Linear Algebra

*(A::AbstractMatrix, B::AbstractMatrix)

Matrix multiplication.

Examples

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

\(A, B)

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

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

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

See also: factorize, pinv.

Examples

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

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

julia> A * X == B
true

A / B

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

See also: rdiv!.

Examples

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

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

julia> isapprox(A, X*B)
true

julia> isapprox(X, A*pinv(B))
true

SingularException

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

PosDefException

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

ZeroPivotException <: Exception

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

dot(x, y)
x ⋅ y

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

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

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

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

Examples

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

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

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

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

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

julia> dot(x, y)
150.0

dot(x, A, y)

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

Examples

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

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

julia> ⋅(1:5, reshape(1:25, 5, 5), 2:6) == dot(1:5, reshape(1:25, 5, 5), 2:6)
true

cross(x, y)
×(x,y)

Compute the cross product of two 3-vectors.

Examples

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

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

julia> cross(a,b)
3-element Vector{Int64}:
 1
 0
 0

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.

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

Examples

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

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

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

Diagonal(V::AbstractVector)

Construct a matrix with V as its diagonal.

See also diag, diagm.

Examples

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

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

Diagonal(A::AbstractMatrix)

Construct a matrix from the diagonal of A.

Examples

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

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

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

Diagonal{T}(undef, n)

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

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

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

Examples

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

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

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

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

Bidiagonal(A, uplo::Symbol)

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

Examples

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

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

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

SymTridiagonal(dv::V, ev::V) where V <: AbstractVector

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

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

Examples

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

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

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

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

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

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

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

SymTridiagonal(A::AbstractMatrix)

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

Examples

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

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

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

julia> SymTridiagonal(B)
2×2 SymTridiagonal{Matrix{Int64}, Vector{Matrix{Int64}}}:
 [1 2; 2 3]  [1 3; 2 4]
 [1 2; 3 4]  [1 2; 2 3]

Tridiagonal(dl::V, d::V, du::V) where V <: AbstractVector

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

Examples

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

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

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

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

Tridiagonal(A)

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

Examples

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

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

Symmetric(A, uplo=:U)

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

Examples

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

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

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

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

Hermitian(A, uplo=:U)

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

Examples

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

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

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

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

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

Hermitian(fill(complex(1,1), 1, 1)) == fill(1, 1, 1)

LowerTriangular(A::AbstractMatrix)

Construct a LowerTriangular view of the matrix A.

Examples

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

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

UpperTriangular(A::AbstractMatrix)

Construct an UpperTriangular view of the matrix A.

Examples

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

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

UnitLowerTriangular(A::AbstractMatrix)

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

Examples

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

julia> UnitLowerTriangular(A)
3×3 UnitLowerTriangular{Float64, Matrix{Float64}}:
 1.0   ⋅    ⋅
 4.0  1.0   ⋅
 7.0  8.0  1.0

UnitUpperTriangular(A::AbstractMatrix)

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

Examples

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

julia> UnitUpperTriangular(A)
3×3 UnitUpperTriangular{Float64, Matrix{Float64}}:
 1.0  2.0  3.0
  ⋅   1.0  6.0
  ⋅    ⋅   1.0

UpperHessenberg(A::AbstractMatrix)

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

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

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

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

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

Examples

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

julia> UpperHessenberg(A)
4×4 UpperHessenberg{Int64, Matrix{Int64}}:
 1   2   3   4
 5   6   7   8
 ⋅  10  11  12
 ⋅   ⋅  15  16

UniformScaling{T<:Number}

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

Examples

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

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

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

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

I

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

Examples

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

julia> [1 2im 3; 1im 2 3] * I
2×3 Matrix{Complex{Int64}}:
 1+0im  0+2im  3+0im
 0+1im  2+0im  3+0im

(I::UniformScaling)(n::Integer)

Construct a Diagonal matrix from a UniformScaling.

Examples

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

julia> (0.7*I)(3)
3×3 Diagonal{Float64, Vector{Float64}}:
 0.7   ⋅    ⋅
  ⋅   0.7   ⋅
  ⋅    ⋅   0.7

LinearAlgebra.Factorization

Abstract type for matrix factorizations a.k.a. matrix decompositions. See online documentation for a list of available matrix factorizations.

LU <: Factorization

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

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

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

Examples

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

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

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

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

julia> l == F.L && u == F.U && p == F.p
true

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

Compute the LU factorization of A.

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

In most cases, if A is a subtype S of AbstractMatrix{T} with an element type T supporting +, -, * and /, the return type is LU{T,S{T}}. If pivoting is chosen (default) the element type should also support abs and <. Pivoting can be turned off by passing pivot = NoPivot().

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

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:

Examples

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

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

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

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

julia> l == F.L && u == F.U && p == F.p
true

lu(A::SparseMatrixCSC; check = true) -> F::UmfpackLU

Compute the LU factorization of a sparse matrix A.

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

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

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

The relation between F and A is

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

F further supports the following functions:

• \
• det

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

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

Examples

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

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

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

julia> lu!(iA)
ERROR: InexactError: Int64(0.6666666666666666)
Stacktrace:
[...]

lu!(F::UmfpackLU, A::SparseMatrixCSC; check=true) -> F::UmfpackLU

Compute the LU factorization of a sparse matrix A, reusing the symbolic factorization of an already existing LU factorization stored in F. The sparse matrix A must have an identical nonzero pattern as the matrix used to create the LU factorization F, otherwise an error is thrown.

