Linear Algebra

# Linear Algebra

## Standard Functions

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

*(x, y...)

Multiplication operator. x*y*z*... calls this function with all arguments, i.e. *(x, y, z, ...).

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\(x, y)

Left division operator: multiplication of y by the inverse of x on the left. Gives floating-point results for integer arguments.

julia> 3 \ 6
2.0

julia> inv(3) * 6
2.0

julia> A = [1 2; 3 4]; x = [5, 6];

julia> A \ x
2-element Array{Float64,1}:
-4.0
4.5

julia> inv(A) * x
2-element Array{Float64,1}:
-4.0
4.5
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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.

Example:

julia> dot(10, ones(10), 1, ones(20), 2)
10.0
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vecdot(x, y)

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

Examples

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

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

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

julia> vecdot(x, y)
150.0
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cross(x, y)
×(x,y)

Compute the cross product of two 3-vectors.

Example

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

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

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

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

Properties of Atype of factorization
Positive-definiteCholesky (see cholfact)
Dense Symmetric/HermitianBunch-Kaufman (see bkfact)
Sparse Symmetric/HermitianLDLt (see ldltfact)
TriangularTriangular
DiagonalDiagonal
BidiagonalBidiagonal
TridiagonalLU (see lufact)
Symmetric real tridiagonalLDLt (see ldltfact)
General squareLU (see lufact)
General non-squareQR (see qrfact)

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

Example

julia> A = Array(Bidiagonal(ones(5, 5), true))
5×5 Array{Float64,2}:
1.0  1.0  0.0  0.0  0.0
0.0  1.0  1.0  0.0  0.0
0.0  0.0  1.0  1.0  0.0
0.0  0.0  0.0  1.0  1.0
0.0  0.0  0.0  0.0  1.0

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

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Diagonal(A::AbstractMatrix)

Constructs a matrix from the diagonal of A.

Example

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

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

Constructs a matrix with V as its diagonal.

Example

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

julia> Diagonal(V)
2×2 Diagonal{Int64}:
1  ⋅
⋅  2
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Bidiagonal(dv, ev, isupper::Bool)

Constructs an upper (isupper=true) or lower (isupper=false) 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). ev's length must be one less than the length of dv.

Example

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

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

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

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

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). ev's length must be one less than the length of dv.

Example

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

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

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

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

Construct a Bidiagonal matrix from the main diagonal of A and its first super- (if isupper=true) or sub-diagonal (if isupper=false).

Example

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

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

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

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

Example

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

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

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

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.

Example

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

julia> du = [4; 5; 6]
3-element Array{Int64,1}:
4
5
6

julia> d = [7; 8; 9; 0]
4-element Array{Int64,1}:
7
8
9
0

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

returns a Tridiagonal array based on (abstract) matrix A, using its first lower diagonal, main diagonal, and first upper diagonal.

Example

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

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

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

Example

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

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

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

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

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Hermitian(A, uplo=:U)

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

Example

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

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

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

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

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LowerTriangular(A::AbstractMatrix)

Construct a LowerTriangular view of the the matrix A.

Example

julia> A = [1.0 2.0 3.0; 4.0 5.0 6.0; 7.0 8.0 9.0]
3×3 Array{Float64,2}:
1.0  2.0  3.0
4.0  5.0  6.0
7.0  8.0  9.0

julia> LowerTriangular(A)
3×3 LowerTriangular{Float64,Array{Float64,2}}:
1.0   ⋅    ⋅
4.0  5.0   ⋅
7.0  8.0  9.0
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UpperTriangular(A::AbstractMatrix)

Construct an UpperTriangular view of the the matrix A.

Example

julia> A = [1.0 2.0 3.0; 4.0 5.0 6.0; 7.0 8.0 9.0]
3×3 Array{Float64,2}:
1.0  2.0  3.0
4.0  5.0  6.0
7.0  8.0  9.0

julia> UpperTriangular(A)
3×3 UpperTriangular{Float64,Array{Float64,2}}:
1.0  2.0  3.0
⋅   5.0  6.0
⋅    ⋅   9.0
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lu(A, pivot=Val{true}) -> L, U, p

Compute the LU factorization of A, such that A[p,:] = L*U. By default, pivoting is used. This can be overridden by passing Val{false} for the second argument.

See also lufact.

Example

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

julia> L, U, p = lu(A)
([1.0 0.0; 0.666667 1.0], [6.0 3.0; 0.0 1.0], [2, 1])

julia> A[p, :] == L * U
true
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lufact(A::SparseMatrixCSC) -> 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 Complex128 respectively and Ti is an integer type (Int32 or Int64).

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

ComponentDescription
F[:L]L (lower triangular) part of LU
F[:U]U (upper triangular) part of LU
F[:p]right permutation Vector
F[:q]left permutation Vector
F[:Rs]Vector of scaling factors
F[:(:)](L,U,p,q,Rs) components

The relation between F and A is

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

F further supports the following functions:

Note

lufact(A::SparseMatrixCSC) uses the UMFPACK library that is part of SuiteSparse. As this library only supports sparse matrices with Float64 or Complex128 elements, lufact converts A into a copy that is of type SparseMatrixCSC{Float64} or SparseMatrixCSC{Complex128} as appropriate.

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lufact(A [,pivot=Val{true}]) -> F::LU

Compute the LU factorization of A.

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

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

ComponentDescription
F[:L]L (lower triangular) part of LU
F[:U]U (upper triangular) part of LU
F[:p](right) permutation Vector
F[:P](right) permutation Matrix

The relationship between F and A is

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

F further supports the following functions:

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

Example

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

julia> F = lufact(A)
Base.LinAlg.LU{Float64,Array{Float64,2}} with factors L and U:
[1.0 0.0; 1.5 1.0]
[4.0 3.0; 0.0 -1.5]

julia> F[:L] * F[:U] == A[F[:p], :]
true
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lufact!(A, pivot=Val{true}) -> LU

lufact! is the same as lufact, 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.

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chol(A) -> U

Compute the Cholesky factorization of a positive definite matrix A and return the UpperTriangular matrix U such that A = U'U.

Example

julia> A = [1. 2.; 2. 50.]
2×2 Array{Float64,2}:
1.0   2.0
2.0  50.0

julia> U = chol(A)
2×2 UpperTriangular{Float64,Array{Float64,2}}:
1.0  2.0
⋅   6.78233

julia> U'U
2×2 Array{Float64,2}:
1.0   2.0
2.0  50.0
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chol(x::Number) -> y

Compute the square root of a non-negative number x.

Example

julia> chol(16)
4.0
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cholfact(A; shift = 0.0, perm = Int[]) -> 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. A fill-reducing permutation is used. F = cholfact(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).

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

Note

This method uses the CHOLMOD library from SuiteSparse, which only supports doubles or complex doubles. Input matrices not of those element types will be converted to SparseMatrixCSC{Float64} or SparseMatrixCSC{Complex128} as appropriate.

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

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cholfact(A, [uplo::Symbol,] Val{false}) -> 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 HermitianStridedMatrix or a perfectly symmetric or Hermitian StridedMatrix. In the latter case, the optional argument uplo may be :L for using the lower part or :U for the upper part of A. The default is to use :U. The triangular Cholesky factor can be obtained from the factorization F with: F[:L] and F[:U]. The following functions are available for Cholesky objects: size, \, inv, and det. A PosDefException exception is thrown in case the matrix is not positive definite.

Example

julia> A = [4. 12. -16.; 12. 37. -43.; -16. -43. 98.]
3×3 Array{Float64,2}:
4.0   12.0  -16.0
12.0   37.0  -43.0
-16.0  -43.0   98.0

julia> C = cholfact(A)
Base.LinAlg.Cholesky{Float64,Array{Float64,2}} with factor:
[2.0 6.0 -8.0; 0.0 1.0 5.0; 0.0 0.0 3.0]

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

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

julia> C[:L] * C[:U] == A
true
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cholfact(A, [uplo::Symbol,] Val{true}; tol = 0.0) -> 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 HermitianStridedMatrix or a perfectly symmetric or Hermitian StridedMatrix. In the latter case, the optional argument uplo may be :L for using the lower part or :U for the upper part of A. The default is to use :U. The triangular Cholesky factor can be obtained from the factorization F with: F[:L] and F[:U]. The following functions are available for PivotedCholesky objects: size, \, inv, det, and rank. The argument tol determines the tolerance for determining the rank. For negative values, the tolerance is the machine precision.

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cholfact!(F::Factor, A; shift = 0.0) -> 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 cholfact.

Note

This method uses the CHOLMOD library from SuiteSparse, which only supports doubles or complex doubles. Input matrices not of those element types will be converted to SparseMatrixCSC{Float64} or SparseMatrixCSC{Complex128} as appropriate.

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cholfact!(A, [uplo::Symbol,] Val{false}) -> Cholesky

The same as cholfact, 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.

Example

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

julia> cholfact!(A)
ERROR: InexactError()
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cholfact!(A, [uplo::Symbol,] Val{true}; tol = 0.0) -> CholeskyPivoted

The same as cholfact, 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.

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

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

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

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

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

Update a Cholesky factorization C with the vector v. If A = C[:U]'C[:U] then CC = cholfact(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.

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

Downdate a Cholesky factorization C with the vector v. If A = C[:U]'C[:U] then CC = cholfact(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.

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ldltfact(A; shift = 0.0, perm=Int[]) -> 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 = ldltfact(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.

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

Note

This method uses the CHOLMOD library from SuiteSparse, which only supports doubles or complex doubles. Input matrices not of those element types will be converted to SparseMatrixCSC{Float64} or SparseMatrixCSC{Complex128} as appropriate.

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

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ldltfact(S::SymTridiagonal) -> LDLt

Compute an LDLt factorization of a real symmetric tridiagonal matrix such that A = 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 = ldltfact(A) is to solve the linear system of equations Ax = b with F\b.

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ldltfact!(F::Factor, A; shift = 0.0) -> 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 ldltfact.

