julia> eigvecs(A, [1.])
3×1 Matrix{Float64}:
0.8320502943378438
4.263514128092366e-17
-0.5547001962252291
eigvecs(A; permute::Bool=true, scale::Bool=true, sortby) -> Matrix

Return a matrix M whose columns are the eigenvectors of A. (The kth eigenvector can be obtained from the slice M[:, k].) The permute, scale, and sortby keywords are the same as for eigen.

Examples

julia> eigvecs([1.0 0.0 0.0; 0.0 3.0 0.0; 0.0 0.0 18.0])
3×3 Matrix{Float64}:
1.0  0.0  0.0
0.0  1.0  0.0
0.0  0.0  1.0
eigvecs(A, B) -> Matrix

Return a matrix M whose columns are the generalized eigenvectors of A and B. (The kth eigenvector can be obtained from the slice M[:, k].)

Examples

julia> A = [1 0; 0 -1]
2×2 Matrix{Int64}:
1   0
0  -1

julia> B = [0 1; 1 0]
2×2 Matrix{Int64}:
0  1
1  0

julia> eigvecs(A, B)
2×2 Matrix{ComplexF64}:
0.0+1.0im   0.0-1.0im
-1.0+0.0im  -1.0-0.0im
LinearAlgebra.eigenFunction
eigen(A; permute::Bool=true, scale::Bool=true, sortby) -> Eigen

Compute the eigenvalue decomposition of A, returning an Eigen factorization object F which contains the eigenvalues in F.values and the eigenvectors in the columns of the matrix F.vectors. (The kth eigenvector can be obtained from the slice F.vectors[:, k].)

Iterating the decomposition produces the components F.values and F.vectors.

The following functions are available for Eigen objects: inv, det, and isposdef.

For general nonsymmetric matrices it is possible to specify how the matrix is balanced before the eigenvector calculation. The option permute=true permutes the matrix to become closer to upper triangular, and scale=true scales the matrix by its diagonal elements to make rows and columns more equal in norm. The default is true for both options.

By default, the eigenvalues and vectors are sorted lexicographically by (real(λ),imag(λ)). A different comparison function by(λ) can be passed to sortby, or you can pass sortby=nothing to leave the eigenvalues in an arbitrary order. Some special matrix types (e.g. Diagonal or SymTridiagonal) may implement their own sorting convention and not accept a sortby keyword.

Examples

julia> F = eigen([1.0 0.0 0.0; 0.0 3.0 0.0; 0.0 0.0 18.0])
Eigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}
values:
3-element Vector{Float64}:
1.0
3.0
18.0
vectors:
3×3 Matrix{Float64}:
1.0  0.0  0.0
0.0  1.0  0.0
0.0  0.0  1.0

julia> F.values
3-element Vector{Float64}:
1.0
3.0
18.0

julia> F.vectors
3×3 Matrix{Float64}:
1.0  0.0  0.0
0.0  1.0  0.0
0.0  0.0  1.0

julia> vals, vecs = F; # destructuring via iteration

julia> vals == F.values && vecs == F.vectors
true
eigen(A, B; sortby) -> GeneralizedEigen

Compute the generalized eigenvalue decomposition of A and B, returning a GeneralizedEigen factorization object F which contains the generalized eigenvalues in F.values and the generalized eigenvectors in the columns of the matrix F.vectors. (The kth generalized eigenvector can be obtained from the slice F.vectors[:, k].)

Iterating the decomposition produces the components F.values and F.vectors.

By default, the eigenvalues and vectors are sorted lexicographically by (real(λ),imag(λ)). A different comparison function by(λ) can be passed to sortby, or you can pass sortby=nothing to leave the eigenvalues in an arbitrary order.

Examples

julia> A = [1 0; 0 -1]
2×2 Matrix{Int64}:
1   0
0  -1

julia> B = [0 1; 1 0]
2×2 Matrix{Int64}:
0  1
1  0

julia> F = eigen(A, B);

julia> F.values
2-element Vector{ComplexF64}:
0.0 - 1.0im
0.0 + 1.0im

julia> F.vectors
2×2 Matrix{ComplexF64}:
0.0+1.0im   0.0-1.0im
-1.0+0.0im  -1.0-0.0im

julia> vals, vecs = F; # destructuring via iteration

julia> vals == F.values && vecs == F.vectors
true
eigen(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> Eigen

Compute the eigenvalue decomposition of A, returning an Eigen factorization object F which contains the eigenvalues in F.values and the eigenvectors in the columns of the matrix F.vectors. (The kth eigenvector can be obtained from the slice F.vectors[:, k].)

Iterating the decomposition produces the components F.values and F.vectors.

The following functions are available for Eigen objects: inv, det, and isposdef.

The UnitRange irange specifies indices of the sorted eigenvalues to search for.

Note

If irange is not 1:n, where n is the dimension of A, then the returned factorization will be a truncated factorization.

eigen(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> Eigen

Compute the eigenvalue decomposition of A, returning an Eigen factorization object F which contains the eigenvalues in F.values and the eigenvectors in the columns of the matrix F.vectors. (The kth eigenvector can be obtained from the slice F.vectors[:, k].)

Iterating the decomposition produces the components F.values and F.vectors.

The following functions are available for Eigen objects: inv, det, and isposdef.

vl is the lower bound of the window of eigenvalues to search for, and vu is the upper bound.

Note

If [vl, vu] does not contain all eigenvalues of A, then the returned factorization will be a truncated factorization.

LinearAlgebra.eigen!Function
eigen!(A; permute, scale, sortby)
eigen!(A, B; sortby)

Same as eigen, but saves space by overwriting the input A (and B), instead of creating a copy.

LinearAlgebra.HessenbergType
Hessenberg <: Factorization

A Hessenberg object represents the Hessenberg factorization QHQ' of a square matrix, or a shift Q(H+μI)Q' thereof, which is produced by the hessenberg function.

LinearAlgebra.hessenbergFunction
hessenberg(A) -> Hessenberg

Compute the Hessenberg decomposition of A and return a Hessenberg object. If F is the factorization object, the unitary matrix can be accessed with F.Q (of type LinearAlgebra.HessenbergQ) and the Hessenberg matrix with F.H (of type UpperHessenberg), either of which may be converted to a regular matrix with Matrix(F.H) or Matrix(F.Q).

If A is Hermitian or real-Symmetric, then the Hessenberg decomposition produces a real-symmetric tridiagonal matrix and F.H is of type SymTridiagonal.

Note that the shifted factorization A+μI = Q (H+μI) Q' can be constructed efficiently by F + μ*I using the UniformScaling object I, which creates a new Hessenberg object with shared storage and a modified shift. The shift of a given F is obtained by F.μ. This is useful because multiple shifted solves (F + μ*I) \ b (for different μ and/or b) can be performed efficiently once F is created.

Iterating the decomposition produces the factors F.Q, F.H, F.μ.

Examples

julia> A = [4. 9. 7.; 4. 4. 1.; 4. 3. 2.]
3×3 Matrix{Float64}:
4.0  9.0  7.0
4.0  4.0  1.0
4.0  3.0  2.0

julia> F = hessenberg(A)
Hessenberg{Float64, UpperHessenberg{Float64, Matrix{Float64}}, Matrix{Float64}, Vector{Float64}, Bool}
Q factor:
3×3 LinearAlgebra.HessenbergQ{Float64, Matrix{Float64}, Vector{Float64}, false}:
1.0   0.0        0.0
0.0  -0.707107  -0.707107
0.0  -0.707107   0.707107
H factor:
3×3 UpperHessenberg{Float64, Matrix{Float64}}:
4.0      -11.3137       -1.41421
-5.65685    5.0           2.0
⋅        -8.88178e-16   1.0

julia> F.Q * F.H * F.Q'
3×3 Matrix{Float64}:
4.0  9.0  7.0
4.0  4.0  1.0
4.0  3.0  2.0

julia> q, h = F; # destructuring via iteration

julia> q == F.Q && h == F.H
true
LinearAlgebra.SchurType
Schur <: Factorization

Matrix factorization type of the Schur factorization of a matrix A. This is the return type of schur(_), the corresponding matrix factorization function.

If F::Schur is the factorization object, the (quasi) triangular Schur factor can be obtained via either F.Schur or F.T and the orthogonal/unitary Schur vectors via F.vectors or F.Z such that A = F.vectors * F.Schur * F.vectors'. The eigenvalues of A can be obtained with F.values.

Iterating the decomposition produces the components F.T, F.Z, and F.values.

Examples

julia> A = [5. 7.; -2. -4.]
2×2 Matrix{Float64}:
5.0   7.0
-2.0  -4.0

julia> F = schur(A)
Schur{Float64, Matrix{Float64}, Vector{Float64}}
T factor:
2×2 Matrix{Float64}:
3.0   9.0
0.0  -2.0
Z factor:
2×2 Matrix{Float64}:
0.961524  0.274721
-0.274721  0.961524
eigenvalues:
2-element Vector{Float64}:
3.0
-2.0

julia> F.vectors * F.Schur * F.vectors'
2×2 Matrix{Float64}:
5.0   7.0
-2.0  -4.0

julia> t, z, vals = F; # destructuring via iteration

julia> t == F.T && z == F.Z && vals == F.values
true
LinearAlgebra.GeneralizedSchurType
GeneralizedSchur <: Factorization

Matrix factorization type of the generalized Schur factorization of two matrices A and B. This is the return type of schur(_, _), the corresponding matrix factorization function.

If F::GeneralizedSchur is the factorization object, the (quasi) triangular Schur factors can be obtained via F.S and F.T, the left unitary/orthogonal Schur vectors via F.left or F.Q, and the right unitary/orthogonal Schur vectors can be obtained with F.right or F.Z such that A=F.left*F.S*F.right' and B=F.left*F.T*F.right'. The generalized eigenvalues of A and B can be obtained with F.α./F.β.

Iterating the decomposition produces the components F.S, F.T, F.Q, F.Z, F.α, and F.β.

LinearAlgebra.schurFunction
schur(A) -> F::Schur

Computes the Schur factorization of the matrix A. The (quasi) triangular Schur factor can be obtained from the Schur object F with either F.Schur or F.T and the orthogonal/unitary Schur vectors can be obtained with F.vectors or F.Z such that A = F.vectors * F.Schur * F.vectors'. The eigenvalues of A can be obtained with F.values.

For real A, the Schur factorization is "quasitriangular", which means that it is upper-triangular except with 2×2 diagonal blocks for any conjugate pair of complex eigenvalues; this allows the factorization to be purely real even when there are complex eigenvalues. To obtain the (complex) purely upper-triangular Schur factorization from a real quasitriangular factorization, you can use Schur{Complex}(schur(A)).

Iterating the decomposition produces the components F.T, F.Z, and F.values.

Examples

julia> A = [5. 7.; -2. -4.]
2×2 Matrix{Float64}:
5.0   7.0
-2.0  -4.0

julia> F = schur(A)
Schur{Float64, Matrix{Float64}, Vector{Float64}}
T factor:
2×2 Matrix{Float64}:
3.0   9.0
0.0  -2.0
Z factor:
2×2 Matrix{Float64}:
0.961524  0.274721
-0.274721  0.961524
eigenvalues:
2-element Vector{Float64}:
3.0
-2.0

julia> F.vectors * F.Schur * F.vectors'
2×2 Matrix{Float64}:
5.0   7.0
-2.0  -4.0

julia> t, z, vals = F; # destructuring via iteration

julia> t == F.T && z == F.Z && vals == F.values
true
schur(A, B) -> F::GeneralizedSchur

Computes the Generalized Schur (or QZ) factorization of the matrices A and B. The (quasi) triangular Schur factors can be obtained from the Schur object F with F.S and F.T, the left unitary/orthogonal Schur vectors can be obtained with F.left or F.Q and the right unitary/orthogonal Schur vectors can be obtained with F.right or F.Z such that A=F.left*F.S*F.right' and B=F.left*F.T*F.right'. The generalized eigenvalues of A and B can be obtained with F.α./F.β.

Iterating the decomposition produces the components F.S, F.T, F.Q, F.Z, F.α, and F.β.

LinearAlgebra.schur!Function
schur!(A) -> F::Schur

Same as schur but uses the input argument A as workspace.

Examples

julia> A = [5. 7.; -2. -4.]
2×2 Matrix{Float64}:
5.0   7.0
-2.0  -4.0

julia> F = schur!(A)
Schur{Float64, Matrix{Float64}, Vector{Float64}}
T factor:
2×2 Matrix{Float64}:
3.0   9.0
0.0  -2.0
Z factor:
2×2 Matrix{Float64}:
0.961524  0.274721
-0.274721  0.961524
eigenvalues:
2-element Vector{Float64}:
3.0
-2.0

julia> A
2×2 Matrix{Float64}:
3.0   9.0
0.0  -2.0
schur!(A::StridedMatrix, B::StridedMatrix) -> F::GeneralizedSchur

Same as schur but uses the input matrices A and B as workspace.

LinearAlgebra.ordschurFunction
ordschur(F::Schur, select::Union{Vector{Bool},BitVector}) -> F::Schur

Reorders the Schur factorization F of a matrix A = Z*T*Z' according to the logical array select returning the reordered factorization F object. The selected eigenvalues appear in the leading diagonal of F.Schur and the corresponding leading columns of F.vectors form an orthogonal/unitary basis of the corresponding right invariant subspace. In the real case, a complex conjugate pair of eigenvalues must be either both included or both excluded via select.

ordschur(F::GeneralizedSchur, select::Union{Vector{Bool},BitVector}) -> F::GeneralizedSchur

Reorders the Generalized Schur factorization F of a matrix pair (A, B) = (Q*S*Z', Q*T*Z') according to the logical array select and returns a GeneralizedSchur object F. The selected eigenvalues appear in the leading diagonal of both F.S and F.T, and the left and right orthogonal/unitary Schur vectors are also reordered such that (A, B) = F.Q*(F.S, F.T)*F.Z' still holds and the generalized eigenvalues of A and B can still be obtained with F.α./F.β.

LinearAlgebra.ordschur!Function
ordschur!(F::Schur, select::Union{Vector{Bool},BitVector}) -> F::Schur

Same as ordschur but overwrites the factorization F.

ordschur!(F::GeneralizedSchur, select::Union{Vector{Bool},BitVector}) -> F::GeneralizedSchur

Same as ordschur but overwrites the factorization F.

LinearAlgebra.SVDType
SVD <: Factorization

Matrix factorization type of the singular value decomposition (SVD) of a matrix A. This is the return type of svd(_), the corresponding matrix factorization function.

If F::SVD is the factorization object, U, S, V and Vt can be obtained via F.U, F.S, F.V and F.Vt, such that A = U * Diagonal(S) * Vt. The singular values in S are sorted in descending order.

Iterating the decomposition produces the components U, S, and V.