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

julia> F = lu(A);

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

julia> lu!(F, B);

julia> F \ ones(2)
2-element Vector{Float64}:
 0.0
 1.0

Cholesky <: Factorization

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

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

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

Iterating the decomposition produces the components L and U.

Examples

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

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

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

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

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

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

julia> l == C.L && u == C.U
true

CholeskyPivoted

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

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

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

Iterating the decomposition produces the components L and U.

Examples

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

julia> A = X * X';

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

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

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

julia> l == C.L && u == C.U
true

cholesky(A::SparseMatrixCSC; shift = 0.0, check = true, perm = nothing) -> CHOLMOD.Factor

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

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

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

Examples

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

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

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

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

julia> L = sparse(C.L);

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

julia> L * L' ≈ A[C.p, C.p]
true

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

julia> P' * L * L' * P ≈ A
true

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

julia> L = sparse(C.L);

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

julia> L * L' ≈ A
true

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

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

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

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

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

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

Examples

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

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

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

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

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

cholesky(A, RowMaximum(); tol = 0.0, check = true) -> CholeskyPivoted

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

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

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

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

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

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

Examples

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

julia> A = X * X';

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

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

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

julia> l == C.L && u == C.U
true

cholesky!(F::CHOLMOD.Factor, A::SparseMatrixCSC; shift = 0.0, check = true) -> CHOLMOD.Factor

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

See also cholesky.

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

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

Examples

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

julia> cholesky!(A)
ERROR: InexactError: Int64(6.782329983125268)
Stacktrace:
[...]

cholesky!(A::AbstractMatrix, RowMaximum(); tol = 0.0, check = true) -> CholeskyPivoted

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

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

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

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

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

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

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

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

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

LDLt <: Factorization

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

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

Examples

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

julia> F = ldlt(S)
LDLt{Float64, SymTridiagonal{Float64, Vector{Float64}}}
L factor:
3×3 UnitLowerTriangular{Float64, SymTridiagonal{Float64, Vector{Float64}}}:
 1.0        ⋅         ⋅
 0.333333  1.0        ⋅
 0.0       0.545455  1.0
D factor:
3×3 Diagonal{Float64, Vector{Float64}}:
 3.0   ⋅        ⋅
  ⋅   3.66667   ⋅
  ⋅    ⋅       3.90909

ldlt(A::SparseMatrixCSC; shift = 0.0, check = true, perm=nothing) -> CHOLMOD.Factor

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

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

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

ldlt(S::SymTridiagonal) -> LDLt

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

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

Examples

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

julia> ldltS = ldlt(S);

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

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

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

ldlt!(F::CHOLMOD.Factor, A::SparseMatrixCSC; shift = 0.0, check = true) -> CHOLMOD.Factor

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

See also ldlt.

ldlt!(S::SymTridiagonal) -> LDLt

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

Examples

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

julia> ldltS = ldlt!(S);

julia> ldltS === S
false

julia> S
3×3 SymTridiagonal{Float64, Vector{Float64}}:
 3.0       0.333333   ⋅
 0.333333  3.66667   0.545455
  ⋅        0.545455  3.90909

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 $au_i$.

QRCompactWY <: Factorization

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

\[A = Q R\]

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

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

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

Iterating the decomposition produces the components Q and R.

The object has two fields:

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

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

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

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

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 $P$.

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

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

Examples

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

julia> qr(A)
SuiteSparse.SPQR.QRSparse{Float64, Int64}
Q factor:
4×4 SuiteSparse.SPQR.QRSparseQ{Float64, Int64}:
 -0.707107   0.0        0.0       -0.707107
  0.0       -0.707107  -0.707107   0.0
  0.0       -0.707107   0.707107   0.0
 -0.707107   0.0        0.0        0.707107
R factor:
2×2 SparseMatrixCSC{Float64, Int64} with 2 stored entries:
 -1.41421    ⋅
   ⋅       -1.41421
Row permutation:
4-element Vector{Int64}:
 1
 3
 4
 2
Column permutation:
2-element Vector{Int64}:
 1
 2

qr(A, pivot = NoPivot(); blocksize) -> F

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

\[A = Q R\]

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

• if pivot == ColumnNorm() 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. When A is not full rank, factorization with (column) pivoting is required to obtain a minimum norm solution.

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

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

Examples

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

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

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

qr!(A, pivot = NoPivot(); blocksize)

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

Examples

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

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

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

julia> qr!(a)
ERROR: InexactError: Int64(3.1622776601683795)
Stacktrace:
[...]

LQ <: Factorization

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

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

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

Examples

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

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

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

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

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

lq(A) -> S::LQ

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

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

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

Examples

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

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

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

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

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

lq!(A) -> LQ

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

BunchKaufman <: Factorization

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

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

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

Examples

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

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

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

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

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

bunchkaufman(A, rook::Bool=false; check = true) -> S::BunchKaufman

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

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

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

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

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

Examples

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

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

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

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

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

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

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

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.

Eigen <: Factorization

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

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

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

Examples

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

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

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

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

julia> vals == F.values && vecs == F.vectors
true

GeneralizedEigen <: Factorization

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

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

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

Examples

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

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

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

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

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

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

julia> vals == F.values && vecs == F.vectors
true

eigvals(A; permute::Bool=true, scale::Bool=true, sortby) -> values

Return the eigenvalues of A.

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

Examples

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

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

For a scalar input, eigvals will return a scalar.

Example

julia> eigvals(-2)
-2

eigvals(A, B) -> values

Compute the generalized eigenvalues of A and B.