Note

This method uses the CHOLMOD library from SuiteSparse, which only supports doubles or complex doubles. Input matrices not of those element types will be converted to SparseMatrixCSC{Float64} or SparseMatrixCSC{Complex128} as appropriate.

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ldltfact!(S::SymTridiagonal) -> LDLt

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

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qr(A, pivot=Val{false}; thin::Bool=true) -> Q, R, [p]

Compute the (pivoted) QR factorization of A such that either A = Q*R or A[:,p] = Q*R. Also see qrfact. The default is to compute a thin factorization. Note that R is not extended with zeros when the full Q is requested.

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qr(v::AbstractVector) -> w, r

Computes the polar decomposition of a vector. Returns w, a unit vector in the direction of v, and r, the norm of v.

See also normalize, normalize!, and LinAlg.qr!.

Example

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

julia> w, r = qr(v)
([0.447214, 0.894427], 2.23606797749979)

julia> w*r == v
true
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LinAlg.qr!(v::AbstractVector) -> w, r

Computes the polar decomposition of a vector. Instead of returning a new vector as qr(v::AbstractVector), this function mutates the input vector v in place. Returns w, a unit vector in the direction of v (this is a mutation of v), and r, the norm of v.

See also normalize, normalize!, and qr.

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qrfact(A, pivot=Val{false}) -> F

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

$A = Q R$

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

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

• 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)

The following functions are available for the QR objects: inv, size, and \. When A is rectangular, \ will return a least squares solution and if the solution is not unique, the one with smallest norm is returned.

Multiplication with respect to either thin or full 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 full which has a named argument thin.

Example

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

julia> F = qrfact(A)
Base.LinAlg.QRCompactWY{Float64,Array{Float64,2}} with factors Q and R:
[-0.6 0.0 0.8; -0.8 0.0 -0.6; 0.0 -1.0 0.0]
[-5.0 10.0; 0.0 -1.0]

julia> F[:Q] * F[:R] == A
true
Note

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

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qrfact(A) -> SPQR.Factorization

Compute the QR factorization of a sparse matrix A. A fill-reducing permutation is used. The main application of this type is to solve least squares problems with \. The function calls the C library SPQR and a few additional functions from the library are wrapped but not exported.

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qrfact!(A, pivot=Val{false})

qrfact! is the same as qrfact 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.

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QR <: Factorization

A QR matrix factorization stored in a packed format, typically obtained from qrfact. 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:

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

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 = eye(m,n) + tril(F.factors,-1).

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

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QRCompactWY <: Factorization

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

$A = Q R$

where $Q$ is an orthogonal/unitary matrix and $R$ is upper triangular. It is similar to the QR format except that the orthogonal/unitary matrix $Q$ is stored in Compact WY format [Schreiber1989], as a lower trapezoidal matrix $V$ and an upper triangular matrix $T$ where

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

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

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 = eye(m,n) + tril(F.factors,-1).

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

Note

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

[Bischof1987]

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

[Schreiber1989]

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

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QRPivoted <: Factorization

A QR matrix factorization with column pivoting in a packed format, typically obtained from qrfact. 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:

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

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 = eye(m,n) + 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 $P$.

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lqfact!(A) -> LQ

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

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lqfact(A) -> LQ

Compute the LQ factorization of A. See also lq.

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lq(A; [thin=true]) -> L, Q

Perform an LQ factorization of A such that A = L*Q. The default is to compute a thin factorization. The LQ factorization is the QR factorization of A.'. L is not extended with zeros if the full Q is requested.

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bkfact(A, uplo::Symbol=:U, symmetric::Bool=issymmetric(A), rook::Bool=false) -> BunchKaufman

Compute the Bunch-Kaufman [Bunch1977] factorization of a symmetric or Hermitian matrix A and return a BunchKaufman object. uplo indicates which triangle of matrix A to reference. If symmetric is true, A is assumed to be symmetric. If symmetric is false, A is assumed to be Hermitian. If rook is true, rook pivoting is used. If rook is false, rook pivoting is not used. The following functions are available for BunchKaufman objects: size, \, inv, issymmetric, ishermitian.

[Bunch1977]

J R Bunch and L Kaufman, Some stable methods for calculating inertia and solving symmetric linear systems, Mathematics of Computation 31:137 (1977), 163-179. url.

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bkfact!(A, uplo::Symbol=:U, symmetric::Bool=issymmetric(A), rook::Bool=false) -> BunchKaufman

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

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eig(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> D, V
eig(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> D, V
eig(A, permute::Bool=true, scale::Bool=true) -> D, V

Computes eigenvalues (D) and eigenvectors (V) of A. See eigfact for details on the irange, vl, and vu arguments (for SymTridiagonal, Hermitian, and Symmetric matrices) and the permute and scale keyword arguments. The eigenvectors are returned columnwise.

Example

julia> eig([1.0 0.0 0.0; 0.0 3.0 0.0; 0.0 0.0 18.0])
([1.0, 3.0, 18.0], [1.0 0.0 0.0; 0.0 1.0 0.0; 0.0 0.0 1.0])

eig is a wrapper around eigfact, extracting all parts of the factorization to a tuple; where possible, using eigfact is recommended.

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eig(A, B) -> D, V

Computes generalized eigenvalues (D) and vectors (V) of A with respect to B.

eig is a wrapper around eigfact, extracting all parts of the factorization to a tuple; where possible, using eigfact is recommended.

Example

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

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

julia> eig(A, B)
(Complex{Float64}[0.0+1.0im, 0.0-1.0im], Complex{Float64}[0.0-1.0im 0.0+1.0im; -1.0-0.0im -1.0+0.0im])
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eigvals(A; permute::Bool=true, scale::Bool=true) -> values

Returns the eigenvalues of A.

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

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eigvals(A, B) -> values

Computes the generalized eigenvalues of A and B.

Example

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

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

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

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

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

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

julia> eigvals(A)
3-element Array{Float64,1}:
-2.14005
1.0
5.14005
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eigvals(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> values

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

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

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

julia> eigvals(A)
3-element Array{Float64,1}:
-2.14005
1.0
5.14005
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eigvals!(A; permute::Bool=true, scale::Bool=true) -> values

Same as eigvals, but saves space by overwriting the input A, instead of creating a copy. The option permute=true permutes the matrix to become closer to upper triangular, and scale=true scales the matrix by its diagonal elements to make rows and columns more equal in norm.

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eigvals!(A, B) -> values

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

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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.

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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.

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

Returns 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.

Example

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

julia> eigmax(A)
1.0

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

julia> eigmax(A)
ERROR: DomainError:
Stacktrace:
 #eigmax#46(::Bool, ::Bool, ::Function, ::Array{Complex{Int64},2}) at ./linalg/eigen.jl:238
 eigmax(::Array{Complex{Int64},2}) at ./linalg/eigen.jl:236
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eigmin(A; permute::Bool=true, scale::Bool=true)

Returns 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.

Example

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

julia> eigmin(A)
-1.0

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

julia> eigmin(A)
ERROR: DomainError:
Stacktrace:
 #eigmin#47(::Bool, ::Bool, ::Function, ::Array{Complex{Int64},2}) at ./linalg/eigen.jl:280
 eigmin(::Array{Complex{Int64},2}) at ./linalg/eigen.jl:278
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eigvecs(A::SymTridiagonal[, eigvals]) -> Matrix

Returns 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.

Example

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

julia> eigvals(A)
3-element Array{Float64,1}:
-2.14005
1.0
5.14005

julia> eigvecs(A)
3×3 Array{Float64,2}:
0.418304  -0.83205      0.364299
-0.656749  -7.39009e-16  0.754109
0.627457   0.5547       0.546448

julia> eigvecs(A, [1.])
3×1 Array{Float64,2}:
0.83205
4.26351e-17
-0.5547
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eigvecs(A; permute::Bool=true, scale::Bool=true) -> Matrix

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

Example

julia> eigvecs([1.0 0.0 0.0; 0.0 3.0 0.0; 0.0 0.0 18.0])
3×3 Array{Float64,2}:
1.0  0.0  0.0
0.0  1.0  0.0
0.0  0.0  1.0
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eigvecs(A, B) -> Matrix

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

Example

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

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

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

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

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.

Example

julia> F = eigfact([1.0 0.0 0.0; 0.0 3.0 0.0; 0.0 0.0 18.0])
Base.LinAlg.Eigen{Float64,Float64,Array{Float64,2},Array{Float64,1}}([1.0, 3.0, 18.0], [1.0 0.0 0.0; 0.0 1.0 0.0; 0.0 0.0 1.0])

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

julia> F[:vectors]
3×3 Array{Float64,2}:
1.0  0.0  0.0
0.0  1.0  0.0
0.0  0.0  1.0
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eigfact(A, B) -> GeneralizedEigen

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

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

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

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

The UnitRangeirange specifies indices of the sorted eigenvalues to search for.

Note

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

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

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

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

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

Note

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

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eigfact!(A, [B])

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

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hessfact(A) -> Hessenberg

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

Example

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

julia> F = hessfact(A);

julia> F[:Q] * F[:H] * F[:Q]'
3×3 Array{Float64,2}:
4.0  9.0  7.0
4.0  4.0  1.0
4.0  3.0  2.0
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hessfact!(A) -> Hessenberg

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

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schurfact(A::StridedMatrix) -> F::Schur

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

Example

julia> A = [-2. 1. 3.; 2. 1. -1.; -7. 2. 7.]
3×3 Array{Float64,2}:
-2.0  1.0   3.0
2.0  1.0  -1.0
-7.0  2.0   7.0

julia> F = schurfact(A)
Base.LinAlg.Schur{Float64,Array{Float64,2}} with factors T and Z:
[2.0 0.801792 6.63509; -8.55988e-11 2.0 8.08286; 0.0 0.0 1.99999]
[0.577351 0.154299 -0.801784; 0.577346 -0.77152 0.267262; 0.577354 0.617211 0.534522]
and values:
Complex{Float64}[2.0+8.28447e-6im, 2.0-8.28447e-6im, 1.99999+0.0im]

julia> F[:vectors] * F[:Schur] * F[:vectors]'
3×3 Array{Float64,2}:
-2.0  1.0   3.0
2.0  1.0  -1.0
-7.0  2.0   7.0
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schurfact(A::StridedMatrix, B::StridedMatrix) -> F::GeneralizedSchur

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

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schurfact!(A::StridedMatrix) -> F::Schur

Same as schurfact but uses the input argument as workspace.