Examples

julia> A = [1. 0. 0. 0. 2.; 0. 0. 3. 0. 0.; 0. 0. 0. 0. 0.; 0. 2. 0. 0. 0.]
4×5 Matrix{Float64}:
1.0  0.0  0.0  0.0  2.0
0.0  0.0  3.0  0.0  0.0
0.0  0.0  0.0  0.0  0.0
0.0  2.0  0.0  0.0  0.0

julia> F = svd(A)
SVD{Float64, Float64, Matrix{Float64}, Vector{Float64}}
U factor:
4×4 Matrix{Float64}:
0.0  1.0   0.0  0.0
1.0  0.0   0.0  0.0
0.0  0.0   0.0  1.0
0.0  0.0  -1.0  0.0
singular values:
4-element Vector{Float64}:
3.0
2.23606797749979
2.0
0.0
Vt factor:
4×5 Matrix{Float64}:
-0.0        0.0  1.0  -0.0  0.0
0.447214   0.0  0.0   0.0  0.894427
0.0       -1.0  0.0   0.0  0.0
0.0        0.0  0.0   1.0  0.0

julia> F.U * Diagonal(F.S) * F.Vt
4×5 Matrix{Float64}:
1.0  0.0  0.0  0.0  2.0
0.0  0.0  3.0  0.0  0.0
0.0  0.0  0.0  0.0  0.0
0.0  2.0  0.0  0.0  0.0

julia> u, s, v = F; # destructuring via iteration

julia> u == F.U && s == F.S && v == F.V
true
LinearAlgebra.GeneralizedSVDType
GeneralizedSVD <: Factorization

Matrix factorization type of the generalized singular value decomposition (SVD) of two matrices A and B, such that A = F.U*F.D1*F.R0*F.Q' and B = F.V*F.D2*F.R0*F.Q'. This is the return type of svd(_, _), the corresponding matrix factorization function.

For an M-by-N matrix A and P-by-N matrix B,

• U is a M-by-M orthogonal matrix,
• V is a P-by-P orthogonal matrix,
• Q is a N-by-N orthogonal matrix,
• D1 is a M-by-(K+L) diagonal matrix with 1s in the first K entries,
• D2 is a P-by-(K+L) matrix whose top right L-by-L block is diagonal,
• R0 is a (K+L)-by-N matrix whose rightmost (K+L)-by-(K+L) block is nonsingular upper block triangular,

K+L is the effective numerical rank of the matrix [A; B].

Iterating the decomposition produces the components U, V, Q, D1, D2, and R0.

The entries of F.D1 and F.D2 are related, as explained in the LAPACK documentation for the generalized SVD and the xGGSVD3 routine which is called underneath (in LAPACK 3.6.0 and newer).

Examples

julia> A = [1. 0.; 0. -1.]
2×2 Matrix{Float64}:
1.0   0.0
0.0  -1.0

julia> B = [0. 1.; 1. 0.]
2×2 Matrix{Float64}:
0.0  1.0
1.0  0.0

julia> F = svd(A, B)
GeneralizedSVD{Float64, Matrix{Float64}, Float64, Vector{Float64}}
U factor:
2×2 Matrix{Float64}:
1.0  0.0
0.0  1.0
V factor:
2×2 Matrix{Float64}:
-0.0  -1.0
1.0   0.0
Q factor:
2×2 Matrix{Float64}:
1.0  0.0
0.0  1.0
D1 factor:
2×2 Matrix{Float64}:
0.707107  0.0
0.0       0.707107
D2 factor:
2×2 Matrix{Float64}:
0.707107  0.0
0.0       0.707107
R0 factor:
2×2 Matrix{Float64}:
1.41421   0.0
0.0      -1.41421

julia> F.U*F.D1*F.R0*F.Q'
2×2 Matrix{Float64}:
1.0   0.0
0.0  -1.0

julia> F.V*F.D2*F.R0*F.Q'
2×2 Matrix{Float64}:
-0.0  1.0
1.0  0.0
LinearAlgebra.svdFunction
svd(A; full::Bool = false, alg::Algorithm = default_svd_alg(A)) -> SVD

Compute the singular value decomposition (SVD) of A and return an SVD object.

U, S, V and Vt can be obtained from the factorization F with F.U, F.S, F.V and F.Vt, such that A = U * Diagonal(S) * Vt. The algorithm produces Vt and hence Vt is more efficient to extract than V. The singular values in S are sorted in descending order.

Iterating the decomposition produces the components U, S, and V.

If full = false (default), a "thin" SVD is returned. For an $M \times N$ matrix A, in the full factorization U is $M \times M$ and V is $N \times N$, while in the thin factorization U is $M \times K$ and V is $N \times K$, where $K = \min(M,N)$ is the number of singular values.

If alg = DivideAndConquer() a divide-and-conquer algorithm is used to calculate the SVD. Another (typically slower but more accurate) option is alg = QRIteration().

Julia 1.3

The alg keyword argument requires Julia 1.3 or later.

Examples

julia> A = rand(4,3);

julia> F = svd(A); # Store the Factorization Object

julia> A ≈ F.U * Diagonal(F.S) * F.Vt
true

julia> U, S, V = F; # destructuring via iteration

julia> A ≈ U * Diagonal(S) * V'
true

julia> Uonly, = svd(A); # Store U only

julia> Uonly == U
true
svd(A, B) -> GeneralizedSVD

Compute the generalized SVD of A and B, returning a GeneralizedSVD factorization object F such that [A;B] = [F.U * F.D1; F.V * F.D2] * F.R0 * F.Q'

• U is a M-by-M orthogonal matrix,
• V is a P-by-P orthogonal matrix,
• Q is a N-by-N orthogonal matrix,
• D1 is a M-by-(K+L) diagonal matrix with 1s in the first K entries,
• D2 is a P-by-(K+L) matrix whose top right L-by-L block is diagonal,
• R0 is a (K+L)-by-N matrix whose rightmost (K+L)-by-(K+L) block is nonsingular upper block triangular,

K+L is the effective numerical rank of the matrix [A; B].

Iterating the decomposition produces the components U, V, Q, D1, D2, and R0.

The generalized SVD is used in applications such as when one wants to compare how much belongs to A vs. how much belongs to B, as in human vs yeast genome, or signal vs noise, or between clusters vs within clusters. (See Edelman and Wang for discussion: https://arxiv.org/abs/1901.00485)

It decomposes [A; B] into [UC; VS]H, where [UC; VS] is a natural orthogonal basis for the column space of [A; B], and H = RQ' is a natural non-orthogonal basis for the rowspace of [A;B], where the top rows are most closely attributed to the A matrix, and the bottom to the B matrix. The multi-cosine/sine matrices C and S provide a multi-measure of how much A vs how much B, and U and V provide directions in which these are measured.

Examples

julia> A = randn(3,2); B=randn(4,2);

julia> F = svd(A, B);

julia> U,V,Q,C,S,R = F;

julia> H = R*Q';

julia> [A; B] ≈ [U*C; V*S]*H
true

julia> [A; B] ≈ [F.U*F.D1; F.V*F.D2]*F.R0*F.Q'
true

julia> Uonly, = svd(A,B);

julia> U == Uonly
true
LinearAlgebra.svd!Function
svd!(A; full::Bool = false, alg::Algorithm = default_svd_alg(A)) -> SVD

svd! is the same as svd, but saves space by overwriting the input A, instead of creating a copy. See documentation of svd for details.

svd!(A, B) -> GeneralizedSVD

svd! is the same as svd, but modifies the arguments A and B in-place, instead of making copies. See documentation of svd for details.

LinearAlgebra.svdvalsFunction
svdvals(A)

Return the singular values of A in descending order.

Examples

julia> A = [1. 0. 0. 0. 2.; 0. 0. 3. 0. 0.; 0. 0. 0. 0. 0.; 0. 2. 0. 0. 0.]
4×5 Matrix{Float64}:
1.0  0.0  0.0  0.0  2.0
0.0  0.0  3.0  0.0  0.0
0.0  0.0  0.0  0.0  0.0
0.0  2.0  0.0  0.0  0.0

julia> svdvals(A)
4-element Vector{Float64}:
3.0
2.23606797749979
2.0
0.0
svdvals(A, B)

Return the generalized singular values from the generalized singular value decomposition of A and B. See also svd.

Examples

julia> A = [1. 0.; 0. -1.]
2×2 Matrix{Float64}:
1.0   0.0
0.0  -1.0

julia> B = [0. 1.; 1. 0.]
2×2 Matrix{Float64}:
0.0  1.0
1.0  0.0

julia> svdvals(A, B)
2-element Vector{Float64}:
1.0
1.0
LinearAlgebra.svdvals!Function
svdvals!(A)

Return the singular values of A, saving space by overwriting the input. See also svdvals and svd. 

svdvals!(A, B)

Return the generalized singular values from the generalized singular value decomposition of A and B, saving space by overwriting A and B. See also svd and svdvals.

LinearAlgebra.GivensType
LinearAlgebra.Givens(i1,i2,c,s) -> G

A Givens rotation linear operator. The fields c and s represent the cosine and sine of the rotation angle, respectively. The Givens type supports left multiplication G*A and conjugated transpose right multiplication A*G'. The type doesn't have a size and can therefore be multiplied with matrices of arbitrary size as long as i2<=size(A,2) for G*A or i2<=size(A,1) for A*G'.

See also givens.

LinearAlgebra.givensFunction
givens(f::T, g::T, i1::Integer, i2::Integer) where {T} -> (G::Givens, r::T)

Computes the Givens rotation G and scalar r such that for any vector x where

x[i1] = f
x[i2] = g

the result of the multiplication

y = G*x

has the property that

y[i1] = r
y[i2] = 0

See also LinearAlgebra.Givens.

givens(A::AbstractArray, i1::Integer, i2::Integer, j::Integer) -> (G::Givens, r)

Computes the Givens rotation G and scalar r such that the result of the multiplication

B = G*A

has the property that

B[i1,j] = r
B[i2,j] = 0

See also LinearAlgebra.Givens.

givens(x::AbstractVector, i1::Integer, i2::Integer) -> (G::Givens, r)

Computes the Givens rotation G and scalar r such that the result of the multiplication

B = G*x

has the property that

B[i1] = r
B[i2] = 0

See also LinearAlgebra.Givens.

LinearAlgebra.triuFunction
triu(M)

Upper triangle of a matrix.

Examples

julia> a = fill(1.0, (4,4))
4×4 Matrix{Float64}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> triu(a)
4×4 Matrix{Float64}:
1.0  1.0  1.0  1.0
0.0  1.0  1.0  1.0
0.0  0.0  1.0  1.0
0.0  0.0  0.0  1.0
triu(M, k::Integer)

Return the upper triangle of M starting from the kth superdiagonal.

Examples

julia> a = fill(1.0, (4,4))
4×4 Matrix{Float64}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> triu(a,3)
4×4 Matrix{Float64}:
0.0  0.0  0.0  1.0
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0

julia> triu(a,-3)
4×4 Matrix{Float64}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
LinearAlgebra.triu!Function
triu!(M)

Upper triangle of a matrix, overwriting M in the process. See also triu.

triu!(M, k::Integer)

Return the upper triangle of M starting from the kth superdiagonal, overwriting M in the process.

Examples

julia> M = [1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5]
5×5 Matrix{Int64}:
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5

julia> triu!(M, 1)
5×5 Matrix{Int64}:
0  2  3  4  5
0  0  3  4  5
0  0  0  4  5
0  0  0  0  5
0  0  0  0  0
LinearAlgebra.trilFunction
tril(M)

Lower triangle of a matrix.

Examples

julia> a = fill(1.0, (4,4))
4×4 Matrix{Float64}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> tril(a)
4×4 Matrix{Float64}:
1.0  0.0  0.0  0.0
1.0  1.0  0.0  0.0
1.0  1.0  1.0  0.0
1.0  1.0  1.0  1.0
tril(M, k::Integer)

Return the lower triangle of M starting from the kth superdiagonal.

Examples

julia> a = fill(1.0, (4,4))
4×4 Matrix{Float64}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> tril(a,3)
4×4 Matrix{Float64}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> tril(a,-3)
4×4 Matrix{Float64}:
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0
1.0  0.0  0.0  0.0
LinearAlgebra.tril!Function
tril!(M)

Lower triangle of a matrix, overwriting M in the process. See also tril.

tril!(M, k::Integer)

Return the lower triangle of M starting from the kth superdiagonal, overwriting M in the process.

Examples

julia> M = [1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5; 1 2 3 4 5]
5×5 Matrix{Int64}:
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5

julia> tril!(M, 2)
5×5 Matrix{Int64}:
1  2  3  0  0
1  2  3  4  0
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5
LinearAlgebra.diagFunction
diag(M, k::Integer=0)

The kth diagonal of a matrix, as a vector.

Examples

julia> A = [1 2 3; 4 5 6; 7 8 9]
3×3 Matrix{Int64}:
1  2  3
4  5  6
7  8  9

julia> diag(A,1)
2-element Vector{Int64}:
2
6
LinearAlgebra.diagmFunction
diagm(kv::Pair{<:Integer,<:AbstractVector}...)
diagm(m::Integer, n::Integer, kv::Pair{<:Integer,<:AbstractVector}...)

Construct a matrix from Pairs of diagonals and vectors. Vector kv.second will be placed on the kv.first diagonal. By default the matrix is square and its size is inferred from kv, but a non-square size m×n (padded with zeros as needed) can be specified by passing m,n as the first arguments. For repeated diagonal indices kv.first the values in the corresponding vectors kv.second will be added.

diagm constructs a full matrix; if you want storage-efficient versions with fast arithmetic, see Diagonal, Bidiagonal Tridiagonal and SymTridiagonal.

Examples

julia> diagm(1 => [1,2,3])
4×4 Matrix{Int64}:
0  1  0  0
0  0  2  0
0  0  0  3
0  0  0  0

julia> diagm(1 => [1,2,3], -1 => [4,5])
4×4 Matrix{Int64}:
0  1  0  0
4  0  2  0
0  5  0  3
0  0  0  0

julia> diagm(1 => [1,2,3], 1 => [1,2,3])
4×4 Matrix{Int64}:
0  2  0  0
0  0  4  0
0  0  0  6
0  0  0  0
diagm(v::AbstractVector)
diagm(m::Integer, n::Integer, v::AbstractVector)

Construct a matrix with elements of the vector as diagonal elements. By default, the matrix is square and its size is given by length(v), but a non-square size m×n can be specified by passing m,n as the first arguments.

Examples

julia> diagm([1,2,3])
3×3 Matrix{Int64}:
1  0  0
0  2  0
0  0  3
LinearAlgebra.rankFunction
rank(::QRSparse{Tv,Ti}) -> Ti

Return the rank of the QR factorization

rank(S::SparseMatrixCSC{Tv,Ti}; [tol::Real]) -> Ti

Calculate rank of S by calculating its QR factorization. Values smaller than tol are considered as zero. See SPQR's manual.

rank(A::AbstractMatrix; atol::Real=0, rtol::Real=atol>0 ? 0 : n*ϵ)
rank(A::AbstractMatrix, rtol::Real)

Compute the rank of a matrix by counting how many singular values of A have magnitude greater than max(atol, rtol*σ₁) where σ₁ is A's largest singular value. atol and rtol are the absolute and relative tolerances, respectively. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of A, and ϵ is the eps of the element type of A.

Julia 1.1

The atol and rtol keyword arguments requires at least Julia 1.1. In Julia 1.0 rtol is available as a positional argument, but this will be deprecated in Julia 2.0.