Examples

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

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

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

eigvals(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> values

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

Examples

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

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

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

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

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

Examples

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

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

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

eigvals!(A; permute::Bool=true, scale::Bool=true, sortby) -> values

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

Examples

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

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

julia> A
2×2 Matrix{Float64}:
 -0.372281  -1.0
  0.0        5.37228

eigvals!(A, B; sortby) -> values

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

Examples

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

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

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

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

julia> B
2×2 Matrix{Float64}:
 1.0  0.0
 0.0  1.0

eigvals!(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> values

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

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

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

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

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

Examples

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

julia> eigmax(A)
1.0

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

julia> eigmax(A)
ERROR: DomainError with Complex{Int64}[0+0im 0+1im; -1+0im 0+0im]:
`A` cannot have complex eigenvalues.
Stacktrace:
[...]

eigmin(A; permute::Bool=true, scale::Bool=true)

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

Examples

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

julia> eigmin(A)
-1.0

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

julia> eigmin(A)
ERROR: DomainError with Complex{Int64}[0+0im 0+1im; -1+0im 0+0im]:
`A` cannot have complex eigenvalues.
Stacktrace:
[...]

eigvecs(A::SymTridiagonal[, eigvals]) -> Matrix

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

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

Examples

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

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

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

julia> eigvecs(A, [1.])
3×1 Matrix{Float64}:
  0.8320502943378438
  4.263514128092366e-17
 -0.5547001962252291

eigvecs(A; permute::Bool=true, scale::Bool=true, `sortby`) -> Matrix

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

Examples

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

eigvecs(A, B) -> Matrix

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

Examples

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

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

julia> eigvecs(A, B)
2×2 Matrix{ComplexF64}:
  0.0+1.0im   0.0-1.0im
 -1.0+0.0im  -1.0-0.0im

eigen(A; permute::Bool=true, scale::Bool=true, sortby) -> Eigen

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

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

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

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

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

Examples

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

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

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

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

julia> vals == F.values && vecs == F.vectors
true

eigen(A, B; sortby) -> GeneralizedEigen

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

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

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

Examples

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

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

julia> F = eigen(A, B);

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

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

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

julia> vals == F.values && vecs == F.vectors
true

eigen(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> Eigen

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

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

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

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

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

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

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

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

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

eigen!(A; permute, scale, sortby)
eigen!(A, B; sortby)

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

Hessenberg <: Factorization

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

hessenberg(A) -> Hessenberg

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

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

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

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

Examples

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

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

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

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

julia> q == F.Q && h == F.H
true

hessenberg!(A) -> Hessenberg

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

Schur <: Factorization

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

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

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

Examples

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

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

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

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

julia> t == F.T && z == F.Z && vals == F.values
true

GeneralizedSchur <: Factorization

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

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

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

schur(A) -> F::Schur

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

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

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

Examples

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

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

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

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

julia> t == F.T && z == F.Z && vals == F.values
true

schur(A, B) -> F::GeneralizedSchur

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

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

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 Matrix{Float64}:
  5.0   7.0
 -2.0  -4.0

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

julia> A
2×2 Matrix{Float64}:
 3.0   9.0
 0.0  -2.0

schur!(A::StridedMatrix, B::StridedMatrix) -> F::GeneralizedSchur

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

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

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

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

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

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

Same as ordschur but overwrites the factorization F.

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

Same as ordschur but overwrites the factorization F.

SVD <: Factorization

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

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

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

Examples

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

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

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

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

julia> u == F.U && s == F.S && v == F.V
true

GeneralizedSVD <: Factorization

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

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

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

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

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

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

Examples

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

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

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

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

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

svd(A; full::Bool = false, alg::Algorithm = default_svd_alg(A)) -> SVD

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

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

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

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

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

Examples

julia> A = rand(4,3);

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

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

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

julia> A ≈ U * Diagonal(S) * V'
true

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

julia> Uonly == U
true

svd(A, B) -> GeneralizedSVD

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

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

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

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

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

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

Examples

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

julia> F = svd(A, B);

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

julia> H = R*Q';

julia> [A; B] ≈ [U*C; V*S]*H
true

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

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

julia> U == Uonly
true

svd!(A; full::Bool = false, alg::Algorithm = default_svd_alg(A)) -> SVD

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

svd!(A, B) -> GeneralizedSVD

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

svdvals(A)

Return the singular values of A in descending order.

Examples

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

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

svdvals(A, B)

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

Examples

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

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

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

svdvals!(A)

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

svdvals!(A, B)

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

LinearAlgebra.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.

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

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

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

the result of the multiplication

y = G*x

has the property that

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

See also LinearAlgebra.Givens.

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

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

B = G*A

has the property that

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

See also LinearAlgebra.Givens.

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

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

B = G*x

has the property that

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

See also LinearAlgebra.Givens.

triu(M)

Upper triangle of a matrix.

Examples

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

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

triu(M, k::Integer)

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

Examples

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

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

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

triu!(M)

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

triu!(M, k::Integer)

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

Examples

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

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

tril(M)

Lower triangle of a matrix.

Examples

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

julia> tril(a)
4×4 Matrix{Float64}:
 1.0  0.0  0.0  0.0
 1.0  1.0  0.0  0.0
 1.0  1.0  1.0  0.0
 1.0  1.0  1.0  1.0

tril(M, k::Integer)

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

Examples

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

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

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

tril!(M)

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

tril!(M, k::Integer)

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

Examples

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

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

diagind(M, k::Integer=0)

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

See also: diag, diagm, Diagonal.

