Linear Algebra

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. Basic operations, such as tr, det, and inv are all supported:

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

julia> tr(A)
3

julia> det(A)
104.0

julia> inv(A)
3×3 Array{Float64,2}:
 -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 Array{Float64,2}:
 -4.0  -17.0
  2.0    2.0

julia> eigvals(A)
2-element Array{Complex{Float64},1}:
 -1.0 + 5.0im
 -1.0 - 5.0im

julia> eigvecs(A)
2×2 Array{Complex{Float64},2}:
  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 Array{Float64,2}:
   1.5   2.0  -4.0
   3.0  -1.0  -6.0
 -10.0   2.3   4.0

julia> factorize(A)
LU{Float64,Array{Float64,2}}
L factor:
3×3 Array{Float64,2}:
  1.0    0.0       0.0
 -0.15   1.0       0.0
 -0.3   -0.132196  1.0
U factor:
3×3 Array{Float64,2}:
 -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 Array{Float64,2}:
  1.5   2.0  -4.0
  2.0  -1.0  -3.0
 -4.0  -3.0   5.0

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

julia> sB\x
3-element Array{Float64,1}:
 -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
LowerTriangularLower triangular 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
LowerTriangularMVMVinv, 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
LowerTriangularTRAAA
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 Array{Int64,2}:
 1  2
 3  4

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

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

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

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

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

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
CholeskyCholesky factorization
CholeskyPivotedPivoted Cholesky factorization
LULU factorization
LUTridiagonalLU factorization for Tridiagonal matrices
QRQR factorization
QRCompactWYCompact WY form of the QR factorization
QRPivotedPivoted QR factorization
HessenbergHessenberg decomposition
EigenSpectral decomposition
SVDSingular value decomposition
GeneralizedSVDGeneralized SVD

Standard Functions

Linear algebra functions in Julia are largely implemented by calling functions from LAPACK. Sparse factorizations call functions from SuiteSparse.

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

Matrix multiplication.

Examples

julia> [1 1; 0 1] * [1 0; 1 1]
2×2 Array{Int64,2}:
 2  1
 1  1
source
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.

Examples

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

julia> X = A \ B
2-element Array{Float64,1}:
 32.0
 18.0

julia> A * X == B
true
source
LinearAlgebra.dotFunction.
dot(x, y)
x ⋅ y

For any iterable containers x and y (including arrays of any dimension) of numbers (or any element type for which dot is defined), compute the dot product (or inner product or scalar product), i.e. the sum of dot(x[i],y[i]), as if they were vectors.

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

Examples

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
source
dot(x, y)
x ⋅ y

Compute the dot product between two vectors. For complex vectors, the first vector is conjugated. When the vectors have equal lengths, calling dot is semantically equivalent to sum(dot(vx,vy) for (vx,vy) in zip(x, y)).

Examples

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

julia> dot([im; im], [1; 1])
0 - 2im
source
LinearAlgebra.crossFunction.
cross(x, y)
×(x,y)

Compute the cross product of two 3-vectors.

Examples

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

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

julia> cross(a,b)
3-element Array{Int64,1}:
 1
 0
 0
source
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 Array{Float64,2}:
 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,Array{Float64,1}}:
 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.

source
Diagonal(A::AbstractMatrix)

Construct a matrix from the diagonal of A.

Examples

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

julia> Diagonal(A)
3×3 Diagonal{Int64,Array{Int64,1}}:
 1  ⋅  ⋅
 ⋅  5  ⋅
 ⋅  ⋅  9
source
Diagonal(V::AbstractVector)

Construct a matrix with V as its diagonal.

Examples

julia> V = [1, 2]
2-element Array{Int64,1}:
 1
 2

julia> Diagonal(V)
2×2 Diagonal{Int64,Array{Int64,1}}:
 1  ⋅
 ⋅  2
source
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 Array{Int64,1}:
 1
 2
 3
 4

julia> ev = [7, 8, 9]
3-element Array{Int64,1}:
 7
 8
 9

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

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

julia> Bidiagonal(A, :L) # contains the main diagonal and first subdiagonal of A
4×4 Bidiagonal{Int64,Array{Int64,1}}:
 1  ⋅  ⋅  ⋅
 2  2  ⋅  ⋅
 ⋅  3  3  ⋅
 ⋅  ⋅  4  4
source
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).

Examples

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

julia> ev = [7, 8, 9]
3-element Array{Int64,1}:
 7
 8
 9

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

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

Examples

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

julia> SymTridiagonal(A)
3×3 SymTridiagonal{Int64,Array{Int64,1}}:
 1  2  ⋅
 2  4  5
 ⋅  5  6
source
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,Array{Int64,1}}:
 7  4  ⋅  ⋅
 1  8  5  ⋅
 ⋅  2  9  6
 ⋅  ⋅  3  0
source
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 Array{Int64,2}:
 1  2  3  4
 1  2  3  4
 1  2  3  4
 1  2  3  4

julia> Tridiagonal(A)
4×4 Tridiagonal{Int64,Array{Int64,1}}:
 1  2  ⋅  ⋅
 1  2  3  ⋅
 ⋅  2  3  4
 ⋅  ⋅  3  4
source
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 Array{Int64,2}:
 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,Array{Int64,2}}:
 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,Array{Int64,2}}:
 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)).

source
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},Array{Complex{Int64},2}}:
 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},Array{Complex{Int64},2}}:
 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)
source
LowerTriangular(A::AbstractMatrix)

Construct a LowerTriangular view of the 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 Array{Float64,2}:
 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,Array{Float64,2}}:
 1.0   ⋅    ⋅
 4.0  5.0   ⋅
 7.0  8.0  9.0
source
UpperTriangular(A::AbstractMatrix)

Construct an UpperTriangular view of the 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 Array{Float64,2}:
 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,Array{Float64,2}}:
 1.0  2.0  3.0
  ⋅   5.0  6.0
  ⋅    ⋅   9.0
source
UniformScaling{T<:Number}

Generically sized uniform scaling operator defined as a scalar times the identity operator, λ*I. See also I.