Examples

julia> rank(Matrix(I, 3, 3))
3

julia> rank(diagm(0 => [1, 0, 2]))
2

julia> rank(diagm(0 => [1, 0.001, 2]), rtol=0.1)
2

julia> rank(diagm(0 => [1, 0.001, 2]), rtol=0.00001)
3

julia> rank(diagm(0 => [1, 0.001, 2]), atol=1.5)
1
LinearAlgebra.normFunction
norm(A, p::Real=2)

For any iterable container A (including arrays of any dimension) of numbers (or any element type for which norm is defined), compute the p-norm (defaulting to p=2) as if A were a vector of the corresponding length.

The p-norm is defined as

$$$\|A\|_p = \left( \sum_{i=1}^n | a_i | ^p \right)^{1/p}$$$

with $a_i$ the entries of $A$, $| a_i |$ the norm of $a_i$, and $n$ the length of $A$. Since the p-norm is computed using the norms of the entries of A, the p-norm of a vector of vectors is not compatible with the interpretation of it as a block vector in general if p != 2.

p can assume any numeric value (even though not all values produce a mathematically valid vector norm). In particular, norm(A, Inf) returns the largest value in abs.(A), whereas norm(A, -Inf) returns the smallest. If A is a matrix and p=2, then this is equivalent to the Frobenius norm.

The second argument p is not necessarily a part of the interface for norm, i.e. a custom type may only implement norm(A) without second argument.

Use opnorm to compute the operator norm of a matrix.

Examples

julia> v = [3, -2, 6]
3-element Vector{Int64}:
3
-2
6

julia> norm(v)
7.0

julia> norm(v, 1)
11.0

julia> norm(v, Inf)
6.0

julia> norm([1 2 3; 4 5 6; 7 8 9])
16.881943016134134

julia> norm([1 2 3 4 5 6 7 8 9])
16.881943016134134

julia> norm(1:9)
16.881943016134134

julia> norm(hcat(v,v), 1) == norm(vcat(v,v), 1) != norm([v,v], 1)
true

julia> norm(hcat(v,v), 2) == norm(vcat(v,v), 2) == norm([v,v], 2)
true

julia> norm(hcat(v,v), Inf) == norm(vcat(v,v), Inf) != norm([v,v], Inf)
true
norm(x::Number, p::Real=2)

For numbers, return $\left( |x|^p \right)^{1/p}$.

Examples

julia> norm(2, 1)
2.0

julia> norm(-2, 1)
2.0

julia> norm(2, 2)
2.0

julia> norm(-2, 2)
2.0

julia> norm(2, Inf)
2.0

julia> norm(-2, Inf)
2.0
LinearAlgebra.opnormFunction
opnorm(A::AbstractMatrix, p::Real=2)

Compute the operator norm (or matrix norm) induced by the vector p-norm, where valid values of p are 1, 2, or Inf. (Note that for sparse matrices, p=2 is currently not implemented.) Use norm to compute the Frobenius norm.

When p=1, the operator norm is the maximum absolute column sum of A:

$$$\|A\|_1 = \max_{1 ≤ j ≤ n} \sum_{i=1}^m | a_{ij} |$$$

with $a_{ij}$ the entries of $A$, and $m$ and $n$ its dimensions.

When p=2, the operator norm is the spectral norm, equal to the largest singular value of A.

When p=Inf, the operator norm is the maximum absolute row sum of A:

$$$\|A\|_\infty = \max_{1 ≤ i ≤ m} \sum _{j=1}^n | a_{ij} |$$$

Examples

julia> A = [1 -2 -3; 2 3 -1]
2×3 Matrix{Int64}:
1  -2  -3
2   3  -1

julia> opnorm(A, Inf)
6.0

julia> opnorm(A, 1)
5.0
opnorm(x::Number, p::Real=2)

For numbers, return $\left( |x|^p \right)^{1/p}$. This is equivalent to norm.

opnorm(A::Adjoint{<:Any,<:AbstracVector}, q::Real=2)
opnorm(A::Transpose{<:Any,<:AbstracVector}, q::Real=2)

For Adjoint/Transpose-wrapped vectors, return the operator $q$-norm of A, which is equivalent to the p-norm with value p = q/(q-1). They coincide at p = q = 2. Use norm to compute the p norm of A as a vector.

The difference in norm between a vector space and its dual arises to preserve the relationship between duality and the dot product, and the result is consistent with the operator p-norm of a 1 × n matrix.

Examples

julia> v = [1; im];

julia> vc = v';

julia> opnorm(vc, 1)
1.0

julia> norm(vc, 1)
2.0

julia> norm(v, 1)
2.0

julia> opnorm(vc, 2)
1.4142135623730951

julia> norm(vc, 2)
1.4142135623730951

julia> norm(v, 2)
1.4142135623730951

julia> opnorm(vc, Inf)
2.0

julia> norm(vc, Inf)
1.0

julia> norm(v, Inf)
1.0
LinearAlgebra.normalizeFunction
normalize(a, p::Real=2)

Normalize a so that its p-norm equals unity, i.e. norm(a, p) == 1. For scalars, this is similar to sign(a), except normalize(0) = NaN. See also normalize!, norm, and sign.

Examples

julia> a = [1,2,4];

julia> b = normalize(a)
3-element Vector{Float64}:
0.2182178902359924
0.4364357804719848
0.8728715609439696

julia> norm(b)
1.0

julia> c = normalize(a, 1)
3-element Vector{Float64}:
0.14285714285714285
0.2857142857142857
0.5714285714285714

julia> norm(c, 1)
1.0

julia> a = [1 2 4 ; 1 2 4]
2×3 Matrix{Int64}:
1  2  4
1  2  4

julia> norm(a)
6.48074069840786

julia> normalize(a)
2×3 Matrix{Float64}:
0.154303  0.308607  0.617213
0.154303  0.308607  0.617213

julia> normalize(3, 1)
1.0

julia> normalize(-8, 1)
-1.0

julia> normalize(0, 1)
NaN
LinearAlgebra.condFunction
cond(M, p::Real=2)

Condition number of the matrix M, computed using the operator p-norm. Valid values for p are 1, 2 (default), or Inf.

LinearAlgebra.condskeelFunction
condskeel(M, [x, p::Real=Inf])
$$$\kappa_S(M, p) = \left\Vert \left\vert M \right\vert \left\vert M^{-1} \right\vert \right\Vert_p \\ \kappa_S(M, x, p) = \frac{\left\Vert \left\vert M \right\vert \left\vert M^{-1} \right\vert \left\vert x \right\vert \right\Vert_p}{\left \Vert x \right \Vert_p}$$$

Skeel condition number $\kappa_S$ of the matrix M, optionally with respect to the vector x, as computed using the operator p-norm. $\left\vert M \right\vert$ denotes the matrix of (entry wise) absolute values of $M$; $\left\vert M \right\vert_{ij} = \left\vert M_{ij} \right\vert$. Valid values for p are 1, 2 and Inf (default).

This quantity is also known in the literature as the Bauer condition number, relative condition number, or componentwise relative condition number.

LinearAlgebra.trFunction
tr(M)

Matrix trace. Sums the diagonal elements of M.

Examples

julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
1  2
3  4

julia> tr(A)
5
LinearAlgebra.logdetFunction
logdet(M)

Log of matrix determinant. Equivalent to log(det(M)), but may provide increased accuracy and/or speed.

Examples

julia> M = [1 0; 2 2]
2×2 Matrix{Int64}:
1  0
2  2

julia> logdet(M)
0.6931471805599453

julia> logdet(Matrix(I, 3, 3))
0.0
LinearAlgebra.logabsdetFunction
logabsdet(M)

Log of absolute value of matrix determinant. Equivalent to (log(abs(det(M))), sign(det(M))), but may provide increased accuracy and/or speed.

Examples

julia> A = [-1. 0.; 0. 1.]
2×2 Matrix{Float64}:
-1.0  0.0
0.0  1.0

julia> det(A)
-1.0

julia> logabsdet(A)
(0.0, -1.0)

julia> B = [2. 0.; 0. 1.]
2×2 Matrix{Float64}:
2.0  0.0
0.0  1.0

julia> det(B)
2.0

julia> logabsdet(B)
(0.6931471805599453, 1.0)
Base.invMethod
inv(M)

Matrix inverse. Computes matrix N such that M * N = I, where I is the identity matrix. Computed by solving the left-division N = M \ I.

Examples

julia> M = [2 5; 1 3]
2×2 Matrix{Int64}:
2  5
1  3

julia> N = inv(M)
2×2 Matrix{Float64}:
3.0  -5.0
-1.0   2.0

julia> M*N == N*M == Matrix(I, 2, 2)
true
LinearAlgebra.pinvFunction
pinv(M; atol::Real=0, rtol::Real=atol>0 ? 0 : n*ϵ)
pinv(M, rtol::Real) = pinv(M; rtol=rtol) # to be deprecated in Julia 2.0

Computes the Moore-Penrose pseudoinverse.

For matrices M with floating point elements, it is convenient to compute the pseudoinverse by inverting only singular values greater than max(atol, rtol*σ₁) where σ₁ is the largest singular value of M.

The optimal choice of absolute (atol) and relative tolerance (rtol) varies both with the value of M and the intended application of the pseudoinverse. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of M, and ϵ is the eps of the element type of M.

For inverting dense ill-conditioned matrices in a least-squares sense, rtol = sqrt(eps(real(float(oneunit(eltype(M)))))) is recommended.

For more information, see [issue8859], [B96], [S84], [KY88].

Examples

julia> M = [1.5 1.3; 1.2 1.9]
2×2 Matrix{Float64}:
1.5  1.3
1.2  1.9

julia> N = pinv(M)
2×2 Matrix{Float64}:
1.47287   -1.00775
-0.930233   1.16279

julia> M * N
2×2 Matrix{Float64}:
1.0          -2.22045e-16
4.44089e-16   1.0
LinearAlgebra.nullspaceFunction
nullspace(M; atol::Real=0, rtol::Real=atol>0 ? 0 : n*ϵ)
nullspace(M, rtol::Real) = nullspace(M; rtol=rtol) # to be deprecated in Julia 2.0

Computes a basis for the nullspace of M by including the singular vectors of M whose singular values have magnitudes smaller than max(atol, rtol*σ₁), where σ₁ is M's largest singular value.

By default, the relative tolerance rtol is n*ϵ, where n is the size of the smallest dimension of M, and ϵ is the eps of the element type of M.

Examples

julia> M = [1 0 0; 0 1 0; 0 0 0]
3×3 Matrix{Int64}:
1  0  0
0  1  0
0  0  0

julia> nullspace(M)
3×1 Matrix{Float64}:
0.0
0.0
1.0

julia> nullspace(M, rtol=3)
3×3 Matrix{Float64}:
0.0  1.0  0.0
1.0  0.0  0.0
0.0  0.0  1.0

julia> nullspace(M, atol=0.95)
3×1 Matrix{Float64}:
0.0
0.0
1.0
Base.kronFunction
kron(A, B)

Computes the Kronecker product of two vectors, matrices or numbers.

For real vectors v and w, the Kronecker product is related to the outer product by kron(v,w) == vec(w * transpose(v)) or w * transpose(v) == reshape(kron(v,w), (length(w), length(v))). Note how the ordering of v and w differs on the left and right of these expressions (due to column-major storage). For complex vectors, the outer product w * v' also differs by conjugation of v.

Examples

julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
1  2
3  4

julia> B = [im 1; 1 -im]
2×2 Matrix{Complex{Int64}}:
0+1im  1+0im
1+0im  0-1im

julia> kron(A, B)
4×4 Matrix{Complex{Int64}}:
0+1im  1+0im  0+2im  2+0im
1+0im  0-1im  2+0im  0-2im
0+3im  3+0im  0+4im  4+0im
3+0im  0-3im  4+0im  0-4im

julia> v = [1, 2]; w = [3, 4, 5];

julia> w*transpose(v)
3×2 Matrix{Int64}:
3   6
4   8
5  10

julia> reshape(kron(v,w), (length(w), length(v)))
3×2 Matrix{Int64}:
3   6
4   8
5  10
Base.kron!Function
kron!(C, A, B)

Computes the Kronecker product of A and B and stores the result in C, overwriting the existing content of C. This is the in-place version of kron.

Julia 1.6

This function requires Julia 1.6 or later.

Base.expMethod
exp(A::AbstractMatrix)

Compute the matrix exponential of A, defined by

$$$e^A = \sum_{n=0}^{\infty} \frac{A^n}{n!}.$$$

For symmetric or Hermitian A, an eigendecomposition (eigen) is used, otherwise the scaling and squaring algorithm (see [H05]) is chosen.

Examples

julia> A = Matrix(1.0I, 2, 2)
2×2 Matrix{Float64}:
1.0  0.0
0.0  1.0

julia> exp(A)
2×2 Matrix{Float64}:
2.71828  0.0
0.0      2.71828
Base.cisMethod
cis(A::AbstractMatrix)

More efficient method for exp(im*A) of square matrix A (especially if A is Hermitian or real-Symmetric).

See also cispi, sincos, exp.

Julia 1.7

Support for using cis with matrices was added in Julia 1.7.

Examples

julia> cis([π 0; 0 π]) ≈ -I
true
Base.:^Method
^(A::AbstractMatrix, p::Number)

Matrix power, equivalent to $\exp(p\log(A))$

Examples

julia> [1 2; 0 3]^3
2×2 Matrix{Int64}:
1  26
0  27
Base.:^Method
^(b::Number, A::AbstractMatrix)

Matrix exponential, equivalent to $\exp(\log(b)A)$.

Julia 1.1

Support for raising Irrational numbers (like ℯ) to a matrix was added in Julia 1.1.

Examples

julia> 2^[1 2; 0 3]
2×2 Matrix{Float64}:
2.0  6.0
0.0  8.0

julia> ℯ^[1 2; 0 3]
2×2 Matrix{Float64}:
2.71828  17.3673
0.0      20.0855
Base.logMethod
log(A::AbstractMatrix)

If A has no negative real eigenvalue, compute the principal matrix logarithm of A, i.e. the unique matrix $X$ such that $e^X = A$ and $-\pi < Im(\lambda) < \pi$ for all the eigenvalues $\lambda$ of $X$. If A has nonpositive eigenvalues, a nonprincipal matrix function is returned whenever possible.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used, if A is triangular an improved version of the inverse scaling and squaring method is employed (see [AH12] and [AHR13]). If A is real with no negative eigenvalues, then the real Schur form is computed. Otherwise, the complex Schur form is computed. Then the upper (quasi-)triangular algorithm in [AHR13] is used on the upper (quasi-)triangular factor.

Examples

julia> A = Matrix(2.7182818*I, 2, 2)
2×2 Matrix{Float64}:
2.71828  0.0
0.0      2.71828

julia> log(A)
2×2 Matrix{Float64}:
1.0  0.0
0.0  1.0
Base.sqrtMethod
sqrt(x)

Return $\sqrt{x}$. Throws DomainError for negative Real arguments. Use complex negative arguments instead. The prefix operator √ is equivalent to sqrt.