Examples

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

julia> diagind(A,-1)
2:4:6

diag(M, k::Integer=0)

The kth diagonal of a matrix, as a vector.

See also diagm, diagind, Diagonal, isdiag.

Examples

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

julia> diag(A,1)
2-element Vector{Int64}:
 2
 6

diagm(kv::Pair{<:Integer,<:AbstractVector}...)
diagm(m::Integer, n::Integer, kv::Pair{<:Integer,<:AbstractVector}...)

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

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

Examples

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

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

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

diagm(v::AbstractVector)
diagm(m::Integer, n::Integer, v::AbstractVector)

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

Examples

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

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

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

Examples

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

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

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

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

julia> rank(diagm(0 => [1, 0.001, 2]), atol=1.5)
1

norm(A, p::Real=2)

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

The p-norm is defined as

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

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

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

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

Use opnorm to compute the operator norm of a matrix.

Examples

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

julia> norm(v)
7.0

julia> norm(v, 1)
11.0

julia> norm(v, Inf)
6.0

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

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

julia> norm(1:9)
16.881943016134134

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

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

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

norm(x::Number, p::Real=2)

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

Examples

julia> norm(2, 1)
2.0

julia> norm(-2, 1)
2.0

julia> norm(2, 2)
2.0

julia> norm(-2, 2)
2.0

julia> norm(2, Inf)
2.0

julia> norm(-2, Inf)
2.0

opnorm(A::AbstractMatrix, p::Real=2)

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

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

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

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

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

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

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

Examples

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

julia> opnorm(A, Inf)
6.0

julia> opnorm(A, 1)
5.0

opnorm(x::Number, p::Real=2)

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

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

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

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

Examples

julia> v = [1; im];

julia> vc = v';

julia> opnorm(vc, 1)
1.0

julia> norm(vc, 1)
2.0

julia> norm(v, 1)
2.0

julia> opnorm(vc, 2)
1.4142135623730951

julia> norm(vc, 2)
1.4142135623730951

julia> norm(v, 2)
1.4142135623730951

julia> opnorm(vc, Inf)
2.0

julia> norm(vc, Inf)
1.0

julia> norm(v, Inf)
1.0

normalize!(a::AbstractArray, p::Real=2)

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

normalize(a::AbstractArray, p::Real=2)

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

Examples

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

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

julia> norm(b)
1.0

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

julia> norm(c, 1)
1.0

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

julia> norm(a)
6.48074069840786

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

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.

condskeel(M, [x, p::Real=Inf])

\[\kappa_S(M, p) = \left\Vert \left\vert M \right\vert \left\vert M^{-1} \right\vert \right\Vert_p \\ \kappa_S(M, x, p) = \frac{\left\Vert \left\vert M \right\vert \left\vert M^{-1} \right\vert \left\vert x \right\vert \right\Vert_p}{\left \Vert x \right \Vert_p}\]

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

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

tr(M)

Matrix trace. Sums the diagonal elements of M.

Examples

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

julia> tr(A)
5

det(M)

Matrix determinant.

See also: logdet and logabsdet.

Examples

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

julia> det(M)
2.0

logdet(M)

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

Examples

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

julia> logdet(M)
0.6931471805599453

julia> logdet(Matrix(I, 3, 3))
0.0

logabsdet(M)

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

Examples

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

julia> det(A)
-1.0

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

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

julia> det(B)
2.0

julia> logabsdet(B)
(0.6931471805599453, 1.0)

inv(M)

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

Examples

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

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

julia> M*N == N*M == Matrix(I, 2, 2)
true

pinv(M; atol::Real=0, rtol::Real=atol>0 ? 0 : n*ϵ)
pinv(M, rtol::Real) = pinv(M; rtol=rtol) # to be deprecated in Julia 2.0

Computes the Moore-Penrose pseudoinverse.

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

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

For inverting dense ill-conditioned matrices in a least-squares sense, rtol = sqrt(eps(real(float(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 Matrix{Float64}:
 1.5  1.3
 1.2  1.9

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

julia> M * N
2×2 Matrix{Float64}:
 1.0          -2.22045e-16
 4.44089e-16   1.0

nullspace(M; atol::Real=0, rtol::Real=atol>0 ? 0 : n*ϵ)
nullspace(M, rtol::Real) = nullspace(M; rtol=rtol) # to be deprecated in Julia 2.0

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

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

Examples

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

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

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

julia> nullspace(M, atol=0.95)
3×1 Matrix{Float64}:
 0.0
 0.0
 1.0

kron(A, B)

Kronecker tensor product of two vectors or two matrices.

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

Examples

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

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

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

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

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

julia> reshape(kron(v,w), (length(w), length(v)))
3×2 Matrix{Int64}:
 3   6
 4   8
 5  10

kron!(C, A, B)

kron! is the in-place version of kron. Computes kron(A, B) and stores the result in C overwriting the existing value of C.

exp(A::AbstractMatrix)

Compute the matrix exponential of A, defined by

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

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

Examples

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

julia> exp(A)
2×2 Matrix{Float64}:
 2.71828  0.0
 0.0      2.71828

cis(A::AbstractMatrix)

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

See also cispi, sincos, exp.