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

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

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schur(A::StridedMatrix) -> T::Matrix, Z::Matrix, λ::Vector

Computes the Schur factorization of the matrix A. The methods return the (quasi) triangular Schur factor T and the orthogonal/unitary Schur vectors Z such that A = Z*T*Z'. The eigenvalues of A are returned in the vector λ.

Example

julia> A = [-2. 1. 3.; 2. 1. -1.; -7. 2. 7.]
3×3 Array{Float64,2}:
-2.0  1.0   3.0
2.0  1.0  -1.0
-7.0  2.0   7.0

julia> T, Z, lambda = schur(A)
([2.0 0.801792 6.63509; -8.55988e-11 2.0 8.08286; 0.0 0.0 1.99999], [0.577351 0.154299 -0.801784; 0.577346 -0.77152 0.267262; 0.577354 0.617211 0.534522], Complex{Float64}[2.0+8.28447e-6im, 2.0-8.28447e-6im, 1.99999+0.0im])

julia> Z * T * Z'
3×3 Array{Float64,2}:
-2.0  1.0   3.0
2.0  1.0  -1.0
-7.0  2.0   7.0
source
schur(A::StridedMatrix, B::StridedMatrix) -> S::StridedMatrix, T::StridedMatrix, Q::StridedMatrix, Z::StridedMatrix, α::Vector, β::Vector
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ordschur(F::Schur, select::Union{Vector{Bool},BitVector}) -> F::Schur

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

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ordschur(T::StridedMatrix, Z::StridedMatrix, select::Union{Vector{Bool},BitVector}) -> T::StridedMatrix, Z::StridedMatrix, λ::Vector

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

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

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

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ordschur(S::StridedMatrix, T::StridedMatrix, Q::StridedMatrix, Z::StridedMatrix, select) -> S::StridedMatrix, T::StridedMatrix, Q::StridedMatrix, Z::StridedMatrix, α::Vector, β::Vector

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

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

Same as ordschur but overwrites the factorization F.

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ordschur!(T::StridedMatrix, Z::StridedMatrix, select::Union{Vector{Bool},BitVector}) -> T::StridedMatrix, Z::StridedMatrix, λ::Vector

Same as ordschur but overwrites the input arguments.

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

Same as ordschur but overwrites the factorization F.

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ordschur!(S::StridedMatrix, T::StridedMatrix, Q::StridedMatrix, Z::StridedMatrix, select) -> S::StridedMatrix, T::StridedMatrix, Q::StridedMatrix, Z::StridedMatrix, α::Vector, β::Vector

Same as ordschur but overwrites the factorization the input arguments.

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svdfact(A; thin::Bool=true) -> 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*diagm(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.

If thin=true (default), a thin SVD is returned. For a $M \times N$ matrix A, U is $M \times M$ for a full SVD (thin=false) and $M \times \min(M, N)$ for a thin SVD.

Example

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

julia> F = svdfact(A)
Base.LinAlg.SVD{Float64,Float64,Array{Float64,2}}([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], [3.0, 2.23607, 2.0, 0.0], [-0.0 0.0 … -0.0 0.0; 0.447214 0.0 … 0.0 0.894427; -0.0 1.0 … -0.0 0.0; 0.0 0.0 … 1.0 0.0])

julia> F[:U] * diagm(F[:S]) * F[:Vt]
4×5 Array{Float64,2}:
1.0  0.0  0.0  0.0  2.0
0.0  0.0  3.0  0.0  0.0
0.0  0.0  0.0  0.0  0.0
0.0  2.0  0.0  0.0  0.0
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svdfact(A, B) -> GeneralizedSVD

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

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

• F[:U] is a M-by-M orthogonal matrix,

• F[:V] is a P-by-P orthogonal matrix,

• F[:Q] is a N-by-N orthogonal matrix,

• F[:R0] is a (K+L)-by-N matrix whose rightmost (K+L)-by-(K+L) block is nonsingular upper block triangular,

• F[:D1] is a M-by-(K+L) diagonal matrix with 1s in the first K entries,

• F[:D2] is a P-by-(K+L) matrix whose top right L-by-L block is diagonal,

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

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).

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svdfact!(A, thin::Bool=true) -> SVD

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

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svdfact!(A, B) -> GeneralizedSVD

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

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svd(A; thin::Bool=true) -> U, S, V

Computes the SVD of A, returning U, vector S, and V such that A == U*diagm(S)*V'. The singular values in S are sorted in descending order.

If thin=true (default), a thin SVD is returned. For a $M \times N$ matrix A, U is $M \times M$ for a full SVD (thin=false) and $M \times \min(M, N)$ for a thin SVD.

svd is a wrapper around svdfact, extracting all parts of the SVD factorization to a tuple. Direct use of svdfact is therefore more efficient.

Example

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

julia> U, S, V = svd(A)
([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], [3.0, 2.23607, 2.0, 0.0], [-0.0 0.447214 -0.0 0.0; 0.0 0.0 1.0 0.0; … ; -0.0 0.0 -0.0 1.0; 0.0 0.894427 0.0 0.0])

julia> U*diagm(S)*V'
4×5 Array{Float64,2}:
1.0  0.0  0.0  0.0  2.0
0.0  0.0  3.0  0.0  0.0
0.0  0.0  0.0  0.0  0.0
0.0  2.0  0.0  0.0  0.0
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svd(A, B) -> U, V, Q, D1, D2, R0

Wrapper around svdfact extracting all parts of the factorization to a tuple. Direct use of svdfact is therefore generally more efficient. The function returns the generalized SVD of A and B, returning U, V, Q, D1, D2, and R0 such that A = U*D1*R0*Q' and B = V*D2*R0*Q'.

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svdvals(A)

Returns the singular values of A in descending order.

Example

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

julia> svdvals(A)
4-element Array{Float64,1}:
3.0
2.23607
2.0
0.0
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svdvals(A, B)

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

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LinAlg.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

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givens{T}(f::T, g::T, i1::Integer, i2::Integer) -> (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: LinAlg.Givens

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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: LinAlg.Givens

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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: LinAlg.Givens

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triu(M)

Upper triangle of a matrix.

Example

julia> a = ones(4,4)
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> triu(a)
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
0.0  1.0  1.0  1.0
0.0  0.0  1.0  1.0
0.0  0.0  0.0  1.0
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triu(M, k::Integer)

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

Example

julia> a = ones(4,4)
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> triu(a,3)
4×4 Array{Float64,2}:
0.0  0.0  0.0  1.0
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0

julia> triu(a,-3)
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
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triu!(M)

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

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triu!(M, k::Integer)

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

Example

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

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

Lower triangle of a matrix.

Example

julia> a = ones(4,4)
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> tril(a)
4×4 Array{Float64,2}:
1.0  0.0  0.0  0.0
1.0  1.0  0.0  0.0
1.0  1.0  1.0  0.0
1.0  1.0  1.0  1.0
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tril(M, k::Integer)

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

Example

julia> a = ones(4,4)
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> tril(a,3)
4×4 Array{Float64,2}:
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0

julia> tril(a,-3)
4×4 Array{Float64,2}:
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0
1.0  0.0  0.0  0.0
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tril!(M)

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

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tril!(M, k::Integer)

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

Example

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

julia> tril!(M, 2)
5×5 Array{Int64,2}:
1  2  3  0  0
1  2  3  4  0
1  2  3  4  5
1  2  3  4  5
1  2  3  4  5
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diagind(M, k::Integer=0)

A Range giving the indices of the kth diagonal of the matrix M.

Example

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

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

The kth diagonal of a matrix, as a vector. Use diagm to construct a diagonal matrix.

Example

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

julia> diag(A,1)
2-element Array{Int64,1}:
2
6
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diagm(v, k::Integer=0)

Construct a matrix by placing v on the kth diagonal.

Example

julia> diagm([1,2,3],1)
4×4 Array{Int64,2}:
0  1  0  0
0  0  2  0
0  0  0  3
0  0  0  0
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scale!(A, b)
scale!(b, A)

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

If A is a matrix and b is a vector, then scale!(A,b) scales each column i of A by b[i] (similar to A*Diagonal(b)), while scale!(b,A) scales each row i of A by b[i] (similar to Diagonal(b)*A), again operating in-place on A. An InexactError exception is thrown if the scaling produces a number not representable by the element type of A, e.g. for integer types.

Example

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

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

julia> scale!(a,b)
2×2 Array{Int64,2}:
1  4
3  8

julia> a = [1 2; 3 4];

julia> b = [1; 2];

julia> scale!(b,a)
2×2 Array{Int64,2}:
1  2
6  8
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rank(M[, tol::Real])

Compute the rank of a matrix by counting how many singular values of M have magnitude greater than tol. By default, the value of tol is the largest dimension of M multiplied by the eps of the eltype of M.

Example

julia> rank(eye(3))
3

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

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

julia> rank(diagm([1, 0.001, 2]), 0.00001)
3
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norm(A::AbstractArray, p::Real=2)

Compute the p-norm of a vector or the operator norm of a matrix A, defaulting to the 2-norm.

norm(A::AbstractVector, p::Real=2)

For vectors, this is equivalent to vecnorm and equal to:

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

with $a_i$ the entries of $A$ and $n$ its length.

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.

Example

julia> v = [3, -2, 6]
3-element Array{Int64,1}:
3
-2
6

julia> norm(v)
7.0

julia> norm(v, Inf)
6.0
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norm(A::AbstractMatrix, p::Real=2)

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

When p=1, the matrix 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 matrix norm is the spectral norm, equal to the largest singular value of A.