Examples

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

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

julia> J*A
2×2 Array{Float64,2}:
 2.0  4.0
 6.0  8.0
source
LinearAlgebra.luFunction.
lu(A, pivot=Val(true); 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}}. If pivoting is chosen (default) the element type should also support abs and <.

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 Array{Int64,2}:
 4  3
 6  3

julia> F = lu(A)
LU{Float64,Array{Float64,2}}
L factor:
2×2 Array{Float64,2}:
 1.0  0.0
 1.5  1.0
U factor:
2×2 Array{Float64,2}:
 4.0   3.0
 0.0  -1.5

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
source
LinearAlgebra.lu!Function.
lu!(A, pivot=Val(true); 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 Array{Float64,2}:
 4.0  3.0
 6.0  3.0

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

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

julia> lu!(iA)
ERROR: InexactError: Int64(Int64, 0.6666666666666666)
Stacktrace:
[...]
source
cholesky(A, Val(false); 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 StridedMatrix or a perfectly symmetric or Hermitian StridedMatrix. The triangular Cholesky factor can be obtained from the factorization F with: F.L and F.U. The following functions are available for Cholesky objects: size, \, inv, det, logdet and isposdef.

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 Array{Float64,2}:
   4.0   12.0  -16.0
  12.0   37.0  -43.0
 -16.0  -43.0   98.0

julia> C = cholesky(A)
Cholesky{Float64,Array{Float64,2}}
U factor:
3×3 UpperTriangular{Float64,Array{Float64,2}}:
 2.0  6.0  -8.0
  ⋅   1.0   5.0
  ⋅    ⋅    3.0

julia> C.U
3×3 UpperTriangular{Float64,Array{Float64,2}}:
 2.0  6.0  -8.0
  ⋅   1.0   5.0
  ⋅    ⋅    3.0

julia> C.L
3×3 LowerTriangular{Float64,Array{Float64,2}}:
  2.0   ⋅    ⋅
  6.0  1.0   ⋅
 -8.0  5.0  3.0

julia> C.L * C.U == A
true
source
cholesky(A, Val(true); 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 StridedMatrix or a perfectly symmetric or Hermitian StridedMatrix. The triangular Cholesky factor can be obtained from the factorization F with: F.L and F.U. The following functions are available for PivotedCholesky 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.

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.

source
cholesky!(A, Val(false); 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 Array{Int64,2}:
 1   2
 2  50

julia> cholesky!(A)
ERROR: InexactError: Int64(Int64, 6.782329983125268)
Stacktrace:
[...]
source
cholesky!(A, Val(true); 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.

source
lowrankupdate(C::Cholesky, v::StridedVector) -> 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.

source
lowrankdowndate(C::Cholesky, v::StridedVector) -> 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.

source
lowrankupdate!(C::Cholesky, v::StridedVector) -> 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.

source
lowrankdowndate!(C::Cholesky, v::StridedVector) -> 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.

source
LinearAlgebra.ldltFunction.
ldlt(S::SymTridiagonal) -> LDLt

Compute an LDLt 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.

Examples

julia> S = SymTridiagonal([3., 4., 5.], [1., 2.])
3×3 SymTridiagonal{Float64,Array{Float64,1}}:
 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 Array{Float64,1}:
 1.7906976744186047
 0.627906976744186
 1.3488372093023255

julia> S \ b
3-element Array{Float64,1}:
 1.7906976744186047
 0.627906976744186
 1.3488372093023255
source
LinearAlgebra.ldlt!Function.
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,Array{Float64,1}}:
 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,Array{Float64,1}}:
 3.0       0.333333   ⋅
 0.333333  3.66667   0.545455
  ⋅        0.545455  3.90909
source
LinearAlgebra.qrFunction.
qr(A, pivot=Val(false)) -> 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:

  • if pivot == Val(true) then F is a QRPivoted object,

  • otherwise if the element type of A is a BLAS type (Float32, Float64, ComplexF32 or ComplexF64), then F is a QRCompactWY object,

  • otherwise F is a QR object.

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.

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*Matrix(I,m,m). If m<=n, then Matrix(F.Q) yields an m×m orthogonal matrix.

Examples

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

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

source
LinearAlgebra.qr!Function.
qr!(A, pivot=Val(false))

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.

Examples

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

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

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

julia> qr!(a)
ERROR: InexactError: Int64(Int64, -3.1622776601683795)
Stacktrace:
[...]
source
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 $au_i$.

source
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], as a lower trapezoidal matrix $V$ and an upper triangular matrix $T$ where

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

such that $v_i$ is the $i$th column of $V$, and $au_i$ is the $i$th diagonal element of $T$.

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 square matrix with min(m,n) columns, whose upper triangular part gives the matrix $T$ above (the subdiagonal elements are ignored).

Note

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

[Bischof1987]

C Bischof and C Van Loan, "The WY representation for products of Householder matrices", SIAM J Sci Stat Comput 8 (1987), s2-s13. doi:10.1137/0908009

[Schreiber1989]

R Schreiber and C Van Loan, "A storage-efficient WY representation for products of Householder transformations", SIAM J Sci Stat Comput 10 (1989), 53-57. doi:10.1137/0910005

source
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 $P$.

source
LinearAlgebra.lq!Function.
lq!(A) -> LQ

Compute the LQ factorization of A, using the input matrix as a workspace. See also lq.

source
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).