Examples

julia> cis([π 0; 0 π]) ≈ -I
true

^(A::AbstractMatrix, p::Number)

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

Examples

julia> [1 2; 0 3]^3
2×2 Matrix{Int64}:
 1  26
 0  27

^(b::Number, A::AbstractMatrix)

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

Examples

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

julia> ℯ^[1 2; 0 3]
2×2 Matrix{Float64}:
 2.71828  17.3673
 0.0      20.0855

log(A::StridedMatrix)

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

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

Examples

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

julia> log(A)
2×2 Matrix{Float64}:
 1.0  0.0
 0.0  1.0

sqrt(A::AbstractMatrix)

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

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

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

Examples

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

julia> sqrt(A)
2×2 Matrix{Float64}:
 2.0  0.0
 0.0  2.0

cos(A::AbstractMatrix)

Compute the matrix cosine of a square matrix A.

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

Examples

julia> cos(fill(1.0, (2,2)))
2×2 Matrix{Float64}:
  0.291927  -0.708073
 -0.708073   0.291927

sin(A::AbstractMatrix)

Compute the matrix sine of a square matrix A.

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

Examples

julia> sin(fill(1.0, (2,2)))
2×2 Matrix{Float64}:
 0.454649  0.454649
 0.454649  0.454649

sincos(A::AbstractMatrix)

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

Examples

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

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

julia> C
2×2 Matrix{Float64}:
  0.291927  -0.708073
 -0.708073   0.291927

tan(A::AbstractMatrix)

Compute the matrix tangent of a square matrix A.

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

Examples

julia> tan(fill(1.0, (2,2)))
2×2 Matrix{Float64}:
 -1.09252  -1.09252
 -1.09252  -1.09252

sec(A::AbstractMatrix)

Compute the matrix secant of a square matrix A.

csc(A::AbstractMatrix)

Compute the matrix cosecant of a square matrix A.

cot(A::AbstractMatrix)

Compute the matrix cotangent of a square matrix A.

cosh(A::AbstractMatrix)

Compute the matrix hyperbolic cosine of a square matrix A.

sinh(A::AbstractMatrix)

Compute the matrix hyperbolic sine of a square matrix A.

tanh(A::AbstractMatrix)

Compute the matrix hyperbolic tangent of a square matrix A.

sech(A::AbstractMatrix)

Compute the matrix hyperbolic secant of square matrix A.

csch(A::AbstractMatrix)

Compute the matrix hyperbolic cosecant of square matrix A.

coth(A::AbstractMatrix)

Compute the matrix hyperbolic cotangent of square matrix A.

acos(A::AbstractMatrix)

Compute the inverse matrix cosine of a square matrix A.

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

Examples

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

asin(A::AbstractMatrix)

Compute the inverse matrix sine of a square matrix A.

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

Examples

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

atan(A::AbstractMatrix)

Compute the inverse matrix tangent of a square matrix A.

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

Examples

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

asec(A::AbstractMatrix)

Compute the inverse matrix secant of A.

acsc(A::AbstractMatrix)

Compute the inverse matrix cosecant of A.

acot(A::AbstractMatrix)

Compute the inverse matrix cotangent of A.

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].

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].

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].

asech(A::AbstractMatrix)

Compute the inverse matrix hyperbolic secant of A.

acsch(A::AbstractMatrix)

Compute the inverse matrix hyperbolic cosecant of A.

acoth(A::AbstractMatrix)

Compute the inverse matrix hyperbolic cotangent of A.

lyap(A, C)

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

Examples

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

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

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

julia> A*X + X*A' + B
2×2 Matrix{Float64}:
 0.0          6.66134e-16
 6.66134e-16  8.88178e-16

sylvester(A, B, C)

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

Examples

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

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

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

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

julia> A*X + X*B + C
2×2 Matrix{Float64}:
  2.66454e-15  1.77636e-15
 -3.77476e-15  4.44089e-16

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

issymmetric(A) -> Bool

Test whether a matrix is symmetric.

Examples

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

julia> issymmetric(a)
true

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

julia> issymmetric(b)
false

isposdef(A) -> Bool

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

See also isposdef!, cholesky.

Examples

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

julia> isposdef(A)
true

isposdef!(A) -> Bool

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

Examples

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

julia> isposdef!(A)
true

julia> A
2×2 Matrix{Float64}:
 1.0  2.0
 2.0  6.78233

istril(A::AbstractMatrix, k::Integer = 0) -> Bool

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

Examples

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

julia> istril(a)
false

julia> istril(a, 1)
true

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

julia> istril(b)
true

julia> istril(b, -1)
false

istriu(A::AbstractMatrix, k::Integer = 0) -> Bool

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

Examples

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

julia> istriu(a)
false

julia> istriu(a, -1)
true

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

julia> istriu(b)
true

julia> istriu(b, 1)
false

isdiag(A) -> Bool

Test whether a matrix is diagonal.

Examples

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

julia> isdiag(a)
false

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

julia> isdiag(b)
true

ishermitian(A) -> Bool

Test whether a matrix is Hermitian.

Examples

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

julia> ishermitian(a)
true

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

julia> ishermitian(b)
true

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 Matrix{Complex{Int64}}:
 3+2im  9+2im
 8+7im  4+6im

julia> transpose(A)
2×2 transpose(::Matrix{Complex{Int64}}) with eltype Complex{Int64}:
 3+2im  8+7im
 9+2im  4+6im

transpose!(dest,src)

Transpose array src and store the result in the preallocated array dest, which should have a size corresponding to (size(src,2),size(src,1)). No in-place transposition is supported and unexpected results will happen if src and dest have overlapping memory regions.