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

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

Example

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

julia> norm(A, Inf)
6.0
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norm(x::Number, p::Real=2)

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

source
norm(A::RowVector, q::Real=2)

For row vectors, return the $q$-norm of A, which is equivalent to the p-norm with value p = q/(q-1). They coincide at p = q = 2.

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

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vecnorm(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 $A$ and $n$ its length.

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

Example

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

julia> vecnorm([1 2 3 4 5 6 7 8 9])
16.881943016134134
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vecnorm(x::Number, p::Real=2)

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

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

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

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

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

Examples

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

julia> b = normalize(a)
3-element Array{Float64,1}:
0.218218
0.436436
0.872872

julia> norm(b)
1.0

julia> c = normalize(a, 1)
3-element Array{Float64,1}:
0.142857
0.285714
0.571429

julia> norm(c, 1)
1.0
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cond(M, p::Real=2)

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

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

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

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

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trace(M)

Matrix trace. Sums the diagonal elements of M.

Example

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

julia> trace(A)
5
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det(M)

Matrix determinant.

Example

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

julia> det(M)
2.0
source
logdet(M)

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

Examples

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

julia> logdet(M)
0.6931471805599453

julia> logdet(eye(3))
0.0
source
logabsdet(M)

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

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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.

Example

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

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

julia> M*N == N*M == eye(2)
true
source
pinv(M[, tol::Real])

Computes the Moore-Penrose pseudoinverse.

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

The optimal choice of tol varies both with the value of M and the intended application of the pseudoinverse. The default value of tol is eps(real(float(one(eltype(M)))))*maximum(size(A)), which is essentially machine epsilon for the real part of a matrix element multiplied by the larger matrix dimension. For inverting dense ill-conditioned matrices in a least-squares sense, tol = sqrt(eps(real(float(one(eltype(M)))))) is recommended.

Example

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

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

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

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

[B96]

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

[S84]

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

[KY88]

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

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nullspace(M)

Basis for nullspace of M.

Example

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

julia> nullspace(M)
3×1 Array{Float64,2}:
0.0
0.0
1.0
source
repmat(A, m::Integer, n::Integer=1)

Construct a matrix by repeating the given matrix (or vector) m times in dimension 1 and n times in dimension 2.

Examples

julia> repmat([1, 2, 3], 2)
6-element Array{Int64,1}:
1
2
3
1
2
3

julia> repmat([1, 2, 3], 2, 3)
6×3 Array{Int64,2}:
1  1  1
2  2  2
3  3  3
1  1  1
2  2  2
3  3  3
source
repeat(A::AbstractArray; inner=ntuple(x->1, ndims(A)), outer=ntuple(x->1, ndims(A)))

Construct an array by repeating the entries of A. The i-th element of inner specifies the number of times that the individual entries of the i-th dimension of A should be repeated. The i-th element of outer specifies the number of times that a slice along the i-th dimension of A should be repeated. If inner or outer are omitted, no repetition is performed.

Examples

julia> repeat(1:2, inner=2)
4-element Array{Int64,1}:
1
1
2
2

julia> repeat(1:2, outer=2)
4-element Array{Int64,1}:
1
2
1
2

julia> repeat([1 2; 3 4], inner=(2, 1), outer=(1, 3))
4×6 Array{Int64,2}:
1  2  1  2  1  2
1  2  1  2  1  2
3  4  3  4  3  4
3  4  3  4  3  4
source
kron(A, B)

Kronecker tensor product of two vectors or two matrices.

Example

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

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

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

Concatenate matrices block-diagonally. Currently only implemented for sparse matrices.

Example

julia> blkdiag(speye(3), 2*speye(2))
5×5 SparseMatrixCSC{Float64,Int64} with 5 stored entries:
[1, 1]  =  1.0
[2, 2]  =  1.0
[3, 3]  =  1.0
[4, 4]  =  2.0
[5, 5]  =  2.0
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linreg(x, y)

Perform simple linear regression using Ordinary Least Squares. Returns a and b such that a + b*x is the closest straight line to the given points (x, y), i.e., such that the squared error between y and a + b*x is minimized.

Examples:

using PyPlot
x = 1.0:12.0
y = [5.5, 6.3, 7.6, 8.8, 10.9, 11.79, 13.48, 15.02, 17.77, 20.81, 22.0, 22.99]
a, b = linreg(x, y)          # Linear regression
plot(x, y, "o")              # Plot (x, y) points
plot(x, a + b*x)             # Plot line determined by linear regression

\, cov, std, mean.

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expm(A)

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 (eigfact) is used, otherwise the scaling and squaring algorithm (see [H05]) is chosen.

[H05]

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

Example

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

julia> expm(A)
2×2 Array{Float64,2}:
2.71828  0.0
0.0      2.71828
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logm(A{T}::StridedMatrix{T})

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

If A is symmetric or Hermitian, its eigendecomposition (eigfact) is used, if A is triangular an improved version of the inverse scaling and squaring method is employed (see [AH12] and [AHR13]). For general matrices, the complex Schur form (schur) is computed and the triangular algorithm is used on the triangular factor.

[AH12]

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

[AHR13]

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

Example

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

julia> logm(A)
2×2 Symmetric{Float64,Array{Float64,2}}:
1.0  0.0
0.0  1.0
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sqrtm(A)

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

If A is symmetric or Hermitian, its eigendecomposition (eigfact) is used to compute the square root. Otherwise, the square root is determined by means of the Björck-Hammarling method [BH83], which computes the complex Schur form (schur) and then the complex square root of the triangular factor.

[BH83]

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

Example

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

julia> sqrtm(A)
2×2 Array{Float64,2}:
2.0  0.0
0.0  2.0
source
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.

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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.

source
issymmetric(A) -> Bool

Test whether a matrix is symmetric.

Examples

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

julia> issymmetric(a)
true

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

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

Test whether a matrix is positive definite.

Example

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

julia> isposdef(A)
true
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isposdef!(A) -> Bool

Test whether a matrix is positive definite, overwriting A in the process.

Example

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

julia> isposdef!(A)
true

julia> A
2×2 Array{Float64,2}:
1.0  2.0
2.0  6.78233
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istril(A) -> Bool

Test whether a matrix is lower triangular.

Examples

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

julia> istril(a)
false

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

julia> istril(b)
true
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istriu(A) -> Bool

Test whether a matrix is upper triangular.

Examples

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

julia> istriu(a)
false

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

julia> istriu(b)
true
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isdiag(A) -> Bool

Test whether a matrix is diagonal.

Examples

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

julia> isdiag(a)
false

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

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

Test whether a matrix is Hermitian.

Examples

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

julia> ishermitian(a)
true

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

julia> ishermitian(b)
true
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RowVector(vector)

A lazy-view wrapper of an AbstractVector, which turns a length-n vector into a 1×n shaped row vector and represents the transpose of a vector (the elements are also transposed recursively). This type is usually constructed (and unwrapped) via the transpose function or .' operator (or related ctranspose or ' operator).

By convention, a vector can be multiplied by a matrix on its left (A * v) whereas a row vector can be multiplied by a matrix on its right (such that v.' * A = (A.' * v).'). It differs from a 1×n-sized matrix by the facts that its transpose returns a vector and the inner product v1.' * v2 returns a scalar, but will otherwise behave similarly.

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ConjArray(array)

A lazy-view wrapper of an AbstractArray, taking the elementwise complex conjugate. This type is usually constructed (and unwrapped) via the conj function (or related ctranspose), but currently this is the default behavior for RowVector only. For other arrays, the ConjArray constructor can be used directly.

Examples

julia> [1+im, 1-im]'
1×2 RowVector{Complex{Int64},ConjArray{Complex{Int64},1,Array{Complex{Int64},1}}}:
1-1im  1+1im

julia> ConjArray([1+im 0; 0 1-im])
2×2 ConjArray{Complex{Int64},2,Array{Complex{Int64},2}}:
1-1im  0+0im
0+0im  1+1im
source
transpose(A::AbstractMatrix)

The transposition operator (.').

Example

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

julia> transpose(A)
3×3 Array{Int64,2}:
1  4  7
2  5  8
3  6  9
source
transpose(v::AbstractVector)

The transposition operator (.').

Example

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

julia> transpose(v)
1×3 RowVector{Int64,Array{Int64,1}}:
1  2  3
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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.

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ctranspose(A)

The conjugate transposition operator (').

Example

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

julia> ctranspose(A)
2×2 Array{Complex{Int64},2}:
3-2im  8-7im
9-2im  4-6im
source
ctranspose!(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.

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eigs(A; nev=6, ncv=max(20,2*nev+1), which=:LM, tol=0.0, maxiter=300, sigma=nothing, ritzvec=true, v0=zeros((0,))) -> (d,[v,],nconv,niter,nmult,resid)

Computes eigenvalues d of A using implicitly restarted Lanczos or Arnoldi iterations for real symmetric or general nonsymmetric matrices respectively.

The following keyword arguments are supported:

• nev: Number of eigenvalues

• ncv: Number of Krylov vectors used in the computation; should satisfy nev+1 <= ncv <= n for real symmetric problems and nev+2 <= ncv <= n for other problems, where n is the size of the input matrix A. The default is ncv = max(20,2*nev+1). Note that these restrictions limit the input matrix A to be of dimension at least 2.