See also: hypot.

Examples

julia> sqrt(big(81))
9.0

julia> sqrt(big(-81))
ERROR: DomainError with -81.0:
NaN result for non-NaN input.
Stacktrace:
[1] sqrt(::BigFloat) at ./mpfr.jl:501
[...]

julia> sqrt(big(complex(-81)))
0.0 + 9.0im

julia> .√(1:4)
4-element Vector{Float64}:
1.0
1.4142135623730951
1.7320508075688772
2.0
source
sqrt(A::AbstractMatrix)

If A has no negative real eigenvalues, compute the principal matrix square root of A, that is the unique matrix $X$ with eigenvalues having positive real part such that $X^2 = A$. Otherwise, a nonprincipal square root is returned.

If A is real-symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the square root. For such matrices, eigenvalues λ that appear to be slightly negative due to roundoff errors are treated as if they were zero. More precisely, matrices with all eigenvalues ≥ -rtol*(max |λ|) are treated as semidefinite (yielding a Hermitian square root), with negative eigenvalues taken to be zero. rtol is a keyword argument to sqrt (in the Hermitian/real-symmetric case only) that defaults to machine precision scaled by size(A,1).

Otherwise, the square root is determined by means of the Björck-Hammarling method [BH83], which computes the complex Schur form (schur) and then the complex square root of the triangular factor. If a real square root exists, then an extension of this method [H87] that computes the real Schur form and then the real square root of the quasi-triangular factor is instead used.

Examples

julia> A = [4 0; 0 4]
2×2 Matrix{Int64}:
4  0
0  4

julia> sqrt(A)
2×2 Matrix{Float64}:
2.0  0.0
0.0  2.0
Base.cosMethod
cos(A::AbstractMatrix)

Compute the matrix cosine of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the cosine. Otherwise, the cosine is determined by calling exp.

Examples

julia> cos(fill(1.0, (2,2)))
2×2 Matrix{Float64}:
0.291927  -0.708073
-0.708073   0.291927
Base.sinMethod
sin(A::AbstractMatrix)

Compute the matrix sine of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the sine. Otherwise, the sine is determined by calling exp.

Examples

julia> sin(fill(1.0, (2,2)))
2×2 Matrix{Float64}:
0.454649  0.454649
0.454649  0.454649
Base.Math.sincosMethod
sincos(A::AbstractMatrix)

Compute the matrix sine and cosine of a square matrix A.

Examples

julia> S, C = sincos(fill(1.0, (2,2)));

julia> S
2×2 Matrix{Float64}:
0.454649  0.454649
0.454649  0.454649

julia> C
2×2 Matrix{Float64}:
0.291927  -0.708073
-0.708073   0.291927
Base.tanMethod
tan(A::AbstractMatrix)

Compute the matrix tangent of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the tangent. Otherwise, the tangent is determined by calling exp.

Examples

julia> tan(fill(1.0, (2,2)))
2×2 Matrix{Float64}:
-1.09252  -1.09252
-1.09252  -1.09252
Base.Math.secMethod
sec(A::AbstractMatrix)

Compute the matrix secant of a square matrix A.

Base.Math.cscMethod
csc(A::AbstractMatrix)

Compute the matrix cosecant of a square matrix A.

Base.Math.cotMethod
cot(A::AbstractMatrix)

Compute the matrix cotangent of a square matrix A.

Base.coshMethod
cosh(A::AbstractMatrix)

Compute the matrix hyperbolic cosine of a square matrix A.

Base.sinhMethod
sinh(A::AbstractMatrix)

Compute the matrix hyperbolic sine of a square matrix A.

Base.tanhMethod
tanh(A::AbstractMatrix)

Compute the matrix hyperbolic tangent of a square matrix A.

Base.Math.sechMethod
sech(A::AbstractMatrix)

Compute the matrix hyperbolic secant of square matrix A.

Base.Math.cschMethod
csch(A::AbstractMatrix)

Compute the matrix hyperbolic cosecant of square matrix A.

Base.Math.cothMethod
coth(A::AbstractMatrix)

Compute the matrix hyperbolic cotangent of square matrix A.

Base.acosMethod
acos(A::AbstractMatrix)

Compute the inverse matrix cosine of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the inverse cosine. Otherwise, the inverse cosine is determined by using log and sqrt. For the theory and logarithmic formulas used to compute this function, see [AH16_1].

Examples

julia> acos(cos([0.5 0.1; -0.2 0.3]))
2×2 Matrix{ComplexF64}:
0.5-8.32667e-17im  0.1+0.0im
-0.2+2.63678e-16im  0.3-3.46945e-16im
Base.asinMethod
asin(A::AbstractMatrix)

Compute the inverse matrix sine of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the inverse sine. Otherwise, the inverse sine is determined by using log and sqrt. For the theory and logarithmic formulas used to compute this function, see [AH16_2].

Examples

julia> asin(sin([0.5 0.1; -0.2 0.3]))
2×2 Matrix{ComplexF64}:
0.5-4.16334e-17im  0.1-5.55112e-17im
-0.2+9.71445e-17im  0.3-1.249e-16im
Base.atanMethod
atan(A::AbstractMatrix)

Compute the inverse matrix tangent of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the inverse tangent. Otherwise, the inverse tangent is determined by using log. For the theory and logarithmic formulas used to compute this function, see [AH16_3].

Examples

julia> atan(tan([0.5 0.1; -0.2 0.3]))
2×2 Matrix{ComplexF64}:
0.5+1.38778e-17im  0.1-2.77556e-17im
-0.2+6.93889e-17im  0.3-4.16334e-17im
Base.Math.asecMethod
asec(A::AbstractMatrix)

Compute the inverse matrix secant of A.

Base.Math.acscMethod
acsc(A::AbstractMatrix)

Compute the inverse matrix cosecant of A.

Base.Math.acotMethod
acot(A::AbstractMatrix)

Compute the inverse matrix cotangent of A.

Base.acoshMethod
acosh(A::AbstractMatrix)

Compute the inverse hyperbolic matrix cosine of a square matrix A. For the theory and logarithmic formulas used to compute this function, see [AH16_4].

Base.asinhMethod
asinh(A::AbstractMatrix)

Compute the inverse hyperbolic matrix sine of a square matrix A. For the theory and logarithmic formulas used to compute this function, see [AH16_5].

Base.atanhMethod
atanh(A::AbstractMatrix)

Compute the inverse hyperbolic matrix tangent of a square matrix A. For the theory and logarithmic formulas used to compute this function, see [AH16_6].

Base.Math.asechMethod
asech(A::AbstractMatrix)

Compute the inverse matrix hyperbolic secant of A.

Base.Math.acschMethod
acsch(A::AbstractMatrix)

Compute the inverse matrix hyperbolic cosecant of A.

Base.Math.acothMethod
acoth(A::AbstractMatrix)

Compute the inverse matrix hyperbolic cotangent of A.

LinearAlgebra.lyapFunction
lyap(A, C)

Computes the solution X to the continuous Lyapunov equation AX + XA' + C = 0, where no eigenvalue of A has a zero real part and no two eigenvalues are negative complex conjugates of each other.

Examples

julia> A = [3. 4.; 5. 6]
2×2 Matrix{Float64}:
3.0  4.0
5.0  6.0

julia> B = [1. 1.; 1. 2.]
2×2 Matrix{Float64}:
1.0  1.0
1.0  2.0

julia> X = lyap(A, B)
2×2 Matrix{Float64}:
0.5  -0.5
-0.5   0.25

julia> A*X + X*A' ≈ -B
true
LinearAlgebra.sylvesterFunction
sylvester(A, B, C)

Computes the solution X to the Sylvester equation AX + XB + C = 0, where A, B and C have compatible dimensions and A and -B have no eigenvalues with equal real part.

Examples

julia> A = [3. 4.; 5. 6]
2×2 Matrix{Float64}:
3.0  4.0
5.0  6.0

julia> B = [1. 1.; 1. 2.]
2×2 Matrix{Float64}:
1.0  1.0
1.0  2.0

julia> C = [1. 2.; -2. 1]
2×2 Matrix{Float64}:
1.0  2.0
-2.0  1.0

julia> X = sylvester(A, B, C)
2×2 Matrix{Float64}:
-4.46667   1.93333
3.73333  -1.8

julia> A*X + X*B ≈ -C
true
LinearAlgebra.issuccessFunction
issuccess(F::Factorization)

Test that a factorization of a matrix succeeded.

Julia 1.6

issuccess(::CholeskyPivoted) requires Julia 1.6 or later.

julia> F = cholesky([1 0; 0 1]);

julia> issuccess(F)
true

julia> F = lu([1 0; 0 0]; check = false);

julia> issuccess(F)
false
LinearAlgebra.issymmetricFunction
issymmetric(A) -> Bool

Test whether a matrix is symmetric.

Examples

julia> a = [1 2; 2 -1]
2×2 Matrix{Int64}:
1   2
2  -1

julia> issymmetric(a)
true

julia> b = [1 im; -im 1]
2×2 Matrix{Complex{Int64}}:
1+0im  0+1im
0-1im  1+0im

julia> issymmetric(b)
false
LinearAlgebra.isposdef!Function
isposdef!(A) -> Bool

Test whether a matrix is positive definite (and Hermitian) by trying to perform a Cholesky factorization of A, overwriting A in the process. See also isposdef.

Examples

julia> A = [1. 2.; 2. 50.];

julia> isposdef!(A)
true

julia> A
2×2 Matrix{Float64}:
1.0  2.0
2.0  6.78233
LinearAlgebra.istrilFunction
istril(A::AbstractMatrix, k::Integer = 0) -> Bool

Test whether A is lower triangular starting from the kth superdiagonal.

Examples

julia> a = [1 2; 2 -1]
2×2 Matrix{Int64}:
1   2
2  -1

julia> istril(a)
false

julia> istril(a, 1)
true

julia> b = [1 0; -im -1]
2×2 Matrix{Complex{Int64}}:
1+0im   0+0im
0-1im  -1+0im

julia> istril(b)
true

julia> istril(b, -1)
false
LinearAlgebra.istriuFunction
istriu(A::AbstractMatrix, k::Integer = 0) -> Bool

Test whether A is upper triangular starting from the kth superdiagonal.

Examples

julia> a = [1 2; 2 -1]
2×2 Matrix{Int64}:
1   2
2  -1

julia> istriu(a)
false

julia> istriu(a, -1)
true

julia> b = [1 im; 0 -1]
2×2 Matrix{Complex{Int64}}:
1+0im   0+1im
0+0im  -1+0im

julia> istriu(b)
true

julia> istriu(b, 1)
false
LinearAlgebra.isdiagFunction
isdiag(A) -> Bool

Test whether a matrix is diagonal in the sense that iszero(A[i,j]) is true unless i == j. Note that it is not necessary for A to be square; if you would also like to check that, you need to check that size(A, 1) == size(A, 2).

Examples

julia> a = [1 2; 2 -1]
2×2 Matrix{Int64}:
1   2
2  -1

julia> isdiag(a)
false

julia> b = [im 0; 0 -im]
2×2 Matrix{Complex{Int64}}:
0+1im  0+0im
0+0im  0-1im

julia> isdiag(b)
true

julia> c = [1 0 0; 0 2 0]
2×3 Matrix{Int64}:
1  0  0
0  2  0

julia> isdiag(c)
true

julia> d = [1 0 0; 0 2 3]
2×3 Matrix{Int64}:
1  0  0
0  2  3

julia> isdiag(d)
false
LinearAlgebra.ishermitianFunction
ishermitian(A) -> Bool

Test whether a matrix is Hermitian.

Examples

julia> a = [1 2; 2 -1]
2×2 Matrix{Int64}:
1   2
2  -1

julia> ishermitian(a)
true

julia> b = [1 im; -im 1]
2×2 Matrix{Complex{Int64}}:
1+0im  0+1im
0-1im  1+0im

julia> ishermitian(b)
true
Base.transposeFunction
transpose(A)

Lazy transpose. Mutating the returned object should appropriately mutate A. Often, but not always, yields Transpose(A), where Transpose is a lazy transpose wrapper. Note that this operation is recursive.

This operation is intended for linear algebra usage - for general data manipulation see permutedims, which is non-recursive.

Examples

julia> A = [3 2; 0 0]
2×2 Matrix{Int64}:
3  2
0  0

julia> B = transpose(A)
2×2 transpose(::Matrix{Int64}) with eltype Int64:
3  0
2  0

julia> B isa Transpose
true

julia> transpose(B) === A # the transpose of a transpose unwraps the parent
true

julia> Transpose(B) # however, the constructor always wraps its argument
2×2 transpose(transpose(::Matrix{Int64})) with eltype Int64:
3  2
0  0

julia> B[1,2] = 4; # modifying B will modify A automatically

julia> A
2×2 Matrix{Int64}:
3  2
4  0

For complex matrices, the adjoint operation is equivalent to a conjugate-transpose.

julia> A = reshape([Complex(x, x) for x in 1:4], 2, 2)
2×2 Matrix{Complex{Int64}}:
1+1im  3+3im
2+2im  4+4im

julia> adjoint(A) == conj(transpose(A))
true

The transpose of an AbstractVector is a row-vector:

julia> v = [1,2,3]
3-element Vector{Int64}:
1
2
3

julia> transpose(v) # returns a row-vector
1×3 transpose(::Vector{Int64}) with eltype Int64:
1  2  3

julia> transpose(v) * v # compute the dot product
14

For a matrix of matrices, the individual blocks are recursively operated on:

julia> C = [1 3; 2 4]
2×2 Matrix{Int64}:
1  3
2  4

julia> D = reshape([C, 2C, 3C, 4C], 2, 2) # construct a block matrix
2×2 Matrix{Matrix{Int64}}:
[1 3; 2 4]  [3 9; 6 12]
[2 6; 4 8]  [4 12; 8 16]

julia> transpose(D) # blocks are recursively transposed
2×2 transpose(::Matrix{Matrix{Int64}}) with eltype Transpose{Int64, Matrix{Int64}}:
[1 2; 3 4]   [2 4; 6 8]
[3 6; 9 12]  [4 8; 12 16]
LinearAlgebra.transpose!Function
transpose!(X::AbstractSparseMatrixCSC{Tv,Ti}, A::AbstractSparseMatrixCSC{Tv,Ti}) where {Tv,Ti}

Transpose the matrix A and stores it in the matrix X. size(X) must be equal to size(transpose(A)). No additonal memory is allocated other than resizing the rowval and nzval of X, if needed.

See halfperm!

transpose!(dest,src)

Transpose array src and store the result in the preallocated array dest, which should have a size corresponding to (size(src,2),size(src,1)). No in-place transposition is supported and unexpected results will happen if src and dest have overlapping memory regions.