Examples

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

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

julia> transpose!(B, A);

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

julia> A
2×2 Matrix{Complex{Int64}}:
 3+2im  9+2im
 8+7im  4+6im

Transpose

Lazy wrapper type for a transpose view of the underlying linear algebra object, usually an AbstractVector/AbstractMatrix, but also some Factorization, for instance. Usually, the Transpose constructor should not be called directly, use transpose instead. To materialize the view use copy.

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

Examples

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

julia> transpose(A)
2×2 transpose(::Matrix{Complex{Int64}}) with eltype Complex{Int64}:
 3+2im  8+7im
 9+2im  4+6im

A'
adjoint(A)

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

For number types, adjoint returns the complex conjugate, and therefore it is equivalent to the identity function for real numbers.

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

Examples

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

julia> adjoint(A)
2×2 adjoint(::Matrix{Complex{Int64}}) with eltype Complex{Int64}:
 3-2im  8-7im
 9-2im  4-6im

julia> x = [3, 4im]
2-element Vector{Complex{Int64}}:
 3 + 0im
 0 + 4im

julia> x'x
25 + 0im

adjoint!(dest,src)

Conjugate transpose array src and store the result in the preallocated array dest, which should have a size corresponding to (size(src,2),size(src,1)). No in-place transposition is supported and unexpected results will happen if src and dest have overlapping memory regions.

Examples

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

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

julia> adjoint!(B, A);

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

julia> A
2×2 Matrix{Complex{Int64}}:
 3+2im  9+2im
 8+7im  4+6im

Adjoint

Lazy wrapper type for an adjoint view of the underlying linear algebra object, usually an AbstractVector/AbstractMatrix, but also some Factorization, for instance. Usually, the Adjoint constructor should not be called directly, use adjoint instead. To materialize the view use copy.

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

Examples

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

julia> adjoint(A)
2×2 adjoint(::Matrix{Complex{Int64}}) with eltype Complex{Int64}:
 3-2im  8-7im
 9-2im  4-6im

copy(A::Transpose)
copy(A::Adjoint)

Eagerly evaluate the lazy matrix transpose/adjoint. Note that the transposition is applied recursively to elements.

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

Examples

julia> A = [1 2im; -3im 4]
2×2 Matrix{Complex{Int64}}:
 1+0im  0+2im
 0-3im  4+0im

julia> T = transpose(A)
2×2 transpose(::Matrix{Complex{Int64}}) with eltype Complex{Int64}:
 1+0im  0-3im
 0+2im  4+0im

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

stride1(A) -> Int

Return the distance between successive array elements in dimension 1 in units of element size.

Examples

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

julia> LinearAlgebra.stride1(A)
1

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

julia> LinearAlgebra.stride1(B)
2

LinearAlgebra.checksquare(A)

Check that a matrix is square, then return its common dimension. For multiple arguments, return a vector.

Examples

julia> A = fill(1, (4,4)); B = fill(1, (5,5));

julia> LinearAlgebra.checksquare(A, B)
2-element Vector{Int64}:
 4
 5

LinearAlgebra.peakflops(n::Integer=2000; parallel::Bool=false)

peakflops computes the peak flop rate of the computer by using double precision gemm!. By default, if no arguments are specified, it multiplies a matrix of size n x n, where n = 2000. If the underlying BLAS is using multiple threads, higher flop rates are realized. The number of BLAS threads can be set with BLAS.set_num_threads(n).

If the keyword argument parallel is set to true, peakflops is run in parallel on all the worker processors. The flop rate of the entire parallel computer is returned. When running in parallel, only 1 BLAS thread is used. The argument n still refers to the size of the problem that is solved on each processor.

mul!(Y, A, B) -> Y

Calculates the matrix-matrix or matrix-vector product $AB$ and stores the result in Y, overwriting the existing value of Y. Note that Y must not be aliased with either A or B.

Examples

julia> A=[1.0 2.0; 3.0 4.0]; B=[1.0 1.0; 1.0 1.0]; Y = similar(B); mul!(Y, A, B);

julia> Y
2×2 Matrix{Float64}:
 3.0  3.0
 7.0  7.0

Implementation

For custom matrix and vector types, it is recommended to implement 5-argument mul! rather than implementing 3-argument mul! directly if possible.

mul!(C, A, B, α, β) -> C

Combined inplace matrix-matrix or matrix-vector multiply-add $A B α + C β$. The result is stored in C by overwriting it. Note that C must not be aliased with either A or B.

Examples

julia> A=[1.0 2.0; 3.0 4.0]; B=[1.0 1.0; 1.0 1.0]; C=[1.0 2.0; 3.0 4.0];

julia> mul!(C, A, B, 100.0, 10.0) === C
true

julia> C
2×2 Matrix{Float64}:
 310.0  320.0
 730.0  740.0

lmul!(a::Number, B::AbstractArray)

Scale an array B by a scalar a overwriting B in-place. Use rmul! to multiply scalar from right. The scaling operation respects the semantics of the multiplication * between a and an element of B. In particular, this also applies to multiplication involving non-finite numbers such as NaN and ±Inf.

Examples

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

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

julia> lmul!(0.0, [Inf])
1-element Vector{Float64}:
 NaN

lmul!(A, B)

Calculate the matrix-matrix product $AB$, overwriting B, and return the result. Here, A must be of special matrix type, like, e.g., Diagonal, UpperTriangular or LowerTriangular, or of some orthogonal type, see QR.