Examples

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

julia> S = lq(A)
LQ{Float64,Array{Float64,2}} with factors L and Q:
[-8.60233 0.0; 4.41741 -0.697486]
[-0.581238 -0.813733; -0.813733 0.581238]

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

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

julia> l == S.L &&  q == S.Q
true
source
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.

[Bunch1977]

J R Bunch and L Kaufman, Some stable methods for calculating inertia

and solving symmetric linear systems, Mathematics of Computation 31:137 (1977), 163-179. url.

Examples

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

julia> S = bunchkaufman(A)
BunchKaufman{Float64,Array{Float64,2}}
D factor:
2×2 Tridiagonal{Float64,Array{Float64,1}}:
 -0.333333  0.0
  0.0       3.0
U factor:
2×2 UnitUpperTriangular{Float64,Array{Float64,2}}:
 1.0  0.666667
  ⋅   1.0
permutation:
2-element Array{Int64,1}:
 1
 2

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

julia> d == S.D && u == S.U && p == S.p
true
source
bunchkaufman!(A, rook::Bool=false; check = true) -> BunchKaufman

bunchkaufman! is the same as bunchkaufman, but saves space by overwriting the input A, instead of creating a copy.

source
LinearAlgebra.eigvalsFunction.
eigvals(A; permute::Bool=true, scale::Bool=true) -> 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 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.

Examples

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

julia> eigvals(diag_matrix)
2-element Array{Float64,1}:
 1.0
 4.0
source

For a scalar input, eigvals will return a scalar.

Example

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

Computes the generalized eigenvalues of A and B.

Examples

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

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

julia> eigvals(A,B)
2-element Array{Complex{Float64},1}:
 0.0 + 1.0im
 0.0 - 1.0im
source
eigvals(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> values

Returns 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.

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

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

julia> eigvals(A)
3-element Array{Float64,1}:
 -2.1400549446402604
  1.0000000000000002
  5.140054944640259
source
eigvals(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> values

Returns 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.

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

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

julia> eigvals(A)
3-element Array{Float64,1}:
 -2.1400549446402604
  1.0000000000000002
  5.140054944640259
source
eigvals!(A; permute::Bool=true, scale::Bool=true) -> values

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

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 Array{Float64,2}:
 1.0  2.0
 3.0  4.0

julia> eigvals!(A)
2-element Array{Float64,1}:
 -0.3722813232690143
  5.372281323269014

julia> A
2×2 Array{Float64,2}:
 -0.372281  -1.0
  0.0        5.37228
source
eigvals!(A, B) -> 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 Array{Float64,2}:
 1.0   0.0
 0.0  -1.0

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

julia> eigvals!(A, B)
2-element Array{Complex{Float64},1}:
 0.0 + 1.0im
 0.0 - 1.0im

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

julia> B
2×2 Array{Float64,2}:
 1.0  0.0
 0.0  1.0
source
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.

source
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.

source
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 Array{Complex{Int64},2}:
 0+0im  0+1im
 0-1im  0+0im

julia> eigmax(A)
1.0

julia> A = [0 im; -1 0]
2×2 Array{Complex{Int64},2}:
  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:
[...]
source
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 Array{Complex{Int64},2}:
 0+0im  0+1im
 0-1im  0+0im

julia> eigmin(A)
-1.0

julia> A = [0 im; -1 0]
2×2 Array{Complex{Int64},2}:
  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:
[...]
source
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,Array{Float64,1}}:
 1.0  2.0   ⋅
 2.0  2.0  3.0
  ⋅   3.0  1.0

julia> eigvals(A)
3-element Array{Float64,1}:
 -2.1400549446402604
  1.0000000000000002
  5.140054944640259

julia> eigvecs(A)
3×3 Array{Float64,2}:
  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 Array{Float64,2}:
  0.8320502943378438
  4.263514128092366e-17
 -0.5547001962252291
source
eigvecs(A; permute::Bool=true, scale::Bool=true) -> 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 and scale 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 Array{Float64,2}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0
source
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 Array{Int64,2}:
 1   0
 0  -1

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

julia> eigvecs(A, B)
2×2 Array{Complex{Float64},2}:
  0.0-1.0im   0.0+1.0im
 -1.0-0.0im  -1.0+0.0im
source
LinearAlgebra.eigenFunction.
eigen(A; permute::Bool=true, scale::Bool=true) -> Eigen

Computes 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.

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,Array{Float64,2},Array{Float64,1}}
eigenvalues:
3-element Array{Float64,1}:
  1.0
  3.0
 18.0
eigenvectors:
3×3 Array{Float64,2}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0

julia> F.values
3-element Array{Float64,1}:
  1.0
  3.0
 18.0

julia> F.vectors
3×3 Array{Float64,2}:
 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
source
eigen(A, B) -> GeneralizedEigen

Computes 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.

Examples

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

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

julia> F = eigen(A, B);

julia> F.values
2-element Array{Complex{Float64},1}:
 0.0 + 1.0im
 0.0 - 1.0im

julia> F.vectors
2×2 Array{Complex{Float64},2}:
  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
source
eigen(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> Eigen

Computes 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.