Examples

julia> A = [3+2im 9+2im; 8+7im  4+6im]
2×2 Matrix{Complex{Int64}}:
3+2im  9+2im
8+7im  4+6im

julia> B = zeros(Complex{Int64}, 2, 2)
2×2 Matrix{Complex{Int64}}:
0+0im  0+0im
0+0im  0+0im

julia> transpose!(B, A);

julia> B
2×2 Matrix{Complex{Int64}}:
3+2im  8+7im
9+2im  4+6im

julia> A
2×2 Matrix{Complex{Int64}}:
3+2im  9+2im
8+7im  4+6im
LinearAlgebra.TransposeType
Transpose

Lazy wrapper type for a transpose view of the underlying linear algebra object, usually an AbstractVector/AbstractMatrix, but also some Factorization, for instance. Usually, the Transpose constructor should not be called directly, use transpose instead. To materialize the view use copy.

This type is intended for linear algebra usage - for general data manipulation see permutedims.

Examples

julia> A = [2 3; 0 0]
2×2 Matrix{Int64}:
2  3
0  0

julia> Transpose(A)
2×2 transpose(::Matrix{Int64}) with eltype Int64:
2  0
3  0
Base.adjointFunction
A'
adjoint(A)

Lazy adjoint (conjugate transposition). Note that adjoint is applied recursively to elements.

For number types, adjoint returns the complex conjugate, and therefore it is equivalent to the identity function for real numbers.

This operation is intended for linear algebra usage - for general data manipulation see permutedims.

Examples

julia> A = [3+2im 9+2im; 0  0]
2×2 Matrix{Complex{Int64}}:
3+2im  9+2im
0+0im  0+0im

julia> B = A' # equivalently adjoint(A)
2×2 adjoint(::Matrix{Complex{Int64}}) with eltype Complex{Int64}:
3-2im  0+0im
9-2im  0+0im

julia> B isa Adjoint
true

julia> adjoint(B) === A # the adjoint of an adjoint unwraps the parent
true

julia> Adjoint(B) # however, the constructor always wraps its argument
3+2im  9+2im
0+0im  0+0im

julia> B[1,2] = 4 + 5im; # modifying B will modify A automatically

julia> A
2×2 Matrix{Complex{Int64}}:
3+2im  9+2im
4-5im  0+0im

For real matrices, the adjoint operation is equivalent to a transpose.

julia> A = reshape([x for x in 1:4], 2, 2)
2×2 Matrix{Int64}:
1  3
2  4

julia> A'
2×2 adjoint(::Matrix{Int64}) with eltype Int64:
1  2
3  4

julia> adjoint(A) == transpose(A)
true

The adjoint of an AbstractVector is a row-vector:

julia> x = [3, 4im]
2-element Vector{Complex{Int64}}:
3 + 0im
0 + 4im

julia> x'
1×2 adjoint(::Vector{Complex{Int64}}) with eltype Complex{Int64}:
3+0im  0-4im

julia> x'x # compute the dot product, equivalently x' * x
25 + 0im

For a matrix of matrices, the individual blocks are recursively operated on:

julia> A = reshape([x + im*x for x in 1:4], 2, 2)
2×2 Matrix{Complex{Int64}}:
1+1im  3+3im
2+2im  4+4im

julia> C = reshape([A, 2A, 3A, 4A], 2, 2)
2×2 Matrix{Matrix{Complex{Int64}}}:
[1+1im 3+3im; 2+2im 4+4im]  [3+3im 9+9im; 6+6im 12+12im]
[2+2im 6+6im; 4+4im 8+8im]  [4+4im 12+12im; 8+8im 16+16im]

julia> C'
[1-1im 2-2im; 3-3im 4-4im]    [2-2im 4-4im; 6-6im 8-8im]
[3-3im 6-6im; 9-9im 12-12im]  [4-4im 8-8im; 12-12im 16-16im]
LinearAlgebra.adjoint!Function
adjoint!(X::AbstractSparseMatrixCSC{Tv,Ti}, A::AbstractSparseMatrixCSC{Tv,Ti}) where {Tv,Ti}

Transpose the matrix A and stores the adjoint of the elements in the matrix X. size(X) must be equal to size(transpose(A)). No additonal memory is allocated other than resizing the rowval and nzval of X, if needed.

See halfperm!

adjoint!(dest,src)

Conjugate transpose array src and store the result in the preallocated array dest, which should have a size corresponding to (size(src,2),size(src,1)). No in-place transposition is supported and unexpected results will happen if src and dest have overlapping memory regions.

Examples

julia> A = [3+2im 9+2im; 8+7im  4+6im]
2×2 Matrix{Complex{Int64}}:
3+2im  9+2im
8+7im  4+6im

julia> B = zeros(Complex{Int64}, 2, 2)
2×2 Matrix{Complex{Int64}}:
0+0im  0+0im
0+0im  0+0im

julia> B
2×2 Matrix{Complex{Int64}}:
3-2im  8-7im
9-2im  4-6im

julia> A
2×2 Matrix{Complex{Int64}}:
3+2im  9+2im
8+7im  4+6im
LinearAlgebra.AdjointType
Adjoint

Lazy wrapper type for an adjoint view of the underlying linear algebra object, usually an AbstractVector/AbstractMatrix, but also some Factorization, for instance. Usually, the Adjoint constructor should not be called directly, use adjoint instead. To materialize the view use copy.

This type is intended for linear algebra usage - for general data manipulation see permutedims.

Examples

julia> A = [3+2im 9+2im; 0 0]
2×2 Matrix{Complex{Int64}}:
3+2im  9+2im
0+0im  0+0im

2×2 adjoint(::Matrix{Complex{Int64}}) with eltype Complex{Int64}:
3-2im  0+0im
9-2im  0+0im
Base.copyMethod
copy(A::Transpose)
copy(A::Adjoint)

Eagerly evaluate the lazy matrix transpose/adjoint. Note that the transposition is applied recursively to elements.

This operation is intended for linear algebra usage - for general data manipulation see permutedims, which is non-recursive.

Examples

julia> A = [1 2im; -3im 4]
2×2 Matrix{Complex{Int64}}:
1+0im  0+2im
0-3im  4+0im

julia> T = transpose(A)
2×2 transpose(::Matrix{Complex{Int64}}) with eltype Complex{Int64}:
1+0im  0-3im
0+2im  4+0im

julia> copy(T)
2×2 Matrix{Complex{Int64}}:
1+0im  0-3im
0+2im  4+0im
LinearAlgebra.stride1Function
stride1(A) -> Int

Return the distance between successive array elements in dimension 1 in units of element size.

Examples

julia> A = [1,2,3,4]
4-element Vector{Int64}:
1
2
3
4

julia> LinearAlgebra.stride1(A)
1

julia> B = view(A, 2:2:4)
2-element view(::Vector{Int64}, 2:2:4) with eltype Int64:
2
4

julia> LinearAlgebra.stride1(B)
2
LinearAlgebra.checksquareFunction
LinearAlgebra.checksquare(A)

Check that a matrix is square, then return its common dimension. For multiple arguments, return a vector.

Examples

julia> A = fill(1, (4,4)); B = fill(1, (5,5));

julia> LinearAlgebra.checksquare(A, B)
2-element Vector{Int64}:
4
5
LinearAlgebra.peakflopsFunction
LinearAlgebra.peakflops(n::Integer=2000; parallel::Bool=false)

peakflops computes the peak flop rate of the computer by using double precision gemm!. By default, if no arguments are specified, it multiplies a matrix of size n x n, where n = 2000. If the underlying BLAS is using multiple threads, higher flop rates are realized. The number of BLAS threads can be set with BLAS.set_num_threads(n).

If the keyword argument parallel is set to true, peakflops is run in parallel on all the worker processors. The flop rate of the entire parallel computer is returned. When running in parallel, only 1 BLAS thread is used. The argument n still refers to the size of the problem that is solved on each processor.

Julia 1.1

This function requires at least Julia 1.1. In Julia 1.0 it is available from the standard library InteractiveUtils.

## Low-level matrix operations

In many cases there are in-place versions of matrix operations that allow you to supply a pre-allocated output vector or matrix. This is useful when optimizing critical code in order to avoid the overhead of repeated allocations. These in-place operations are suffixed with ! below (e.g. mul!) according to the usual Julia convention.

LinearAlgebra.mul!Function
mul!(Y, A, B) -> Y

Calculates the matrix-matrix or matrix-vector product $AB$ and stores the result in Y, overwriting the existing value of Y. Note that Y must not be aliased with either A or B.

Examples

julia> A=[1.0 2.0; 3.0 4.0]; B=[1.0 1.0; 1.0 1.0]; Y = similar(B); mul!(Y, A, B);

julia> Y
2×2 Matrix{Float64}:
3.0  3.0
7.0  7.0

Implementation

For custom matrix and vector types, it is recommended to implement 5-argument mul! rather than implementing 3-argument mul! directly if possible.

mul!(C, A, B, α, β) -> C

Combined inplace matrix-matrix or matrix-vector multiply-add $A B α + C β$. The result is stored in C by overwriting it. Note that C must not be aliased with either A or B.

Julia 1.3

Five-argument mul! requires at least Julia 1.3.

Examples

julia> A=[1.0 2.0; 3.0 4.0]; B=[1.0 1.0; 1.0 1.0]; C=[1.0 2.0; 3.0 4.0];

julia> mul!(C, A, B, 100.0, 10.0) === C
true

julia> C
2×2 Matrix{Float64}:
310.0  320.0
730.0  740.0
LinearAlgebra.lmul!Function
lmul!(a::Number, B::AbstractArray)

Scale an array B by a scalar a overwriting B in-place. Use rmul! to multiply scalar from right. The scaling operation respects the semantics of the multiplication * between a and an element of B. In particular, this also applies to multiplication involving non-finite numbers such as NaN and ±Inf.

Julia 1.1

Prior to Julia 1.1, NaN and ±Inf entries in B were treated inconsistently.

Examples

julia> B = [1 2; 3 4]
2×2 Matrix{Int64}:
1  2
3  4

julia> lmul!(2, B)
2×2 Matrix{Int64}:
2  4
6  8

julia> lmul!(0.0, [Inf])
1-element Vector{Float64}:
NaN
lmul!(A, B)

Calculate the matrix-matrix product $AB$, overwriting B, and return the result. Here, A must be of special matrix type, like, e.g., Diagonal, UpperTriangular or LowerTriangular, or of some orthogonal type, see QR.

Examples

julia> B = [0 1; 1 0];

julia> A = UpperTriangular([1 2; 0 3]);

julia> lmul!(A, B);

julia> B
2×2 Matrix{Int64}:
2  1
3  0

julia> B = [1.0 2.0; 3.0 4.0];

julia> F = qr([0 1; -1 0]);

julia> lmul!(F.Q, B)
2×2 Matrix{Float64}:
3.0  4.0
1.0  2.0
LinearAlgebra.rmul!Function
rmul!(A::AbstractArray, b::Number)

Scale an array A by a scalar b overwriting A in-place. Use lmul! to multiply scalar from left. The scaling operation respects the semantics of the multiplication * between an element of A and b. In particular, this also applies to multiplication involving non-finite numbers such as NaN and ±Inf.

Julia 1.1

Prior to Julia 1.1, NaN and ±Inf entries in A were treated inconsistently.

Examples

julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
1  2
3  4

julia> rmul!(A, 2)
2×2 Matrix{Int64}:
2  4
6  8

julia> rmul!([NaN], 0.0)
1-element Vector{Float64}:
NaN
rmul!(A, B)

Calculate the matrix-matrix product $AB$, overwriting A, and return the result. Here, B must be of special matrix type, like, e.g., Diagonal, UpperTriangular or LowerTriangular, or of some orthogonal type, see QR.

Examples

julia> A = [0 1; 1 0];

julia> B = UpperTriangular([1 2; 0 3]);

julia> rmul!(A, B);

julia> A
2×2 Matrix{Int64}:
0  3
1  2

julia> A = [1.0 2.0; 3.0 4.0];

julia> F = qr([0 1; -1 0]);

julia> rmul!(A, F.Q)
2×2 Matrix{Float64}:
2.0  1.0
4.0  3.0
LinearAlgebra.ldiv!Function
ldiv!(Y, A, B) -> Y

Compute A \ B in-place and store the result in Y, returning the result.

The argument A should not be a matrix. Rather, instead of matrices it should be a factorization object (e.g. produced by factorize or cholesky). The reason for this is that factorization itself is both expensive and typically allocates memory (although it can also be done in-place via, e.g., lu!), and performance-critical situations requiring ldiv! usually also require fine-grained control over the factorization of A.

Note

Certain structured matrix types, such as Diagonal and UpperTriangular, are permitted, as these are already in a factorized form

Examples

julia> A = [1 2.2 4; 3.1 0.2 3; 4 1 2];

julia> X = [1; 2.5; 3];

julia> Y = zero(X);

julia> ldiv!(Y, qr(A), X);

julia> Y
3-element Vector{Float64}:
0.7128099173553719
-0.051652892561983674
0.10020661157024757

julia> A\X
3-element Vector{Float64}:
0.7128099173553719
-0.05165289256198333
0.10020661157024785
ldiv!(A, B)

Compute A \ B in-place and overwriting B to store the result.

The argument A should not be a matrix. Rather, instead of matrices it should be a factorization object (e.g. produced by factorize or cholesky). The reason for this is that factorization itself is both expensive and typically allocates memory (although it can also be done in-place via, e.g., lu!), and performance-critical situations requiring ldiv! usually also require fine-grained control over the factorization of A.

Note

Certain structured matrix types, such as Diagonal and UpperTriangular, are permitted, as these are already in a factorized form

Examples

julia> A = [1 2.2 4; 3.1 0.2 3; 4 1 2];

julia> X = [1; 2.5; 3];

julia> Y = copy(X);

julia> ldiv!(qr(A), X);

julia> X
3-element Vector{Float64}:
0.7128099173553719
-0.051652892561983674
0.10020661157024757

julia> A\Y
3-element Vector{Float64}:
0.7128099173553719
-0.05165289256198333
0.10020661157024785
ldiv!(a::Number, B::AbstractArray)

Divide each entry in an array B by a scalar a overwriting B in-place. Use rdiv! to divide scalar from right.

Examples

julia> B = [1.0 2.0; 3.0 4.0]
2×2 Matrix{Float64}:
1.0  2.0
3.0  4.0

julia> ldiv!(2.0, B)
2×2 Matrix{Float64}:
0.5  1.0
1.5  2.0
LinearAlgebra.rdiv!Function
rdiv!(A, B)

Compute A / B in-place and overwriting A to store the result.

The argument B should not be a matrix. Rather, instead of matrices it should be a factorization object (e.g. produced by factorize or cholesky). The reason for this is that factorization itself is both expensive and typically allocates memory (although it can also be done in-place via, e.g., lu!), and performance-critical situations requiring rdiv! usually also require fine-grained control over the factorization of B.

Note

Certain structured matrix types, such as Diagonal and UpperTriangular, are permitted, as these are already in a factorized form

rdiv!(A::AbstractArray, b::Number)

Divide each entry in an array A by a scalar b overwriting A in-place. Use ldiv! to divide scalar from left.

Examples

julia> A = [1.0 2.0; 3.0 4.0]
2×2 Matrix{Float64}:
1.0  2.0
3.0  4.0

julia> rdiv!(A, 2.0)
2×2 Matrix{Float64}:
0.5  1.0
1.5  2.0

## BLAS functions

In Julia (as in much of scientific computation), dense linear-algebra operations are based on the LAPACK library, which in turn is built on top of basic linear-algebra building-blocks known as the BLAS. There are highly optimized implementations of BLAS available for every computer architecture, and sometimes in high-performance linear algebra routines it is useful to call the BLAS functions directly.