• which: type of eigenvalues to compute. See the note below.

whichtype of eigenvalues
:LMeigenvalues of largest magnitude (default)
:SMeigenvalues of smallest magnitude
:LReigenvalues of largest real part
:SReigenvalues of smallest real part
:LIeigenvalues of largest imaginary part (nonsymmetric or complex A only)
:SIeigenvalues of smallest imaginary part (nonsymmetric or complex A only)
:BEcompute half of the eigenvalues from each end of the spectrum, biased in favor of the high end. (real symmetric A only)
• tol: parameter defining the relative tolerance for convergence of Ritz values (eigenvalue estimates). A Ritz value $θ$ is considered converged when its associated residual is less than or equal to the product of tol and $max(ɛ^{2/3}, |θ|)$, where ɛ = eps(real(eltype(A)))/2 is LAPACK's machine epsilon. The residual associated with $θ$ and its corresponding Ritz vector $v$ is defined as the norm $||Av - vθ||$. The specified value of tol should be positive; otherwise, it is ignored and $ɛ$ is used instead. Default: $ɛ$.

• maxiter: Maximum number of iterations (default = 300)

• sigma: Specifies the level shift used in inverse iteration. If nothing (default), defaults to ordinary (forward) iterations. Otherwise, find eigenvalues close to sigma using shift and invert iterations.

• ritzvec: Returns the Ritz vectors v (eigenvectors) if true

• v0: starting vector from which to start the iterations

eigs returns the nev requested eigenvalues in d, the corresponding Ritz vectors v (only if ritzvec=true), the number of converged eigenvalues nconv, the number of iterations niter and the number of matrix vector multiplications nmult, as well as the final residual vector resid.

Example

julia> A = spdiagm(1:4);

julia> λ, ϕ = eigs(A, nev = 2);

julia> λ
2-element Array{Float64,1}:
4.0
3.0
Note

The sigma and which keywords interact: the description of eigenvalues searched for by which do not necessarily refer to the eigenvalues of A, but rather the linear operator constructed by the specification of the iteration mode implied by sigma.

sigmaiteration modewhich refers to eigenvalues of
nothingordinary (forward)$A$
real or complexinverse with level shift sigma$(A - \sigma I )^{-1}$
Note

Although tol has a default value, the best choice depends strongly on the matrix A. We recommend that users _always_ specify a value for tol which suits their specific needs.

For details of how the errors in the computed eigenvalues are estimated, see:

• B. N. Parlett, "The Symmetric Eigenvalue Problem", SIAM: Philadelphia, 2/e (1998), Ch. 13.2, "Accessing Accuracy in Lanczos Problems", pp. 290-292 ff.

• R. B. Lehoucq and D. C. Sorensen, "Deflation Techniques for an Implicitly Restarted Arnoldi Iteration", SIAM Journal on Matrix Analysis and Applications (1996), 17(4), 789–821. doi:10.1137/S0895479895281484

source
eigs(A, B; nev=6, ncv=max(20,2*nev+1), which=:LM, tol=0.0, maxiter=300, sigma=nothing, ritzvec=true, v0=zeros((0,))) -> (d,[v,],nconv,niter,nmult,resid)

Computes generalized eigenvalues d of A and B using implicitly restarted Lanczos or Arnoldi iterations for real symmetric or general nonsymmetric matrices respectively.

The following keyword arguments are supported:

• nev: Number of eigenvalues

• ncv: Number of Krylov vectors used in the computation; should satisfy nev+1 <= ncv <= n for real symmetric problems and nev+2 <= ncv <= n for other problems, where n is the size of the input matrices A and B. The default is ncv = max(20,2*nev+1). Note that these restrictions limit the input matrix A to be of dimension at least 2.

• which: type of eigenvalues to compute. See the note below.

whichtype of eigenvalues
:LMeigenvalues of largest magnitude (default)
:SMeigenvalues of smallest magnitude
:LReigenvalues of largest real part
:SReigenvalues of smallest real part
:LIeigenvalues of largest imaginary part (nonsymmetric or complex A only)
:SIeigenvalues of smallest imaginary part (nonsymmetric or complex A only)
:BEcompute half of the eigenvalues from each end of the spectrum, biased in favor of the high end. (real symmetric A only)
• tol: relative tolerance used in the convergence criterion for eigenvalues, similar to tol in the eigs(A) method for the ordinary eigenvalue problem, but effectively for the eigenvalues of $B^{-1} A$ instead of $A$. See the documentation for the ordinary eigenvalue problem in eigs(A) and the accompanying note about tol.

• maxiter: Maximum number of iterations (default = 300)

• sigma: Specifies the level shift used in inverse iteration. If nothing (default), defaults to ordinary (forward) iterations. Otherwise, find eigenvalues close to sigma using shift and invert iterations.

• ritzvec: Returns the Ritz vectors v (eigenvectors) if true

• v0: starting vector from which to start the iterations

eigs returns the nev requested eigenvalues in d, the corresponding Ritz vectors v (only if ritzvec=true), the number of converged eigenvalues nconv, the number of iterations niter and the number of matrix vector multiplications nmult, as well as the final residual vector resid.

Example

julia> A = speye(4, 4); B = spdiagm(1:4);

julia> λ, ϕ = eigs(A, B, nev = 2);

julia> λ
2-element Array{Float64,1}:
1.0
0.5
Note

The sigma and which keywords interact: the description of eigenvalues searched for by which do not necessarily refer to the eigenvalue problem $Av = Bv\lambda$, but rather the linear operator constructed by the specification of the iteration mode implied by sigma.

sigmaiteration modewhich refers to the problem
nothingordinary (forward)$Av = Bv\lambda$
real or complexinverse with level shift sigma$(A - \sigma B )^{-1}B = v\nu$
source
svds(A; nsv=6, ritzvec=true, tol=0.0, maxiter=1000, ncv=2*nsv, u0=zeros((0,)), v0=zeros((0,))) -> (SVD([left_sv,] s, [right_sv,]), nconv, niter, nmult, resid)

Computes the largest singular values s of A using implicitly restarted Lanczos iterations derived from eigs.

Inputs

• A: Linear operator whose singular values are desired. A may be represented as a subtype of AbstractArray, e.g., a sparse matrix, or any other type supporting the four methods size(A), eltype(A), A * vector, and A' * vector.

• nsv: Number of singular values. Default: 6.

• ritzvec: If true, return the left and right singular vectors left_sv and right_sv. If false, omit the singular vectors. Default: true.

• tol: tolerance, see eigs.

• maxiter: Maximum number of iterations, see eigs. Default: 1000.

• ncv: Maximum size of the Krylov subspace, see eigs (there called nev). Default: 2*nsv.

• u0: Initial guess for the first left Krylov vector. It may have length m (the first dimension of A), or 0.

• v0: Initial guess for the first right Krylov vector. It may have length n (the second dimension of A), or 0.

Outputs

• svd: An SVD object containing the left singular vectors, the requested values, and the right singular vectors. If ritzvec = false, the left and right singular vectors will be empty.

• nconv: Number of converged singular values.

• niter: Number of iterations.

• nmult: Number of matrix–vector products used.

• resid: Final residual vector.

Example

julia> A = spdiagm(1:4);

julia> s = svds(A, nsv = 2);

julia> s[:S]
2-element Array{Float64,1}:
4.0
3.0
Implementation

svds(A) is formally equivalent to calling eigs to perform implicitly restarted Lanczos tridiagonalization on the Hermitian matrix $\begin{pmatrix} 0 & A^\prime \\ A & 0 \end{pmatrix}$, whose eigenvalues are plus and minus the singular values of $A$.

source
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.

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## Low-level matrix operations

Matrix operations involving transpositions operations like A' \ B are converted by the Julia parser into calls to specially named functions like Ac_ldiv_B. If you want to overload these operations for your own types, then it is useful to know the names of these functions.

Also, in many cases there are in-place versions of matrix operations that allow you to supply a pre-allocated output vector or matrix. This is useful when optimizing critical code in order to avoid the overhead of repeated allocations. These in-place operations are suffixed with ! below (e.g. A_mul_B!) according to the usual Julia convention.

A_ldiv_B!([Y,] A, B) -> Y

Compute A \ B in-place and store the result in Y, returning the result. If only two arguments are passed, then A_ldiv_B!(A, B) overwrites B with 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 cholfact). 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., lufact!), and performance-critical situations requiring A_ldiv_B! usually also require fine-grained control over the factorization of A.

source
A_ldiv_Bc(A, B)

For matrices or vectors $A$ and $B$, calculates $A$ \ $Bᴴ$.

source
A_ldiv_Bt(A, B)

For matrices or vectors $A$ and $B$, calculates $A$ \ $Bᵀ$.

source
A_mul_B!(Y, A, B) -> Y

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

Example

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

julia> Y
2×2 Array{Float64,2}:
3.0  3.0
7.0  7.0
source
A_mul_Bc(A, B)

For matrices or vectors $A$ and $B$, calculates $A⋅Bᴴ$.

source
A_mul_Bt(A, B)

For matrices or vectors $A$ and $B$, calculates $A⋅Bᵀ$.

source
A_rdiv_Bc(A, B)

For matrices or vectors $A$ and $B$, calculates $A / Bᴴ$.

source
A_rdiv_Bt(A, B)

For matrices or vectors $A$ and $B$, calculates $A / Bᵀ$.

source
Ac_ldiv_B(A, B)

For matrices or vectors $A$ and $B$, calculates $Aᴴ$ \ $B$.

source
Ac_ldiv_B!([Y,] A, B) -> Y

Similar to A_ldiv_B!, but return $Aᴴ$ \ $B$, computing the result in-place in Y (or overwriting B if Y is not supplied).

source
Ac_ldiv_Bc(A, B)

For matrices or vectors $A$ and $B$, calculates $Aᴴ$ \ $Bᴴ$.

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Ac_mul_B(A, B)

For matrices or vectors $A$ and $B$, calculates $Aᴴ⋅B$.

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Ac_mul_Bc(A, B)

For matrices or vectors $A$ and $B$, calculates $Aᴴ Bᴴ$.

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Ac_rdiv_B(A, B)

For matrices or vectors $A$ and $B$, calculates $Aᴴ / B$.

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Ac_rdiv_Bc(A, B)

For matrices or vectors $A$ and $B$, calculates $Aᴴ / Bᴴ$.