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

Computes 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.

source
LinearAlgebra.eigen!Function.
eigen!(A, [B])

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

source
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 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 = [4. 9. 7.; 4. 4. 1.; 4. 3. 2.]
3×3 Array{Float64,2}:
 4.0  9.0  7.0
 4.0  4.0  1.0
 4.0  3.0  2.0

julia> F = hessenberg(A);

julia> F.Q * F.H * F.Q'
3×3 Array{Float64,2}:
 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
source
hessenberg!(A) -> Hessenberg

hessenberg! is the same as hessenberg, but saves space by overwriting the input A, instead of creating a copy.

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

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

Examples

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

julia> F = schur!(A)
Schur{Float64,Array{Float64,2}}
T factor:
2×2 Array{Float64,2}:
 3.0   9.0
 0.0  -2.0
Z factor:
2×2 Array{Float64,2}:
  0.961524  0.274721
 -0.274721  0.961524
eigenvalues:
2-element Array{Float64,1}:
  3.0
 -2.0

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

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

source
LinearAlgebra.schurFunction.
schur(A::StridedMatrix) -> 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.

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

Examples

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

julia> F = schur(A)
Schur{Float64,Array{Float64,2}}
T factor:
2×2 Array{Float64,2}:
 3.0   9.0
 0.0  -2.0
Z factor:
2×2 Array{Float64,2}:
  0.961524  0.274721
 -0.274721  0.961524
eigenvalues:
2-element Array{Float64,1}:
  3.0
 -2.0

julia> F.vectors * F.Schur * F.vectors'
2×2 Array{Float64,2}:
  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
source
schur(A::StridedMatrix, B::StridedMatrix) -> 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.

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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.

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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.

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

Same as ordschur but overwrites the factorization F.

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

Same as ordschur but overwrites the factorization F.

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LinearAlgebra.svdFunction.
svd(A; full::Bool = false) -> 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 a $M \times N$ 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.

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 Array{Float64,2}:
 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);

julia> F.U * Diagonal(F.S) * F.Vt
4×5 Array{Float64,2}:
 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
source
svd(A, B) -> GeneralizedSVD

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

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 Array{Float64,2}:
 1.0   0.0
 0.0  -1.0

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

julia> F = svd(A, B);

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

julia> F.V*F.D2*F.R0*F.Q'
2×2 Array{Float64,2}:
 0.0  1.0
 1.0  0.0
source
LinearAlgebra.svd!Function.
svd!(A; full::Bool = false) -> SVD

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

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 Array{Float64,2}:
 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);

julia> F.U * Diagonal(F.S) * F.Vt
4×5 Array{Float64,2}:
 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> A
4×5 Array{Float64,2}:
 -2.23607   0.0   0.0  0.0  0.618034
  0.0      -3.0   1.0  0.0  0.0
  0.0       0.0   0.0  0.0  0.0
  0.0       0.0  -2.0  0.0  0.0
source
svd!(A, B) -> GeneralizedSVD

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

Examples

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

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

julia> F = svd!(A, B);

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

julia> F.V*F.D2*F.R0*F.Q'
2×2 Array{Float64,2}:
 0.0  1.0
 1.0  0.0

julia> A
2×2 Array{Float64,2}:
 1.41421   0.0
 0.0      -1.41421

julia> B
2×2 Array{Float64,2}:
 1.0  -0.0
 0.0  -1.0
source
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 Array{Float64,2}:
 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 Array{Float64,1}:
 3.0
 2.23606797749979
 2.0
 0.0
source
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 Array{Float64,2}:
 1.0   0.0
 0.0  -1.0

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

julia> svdvals(A, B)
2-element Array{Float64,1}:
 1.0
 1.0
source
svdvals!(A)

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

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 Array{Float64,2}:
 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 Array{Float64,1}:
 3.0
 2.23606797749979
 2.0
 0.0

julia> A
4×5 Array{Float64,2}:
 -2.23607   0.0   0.0  0.0  0.618034
  0.0      -3.0   1.0  0.0  0.0
  0.0       0.0   0.0  0.0  0.0
  0.0       0.0  -2.0  0.0  0.0
source
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.

Examples

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

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

julia> svdvals!(A, B)
2-element Array{Float64,1}:
 1.0
 1.0

julia> A
2×2 Array{Float64,2}:
 1.41421   0.0
 0.0      -1.41421

julia> B
2×2 Array{Float64,2}:
 1.0  -0.0
 0.0  -1.0
source
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

source
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

source
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

source
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

source
LinearAlgebra.triuFunction.
triu(M)

Upper triangle of a matrix.

Examples

julia> a = fill(1.0, (4,4))
4×4 Array{Float64,2}:
 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 Array{Float64,2}:
 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
source
triu(M, k::Integer)

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

Examples

julia> a = fill(1.0, (4,4))
4×4 Array{Float64,2}:
 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 Array{Float64,2}:
 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 Array{Float64,2}:
 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
source
LinearAlgebra.triu!Function.
triu!(M)

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

source
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 Array{Int64,2}:
 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 Array{Int64,2}:
 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
source
LinearAlgebra.trilFunction.
tril(M)

Lower triangle of a matrix.

Examples

julia> a = fill(1.0, (4,4))
4×4 Array{Float64,2}:
 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 Array{Float64,2}:
 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
source
tril(M, k::Integer)

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

Examples

julia> a = fill(1.0, (4,4))
4×4 Array{Float64,2}:
 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 Array{Float64,2}:
 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 Array{Float64,2}:
 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
source
LinearAlgebra.tril!Function.
tril!(M)

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

source
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 Array{Int64,2}:
 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 Array{Int64,2}:
 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
source
LinearAlgebra.diagindFunction.
diagind(M, k::Integer=0)

An AbstractRange giving the indices of the kth diagonal of the matrix M.