LinearAlgebra.BLAS provides wrappers for some of the BLAS functions. Those BLAS functions that overwrite one of the input arrays have names ending in '!'. Usually, a BLAS function has four methods defined, for Float32, Float64, ComplexF32, and ComplexF64 arrays.

### BLAS character arguments

Many BLAS functions accept arguments that determine whether to transpose an argument (trans), which triangle of a matrix to reference (uplo or ul), whether the diagonal of a triangular matrix can be assumed to be all ones (dA) or which side of a matrix multiplication the input argument belongs on (side). The possibilities are:

#### Multiplication order

sideMeaning
'L'The argument goes on the left side of a matrix-matrix operation.
'R'The argument goes on the right side of a matrix-matrix operation.

#### Triangle referencing

uplo/ulMeaning
'U'Only the upper triangle of the matrix will be used.
'L'Only the lower triangle of the matrix will be used.

#### Transposition operation

trans/tXMeaning
'N'The input matrix X is not transposed or conjugated.
'T'The input matrix X will be transposed.
'C'The input matrix X will be conjugated and transposed.

#### Unit diagonal

diag/dXMeaning
'N'The diagonal values of the matrix X will be read.
'U'The diagonal of the matrix X is assumed to be all ones.
LinearAlgebra.BLAS.set_num_threadsFunction
set_num_threads(n::Integer)
set_num_threads(::Nothing)

Set the number of threads the BLAS library should use equal to n::Integer.

Also accepts nothing, in which case julia tries to guess the default number of threads. Passing nothing is discouraged and mainly exists for historical reasons.

BLAS functions can be divided into three groups, also called three levels, depending on when they were first proposed, the type of input parameters, and the complexity of the operation.

### Level 1 BLAS functions

The level 1 BLAS functions were first proposed in [(Lawson, 1979)][Lawson-1979] and define operations between scalars and vectors.

[Lawson-1979]: https://dl.acm.org/doi/10.1145/355841.355847

LinearAlgebra.BLAS.rot!Function
rot!(n, X, incx, Y, incy, c, s)

Overwrite X with c*X + s*Y and Y with -conj(s)*X + c*Y for the first n elements of array X with stride incx and first n elements of array Y with stride incy. Returns X and Y.

Julia 1.5

rot! requires at least Julia 1.5.

LinearAlgebra.BLAS.scal!Function
scal!(n, a, X, incx)
scal!(a, X)

Overwrite X with a*X for the first n elements of array X with stride incx. Returns X.

If n and incx are not provided, length(X) and stride(X,1) are used.

LinearAlgebra.BLAS.scalFunction
scal(n, a, X, incx)
scal(a, X)

Return X scaled by a for the first n elements of array X with stride incx.

If n and incx are not provided, length(X) and stride(X,1) are used.

LinearAlgebra.BLAS.blascopy!Function
blascopy!(n, X, incx, Y, incy)

Copy n elements of array X with stride incx to array Y with stride incy. Returns Y.

LinearAlgebra.BLAS.dotFunction
dot(n, X, incx, Y, incy)

Dot product of two vectors consisting of n elements of array X with stride incx and n elements of array Y with stride incy.

Examples

julia> BLAS.dot(10, fill(1.0, 10), 1, fill(1.0, 20), 2)
10.0
LinearAlgebra.BLAS.dotuFunction
dotu(n, X, incx, Y, incy)

Dot function for two complex vectors consisting of n elements of array X with stride incx and n elements of array Y with stride incy.

Examples

julia> BLAS.dotu(10, fill(1.0im, 10), 1, fill(1.0+im, 20), 2)
-10.0 + 10.0im
LinearAlgebra.BLAS.dotcFunction
dotc(n, X, incx, U, incy)

Dot function for two complex vectors, consisting of n elements of array X with stride incx and n elements of array U with stride incy, conjugating the first vector.

Examples

julia> BLAS.dotc(10, fill(1.0im, 10), 1, fill(1.0+im, 20), 2)
10.0 - 10.0im
LinearAlgebra.BLAS.nrm2Function
nrm2(n, X, incx)

2-norm of a vector consisting of n elements of array X with stride incx.

Examples

julia> BLAS.nrm2(4, fill(1.0, 8), 2)
2.0

julia> BLAS.nrm2(1, fill(1.0, 8), 2)
1.0
LinearAlgebra.BLAS.asumFunction
asum(n, X, incx)

Sum of the magnitudes of the first n elements of array X with stride incx.

For a real array, the magnitude is the absolute value. For a complex array, the magnitude is the sum of the absolute value of the real part and the absolute value of the imaginary part.

Examples

julia> BLAS.asum(5, fill(1.0im, 10), 2)
5.0

julia> BLAS.asum(2, fill(1.0im, 10), 5)
2.0
LinearAlgebra.BLAS.iamaxFunction
iamax(n, dx, incx)
iamax(dx)

Find the index of the element of dx with the maximum absolute value. n is the length of dx, and incx is the stride. If n and incx are not provided, they assume default values of n=length(dx) and incx=stride1(dx).

### Level 2 BLAS functions

The level 2 BLAS functions were published in [(Dongarra, 1988)][Dongarra-1988], and define matrix-vector operations.

[Dongarra-1988]: https://dl.acm.org/doi/10.1145/42288.42291

return a vector

LinearAlgebra.BLAS.gemv!Function
gemv!(tA, alpha, A, x, beta, y)

Update the vector y as alpha*A*x + beta*y or alpha*A'x + beta*y according to tA. alpha and beta are scalars. Return the updated y.

LinearAlgebra.BLAS.gbmv!Function
gbmv!(trans, m, kl, ku, alpha, A, x, beta, y)

Update vector y as alpha*A*x + beta*y or alpha*A'*x + beta*y according to trans. The matrix A is a general band matrix of dimension m by size(A,2) with kl sub-diagonals and ku super-diagonals. alpha and beta are scalars. Return the updated y.

LinearAlgebra.BLAS.gbmvFunction
gbmv(trans, m, kl, ku, alpha, A, x)

Return alpha*A*x or alpha*A'*x according to trans. The matrix A is a general band matrix of dimension m by size(A,2) with kl sub-diagonals and ku super-diagonals, and alpha is a scalar.

LinearAlgebra.BLAS.hemv!Function
hemv!(ul, alpha, A, x, beta, y)

Update the vector y as alpha*A*x + beta*y. A is assumed to be Hermitian. Only the ul triangle of A is used. alpha and beta are scalars. Return the updated y.

LinearAlgebra.BLAS.hemvMethod
hemv(ul, alpha, A, x)

Return alpha*A*x. A is assumed to be Hermitian. Only the ul triangle of A is used. alpha is a scalar.

LinearAlgebra.BLAS.hpmv!Function
hpmv!(uplo, α, AP, x, β, y)

Update vector y as α*A*x + β*y, where A is a Hermitian matrix provided in packed format AP.

With uplo = 'U', the array AP must contain the upper triangular part of the Hermitian matrix packed sequentially, column by column, so that AP[1] contains A[1, 1], AP[2] and AP[3] contain A[1, 2] and A[2, 2] respectively, and so on.

With uplo = 'L', the array AP must contain the lower triangular part of the Hermitian matrix packed sequentially, column by column, so that AP[1] contains A[1, 1], AP[2] and AP[3] contain A[2, 1] and A[3, 1] respectively, and so on.

The scalar inputs α and β must be complex or real numbers.

The array inputs x, y and AP must all be of ComplexF32 or ComplexF64 type.

Return the updated y.

Julia 1.5

hpmv! requires at least Julia 1.5.

LinearAlgebra.BLAS.symv!Function
symv!(ul, alpha, A, x, beta, y)

Update the vector y as alpha*A*x + beta*y. A is assumed to be symmetric. Only the ul triangle of A is used. alpha and beta are scalars. Return the updated y.

LinearAlgebra.BLAS.symvMethod
symv(ul, alpha, A, x)

Return alpha*A*x. A is assumed to be symmetric. Only the ul triangle of A is used. alpha is a scalar.

LinearAlgebra.BLAS.sbmv!Function
sbmv!(uplo, k, alpha, A, x, beta, y)

Update vector y as alpha*A*x + beta*y where A is a symmetric band matrix of order size(A,2) with k super-diagonals stored in the argument A. The storage layout for A is described the reference BLAS module, level-2 BLAS at http://www.netlib.org/lapack/explore-html/. Only the uplo triangle of A is used.

Return the updated y.

LinearAlgebra.BLAS.sbmvMethod
sbmv(uplo, k, alpha, A, x)

Return alpha*A*x where A is a symmetric band matrix of order size(A,2) with k super-diagonals stored in the argument A. Only the uplo triangle of A is used.

LinearAlgebra.BLAS.sbmvMethod
sbmv(uplo, k, A, x)

Return A*x where A is a symmetric band matrix of order size(A,2) with k super-diagonals stored in the argument A. Only the uplo triangle of A is used.

LinearAlgebra.BLAS.spmv!Function
spmv!(uplo, α, AP, x, β, y)

Update vector y as α*A*x + β*y, where A is a symmetric matrix provided in packed format AP.

With uplo = 'U', the array AP must contain the upper triangular part of the symmetric matrix packed sequentially, column by column, so that AP[1] contains A[1, 1], AP[2] and AP[3] contain A[1, 2] and A[2, 2] respectively, and so on.

With uplo = 'L', the array AP must contain the lower triangular part of the symmetric matrix packed sequentially, column by column, so that AP[1] contains A[1, 1], AP[2] and AP[3] contain A[2, 1] and A[3, 1] respectively, and so on.

The scalar inputs α and β must be real.

The array inputs x, y and AP must all be of Float32 or Float64 type.

Return the updated y.

Julia 1.5

spmv! requires at least Julia 1.5.

return a matrix

LinearAlgebra.BLAS.ger!Function
ger!(alpha, x, y, A)

Rank-1 update of the matrix A with vectors x and y as alpha*x*y' + A.

LinearAlgebra.BLAS.her!Function
her!(uplo, alpha, x, A)

Methods for complex arrays only. Rank-1 update of the Hermitian matrix A with vector x as alpha*x*x' + A. uplo controls which triangle of A is updated. Returns A.

LinearAlgebra.BLAS.syr!Function
syr!(uplo, alpha, x, A)

Rank-1 update of the symmetric matrix A with vector x as alpha*x*transpose(x) + A. uplo controls which triangle of A is updated. Returns A.

LinearAlgebra.BLAS.spr!Function
spr!(uplo, α, x, AP)

Update matrix A as A+α*x*x', where A is a symmetric matrix provided in packed format AP and x is a vector.

With uplo = 'U', the array AP must contain the upper triangular part of the symmetric matrix packed sequentially, column by column, so that AP[1] contains A[1, 1], AP[2] and AP[3] contain A[1, 2] and A[2, 2] respectively, and so on.

With uplo = 'L', the array AP must contain the lower triangular part of the symmetric matrix packed sequentially, column by column, so that AP[1] contains A[1, 1], AP[2] and AP[3] contain A[2, 1] and A[3, 1] respectively, and so on.

The scalar input α must be real.

The array inputs x and AP must all be of Float32 or Float64 type. Return the updated AP.

Julia 1.8

spr! requires at least Julia 1.8.

### Level 3 BLAS functions

The level 3 BLAS functions were published in [(Dongarra, 1990)][Dongarra-1990], and define matrix-matrix operations.

[Dongarra-1990]: https://dl.acm.org/doi/10.1145/77626.79170

LinearAlgebra.BLAS.gemm!Function
gemm!(tA, tB, alpha, A, B, beta, C)

Update C as alpha*A*B + beta*C or the other three variants according to tA and tB. Return the updated C.

LinearAlgebra.BLAS.symm!Function
symm!(side, ul, alpha, A, B, beta, C)

Update C as alpha*A*B + beta*C or alpha*B*A + beta*C according to side. A is assumed to be symmetric. Only the ul triangle of A is used. Return the updated C.

LinearAlgebra.BLAS.hemm!Function
hemm!(side, ul, alpha, A, B, beta, C)

Update C as alpha*A*B + beta*C or alpha*B*A + beta*C according to side. A is assumed to be Hermitian. Only the ul triangle of A is used. Return the updated C.

LinearAlgebra.BLAS.syrk!Function
syrk!(uplo, trans, alpha, A, beta, C)

Rank-k update of the symmetric matrix C as alpha*A*transpose(A) + beta*C or alpha*transpose(A)*A + beta*C according to trans. Only the uplo triangle of C is used. Return C.

LinearAlgebra.BLAS.herk!Function
herk!(uplo, trans, alpha, A, beta, C)

Methods for complex arrays only. Rank-k update of the Hermitian matrix C as alpha*A*A' + beta*C or alpha*A'*A + beta*C according to trans. Only the uplo triangle of C is updated. Returns C.

LinearAlgebra.BLAS.syr2k!Function
syr2k!(uplo, trans, alpha, A, B, beta, C)

Rank-2k update of the symmetric matrix C as alpha*A*transpose(B) + alpha*B*transpose(A) + beta*C or alpha*transpose(A)*B + alpha*transpose(B)*A + beta*C according to trans. Only the uplo triangle of C is used. Returns C.

LinearAlgebra.BLAS.syr2kFunction
syr2k(uplo, trans, alpha, A, B)

Returns the uplo triangle of alpha*A*transpose(B) + alpha*B*transpose(A) or alpha*transpose(A)*B + alpha*transpose(B)*A, according to trans.

syr2k(uplo, trans, A, B)

Return the uplo triangle of A*transpose(B) + B*transpose(A) or transpose(A)*B + transpose(B)*A, according to trans.

LinearAlgebra.BLAS.her2k!Function
her2k!(uplo, trans, alpha, A, B, beta, C)

Rank-2k update of the Hermitian matrix C as alpha*A*B' + alpha*B*A' + beta*C or alpha*A'*B + alpha*B'*A + beta*C according to trans. The scalar beta has to be real. Only the uplo triangle of C is used. Return C.

## LAPACK functions

LinearAlgebra.LAPACK provides wrappers for some of the LAPACK functions for linear algebra. Those functions that overwrite one of the input arrays have names ending in '!'.

Usually a function has 4 methods defined, one each for Float64, Float32, ComplexF64 and ComplexF32 arrays.

Note that the LAPACK API provided by Julia can and will change in the future. Since this API is not user-facing, there is no commitment to support/deprecate this specific set of functions in future releases.

LinearAlgebra.LAPACK.gbtrf!Function
gbtrf!(kl, ku, m, AB) -> (AB, ipiv)

Compute the LU factorization of a banded matrix AB. kl is the first subdiagonal containing a nonzero band, ku is the last superdiagonal containing one, and m is the first dimension of the matrix AB. Returns the LU factorization in-place and ipiv, the vector of pivots used.