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At_ldiv_B(A, B)

For matrices or vectors $A$ and $B$, calculates $Aᵀ$ \ $B$.

source
At_ldiv_B!([Y,] A, B) -> Y

Similar to A_ldiv_B!, but return $Aᵀ$ \ $B$, computing the result in-place in Y (or overwriting B if Y is not supplied).

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At_ldiv_Bt(A, B)

For matrices or vectors $A$ and $B$, calculates $Aᵀ$ \ $Bᵀ$.

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At_mul_B(A, B)

For matrices or vectors $A$ and $B$, calculates $Aᵀ⋅B$.

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At_mul_Bt(A, B)

For matrices or vectors $A$ and $B$, calculates $Aᵀ⋅Bᵀ$.

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At_rdiv_B(A, B)

For matrices or vectors $A$ and $B$, calculates $Aᵀ / B$.

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At_rdiv_Bt(A, B)

For matrices or vectors $A$ and $B$, calculates $Aᵀ / Bᵀ$.

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## BLAS Functions

In Julia (as in much of scientific computation), dense linear-algebra operations are based on the LAPACK library, which in turn is built on top of basic linear-algebra building-blocks known as the BLAS. There are highly optimized implementations of BLAS available for every computer architecture, and sometimes in high-performance linear algebra routines it is useful to call the BLAS functions directly.

Base.LinAlg.BLAS provides wrappers for some of the BLAS functions. Those BLAS functions that overwrite one of the input arrays have names ending in '!'. Usually, a BLAS function has four methods defined, for Float64, Float32, Complex128, and Complex64 arrays.

### BLAS Character Arguments

Many BLAS functions accept arguments that determine whether to transpose an argument (trans), which triangle of a matrix to reference (uplo or ul), whether the diagonal of a triangular matrix can be assumed to be all ones (dA) or which side of a matrix multiplication the input argument belongs on (side). The possiblities are:

#### Multplication Order

sideMeaning
'L'The argument goes on the left side of a matrix-matrix operation.
'R'The argument goes on the right side of a matrix-matrix operation.

#### Triangle Referencing

uplo/ulMeaning
'U'Only the upper triangle of the matrix will be used.
'L'Only the lower triangle of the matrix will be used.

#### Transposition Operation

trans/tXMeaning
'N'The input matrix X is not transposed or conjugated.
'T'The input matrix X will be transposed.
'C'The input matrix X will be conjugated and transposed.

#### Unit Diagonal

diag/dXMeaning
'N'The diagonal values of the matrix X will be read.
'U'The diagonal of the matrix X is assumed to be all ones.
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.

Example:

julia> Base.BLAS.dotu(10, im*ones(10), 1, complex.(ones(20), ones(20)), 2)
-10.0 + 10.0im
source
dotc(n, X, incx, U, incy)

Dot function for two complex vectors, consisting of n elements of array X with stride incx and n elements of array U with stride incy, conjugating the first vector.

Example:

julia> Base.BLAS.dotc(10, im*ones(10), 1, complex.(ones(20), ones(20)), 2)
10.0 - 10.0im
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blascopy!(n, X, incx, Y, incy)

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

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nrm2(n, X, incx)

2-norm of a vector consisting of n elements of array X with stride incx.

Example:

julia> Base.BLAS.nrm2(4, ones(8), 2)
2.0

julia> Base.BLAS.nrm2(1, ones(8), 2)
1.0
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asum(n, X, incx)

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

Example:

julia> Base.BLAS.asum(5, im*ones(10), 2)
5.0

julia> Base.BLAS.asum(2, im*ones(10), 5)
2.0
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axpy!(a, X, Y)

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

Example:

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

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

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

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

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scal(n, a, X, incx)

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

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ger!(alpha, x, y, A)

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

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syr!(uplo, alpha, x, A)

Rank-1 update of the symmetric matrix A with vector x as alpha*x*x.' + A. uplo controls which triangle of A is updated. Returns A.

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syrk!(uplo, trans, alpha, A, beta, C)

Rank-k update of the symmetric matrix C as alpha*A*A.' + beta*C or alpha*A.'*A + beta*C according to trans. Only the uplo triangle of C is used. Returns C.

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syrk(uplo, trans, alpha, A)

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

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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.

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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.

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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.

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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. Returns the updated y.

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gbmv(trans, m, kl, ku, alpha, A, x)

Returns 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.

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

Update vector y as alpha*A*x + beta*y where A is a a symmetric band matrix of order size(A,2) with k super-diagonals stored in the argument A. The storage layout for A is described the reference BLAS module, level-2 BLAS at http://www.netlib.org/lapack/explore-html/. Only the uplo triangle of A is used.

Returns the updated y.

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sbmv(uplo, k, alpha, A, x)

Returns 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.

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sbmv(uplo, k, A, x)

Returns 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.

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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. Returns the updated C.

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

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

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gemm(tA, tB, A, B)

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

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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. Returns the updated y.

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gemv(tA, alpha, A, x)

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

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gemv(tA, A, x)

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

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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. Returns the updated C.

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

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

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symm(side, ul, A, B)

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

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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. Returns the updated y.

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symv(ul, alpha, A, x)

Returns alpha*A*x. A is assumed to be symmetric. Only the ul triangle of A is used. alpha is a scalar.

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symv(ul, A, x)

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

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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.

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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.

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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.

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

Returns 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.

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

Returns 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.

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

Returns 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.

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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. Returns the updated b.

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

Returns 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.

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set_num_threads(n)

Set the number of threads the BLAS library should use.

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I

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

Example

julia> ones(5, 6) * I == ones(5, 6)
true

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

## LAPACK Functions

Base.LinAlg.LAPACK provides wrappers for some of the LAPACK functions for linear algebra. Those functions that overwrite one of the input arrays have names ending in '!'.

Usually a function has 4 methods defined, one each for Float64, Float32, Complex128 and Complex64 arrays.

Note that the LAPACK API provided by Julia can and will change in the future. Since this API is not user-facing, there is no commitment to support/deprecate this specific set of functions in future releases.

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.

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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.

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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.

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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).

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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.

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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.

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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.

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geqlf!(A, tau)

Compute the QL factorization of A, A = QL. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

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geqlf!(A) -> (A, tau)

Compute the QL factorization of A, A = QL.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

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geqrf!(A, tau)

Compute the QR factorization of A, A = QR. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

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geqrf!(A) -> (A, tau)

Compute the QR factorization of A, A = QR.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

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geqp3!(A, jpvt, tau)

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

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

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

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

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

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

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

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

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gerqf!(A, tau)

Compute the RQ factorization of A, A = RQ. tau contains scalars which parameterize the elementary reflectors of the factorization. tau must have length greater than or equal to the smallest dimension of A.

Returns A and tau modified in-place.

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gerqf!(A) -> (A, tau)

Compute the RQ factorization of A, A = RQ.

Returns A, modified in-place, and tau, which contains scalars which parameterize the elementary reflectors of the factorization.

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geqrt!(A, T)

Compute the blocked QR factorization of A, A = QR. T contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization. The first dimension of T sets the block size and it must be between 1 and n. The second dimension of T must equal the smallest dimension of A.

Returns A and T modified in-place.

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geqrt!(A, nb) -> (A, T)

Compute the blocked QR factorization of A, A = QR. nb sets the block size and it must be between 1 and n, the second dimension of A.

Returns A, modified in-place, and T, which contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization.

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geqrt3!(A, T)

Recursively computes the blocked QR factorization of A, A = QR. T contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization. The first dimension of T sets the block size and it must be between 1 and n. The second dimension of T must equal the smallest dimension of A.

Returns A and T modified in-place.

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geqrt3!(A) -> (A, T)

Recursively computes the blocked QR factorization of A, A = QR.

Returns A, modified in-place, and T, which contains upper triangular block reflectors which parameterize the elementary reflectors of the factorization.

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getrf!(A) -> (A, ipiv, info)

Compute the pivoted LU factorization of A, A = LU.

Returns A, modified in-place, ipiv, the pivoting information, and an info code which indicates success (info = 0), a singular value in U (info = i, in which case U[i,i] is singular), or an error code (info < 0).

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tzrzf!(A) -> (A, tau)

Transforms the upper trapezoidal matrix A to upper triangular form in-place. Returns A and tau, the scalar parameters for the elementary reflectors of the transformation.

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ormrz!(side, trans, A, tau, C)

Multiplies the matrix C by Q from the transformation supplied by tzrzf!. Depending on side or trans the multiplication can be left-sided (side = L, Q*C) or right-sided (side = R, C*Q) and Q can be unmodified (trans = N), transposed (trans = T), or conjugate transposed (trans = C). Returns matrix C which is modified in-place with the result of the multiplication.

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gels!(trans, A, B) -> (F, B, ssr)

Solves the linear equation A * X = B, A.' * X =B, or A' * X = B using a QR or LQ factorization. Modifies the matrix/vector B in place with the solution. A is overwritten with its QR or LQ factorization. trans may be one of N (no modification), T (transpose), or C (conjugate transpose). gels! searches for the minimum norm/least squares solution. A may be under or over determined. The solution is returned in B.

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gesv!(A, B) -> (B, A, ipiv)

Solves the linear equation A * X = B where A is a square matrix using the LU factorization of A. A is overwritten with its LU factorization and B is overwritten with the solution X. ipiv contains the pivoting information for the LU factorization of A.

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getrs!(trans, A, ipiv, B)

Solves the linear equation A * X = B, A.' * X =B, or A' * X = B for square A. Modifies the matrix/vector B in place with the solution. A is the LU factorization from getrf!, with ipiv the pivoting information. trans may be one of N (no modification), T (transpose), or C (conjugate transpose).

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getri!(A, ipiv)

Computes the inverse of A, using its LU factorization found by getrf!. ipiv is the pivot information output and A contains the LU factorization of getrf!. A is overwritten with its inverse.