Examples

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

julia> diagind(A,-1)
2:4:6
source
LinearAlgebra.diagFunction.
diag(M, k::Integer=0)

The kth diagonal of a matrix, as a vector.

See also: diagm

Examples

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

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

Construct a square matrix from Pairs of diagonals and vectors. Vector kv.second will be placed on the kv.first diagonal. 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 Array{Int64,2}:
 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 Array{Int64,2}:
 0  1  0  0
 4  0  2  0
 0  5  0  3
 0  0  0  0
source
LinearAlgebra.rankFunction.
rank(A[, tol::Real])

Compute the rank of a matrix by counting how many singular values of A have magnitude greater than tol*σ₁ where σ₁ is A's largest singular values. By default, the value of tol is the smallest dimension of A multiplied by the eps of the eltype of A.

Examples

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

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

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

julia> rank(diagm(0 => [1, 0.001, 2]), 0.00001)
3
source
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 Array{Int64,1}:
  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
source
norm(x::Number, p::Real=2)

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

Examples

julia> norm(2, 1)
2

julia> norm(-2, 1)
2

julia> norm(2, 2)
2

julia> norm(-2, 2)
2

julia> norm(2, Inf)
2

julia> norm(-2, Inf)
2
source
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 Array{Int64,2}:
 1  -2  -3
 2   3  -1

julia> opnorm(A, Inf)
6.0

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

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

source
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
source
normalize!(v::AbstractVector, p::Real=2)

Normalize the vector v in-place so that its p-norm equals unity, i.e. norm(v, p) == 1. See also normalize and norm.

source
normalize(v::AbstractVector, p::Real=2)

Normalize the vector v so that its p-norm equals unity, i.e. norm(v, p) == vecnorm(v, p) == 1. See also normalize! and norm.

Examples

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

julia> b = normalize(a)
3-element Array{Float64,1}:
 0.2182178902359924
 0.4364357804719848
 0.8728715609439696

julia> norm(b)
1.0

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

julia> norm(c, 1)
1.0
source
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.

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

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.

source
LinearAlgebra.trFunction.
tr(M)

Matrix trace. Sums the diagonal elements of M.

Examples

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

julia> tr(A)
5
source
LinearAlgebra.detFunction.
det(M)

Matrix determinant.

Examples

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

julia> det(M)
2.0
source
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 Array{Int64,2}:
 1  0
 2  2

julia> logdet(M)
0.6931471805599453

julia> logdet(Matrix(I, 3, 3))
0.0
source
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 Array{Float64,2}:
 -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 Array{Float64,2}:
 2.0  0.0
 0.0  1.0

julia> det(B)
2.0

julia> logabsdet(B)
(0.6931471805599453, 1.0)
source
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 Array{Int64,2}:
 2  5
 1  3

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

julia> M*N == N*M == Matrix(I, 2, 2)
true
source
LinearAlgebra.pinvFunction.
pinv(M[, tol::Real])

Computes the Moore-Penrose pseudoinverse.

For matrices M with floating point elements, it is convenient to compute the pseudoinverse by inverting only singular values above a given threshold, tol.

The optimal choice of tol varies both with the value of M and the intended application of the pseudoinverse. The default value of tol is eps(real(float(one(eltype(M)))))*minimum(size(M)), which is essentially machine epsilon for the real part of a matrix element multiplied by the larger matrix dimension. For inverting dense ill-conditioned matrices in a least-squares sense, tol = sqrt(eps(real(float(one(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 Array{Float64,2}:
 1.5  1.3
 1.2  1.9

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

julia> M * N
2×2 Array{Float64,2}:
 1.0          -2.22045e-16
 4.44089e-16   1.0
[issue8859]

Issue 8859, "Fix least squares", https://github.com/JuliaLang/julia/pull/8859

[B96]

Åke Björck, "Numerical Methods for Least Squares Problems", SIAM Press, Philadelphia, 1996, "Other Titles in Applied Mathematics", Vol. 51. doi:10.1137/1.9781611971484

[S84]

G. W. Stewart, "Rank Degeneracy", SIAM Journal on Scientific and Statistical Computing, 5(2), 1984, 403-413. doi:10.1137/0905030

[KY88]

Konstantinos Konstantinides and Kung Yao, "Statistical analysis of effective singular values in matrix rank determination", IEEE Transactions on Acoustics, Speech and Signal Processing, 36(5), 1988, 757-763. doi:10.1109/29.1585

source
nullspace(M[, tol::Real])

Computes a basis for the nullspace of M by including the singular vectors of A whose singular have magnitude are greater than tol*σ₁, where σ₁ is A's largest singular values. By default, the value of tol is the smallest dimension of A multiplied by the eps of the eltype of A.

Examples

julia> M = [1 0 0; 0 1 0; 0 0 0]
3×3 Array{Int64,2}:
 1  0  0
 0  1  0
 0  0  0

julia> nullspace(M)
3×1 Array{Float64,2}:
 0.0
 0.0
 1.0

julia> nullspace(M, 2)
3×3 Array{Float64,2}:
 0.0  1.0  0.0
 1.0  0.0  0.0
 0.0  0.0  1.0
source
Base.kronFunction.
kron(A, B)

Kronecker tensor product of two vectors or two matrices.

Examples

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

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

julia> kron(A, B)
4×4 Array{Complex{Int64},2}:
 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
source
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.

[H05]

Nicholas J. Higham, "The squaring and scaling method for the matrix exponential revisited", SIAM Journal on Matrix Analysis and Applications, 26(4), 2005, 1179-1193. doi:10.1137/090768539

Examples

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

julia> exp(A)
2×2 Array{Float64,2}:
 2.71828  0.0
 0.0      2.71828
source
Base.logMethod.
log(A{T}::StridedMatrix{T})

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]). For general matrices, the complex Schur form (schur) is computed and the triangular algorithm is used on the triangular factor.