Examples

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

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

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

julia> B
2×2 Matrix{Int64}:
 2  1
 3  0

julia> B = [1.0 2.0; 3.0 4.0];

julia> F = qr([0 1; -1 0]);

julia> lmul!(F.Q, B)
2×2 Matrix{Float64}:
 3.0  4.0
 1.0  2.0

rmul!(A::AbstractArray, b::Number)

Scale an array A by a scalar b overwriting A in-place. Use lmul! to multiply scalar from left. The scaling operation respects the semantics of the multiplication * between an element of A and b. In particular, this also applies to multiplication involving non-finite numbers such as NaN and ±Inf.

Examples

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

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

julia> rmul!([NaN], 0.0)
1-element Vector{Float64}:
 NaN

rmul!(A, B)

Calculate the matrix-matrix product $AB$, overwriting A, and return the result. Here, B must be of special matrix type, like, e.g., Diagonal, UpperTriangular or LowerTriangular, or of some orthogonal type, see QR.

Examples

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

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

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

julia> A
2×2 Matrix{Int64}:
 0  3
 1  2

julia> A = [1.0 2.0; 3.0 4.0];

julia> F = qr([0 1; -1 0]);

julia> rmul!(A, F.Q)
2×2 Matrix{Float64}:
 2.0  1.0
 4.0  3.0

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 Vector{Float64}:
  0.7128099173553719
 -0.051652892561983674
  0.10020661157024757

julia> A\X
3-element Vector{Float64}:
  0.7128099173553719
 -0.05165289256198333
  0.10020661157024785

ldiv!(A, B)

Compute A \ B in-place and overwriting B to store the result.

The argument A should not be a matrix. Rather, instead of matrices it should be a factorization object (e.g. produced by factorize or cholesky). The reason for this is that factorization itself is both expensive and typically allocates memory (although it can also be done in-place via, e.g., lu!), and performance-critical situations requiring ldiv! usually also require fine-grained control over the factorization of A.

Examples

julia> A = [1 2.2 4; 3.1 0.2 3; 4 1 2];

julia> X = [1; 2.5; 3];

julia> Y = copy(X);

julia> ldiv!(qr(A), X);

julia> X
3-element Vector{Float64}:
  0.7128099173553719
 -0.051652892561983674
  0.10020661157024757

julia> A\Y
3-element Vector{Float64}:
  0.7128099173553719
 -0.05165289256198333
  0.10020661157024785

ldiv!(a::Number, B::AbstractArray)

Divide each entry in an array B by a scalar a overwriting B in-place. Use rdiv! to divide scalar from right.

Examples

julia> B = [1.0 2.0; 3.0 4.0]
2×2 Matrix{Float64}:
 1.0  2.0
 3.0  4.0

julia> ldiv!(2.0, B)
2×2 Matrix{Float64}:
 0.5  1.0
 1.5  2.0

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.

rdiv!(A::AbstractArray, b::Number)

Divide each entry in an array A by a scalar b overwriting A in-place. Use ldiv! to divide scalar from left.

Examples

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

julia> rdiv!(A, 2.0)
2×2 Matrix{Float64}:
 0.5  1.0
 1.5  2.0

Interface to BLAS subroutines.

dot(n, X, incx, Y, incy)

Dot product of two vectors consisting of n elements of array X with stride incx and n elements of array Y with stride incy.

Examples

julia> BLAS.dot(10, fill(1.0, 10), 1, fill(1.0, 20), 2)
10.0

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

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

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

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

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

asum(n, X, incx)

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

For a real array, the magnitude is the absolute value. For a complex array, the magnitude is the sum of the absolute value of the real part and the absolute value of the imaginary part.

Examples

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

julia> BLAS.asum(2, fill(1.0im, 10), 5)
2.0

axpy!(a, X, Y)

Overwrite Y with X*a + 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 Vector{Int64}:
  6
  9
 12

axpby!(a, X, b, Y)

Overwrite Y with X*a + Y*b, where a and b are scalars. Return Y.

Examples

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

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

julia> BLAS.axpby!(2., x, 3., y)
3-element Vector{Float64}:
 14.0
 19.0
 24.0

scal!(n, a, X, incx)
scal!(a, X)

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

If n and incx are not provided, length(X) and stride(X,1) are used.

scal(n, a, X, incx)
scal(a, X)

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

If n and incx are not provided, length(X) and stride(X,1) are used.

iamax(n, dx, incx)
iamax(dx)

Find the index of the element of dx with the maximum absolute value. n is the length of dx, and incx is the stride. If n and incx are not provided, they assume default values of n=length(dx) and incx=stride1(dx).

ger!(alpha, x, y, A)

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

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.

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.

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.

syr2k!(uplo, trans, alpha, A, B, beta, C)

Rank-2k update of the symmetric matrix C as alpha*A*transpose(B) + alpha*B*transpose(A) + beta*C or alpha*transpose(A)*B + alpha*transpose(B)*A + beta*C according to trans. Only the uplo triangle of C is used. Returns C.

syr2k(uplo, trans, alpha, A, B)

Returns the uplo triangle of alpha*A*transpose(B) + alpha*B*transpose(A) or alpha*transpose(A)*B + alpha*transpose(B)*A, according to trans.

syr2k(uplo, trans, A, B)

Returns the uplo triangle of A*transpose(B) + B*transpose(A) or transpose(A)*B + transpose(B)*A, according to trans.

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.

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.