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gesvx!(fact, trans, A, AF, ipiv, equed, R, C, B) -> (X, equed, R, C, B, rcond, ferr, berr, work)

Solves the linear equation A * X = B (trans = N), A.' * X =B (trans = T), or 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 diagm(R) from the left; C, meaning A was multiplied by diagm(C) from the right; or B, meaning A was multiplied by diagm(R) from the left and diagm(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 diagm(R)*B (if trans = N and equed = R,B) or diagm(C)*B (if trans = T,C and equed = C,B); rcond, the reciprocal condition number of A after equilbrating; ferr, the forward error bound for each solution vector in X; berr, the forward error bound for each solution vector in X; and work, the reciprocal pivot growth factor.

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gesvx!(A, B)

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

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gelsd!(A, B, rcond) -> (B, rnk)

Computes the least norm solution of A * X = B by finding the SVD factorization of A, then dividing-and-conquering the problem. B is overwritten with the solution X. Singular values below rcond will be treated as zero. Returns the solution in B and the effective rank of A in rnk.

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gelsy!(A, B, rcond) -> (B, rnk)

Computes the least norm solution of A * X = B by finding the full QR factorization of A, then dividing-and-conquering the problem. B is overwritten with the solution X. Singular values below rcond will be treated as zero. Returns the solution in B and the effective rank of A in rnk.

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gglse!(A, c, B, d) -> (X,res)

Solves the equation A * x = c where x is subject to the equality constraint B * x = d. Uses the formula ||c - A*x||^2 = 0 to solve. Returns X and the residual sum-of-squares.

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geev!(jobvl, jobvr, A) -> (W, VL, VR)

Finds the eigensystem of A. If jobvl = N, the left eigenvectors of A aren't computed. If jobvr = N, the right eigenvectors of A aren't computed. If jobvl = V or jobvr = V, the corresponding eigenvectors are computed. Returns the eigenvalues in W, the right eigenvectors in VR, and the left eigenvectors in VL.

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gesdd!(job, A) -> (U, S, VT)

Finds the singular value decomposition of A, A = U * S * V', using a divide and conquer approach. If job = A, all the columns of U and the rows of V' are computed. If job = N, no columns of U or rows of V' are computed. If job = O, A is overwritten with the columns of (thin) U and the rows of (thin) V'. If job = S, the columns of (thin) U and the rows of (thin) V' are computed and returned separately.

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gesvd!(jobu, jobvt, A) -> (U, S, VT)

Finds the singular value decomposition of A, A = U * S * V'. If jobu = A, all the columns of U are computed. If jobvt = A all the rows of V' are computed. If jobu = N, no columns of U are computed. If jobvt = N no rows of V' are computed. If jobu = O, A is overwritten with the columns of (thin) U. If jobvt = O, A is overwritten with the rows of (thin) V'. If jobu = S, the columns of (thin) U are computed and returned separately. If jobvt = S the rows of (thin) V' are computed and returned separately. jobu and jobvt can't both be O.

Returns U, S, and Vt, where S are the singular values of A.

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ggsvd!(jobu, jobv, jobq, A, B) -> (U, V, Q, alpha, beta, k, l, R)

Finds the generalized singular value decomposition of A and B, U'*A*Q = D1*R and V'*B*Q = D2*R. D1 has alpha on its diagonal and D2 has beta on its diagonal. If jobu = U, the orthogonal/unitary matrix U is computed. If jobv = V the orthogonal/unitary matrix V is computed. If jobq = Q, the orthogonal/unitary matrix Q is computed. If jobu, jobv or jobq is N, that matrix is not computed. This function is only available in LAPACK versions prior to 3.6.0.

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ggsvd3!(jobu, jobv, jobq, A, B) -> (U, V, Q, alpha, beta, k, l, R)

Finds the generalized singular value decomposition of A and B, U'*A*Q = D1*R and V'*B*Q = D2*R. D1 has alpha on its diagonal and D2 has beta on its diagonal. If jobu = U, the orthogonal/unitary matrix U is computed. If jobv = V the orthogonal/unitary matrix V is computed. If jobq = Q, the orthogonal/unitary matrix Q is computed. If jobu, jobv, or jobq is N, that matrix is not computed. This function requires LAPACK 3.6.0.

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geevx!(balanc, jobvl, jobvr, sense, A) -> (A, w, VL, VR, ilo, ihi, scale, abnrm, rconde, rcondv)

Finds the eigensystem of A with matrix balancing. If jobvl = N, the left eigenvectors of A aren't computed. If jobvr = N, the right eigenvectors of A aren't computed. If jobvl = V or jobvr = V, the corresponding eigenvectors are computed. If balanc = N, no balancing is performed. If balanc = P, A is permuted but not scaled. If balanc = S, A is scaled but not permuted. If balanc = B, A is permuted and scaled. If sense = N, no reciprocal condition numbers are computed. If sense = E, reciprocal condition numbers are computed for the eigenvalues only. If sense = V, reciprocal condition numbers are computed for the right eigenvectors only. If sense = B, reciprocal condition numbers are computed for the right eigenvectors and the eigenvectors. If sense = E,B, the right and left eigenvectors must be computed.

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ggev!(jobvl, jobvr, A, B) -> (alpha, beta, vl, vr)

Finds the generalized eigendecomposition of A and B. If jobvl = N, the left eigenvectors aren't computed. If jobvr = N, the right eigenvectors aren't computed. If jobvl = V or jobvr = V, the corresponding eigenvectors are computed.

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gtsv!(dl, d, du, B)

Solves the equation A * X = B where A is a tridiagonal matrix with dl on the subdiagonal, d on the diagonal, and du on the superdiagonal.

Overwrites B with the solution X and returns it.

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gttrf!(dl, d, du) -> (dl, d, du, du2, ipiv)

Finds the LU factorization of a tridiagonal matrix with dl on the subdiagonal, d on the diagonal, and du on the superdiagonal.

Modifies dl, d, and du in-place and returns them and the second superdiagonal du2 and the pivoting vector ipiv.

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gttrs!(trans, dl, d, du, du2, ipiv, B)

Solves the equation A * X = B (trans = N), A.' * X = B (trans = T), or A' * X = B (trans = C) using the LU factorization computed by gttrf!. B is overwritten with the solution X.

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orglq!(A, tau, k = length(tau))

Explicitly finds the matrix Q of a LQ factorization after calling gelqf! on A. Uses the output of gelqf!. A is overwritten by Q.

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orgqr!(A, tau, k = length(tau))

Explicitly finds the matrix Q of a QR factorization after calling geqrf! on A. Uses the output of geqrf!. A is overwritten by Q.

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orgql!(A, tau, k = length(tau))

Explicitly finds the matrix Q of a QL factorization after calling geqlf! on A. Uses the output of geqlf!. A is overwritten by Q.

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orgrq!(A, tau, k = length(tau))

Explicitly finds the matrix Q of a RQ factorization after calling gerqf! on A. Uses the output of gerqf!. A is overwritten by Q.

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ormlq!(side, trans, A, tau, C)

Computes Q * C (trans = N), Q.' * C (trans = T), Q' * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a LQ factorization of A computed using gelqf!. C is overwritten.

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ormqr!(side, trans, A, tau, C)

Computes Q * C (trans = N), Q.' * C (trans = T), Q' * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QR factorization of A computed using geqrf!. C is overwritten.

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ormql!(side, trans, A, tau, C)

Computes Q * C (trans = N), Q.' * C (trans = T), Q' * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QL factorization of A computed using geqlf!. C is overwritten.

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ormrq!(side, trans, A, tau, C)

Computes Q * C (trans = N), Q.' * C (trans = T), Q' * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a RQ factorization of A computed using gerqf!. C is overwritten.

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gemqrt!(side, trans, V, T, C)

Computes Q * C (trans = N), Q.' * C (trans = T), Q' * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QR factorization of A computed using geqrt!. C is overwritten.

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posv!(uplo, A, B) -> (A, B)

Finds the solution to A * X = B where A is a symmetric or Hermitian positive definite matrix. If uplo = U the upper Cholesky decomposition of A is computed. If uplo = L the lower Cholesky decomposition of A is computed. A is overwritten by its Cholesky decomposition. B is overwritten with the solution X.

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potrf!(uplo, A)

Computes the Cholesky (upper if uplo = U, lower if uplo = L) decomposition of positive-definite matrix A. A is overwritten and returned with an info code.

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potri!(uplo, A)

Computes the inverse of positive-definite matrix A after calling potrf! to find its (upper if uplo = U, lower if uplo = L) Cholesky decomposition.

A is overwritten by its inverse and returned.

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potrs!(uplo, A, B)

Finds the solution to A * X = B where A is a symmetric or Hermitian positive definite matrix whose Cholesky decomposition was computed by potrf!. If uplo = U the upper Cholesky decomposition of A was computed. If uplo = L the lower Cholesky decomposition of A was computed. B is overwritten with the solution X.

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pstrf!(uplo, A, tol) -> (A, piv, rank, info)

Computes the (upper if uplo = U, lower if uplo = L) pivoted Cholesky decomposition of positive-definite matrix A with a user-set tolerance tol. A is overwritten by its Cholesky decomposition.

Returns A, the pivots piv, the rank of A, and an info code. If info = 0, the factorization succeeded. If info = i > 0, then A is indefinite or rank-deficient.

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ptsv!(D, E, B)

Solves A * X = B for positive-definite tridiagonal A. D is the diagonal of A and E is the off-diagonal. B is overwritten with the solution X and returned.

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pttrf!(D, E)

Computes the LDLt factorization of a positive-definite tridiagonal matrix with D as diagonal and E as off-diagonal. D and E are overwritten and returned.

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pttrs!(D, E, B)

Solves A * X = B for positive-definite tridiagonal A with diagonal D and off-diagonal E after computing A's LDLt factorization using pttrf!. B is overwritten with the solution X.

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trtri!(uplo, diag, A)

Finds the inverse of (upper if uplo = U, lower if uplo = L) triangular matrix A. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. A is overwritten with its inverse.