[AH12]

Awad H. Al-Mohy and Nicholas J. Higham, "Improved inverse scaling and squaring algorithms for the matrix logarithm", SIAM Journal on Scientific Computing, 34(4), 2012, C153-C169. doi:10.1137/110852553

[AHR13]

Awad H. Al-Mohy, Nicholas J. Higham and Samuel D. Relton, "Computing the Fréchet derivative of the matrix logarithm and estimating the condition number", SIAM Journal on Scientific Computing, 35(4), 2013, C394-C410. doi:10.1137/120885991

Examples

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

julia> log(A)
2×2 Array{Float64,2}:
 1.0  0.0
 0.0  1.0
source
Base.sqrtMethod.
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 symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the square root. 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.

[BH83]

Åke Björck and Sven Hammarling, "A Schur method for the square root of a matrix", Linear Algebra and its Applications, 52-53, 1983, 127-140. doi:10.1016/0024-3795(83)80010-X

Examples

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

julia> sqrt(A)
2×2 Array{Float64,2}:
 2.0  0.0
 0.0  2.0
source
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 Array{Float64,2}:
  0.291927  -0.708073
 -0.708073   0.291927
source
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 Array{Float64,2}:
 0.454649  0.454649
 0.454649  0.454649
source
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 Array{Float64,2}:
 0.454649  0.454649
 0.454649  0.454649

julia> C
2×2 Array{Float64,2}:
  0.291927  -0.708073
 -0.708073   0.291927
source
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 Array{Float64,2}:
 -1.09252  -1.09252
 -1.09252  -1.09252
source
Base.Math.secMethod.
sec(A::AbstractMatrix)

Compute the matrix secant of a square matrix A.

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

Compute the matrix cosecant of a square matrix A.

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

Compute the matrix cotangent of a square matrix A.

source
Base.coshMethod.
cosh(A::AbstractMatrix)

Compute the matrix hyperbolic cosine of a square matrix A.

source
Base.sinhMethod.
sinh(A::AbstractMatrix)

Compute the matrix hyperbolic sine of a square matrix A.

source
Base.tanhMethod.
tanh(A::AbstractMatrix)

Compute the matrix hyperbolic tangent of a square matrix A.

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

Compute the matrix hyperbolic secant of square matrix A.

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

Compute the matrix hyperbolic cosecant of square matrix A.

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

Compute the matrix hyperbolic cotangent of square matrix A.

source
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].

[AH16_1]

Mary 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

Examples

julia> acos(cos([0.5 0.1; -0.2 0.3]))
2×2 Array{Complex{Float64},2}:
  0.5-5.55112e-17im  0.1-2.77556e-17im
 -0.2+2.498e-16im    0.3-3.46945e-16im
source
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].

[AH16_2]

Mary 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

Examples

julia> asin(sin([0.5 0.1; -0.2 0.3]))
2×2 Array{Complex{Float64},2}:
  0.5-4.16334e-17im  0.1-5.55112e-17im
 -0.2+9.71445e-17im  0.3-1.249e-16im
source
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].

[AH16_3]

Mary 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

Examples

julia> atan(tan([0.5 0.1; -0.2 0.3]))
2×2 Array{Complex{Float64},2}:
  0.5+1.38778e-17im  0.1-2.77556e-17im
 -0.2+6.93889e-17im  0.3-4.16334e-17im
source
Base.Math.asecMethod.
asec(A::AbstractMatrix)

Compute the inverse matrix secant of A.

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

Compute the inverse matrix cosecant of A.

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

Compute the inverse matrix cotangent of A.

source
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].

[AH16_4]

Mary 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

source
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].

[AH16_5]

Mary 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

source
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].

[AH16_6]

Mary 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

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

Compute the inverse matrix hyperbolic secant of A.

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

Compute the inverse matrix hyperbolic cosecant of A.

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

Compute the inverse matrix hyperbolic cotangent of A.

source
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 Array{Float64,2}:
 3.0  4.0
 5.0  6.0

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

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

julia> A*X + X*A' + B
2×2 Array{Float64,2}:
 0.0          6.66134e-16
 6.66134e-16  8.88178e-16
source
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 Array{Float64,2}:
 3.0  4.0
 5.0  6.0

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

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

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

julia> A*X + X*B + C
2×2 Array{Float64,2}:
  2.66454e-15  1.77636e-15
 -3.77476e-15  4.44089e-16
source
issuccess(F::Factorization)

Test that a factorization of a matrix succeeded.

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

julia> LinearAlgebra.issuccess(F)
true

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

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

Test whether a matrix is symmetric.

Examples

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

julia> issymmetric(a)
true

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

julia> issymmetric(b)
false
source
isposdef(A) -> Bool

Test whether a matrix is positive definite (and Hermitian) by trying to perform a Cholesky factorization of A. See also isposdef!

Examples

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

julia> isposdef(A)
true
source
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 Array{Float64,2}:
 1.0  2.0
 2.0  6.78233
source
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 Array{Int64,2}:
 1   2
 2  -1

julia> istril(a)
false

julia> istril(a, 1)
true

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

julia> istril(b)
true

julia> istril(b, -1)
false
source
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 Array{Int64,2}:
 1   2
 2  -1

julia> istriu(a)
false

julia> istriu(a, -1)
true

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

julia> istriu(b)
true

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

Test whether a matrix is diagonal.