LinearAlgebra.LAPACK.gbtrs!Function
gbtrs!(trans, kl, ku, m, AB, ipiv, B)

Solve the equation AB * X = B. trans determines the orientation of AB. It may be N (no transpose), T (transpose), or C (conjugate transpose). kl is the first subdiagonal containing a nonzero band, ku is the last superdiagonal containing one, and m is the first dimension of the matrix AB. ipiv is the vector of pivots returned from gbtrf!. Returns the vector or matrix X, overwriting B in-place.

LinearAlgebra.LAPACK.gebal!Function
gebal!(job, A) -> (ilo, ihi, scale)

Balance the matrix A before computing its eigensystem or Schur factorization. job can be one of N (A will not be permuted or scaled), P (A will only be permuted), S (A will only be scaled), or B (A will be both permuted and scaled). Modifies A in-place and returns ilo, ihi, and scale. If permuting was turned on, A[i,j] = 0 if j > i and 1 < j < ilo or j > ihi. scale contains information about the scaling/permutations performed.

LinearAlgebra.LAPACK.gebak!Function
gebak!(job, side, ilo, ihi, scale, V)

Transform the eigenvectors V of a matrix balanced using gebal! to the unscaled/unpermuted eigenvectors of the original matrix. Modifies V in-place. side can be L (left eigenvectors are transformed) or R (right eigenvectors are transformed).

LinearAlgebra.LAPACK.gebrd!Function
gebrd!(A) -> (A, d, e, tauq, taup)

Reduce A in-place to bidiagonal form A = QBP'. Returns A, containing the bidiagonal matrix B; d, containing the diagonal elements of B; e, containing the off-diagonal elements of B; tauq, containing the elementary reflectors representing Q; and taup, containing the elementary reflectors representing P.

LinearAlgebra.LAPACK.gelqf!Function
gelqf!(A, tau)

Compute the LQ factorization of A, A = LQ. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

gelqf!(A) -> (A, tau)

Compute the LQ factorization of A, A = LQ.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.geqlf!Function
geqlf!(A, tau)

Compute the QL factorization of A, A = QL. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

geqlf!(A) -> (A, tau)

Compute the QL factorization of A, A = QL.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.geqrf!Function
geqrf!(A, tau)

Compute the QR factorization of A, A = QR. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

geqrf!(A) -> (A, tau)

Compute the QR factorization of A, A = QR.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.geqp3!Function
geqp3!(A, [jpvt, tau]) -> (A, tau, jpvt)

Compute the pivoted QR factorization of A, AP = QR using BLAS level 3. P is a pivoting matrix, represented by jpvt. tau stores the elementary reflectors. The arguments jpvt and tau are optional and allow for passing preallocated arrays. When passed, jpvt must have length greater than or equal to n if A is an (m x n) matrix and tau must have length greater than or equal to the smallest dimension of A.

A, jpvt, and tau are modified in-place.

LinearAlgebra.LAPACK.gerqf!Function
gerqf!(A, tau)

Compute the RQ factorization of A, A = RQ. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

gerqf!(A) -> (A, tau)

Compute the RQ factorization of A, A = RQ.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.geqrt!Function
geqrt!(A, T)

Compute the blocked QR factorization of A, A = QR. T contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization. The first dimension of T sets the block size and it must be between 1 and n. The second dimension of T must equal the smallest dimension of A.

Returns A and T modified in-place.

geqrt!(A, nb) -> (A, T)

Compute the blocked QR factorization of A, A = QR. nb sets the block size and it must be between 1 and n, the second dimension of A.

Returns A, modified in-place, and T, which contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.geqrt3!Function
geqrt3!(A, T)

Recursively computes the blocked QR factorization of A, A = QR. T contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization. The first dimension of T sets the block size and it must be between 1 and n. The second dimension of T must equal the smallest dimension of A.

Returns A and T modified in-place.

geqrt3!(A) -> (A, T)

Recursively computes the blocked QR factorization of A, A = QR.

Returns A, modified in-place, and T, which contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.getrf!Function
getrf!(A) -> (A, ipiv, info)

Compute the pivoted LU factorization of A, A = LU.

Returns A, modified in-place, ipiv, the pivoting information, and an info code which indicates success (info = 0), a singular value in U (info = i, in which case U[i,i] is singular), or an error code (info < 0).

LinearAlgebra.LAPACK.tzrzf!Function
tzrzf!(A) -> (A, tau)

Transforms the upper trapezoidal matrix A to upper triangular form in-place. Returns A and tau, the scalar parameters for the elementary reflectors of the transformation.

LinearAlgebra.LAPACK.ormrz!Function
ormrz!(side, trans, A, tau, C)

Multiplies the matrix C by Q from the transformation supplied by tzrzf!. Depending on side or trans the multiplication can be left-sided (side = L, Q*C) or right-sided (side = R, C*Q) and Q can be unmodified (trans = N), transposed (trans = T), or conjugate transposed (trans = C). Returns matrix C which is modified in-place with the result of the multiplication.

LinearAlgebra.LAPACK.gels!Function
gels!(trans, A, B) -> (F, B, ssr)

Solves the linear equation A * X = B, transpose(A) * X = B, or adjoint(A) * X = B using a QR or LQ factorization. Modifies the matrix/vector B in place with the solution. A is overwritten with its QR or LQ factorization. trans may be one of N (no modification), T (transpose), or C (conjugate transpose). gels! searches for the minimum norm/least squares solution. A may be under or over determined. The solution is returned in B.

LinearAlgebra.LAPACK.gesv!Function
gesv!(A, B) -> (B, A, ipiv)

Solves the linear equation A * X = B where A is a square matrix using the LU factorization of A. A is overwritten with its LU factorization and B is overwritten with the solution X. ipiv contains the pivoting information for the LU factorization of A.

LinearAlgebra.LAPACK.getrs!Function
getrs!(trans, A, ipiv, B)

Solves the linear equation A * X = B, transpose(A) * X = B, or adjoint(A) * X = B for square A. Modifies the matrix/vector B in place with the solution. A is the LU factorization from getrf!, with ipiv the pivoting information. trans may be one of N (no modification), T (transpose), or C (conjugate transpose).

LinearAlgebra.LAPACK.getri!Function
getri!(A, ipiv)

Computes the inverse of A, using its LU factorization found by getrf!. ipiv is the pivot information output and A contains the LU factorization of getrf!. A is overwritten with its inverse.

LinearAlgebra.LAPACK.gesvx!Function
gesvx!(fact, trans, A, AF, ipiv, equed, R, C, B) -> (X, equed, R, C, B, rcond, ferr, berr, work)

Solves the linear equation A * X = B (trans = N), transpose(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) using the LU factorization of A. fact may be E, in which case A will be equilibrated and copied to AF; F, in which case AF and ipiv from a previous LU factorization are inputs; or N, in which case A will be copied to AF and then factored. If fact = F, equed may be N, meaning A has not been equilibrated; R, meaning A was multiplied by Diagonal(R) from the left; C, meaning A was multiplied by Diagonal(C) from the right; or B, meaning A was multiplied by Diagonal(R) from the left and Diagonal(C) from the right. If fact = F and equed = R or B the elements of R must all be positive. If fact = F and equed = C or B the elements of C must all be positive.

Returns the solution X; equed, which is an output if fact is not N, and describes the equilibration that was performed; R, the row equilibration diagonal; C, the column equilibration diagonal; B, which may be overwritten with its equilibrated form Diagonal(R)*B (if trans = N and equed = R,B) or Diagonal(C)*B (if trans = T,C and equed = C,B); rcond, the reciprocal condition number of A after equilbrating; ferr, the forward error bound for each solution vector in X; berr, the forward error bound for each solution vector in X; and work, the reciprocal pivot growth factor.

gesvx!(A, B)

The no-equilibration, no-transpose simplification of gesvx!.

LinearAlgebra.LAPACK.gelsd!Function
gelsd!(A, B, rcond) -> (B, rnk)

Computes the least norm solution of A * X = B by finding the SVD factorization of A, then dividing-and-conquering the problem. B is overwritten with the solution X. Singular values below rcond will be treated as zero. Returns the solution in B and the effective rank of A in rnk.

LinearAlgebra.LAPACK.gelsy!Function
gelsy!(A, B, rcond) -> (B, rnk)

Computes the least norm solution of A * X = B by finding the full QR factorization of A, then dividing-and-conquering the problem. B is overwritten with the solution X. Singular values below rcond will be treated as zero. Returns the solution in B and the effective rank of A in rnk.

LinearAlgebra.LAPACK.gglse!Function
gglse!(A, c, B, d) -> (X,res)

Solves the equation A * x = c where x is subject to the equality constraint B * x = d. Uses the formula ||c - A*x||^2 = 0 to solve. Returns X and the residual sum-of-squares.

LinearAlgebra.LAPACK.geev!Function
geev!(jobvl, jobvr, A) -> (W, VL, VR)

Finds the eigensystem of A. If jobvl = N, the left eigenvectors of A aren't computed. If jobvr = N, the right eigenvectors of A aren't computed. If jobvl = V or jobvr = V, the corresponding eigenvectors are computed. Returns the eigenvalues in W, the right eigenvectors in VR, and the left eigenvectors in VL.

LinearAlgebra.LAPACK.gesdd!Function
gesdd!(job, A) -> (U, S, VT)

Finds the singular value decomposition of A, A = U * S * V', using a divide and conquer approach. If job = A, all the columns of U and the rows of V' are computed. If job = N, no columns of U or rows of V' are computed. If job = O, A is overwritten with the columns of (thin) U and the rows of (thin) V'. If job = S, the columns of (thin) U and the rows of (thin) V' are computed and returned separately.

LinearAlgebra.LAPACK.gesvd!Function
gesvd!(jobu, jobvt, A) -> (U, S, VT)

Finds the singular value decomposition of A, A = U * S * V'. If jobu = A, all the columns of U are computed. If jobvt = A all the rows of V' are computed. If jobu = N, no columns of U are computed. If jobvt = N no rows of V' are computed. If jobu = O, A is overwritten with the columns of (thin) U. If jobvt = O, A is overwritten with the rows of (thin) V'. If jobu = S, the columns of (thin) U are computed and returned separately. If jobvt = S the rows of (thin) V' are computed and returned separately. jobu and jobvt can't both be O.

Returns U, S, and Vt, where S are the singular values of A.

LinearAlgebra.LAPACK.ggsvd!Function
ggsvd!(jobu, jobv, jobq, A, B) -> (U, V, Q, alpha, beta, k, l, R)

Finds the generalized singular value decomposition of A and B, U'*A*Q = D1*R and V'*B*Q = D2*R. D1 has alpha on its diagonal and D2 has beta on its diagonal. If jobu = U, the orthogonal/unitary matrix U is computed. If jobv = V the orthogonal/unitary matrix V is computed. If jobq = Q, the orthogonal/unitary matrix Q is computed. If jobu, jobv or jobq is N, that matrix is not computed. This function is only available in LAPACK versions prior to 3.6.0.

LinearAlgebra.LAPACK.ggsvd3!Function
ggsvd3!(jobu, jobv, jobq, A, B) -> (U, V, Q, alpha, beta, k, l, R)

Finds the generalized singular value decomposition of A and B, U'*A*Q = D1*R and V'*B*Q = D2*R. D1 has alpha on its diagonal and D2 has beta on its diagonal. If jobu = U, the orthogonal/unitary matrix U is computed. If jobv = V the orthogonal/unitary matrix V is computed. If jobq = Q, the orthogonal/unitary matrix Q is computed. If jobu, jobv, or jobq is N, that matrix is not computed. This function requires LAPACK 3.6.0.

LinearAlgebra.LAPACK.geevx!Function
geevx!(balanc, jobvl, jobvr, sense, A) -> (A, w, VL, VR, ilo, ihi, scale, abnrm, rconde, rcondv)

Finds the eigensystem of A with matrix balancing. If jobvl = N, the left eigenvectors of A aren't computed. If jobvr = N, the right eigenvectors of A aren't computed. If jobvl = V or jobvr = V, the corresponding eigenvectors are computed. If balanc = N, no balancing is performed. If balanc = P, A is permuted but not scaled. If balanc = S, A is scaled but not permuted. If balanc = B, A is permuted and scaled. If sense = N, no reciprocal condition numbers are computed. If sense = E, reciprocal condition numbers are computed for the eigenvalues only. If sense = V, reciprocal condition numbers are computed for the right eigenvectors only. If sense = B, reciprocal condition numbers are computed for the right eigenvectors and the eigenvectors. If sense = E,B, the right and left eigenvectors must be computed.

LinearAlgebra.LAPACK.ggev!Function
ggev!(jobvl, jobvr, A, B) -> (alpha, beta, vl, vr)

Finds the generalized eigendecomposition of A and B. If jobvl = N, the left eigenvectors aren't computed. If jobvr = N, the right eigenvectors aren't computed. If jobvl = V or jobvr = V, the corresponding eigenvectors are computed.

LinearAlgebra.LAPACK.gtsv!Function
gtsv!(dl, d, du, B)

Solves the equation A * X = B where A is a tridiagonal matrix with dl on the subdiagonal, d on the diagonal, and du on the superdiagonal.

Overwrites B with the solution X and returns it.

LinearAlgebra.LAPACK.gttrf!Function
gttrf!(dl, d, du) -> (dl, d, du, du2, ipiv)

Finds the LU factorization of a tridiagonal matrix with dl on the subdiagonal, d on the diagonal, and du on the superdiagonal.

Modifies dl, d, and du in-place and returns them and the second superdiagonal du2 and the pivoting vector ipiv.

LinearAlgebra.LAPACK.gttrs!Function
gttrs!(trans, dl, d, du, du2, ipiv, B)

Solves the equation A * X = B (trans = N), transpose(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) using the LU factorization computed by gttrf!. B is overwritten with the solution X.

LinearAlgebra.LAPACK.orglq!Function
orglq!(A, tau, k = length(tau))

Explicitly finds the matrix Q of a LQ factorization after calling gelqf! on A. Uses the output of gelqf!. A is overwritten by Q.

LinearAlgebra.LAPACK.orgqr!Function
orgqr!(A, tau, k = length(tau))

Explicitly finds the matrix Q of a QR factorization after calling geqrf! on A. Uses the output of geqrf!. A is overwritten by Q.

LinearAlgebra.LAPACK.orgql!Function
orgql!(A, tau, k = length(tau))

Explicitly finds the matrix Q of a QL factorization after calling geqlf! on A. Uses the output of geqlf!. A is overwritten by Q.

LinearAlgebra.LAPACK.orgrq!Function
orgrq!(A, tau, k = length(tau))

Explicitly finds the matrix Q of a RQ factorization after calling gerqf! on A. Uses the output of gerqf!. A is overwritten by Q.

LinearAlgebra.LAPACK.ormlq!Function
ormlq!(side, trans, A, tau, C)

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a LQ factorization of A computed using gelqf!. C is overwritten.

LinearAlgebra.LAPACK.ormqr!Function
ormqr!(side, trans, A, tau, C)

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QR factorization of A computed using geqrf!. C is overwritten.

LinearAlgebra.LAPACK.ormql!Function
ormql!(side, trans, A, tau, C)

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QL factorization of A computed using geqlf!. C is overwritten.