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trtrs!(uplo, trans, diag, A, B)

Solves A * X = B (trans = N), A.' * X = B (trans = T), or A' * X = B (trans = C) for (upper if uplo = U, lower if uplo = L) triangular matrix A. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. B is overwritten with the solution X.

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trcon!(norm, uplo, diag, A)

Finds the reciprocal condition number of (upper if uplo = U, lower if uplo = L) triangular matrix A. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. If norm = I, the condition number is found in the infinity norm. If norm = O or 1, the condition number is found in the one norm.

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trevc!(side, howmny, select, T, VL = similar(T), VR = similar(T))

Finds the eigensystem of an upper triangular matrix T. If side = R, the right eigenvectors are computed. If side = L, the left eigenvectors are computed. If side = B, both sets are computed. If howmny = A, all eigenvectors are found. If howmny = B, all eigenvectors are found and backtransformed using VL and VR. If howmny = S, only the eigenvectors corresponding to the values in select are computed.

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trrfs!(uplo, trans, diag, A, B, X, Ferr, Berr) -> (Ferr, Berr)

Estimates the error in the solution to A * X = B (trans = N), A.' * X = B (trans = T), A' * X = B (trans = C) for side = L, or the equivalent equations a right-handed side = RX * A after computing X using trtrs!. If uplo = U, A is upper triangular. If uplo = L, A is lower triangular. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. Ferr and Berr are optional inputs. Ferr is the forward error and Berr is the backward error, each component-wise.

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stev!(job, dv, ev) -> (dv, Zmat)

Computes the eigensystem for a symmetric tridiagonal matrix with dv as diagonal and ev as off-diagonal. If job = N only the eigenvalues are found and returned in dv. If job = V then the eigenvectors are also found and returned in Zmat.

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stebz!(range, order, vl, vu, il, iu, abstol, dv, ev) -> (dv, iblock, isplit)

Computes the eigenvalues for a symmetric tridiagonal matrix with dv as diagonal and ev as off-diagonal. If range = A, all the eigenvalues are found. If range = V, the eigenvalues in the half-open interval (vl, vu] are found. If range = I, the eigenvalues with indices between il and iu are found. If order = B, eigvalues are ordered within a block. If order = E, they are ordered across all the blocks. abstol can be set as a tolerance for convergence.

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stegr!(jobz, range, dv, ev, vl, vu, il, iu) -> (w, Z)

Computes the eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) for a symmetric tridiagonal matrix with dv as diagonal and ev as off-diagonal. If range = A, all the eigenvalues are found. If range = V, the eigenvalues in the half-open interval (vl, vu] are found. If range = I, the eigenvalues with indices between il and iu are found. The eigenvalues are returned in w and the eigenvectors in Z.

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stein!(dv, ev_in, w_in, iblock_in, isplit_in)

Computes the eigenvectors for a symmetric tridiagonal matrix with dv as diagonal and ev_in as off-diagonal. w_in specifies the input eigenvalues for which to find corresponding eigenvectors. iblock_in specifies the submatrices corresponding to the eigenvalues in w_in. isplit_in specifies the splitting points between the submatrix blocks.

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syconv!(uplo, A, ipiv) -> (A, work)

Converts a symmetric matrix A (which has been factorized into a triangular matrix) into two matrices L and D. If uplo = U, A is upper triangular. If uplo = L, it is lower triangular. ipiv is the pivot vector from the triangular factorization. A is overwritten by L and D.

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sysv!(uplo, A, B) -> (B, A, ipiv)

Finds the solution to A * X = B for symmetric matrix A. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. B is overwritten by the solution X. A is overwritten by its Bunch-Kaufman factorization. ipiv contains pivoting information about the factorization.

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sytrf!(uplo, A) -> (A, ipiv, info)

Computes the Bunch-Kaufman factorization of a symmetric matrix A. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored.

Returns A, overwritten by the factorization, a pivot vector ipiv, and the error code info which is a non-negative integer. If info is positive the matrix is singular and the diagonal part of the factorization is exactly zero at position info.

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sytri!(uplo, A, ipiv)

Computes the inverse of a symmetric matrix A using the results of sytrf!. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. A is overwritten by its inverse.

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sytrs!(uplo, A, ipiv, B)

Solves the equation A * X = B for a symmetric matrix A using the results of sytrf!. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. B is overwritten by the solution X.

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hesv!(uplo, A, B) -> (B, A, ipiv)

Finds the solution to A * X = B for Hermitian matrix A. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. B is overwritten by the solution X. A is overwritten by its Bunch-Kaufman factorization. ipiv contains pivoting information about the factorization.

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hetrf!(uplo, A) -> (A, ipiv, info)

Computes the Bunch-Kaufman factorization of a Hermitian matrix A. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored.

Returns A, overwritten by the factorization, a pivot vector ipiv, and the error code info which is a non-negative integer. If info is positive the matrix is singular and the diagonal part of the factorization is exactly zero at position info.

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hetri!(uplo, A, ipiv)

Computes the inverse of a Hermitian matrix A using the results of sytrf!. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. A is overwritten by its inverse.

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hetrs!(uplo, A, ipiv, B)

Solves the equation A * X = B for a Hermitian matrix A using the results of sytrf!. If uplo = U, the upper half of A is stored. If uplo = L, the lower half is stored. B is overwritten by the solution X.

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syev!(jobz, uplo, A)

Finds the eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) of a symmetric matrix A. If uplo = U, the upper triangle of A is used. If uplo = L, the lower triangle of A is used.

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syevr!(jobz, range, uplo, A, vl, vu, il, iu, abstol) -> (W, Z)

Finds the eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) of a symmetric matrix A. If uplo = U, the upper triangle of A is used. If uplo = L, the lower triangle of A is used. If range = A, all the eigenvalues are found. If range = V, the eigenvalues in the half-open interval (vl, vu] are found. If range = I, the eigenvalues with indices between il and iu are found. abstol can be set as a tolerance for convergence.

The eigenvalues are returned in W and the eigenvectors in Z.

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sygvd!(itype, jobz, uplo, A, B) -> (w, A, B)

Finds the generalized eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) of a symmetric matrix A and symmetric positive-definite matrix B. If uplo = U, the upper triangles of A and B are used. If uplo = L, the lower triangles of A and B are used. If itype = 1, the problem to solve is A * x = lambda * B * x. If itype = 2, the problem to solve is A * B * x = lambda * x. If itype = 3, the problem to solve is B * A * x = lambda * x.

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bdsqr!(uplo, d, e_, Vt, U, C) -> (d, Vt, U, C)

Computes the singular value decomposition of a bidiagonal matrix with d on the diagonal and e_ on the off-diagonal. If uplo = U, e_ is the superdiagonal. If uplo = L, e_ is the subdiagonal. Can optionally also compute the product Q' * C.

Returns the singular values in d, and the matrix C overwritten with Q' * C.

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bdsdc!(uplo, compq, d, e_) -> (d, e, u, vt, q, iq)

Computes the singular value decomposition of a bidiagonal matrix with d on the diagonal and e_ on the off-diagonal using a divide and conqueq method. If uplo = U, e_ is the superdiagonal. If uplo = L, e_ is the subdiagonal. If compq = N, only the singular values are found. If compq = I, the singular values and vectors are found. If compq = P, the singular values and vectors are found in compact form. Only works for real types.

Returns the singular values in d, and if compq = P, the compact singular vectors in iq.

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gecon!(normtype, A, anorm)

Finds the reciprocal condition number of matrix A. If normtype = I, the condition number is found in the infinity norm. If normtype = O or 1, the condition number is found in the one norm. A must be the result of getrf! and anorm is the norm of A in the relevant norm.

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gehrd!(ilo, ihi, A) -> (A, tau)

Converts a matrix A to Hessenberg form. If A is balanced with gebal! then ilo and ihi are the outputs of gebal!. Otherwise they should be ilo = 1 and ihi = size(A,2). tau contains the elementary reflectors of the factorization.

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orghr!(ilo, ihi, A, tau)

Explicitly finds Q, the orthogonal/unitary matrix from gehrd!. ilo, ihi, A, and tau must correspond to the input/output to gehrd!.

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gees!(jobvs, A) -> (A, vs, w)

Computes the eigenvalues (jobvs = N) or the eigenvalues and Schur vectors (jobvs = V) of matrix A. A is overwritten by its Schur form.

Returns A, vs containing the Schur vectors, and w, containing the eigenvalues.

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gges!(jobvsl, jobvsr, A, B) -> (A, B, alpha, beta, vsl, vsr)

Computes the generalized eigenvalues, generalized Schur form, left Schur vectors (jobsvl = V), or right Schur vectors (jobvsr = V) of A and B.

The generalized eigenvalues are returned in alpha and beta. The left Schur vectors are returned in vsl and the right Schur vectors are returned in vsr.

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

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

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trsen!(compq, job, select, T, Q) -> (T, Q, w)

Reorder the Schur factorization of a matrix and optionally finds reciprocal condition numbers. If job = N, no condition numbers are found. If job = E, only the condition number for this cluster of eigenvalues is found. If job = V, only the condition number for the invariant subspace is found. If job = B then the condition numbers for the cluster and subspace are found. If compq = V the Schur vectors Q are updated. If compq = N the Schur vectors are not modified. select determines which eigenvalues are in the cluster.

Returns T, Q, and reordered eigenvalues in w.

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tgsen!(select, S, T, Q, Z) -> (S, T, alpha, beta, Q, Z)

Reorders the vectors of a generalized Schur decomposition. select specifices the eigenvalues in each cluster.

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trsyl!(transa, transb, A, B, C, isgn=1) -> (C, scale)

Solves the Sylvester matrix equation A * X +/- X * B = scale*C where A and B are both quasi-upper triangular. If transa = N, A is not modified. If transa = T, A is transposed. If transa = C, A is conjugate transposed. Similarly for transb and B. If isgn = 1, the equation A * X + X * B = scale * C is solved. If isgn = -1, the equation A * X - X * B = scale * C is solved.

Returns X (overwriting C) and scale.

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