Examples

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

julia> isdiag(a)
false

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

julia> isdiag(b)
true
source
ishermitian(A) -> Bool

Test whether a matrix is Hermitian.

Examples

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

julia> ishermitian(a)
true

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

julia> ishermitian(b)
true
source
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+2im 9+2im; 8+7im  4+6im]
2×2 Array{Complex{Int64},2}:
 3+2im  9+2im
 8+7im  4+6im

julia> transpose(A)
2×2 Transpose{Complex{Int64},Array{Complex{Int64},2}}:
 3+2im  8+7im
 9+2im  4+6im
source
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 Array{Complex{Int64},2}:
 3+2im  9+2im
 8+7im  4+6im

julia> B = zeros(Complex{Int64}, 2, 2)
2×2 Array{Complex{Int64},2}:
 0+0im  0+0im
 0+0im  0+0im

julia> transpose!(B, A);

julia> B
2×2 Array{Complex{Int64},2}:
 3+2im  8+7im
 9+2im  4+6im

julia> A
2×2 Array{Complex{Int64},2}:
 3+2im  9+2im
 8+7im  4+6im
source
Base.adjointFunction.
adjoint(A)

Lazy adjoint (conjugate transposition) (also postfix '). Note that adjoint is applied recursively to elements.

This operation is intended for linear algebra usage - for general data manipulation see permutedims.

Examples

julia> A = [3+2im 9+2im; 8+7im  4+6im]
2×2 Array{Complex{Int64},2}:
 3+2im  9+2im
 8+7im  4+6im

julia> adjoint(A)
2×2 Adjoint{Complex{Int64},Array{Complex{Int64},2}}:
 3-2im  8-7im
 9-2im  4-6im
source
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 Array{Complex{Int64},2}:
 3+2im  9+2im
 8+7im  4+6im

julia> B = zeros(Complex{Int64}, 2, 2)
2×2 Array{Complex{Int64},2}:
 0+0im  0+0im
 0+0im  0+0im

julia> adjoint!(B, A);

julia> B
2×2 Array{Complex{Int64},2}:
 3-2im  8-7im
 9-2im  4-6im

julia> A
2×2 Array{Complex{Int64},2}:
 3+2im  9+2im
 8+7im  4+6im
source
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 Array{Complex{Int64},2}:
 1+0im  0+2im
 0-3im  4+0im

julia> T = transpose(A)
2×2 Transpose{Complex{Int64},Array{Complex{Int64},2}}:
 1+0im  0-3im
 0+2im  4+0im

julia> copy(T)
2×2 Array{Complex{Int64},2}:
 1+0im  0-3im
 0+2im  4+0im
source
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 Array{Int64,1}:
 1
 2
 3
 4

julia> LinearAlgebra.stride1(A)
1

julia> B = view(A, 2:2:4)
2-element view(::Array{Int64,1}, 2:2:4) with eltype Int64:
 2
 4

julia> LinearAlgebra.stride1(B)
2
source
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 Array{Int64,1}:
 4
 5
source

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 Array{Float64,2}:
 3.0  3.0
 7.0  7.0
source
LinearAlgebra.lmul!Function.
lmul!(a::Number, B::AbstractArray)

Scale an array B by a scalar a overwriting B in-place.

Examples

julia> B = [1 2; 3 4]
2×2 Array{Int64,2}:
 1  2
 3  4

julia> lmul!(2, B)
2×2 Array{Int64,2}:
 2  4
 6  8
source
lmul!(A, B)

Calculate the matrix-matrix product $AB$, overwriting B, and return the result.

Examples

julia> B = [0 1; 1 0];

julia> A = LinearAlgebra.UpperTriangular([1 2; 0 3]);

julia> LinearAlgebra.lmul!(A, B);

julia> B
2×2 Array{Int64,2}:
 2  1
 3  0
source
LinearAlgebra.rmul!Function.
rmul!(A::AbstractArray, b::Number)

Scale an array A by a scalar b overwriting A in-place.

Examples

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

julia> rmul!(A, 2)
2×2 Array{Int64,2}:
 2  4
 6  8
source
rmul!(A, B)

Calculate the matrix-matrix product $AB$, overwriting A, and return the result.

Examples

julia> A = [0 1; 1 0];

julia> B = LinearAlgebra.UpperTriangular([1 2; 0 3]);

julia> LinearAlgebra.rmul!(A, B);

julia> A
2×2 Array{Int64,2}:
 0  3
 1  2
source
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.

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 Array{Float64,1}:
  0.7128099173553719
 -0.051652892561983674
  0.10020661157024757

julia> A\X
3-element Array{Float64,1}:
  0.7128099173553719
 -0.05165289256198333
  0.10020661157024785
source
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.

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 Array{Float64,1}:
  0.7128099173553719
 -0.051652892561983674
  0.10020661157024757

julia> A\Y
3-element Array{Float64,1}:
  0.7128099173553719
 -0.05165289256198333
  0.10020661157024785
source
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.

source

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 Float64, Float32, ComplexF64, and ComplexF32 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:

Multplication 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.BLASModule.

Interface to BLAS subroutines.

source
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
source
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
source
blascopy!(n, X, incx, Y, incy)

Copy n elements of array X with stride incx to array Y with stride incy. Returns Y.

source
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
source
asum(n, X, incx)

Sum of the absolute values of the first n elements of array X with stride incx.

Examples

julia> BLAS.asum(5, fill(1.0im, 10), 2)
5.0

julia> BLAS.asum(2, fill(1.0im, 10), 5)
2.0
source
LinearAlgebra.axpy!Function.
axpy!(a, X, Y)

Overwrite Y with a*X + Y, where a is a scalar. Return Y.