LinearAlgebra.LAPACK.ormrq!Function
ormrq!(side, trans, A, tau, C)

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a RQ factorization of A computed using gerqf!. C is overwritten.

LinearAlgebra.LAPACK.gemqrt!Function
gemqrt!(side, trans, V, T, C)

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QR factorization of A computed using geqrt!. C is overwritten.

LinearAlgebra.LAPACK.posv!Function
posv!(uplo, A, B) -> (A, B)

Finds the solution to A * X = B where A is a symmetric or Hermitian positive definite matrix. If uplo = U the upper Cholesky decomposition of A is computed. If uplo = L the lower Cholesky decomposition of A is computed. A is overwritten by its Cholesky decomposition. B is overwritten with the solution X.

LinearAlgebra.LAPACK.potrf!Function
potrf!(uplo, A)

Computes the Cholesky (upper if uplo = U, lower if uplo = L) decomposition of positive-definite matrix A. A is overwritten and returned with an info code.

LinearAlgebra.LAPACK.potri!Function
potri!(uplo, A)

Computes the inverse of positive-definite matrix A after calling potrf! to find its (upper if uplo = U, lower if uplo = L) Cholesky decomposition.

A is overwritten by its inverse and returned.

LinearAlgebra.LAPACK.potrs!Function
potrs!(uplo, A, B)

Finds the solution to A * X = B where A is a symmetric or Hermitian positive definite matrix whose Cholesky decomposition was computed by potrf!. If uplo = U the upper Cholesky decomposition of A was computed. If uplo = L the lower Cholesky decomposition of A was computed. B is overwritten with the solution X.

LinearAlgebra.LAPACK.pstrf!Function
pstrf!(uplo, A, tol) -> (A, piv, rank, info)

Computes the (upper if uplo = U, lower if uplo = L) pivoted Cholesky decomposition of positive-definite matrix A with a user-set tolerance tol. A is overwritten by its Cholesky decomposition.

Returns A, the pivots piv, the rank of A, and an info code. If info = 0, the factorization succeeded. If info = i > 0, then A is indefinite or rank-deficient.

LinearAlgebra.LAPACK.ptsv!Function
ptsv!(D, E, B)

Solves A * X = B for positive-definite tridiagonal A. D is the diagonal of A and E is the off-diagonal. B is overwritten with the solution X and returned.

LinearAlgebra.LAPACK.pttrf!Function
pttrf!(D, E)

Computes the LDLt factorization of a positive-definite tridiagonal matrix with D as diagonal and E as off-diagonal. D and E are overwritten and returned.

LinearAlgebra.LAPACK.pttrs!Function
pttrs!(D, E, B)

Solves A * X = B for positive-definite tridiagonal A with diagonal D and off-diagonal E after computing A's LDLt factorization using pttrf!. B is overwritten with the solution X.

LinearAlgebra.LAPACK.trtri!Function
trtri!(uplo, diag, A)

Finds the inverse of (upper if uplo = U, lower if uplo = L) triangular matrix A. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. A is overwritten with its inverse.

LinearAlgebra.LAPACK.trtrs!Function
trtrs!(uplo, trans, diag, A, B)

Solves A * X = B (trans = N), transpose(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) for (upper if uplo = U, lower if uplo = L) triangular matrix A. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. B is overwritten with the solution X.

LinearAlgebra.LAPACK.trcon!Function
trcon!(norm, uplo, diag, A)

Finds the reciprocal condition number of (upper if uplo = U, lower if uplo = L) triangular matrix A. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. If norm = I, the condition number is found in the infinity norm. If norm = O or 1, the condition number is found in the one norm.

LinearAlgebra.LAPACK.trevc!Function
trevc!(side, howmny, select, T, VL = similar(T), VR = similar(T))

Finds the eigensystem of an upper triangular matrix T. If side = R, the right eigenvectors are computed. If side = L, the left eigenvectors are computed. If side = B, both sets are computed. If howmny = A, all eigenvectors are found. If howmny = B, all eigenvectors are found and backtransformed using VL and VR. If howmny = S, only the eigenvectors corresponding to the values in select are computed.

LinearAlgebra.LAPACK.trrfs!Function
trrfs!(uplo, trans, diag, A, B, X, Ferr, Berr) -> (Ferr, Berr)

Estimates the error in the solution to A * X = B (trans = N), transpose(A) * X = B (trans = T), adjoint(A) * X = B (trans = C) for side = L, or the equivalent equations a right-handed side = R X * A after computing X using trtrs!. If uplo = U, A is upper triangular. If uplo = L, A is lower triangular. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. Ferr and Berr are optional inputs. Ferr is the forward error and Berr is the backward error, each component-wise.

LinearAlgebra.LAPACK.stev!Function
stev!(job, dv, ev) -> (dv, Zmat)

Computes the eigensystem for a symmetric tridiagonal matrix with dv as diagonal and ev as off-diagonal. If job = N only the eigenvalues are found and returned in dv. If job = V then the eigenvectors are also found and returned in Zmat.

LinearAlgebra.LAPACK.stebz!Function
stebz!(range, order, vl, vu, il, iu, abstol, dv, ev) -> (dv, iblock, isplit)

Computes the eigenvalues for a symmetric tridiagonal matrix with dv as diagonal and ev as off-diagonal. If range = A, all the eigenvalues are found. If range = V, the eigenvalues in the half-open interval (vl, vu] are found. If range = I, the eigenvalues with indices between il and iu are found. If order = B, eigvalues are ordered within a block. If order = E, they are ordered across all the blocks. abstol can be set as a tolerance for convergence.

LinearAlgebra.LAPACK.stegr!Function
stegr!(jobz, range, dv, ev, vl, vu, il, iu) -> (w, Z)

Computes the eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) for a symmetric tridiagonal matrix with dv as diagonal and ev as off-diagonal. If range = A, all the eigenvalues are found. If range = V, the eigenvalues in the half-open interval (vl, vu] are found. If range = I, the eigenvalues with indices between il and iu are found. The eigenvalues are returned in w and the eigenvectors in Z.

LinearAlgebra.LAPACK.stein!Function
stein!(dv, ev_in, w_in, iblock_in, isplit_in)

Computes the eigenvectors for a symmetric tridiagonal matrix with dv as diagonal and ev_in as off-diagonal. w_in specifies the input eigenvalues for which to find corresponding eigenvectors. iblock_in specifies the submatrices corresponding to the eigenvalues in w_in. isplit_in specifies the splitting points between the submatrix blocks.

LinearAlgebra.LAPACK.syconv!Function
syconv!(uplo, A, ipiv) -> (A, work)

Converts a symmetric matrix A (which has been factorized into a triangular matrix) into two matrices L and D. If uplo = U, A is upper triangular. If uplo = L, it is lower triangular. ipiv is the pivot vector from the triangular factorization. A is overwritten by L and D.

LinearAlgebra.LAPACK.sysv!Function
sysv!(uplo, A, B) -> (B, A, ipiv)

Finds the solution to A * X = B for symmetric matrix A. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. B is overwritten by the solution X. A is overwritten by its Bunch-Kaufman factorization. ipiv contains pivoting information about the factorization.

LinearAlgebra.LAPACK.sytrf!Function
sytrf!(uplo, A) -> (A, ipiv, info)

Computes the Bunch-Kaufman factorization of a symmetric matrix A. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored.

Returns A, overwritten by the factorization, a pivot vector ipiv, and the error code info which is a non-negative integer. If info is positive the matrix is singular and the diagonal part of the factorization is exactly zero at position info.

LinearAlgebra.LAPACK.sytri!Function
sytri!(uplo, A, ipiv)

Computes the inverse of a symmetric matrix A using the results of sytrf!. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. A is overwritten by its inverse.

LinearAlgebra.LAPACK.sytrs!Function
sytrs!(uplo, A, ipiv, B)

Solves the equation A * X = B for a symmetric matrix A using the results of sytrf!. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. B is overwritten by the solution X.

LinearAlgebra.LAPACK.hesv!Function
hesv!(uplo, A, B) -> (B, A, ipiv)

Finds the solution to A * X = B for Hermitian matrix A. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. B is overwritten by the solution X. A is overwritten by its Bunch-Kaufman factorization. ipiv contains pivoting information about the factorization.

LinearAlgebra.LAPACK.hetrf!Function
hetrf!(uplo, A) -> (A, ipiv, info)

Computes the Bunch-Kaufman factorization of a Hermitian matrix A. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored.

Returns A, overwritten by the factorization, a pivot vector ipiv, and the error code info which is a non-negative integer. If info is positive the matrix is singular and the diagonal part of the factorization is exactly zero at position info.

LinearAlgebra.LAPACK.hetri!Function
hetri!(uplo, A, ipiv)

Computes the inverse of a Hermitian matrix A using the results of sytrf!. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. A is overwritten by its inverse.

LinearAlgebra.LAPACK.hetrs!Function
hetrs!(uplo, A, ipiv, B)

Solves the equation A * X = B for a Hermitian matrix A using the results of sytrf!. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. B is overwritten by the solution X.

LinearAlgebra.LAPACK.syev!Function
syev!(jobz, uplo, A)

Finds the eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) of a symmetric matrix A. If uplo = U, the upper triangle of A is used. If uplo = L, the lower triangle of A is used.

LinearAlgebra.LAPACK.syevr!Function
syevr!(jobz, range, uplo, A, vl, vu, il, iu, abstol) -> (W, Z)

Finds the eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) of a symmetric matrix A. If uplo = U, the upper triangle of A is used. If uplo = L, the lower triangle of A is used. If range = A, all the eigenvalues are found. If range = V, the eigenvalues in the half-open interval (vl, vu] are found. If range = I, the eigenvalues with indices between il and iu are found. abstol can be set as a tolerance for convergence.

The eigenvalues are returned in W and the eigenvectors in Z.

LinearAlgebra.LAPACK.sygvd!Function
sygvd!(itype, jobz, uplo, A, B) -> (w, A, B)

Finds the generalized eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) of a symmetric matrix A and symmetric positive-definite matrix B. If uplo = U, the upper triangles of A and B are used. If uplo = L, the lower triangles of A and B are used. If itype = 1, the problem to solve is A * x = lambda * B * x. If itype = 2, the problem to solve is A * B * x = lambda * x. If itype = 3, the problem to solve is B * A * x = lambda * x.

LinearAlgebra.LAPACK.bdsqr!Function
bdsqr!(uplo, d, e_, Vt, U, C) -> (d, Vt, U, C)

Computes the singular value decomposition of a bidiagonal matrix with d on the diagonal and e_ on the off-diagonal. If uplo = U, e_ is the superdiagonal. If uplo = L, e_ is the subdiagonal. Can optionally also compute the product Q' * C.

Returns the singular values in d, and the matrix C overwritten with Q' * C.

LinearAlgebra.LAPACK.bdsdc!Function
bdsdc!(uplo, compq, d, e_) -> (d, e, u, vt, q, iq)

Computes the singular value decomposition of a bidiagonal matrix with d on the diagonal and e_ on the off-diagonal using a divide and conqueq method. If uplo = U, e_ is the superdiagonal. If uplo = L, e_ is the subdiagonal. If compq = N, only the singular values are found. If compq = I, the singular values and vectors are found. If compq = P, the singular values and vectors are found in compact form. Only works for real types.

Returns the singular values in d, and if compq = P, the compact singular vectors in iq.

LinearAlgebra.LAPACK.gecon!Function
gecon!(normtype, A, anorm)

Finds the reciprocal condition number of matrix A. If normtype = I, the condition number is found in the infinity norm. If normtype = O or 1, the condition number is found in the one norm. A must be the result of getrf! and anorm is the norm of A in the relevant norm.

LinearAlgebra.LAPACK.gehrd!Function
gehrd!(ilo, ihi, A) -> (A, tau)

Converts a matrix A to Hessenberg form. If A is balanced with gebal! then ilo and ihi are the outputs of gebal!. Otherwise they should be ilo = 1 and ihi = size(A,2). tau contains the elementary reflectors of the factorization.

LinearAlgebra.LAPACK.orghr!Function
orghr!(ilo, ihi, A, tau)

Explicitly finds Q, the orthogonal/unitary matrix from gehrd!. ilo, ihi, A, and tau must correspond to the input/output to gehrd!.

LinearAlgebra.LAPACK.gees!Function
gees!(jobvs, A) -> (A, vs, w)

Computes the eigenvalues (jobvs = N) or the eigenvalues and Schur vectors (jobvs = V) of matrix A. A is overwritten by its Schur form.

Returns A, vs containing the Schur vectors, and w, containing the eigenvalues.

LinearAlgebra.LAPACK.gges!Function
gges!(jobvsl, jobvsr, A, B) -> (A, B, alpha, beta, vsl, vsr)

Computes the generalized eigenvalues, generalized Schur form, left Schur vectors (jobsvl = V), or right Schur vectors (jobvsr = V) of A and B.

The generalized eigenvalues are returned in alpha and beta. The left Schur vectors are returned in vsl and the right Schur vectors are returned in vsr.

LinearAlgebra.LAPACK.trexc!Function
trexc!(compq, ifst, ilst, T, Q) -> (T, Q)
trexc!(ifst, ilst, T, Q) -> (T, Q)

Reorder the Schur factorization T of a matrix, such that the diagonal block of T with row index ifst is moved to row index ilst. If compq = V, the Schur vectors Q are reordered. If compq = N they are not modified. The 4-arg method calls the 5-arg method with compq = V.

LinearAlgebra.LAPACK.trsen!Function
trsen!(job, compq, select, T, Q) -> (T, Q, w, s, sep)
trsen!(select, T, Q) -> (T, Q, w, s, sep)

Reorder the Schur factorization of a matrix and optionally finds reciprocal condition numbers. If job = N, no condition numbers are found. If job = E, only the condition number for this cluster of eigenvalues is found. If job = V, only the condition number for the invariant subspace is found. If job = B then the condition numbers for the cluster and subspace are found. If compq = V the Schur vectors Q are updated. If compq = N the Schur vectors are not modified. select determines which eigenvalues are in the cluster. The 3-arg method calls the 5-arg method with job = N and compq = V.

Returns T, Q, reordered eigenvalues in w, the condition number of the cluster of eigenvalues s, and the condition number of the invariant subspace sep.

LinearAlgebra.LAPACK.tgsen!Function
tgsen!(select, S, T, Q, Z) -> (S, T, alpha, beta, Q, Z)

Reorders the vectors of a generalized Schur decomposition. select specifies the eigenvalues in each cluster.

LinearAlgebra.LAPACK.trsyl!Function
trsyl!(transa, transb, A, B, C, isgn=1) -> (C, scale)

Solves the Sylvester matrix equation A * X +/- X * B = scale*C where A and B are both quasi-upper triangular. If transa = N, A is not modified. If transa = T, A is transposed. If transa = C, A is conjugate transposed. Similarly for transb and B. If isgn = 1, the equation A * X + X * B = scale * C is solved. If isgn = -1, the equation A * X - X * B = scale * C is solved.

Returns X (overwriting C) and scale`.

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• AH16_1Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577
• AH16_2Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577
• AH16_3Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577
• AH16_4Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577
• AH16_5Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577
• AH16_6Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577