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.

her2k!(uplo, trans, alpha, A, B, beta, C)

Rank-2k update of the Hermitian matrix C as alpha*A*B' + alpha*B*A' + beta*C or alpha*A'*B + alpha*B'*A + beta*C according to trans. The scalar beta has to be real. Only the uplo triangle of C is used. Returns C.

her2k(uplo, trans, alpha, A, B)

Returns the uplo triangle of alpha*A*B' + alpha*B*A' or alpha*A'*B + alpha*B'*A, according to trans.

her2k(uplo, trans, A, B)

Returns the uplo triangle of A*B' + B*A' or A'*B + B'*A, according to trans.

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.

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.

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

Update vector y as alpha*A*x + beta*y where A is a symmetric band matrix of order size(A,2) with k super-diagonals stored in the argument A. The storage layout for A is described the reference BLAS module, level-2 BLAS at http://www.netlib.org/lapack/explore-html/. Only the uplo triangle of A is used.

Return the updated y.

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.

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.

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.

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

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

gemm(tA, tB, A, B)

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

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.

gemv(tA, alpha, A, x)

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

gemv(tA, A, x)

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

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.

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.

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.

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.

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.

symv(ul, A, x)

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

hemm!(side, ul, alpha, A, B, beta, C)

Update C as alpha*A*B + beta*C or alpha*B*A + beta*C according to side. A is assumed to be Hermitian. Only the ul triangle of A is used. Return the updated C.

hemm(side, ul, alpha, A, B)

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

hemm(side, ul, A, B)

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

hemv!(ul, alpha, A, x, beta, y)

Update the vector y as alpha*A*x + beta*y. A is assumed to be Hermitian. Only the ul triangle of A is used. alpha and beta are scalars. Return the updated y.

hemv(ul, alpha, A, x)

Return alpha*A*x. A is assumed to be Hermitian. Only the ul triangle of A is used. alpha is a scalar.

hemv(ul, A, x)

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

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.

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.

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.

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.

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.

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.

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.

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.

set_num_threads(n::Integer)
set_num_threads(::Nothing)

Set the number of threads the BLAS library should use equal to n::Integer.

Also accepts nothing, in which case julia tries to guess the default number of threads. Passing nothing is discouraged and mainly exists for historical reasons.

get_num_threads()

Get the number of threads the BLAS library is using.

Interfaces to LAPACK subroutines.

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.

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.

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.

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).

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.

gelqf!(A, tau)

Compute the LQ factorization of A, A = LQ. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

gelqf!(A) -> (A, tau)

Compute the LQ factorization of A, A = LQ.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

geqlf!(A, tau)

Compute the QL factorization of A, A = QL. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

geqlf!(A) -> (A, tau)

Compute the QL factorization of A, A = QL.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

geqrf!(A, tau)

Compute the QR factorization of A, A = QR. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

geqrf!(A) -> (A, tau)

Compute the QR factorization of A, A = QR.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

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

Compute the pivoted QR factorization of A, AP = QR using BLAS level 3. P is a pivoting matrix, represented by jpvt. tau stores the elementary reflectors. The arguments jpvt and tau are optional and allow for passing preallocated arrays. When passed, jpvt must have length greater than or equal to n if A is an (m x n) matrix and tau must have length greater than or equal to the smallest dimension of A.

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

gerqf!(A, tau)

Compute the RQ factorization of A, A = RQ. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

gerqf!(A) -> (A, tau)

Compute the RQ factorization of A, A = RQ.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

geqrt!(A, T)

Compute the blocked QR factorization of A, A = QR. T contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization. The first dimension of T sets the block size and it must be between 1 and n. The second dimension of T must equal the smallest dimension of A.

Returns A and T modified in-place.

geqrt!(A, nb) -> (A, T)

Compute the blocked QR factorization of A, A = QR. nb sets the block size and it must be between 1 and n, the second dimension of A.

Returns A, modified in-place, and T, which contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization.

geqrt3!(A, T)

Recursively computes the blocked QR factorization of A, A = QR. T contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization. The first dimension of T sets the block size and it must be between 1 and n. The second dimension of T must equal the smallest dimension of A.

Returns A and T modified in-place.

geqrt3!(A) -> (A, T)

Recursively computes the blocked QR factorization of A, A = QR.

Returns A, modified in-place, and T, which contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization.

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).

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.

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.

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.

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.

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).

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.

gesvx!(fact, trans, A, AF, ipiv, equed, R, C, B) -> (X, equed, R, C, B, rcond, ferr, berr, work)

Solves the linear equation A * X = B (trans = N), transpose(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) using the LU factorization of A. fact may be E, in which case A will be equilibrated and copied to AF; F, in which case AF and ipiv from a previous LU factorization are inputs; or N, in which case A will be copied to AF and then factored. If fact = F, equed may be N, meaning A has not been equilibrated; R, meaning A was multiplied by Diagonal(R) from the left; C, meaning A was multiplied by Diagonal(C) from the right; or B, meaning A was multiplied by Diagonal(R) from the left and Diagonal(C) from the right. If fact = F and equed = R or B the elements of R must all be positive. If fact = F and equed = C or B the elements of C must all be positive.

Returns the solution X; equed, which is an output if fact is not N, and describes the equilibration that was performed; R, the row equilibration diagonal; C, the column equilibration diagonal; B, which may be overwritten with its equilibrated form Diagonal(R)*B (if trans = N and equed = R,B) or Diagonal(C)*B (if trans = T,C and equed = C,B); rcond, the reciprocal condition number of A after equilbrating; ferr, the forward error bound for each solution vector in X; berr, the forward error bound for each solution vector in X; and work, the reciprocal pivot growth factor.

gesvx!(A, B)

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.