Examples

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

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

julia> BLAS.axpy!(2, x, y)
3-element Array{Int64,1}:
  6
  9
 12
source
scal!(n, a, X, incx)

Overwrite X with a*X for the first n elements of array X with stride incx. Returns X.

source
scal(n, a, X, incx)

Return X scaled by a for the first n elements of array X with stride incx.

source
ger!(alpha, x, y, A)

Rank-1 update of the matrix A with vectors x and y as alpha*x*y' + A.

source
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.

source
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. Returns C.

source
syrk(uplo, trans, alpha, A)

Returns either the upper triangle or the lower triangle of A, according to uplo, of alpha*A*transpose(A) or alpha*transpose(A)*A, according to trans.

source
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.

source
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.

source
herk(uplo, trans, alpha, A)

Methods for complex arrays only. Returns the uplo triangle of alpha*A*A' or alpha*A'*A, according to trans.

source
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.

source
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.

source
sbmv!(uplo, k, alpha, A, x, beta, y)

Update vector y as alpha*A*x + beta*y where A is a 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.

source
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.

source
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.

source
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.

source
gemm(tA, tB, alpha, A, B)

Return alpha*A*B or the other three variants according to tA and tB.

source
gemm(tA, tB, A, B)

Return A*B or the other three variants according to tA and tB.

source
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.

source
gemv(tA, alpha, A, x)

Return alpha*A*x or alpha*A'x according to tA. alpha is a scalar.

source
gemv(tA, A, x)

Return A*x or A'x according to tA.

source
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.

source
symm(side, ul, alpha, A, B)

Return alpha*A*B or alpha*B*A according to side. A is assumed to be symmetric. Only the ul triangle of A is used.

source
symm(side, ul, A, B)

Return A*B or B*A according to side. A is assumed to be symmetric. Only the ul triangle of A is used.

source
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.

source
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.

source
symv(ul, A, x)

Return A*x. A is assumed to be symmetric. Only the ul triangle of A is used.

source
trmm!(side, ul, tA, dA, alpha, A, B)

Update B as alpha*A*B or one of the other three variants determined by side and tA. Only the ul triangle of A is used. dA determines if the diagonal values are read or are assumed to be all ones. Returns the updated B.

source
trmm(side, ul, tA, dA, alpha, A, B)

Returns alpha*A*B or one of the other three variants determined by side and tA. Only the ul triangle of A is used. dA determines if the diagonal values are read or are assumed to be all ones.

source
trsm!(side, ul, tA, dA, alpha, A, B)

Overwrite B with the solution to A*X = alpha*B or one of the other three variants determined by side and tA. Only the ul triangle of A is used. dA determines if the diagonal values are read or are assumed to be all ones. Returns the updated B.

source
trsm(side, ul, tA, dA, alpha, A, B)

Return the solution to A*X = alpha*B or one of the other three variants determined by determined by side and tA. Only the ul triangle of A is used. dA determines if the diagonal values are read or are assumed to be all ones.

source
trmv!(ul, tA, dA, A, b)

Return op(A)*b, where op is determined by tA. Only the ul triangle of A is used. dA determines if the diagonal values are read or are assumed to be all ones. The multiplication occurs in-place on b.

source
trmv(ul, tA, dA, A, b)

Return op(A)*b, where op is determined by tA. Only the ul triangle of A is used. dA determines if the diagonal values are read or are assumed to be all ones.

source
trsv!(ul, tA, dA, A, b)

Overwrite b with the solution to A*x = b or one of the other two variants determined by tA and ul. dA determines if the diagonal values are read or are assumed to be all ones. Return the updated b.

source
trsv(ul, tA, dA, A, b)

Return the solution to A*x = b or one of the other two variants determined by tA and ul. dA determines if the diagonal values are read or are assumed to be all ones.

source
set_num_threads(n)

Set the number of threads the BLAS library should use.

source
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 Array{Complex{Int64},2}:
 1+0im  0+2im  3+0im
 0+1im  2+0im  3+0im
source

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.

Interfaces to LAPACK subroutines.

source
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.

source
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.

source
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.

source
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).

source
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.

source
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.

source
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.

source
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.

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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.

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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.

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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.

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geqp3!(A, jpvt, tau)

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. jpvt must have length length greater than or equal to n if A is an (m x n) matrix. tau must have length greater than or equal to the smallest dimension of A.

A, jpvt, and tau are modified in-place.

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geqp3!(A, jpvt) -> (A, jpvt, tau)

Compute the pivoted QR factorization of A, AP = QR using BLAS level 3. P is a pivoting matrix, represented by jpvt. jpvt must have length greater than or equal to n if A is an (m x n) matrix.

Returns A and jpvt, modified in-place, and tau, which stores the elementary reflectors.

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geqp3!(A) -> (A, jpvt, tau)

Compute the pivoted QR factorization of A, AP = QR using BLAS level 3.

Returns A, modified in-place, jpvt, which represents the pivoting matrix P, and tau, which stores the elementary reflectors.

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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.

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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.

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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.

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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.

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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.

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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.

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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).

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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.

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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.

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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.

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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.

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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).

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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.

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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.

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gesvx!(A, B)

The no-equilibration, no-transpose simplification of gesvx!.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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!.

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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.

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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.

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trexc!(compq, ifst, ilst, T, Q) -> (T, Q)

Reorder the Schur factorization of a matrix. If compq = V, the Schur vectors Q are reordered. If compq = N they are not modified. ifst and ilst specify the reordering of the vectors.

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trsen!(compq, job, 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.

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.

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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.

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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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