Linear Algebra

Standard Functions

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

Base.:* — Method.

*(x, y...)

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

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Base.:\ — Method.

\(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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Base.LinAlg.dot — Function.

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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Base.LinAlg.vecdot — Function.

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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Base.LinAlg.cross — Function.

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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Base.LinAlg.factorize — Function.

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 `A`	type of factorization
Positive-definite	Cholesky (see `cholfact`)
Dense Symmetric/Hermitian	Bunch-Kaufman (see `bkfact`)
Sparse Symmetric/Hermitian	LDLt (see `ldltfact`)
Triangular	Triangular
Diagonal	Diagonal
Bidiagonal	Bidiagonal
Tridiagonal	LU (see `lufact`)
Symmetric real tridiagonal	LDLt (see `ldltfact`)
General square	LU (see `lufact`)
General non-square	QR (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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Base.LinAlg.Diagonal — Type.

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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Base.LinAlg.Bidiagonal — Type.

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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Base.LinAlg.SymTridiagonal — Type.

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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Base.LinAlg.Tridiagonal — Type.

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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Base.LinAlg.Symmetric — Type.

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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Base.LinAlg.Hermitian — Type.

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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Base.LinAlg.LowerTriangular — Type.

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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Base.LinAlg.UpperTriangular — Type.

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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Base.LinAlg.lu — Function.

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.

Component	Description
`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

Component	Description
`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`

Supported function	`LU`	`LU{T,Tridiagonal{T}}`
`/`	✓
`\`	✓	✓
`cond`	✓
`inv`	✓	✓
`det`	✓	✓
`logdet`	✓	✓
`logabsdet`	✓	✓
`size`	✓	✓

See also ldltfact.

Note

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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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Base.LinAlg.qr — Function.

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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Base.LinAlg.qr! — Function.

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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Base.LinAlg.qrfact — Function.

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:

if pivot == Val{true} then F is a QRPivoted object,
otherwise if the element type of A is a BLAS type (Float32, Float64, Complex64 or Complex128), then F is a QRCompactWY object,
otherwise F is a QR object.

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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Base.LinAlg.qrfact! — Function.

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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Base.LinAlg.QR — Type.

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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Base.LinAlg.QRCompactWY — Type.

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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Base.LinAlg.QRPivoted — Type.

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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Base.LinAlg.lqfact! — Function.

lqfact!(A) -> LQ

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

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Base.LinAlg.lqfact — Function.

lqfact(A) -> LQ

Compute the LQ factorization of A. See also lq.

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Base.LinAlg.lq — Function.

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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Base.LinAlg.bkfact — Function.

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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Base.LinAlg.bkfact! — Function.

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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Base.LinAlg.eig — Function.

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])

source

Base.LinAlg.eigvals — Function.

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.

source

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

source

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

Returns the eigenvalues of A. It is possible to calculate only a subset of the eigenvalues by specifying a 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

source

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

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

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

source

Base.LinAlg.eigvals! — Function.

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.

source

eigvals!(A, B) -> values

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

source

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

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

source

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

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

source

Base.LinAlg.eigmax — Function.

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:
 [1] #eigmax#46(::Bool, ::Bool, ::Function, ::Array{Complex{Int64},2}) at ./linalg/eigen.jl:238
 [2] eigmax(::Array{Complex{Int64},2}) at ./linalg/eigen.jl:236

source

Base.LinAlg.eigmin — Function.

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:
 [1] #eigmin#47(::Bool, ::Bool, ::Function, ::Array{Complex{Int64},2}) at ./linalg/eigen.jl:280
 [2] eigmin(::Array{Complex{Int64},2}) at ./linalg/eigen.jl:278

source

Base.LinAlg.eigvecs — Function.

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

source

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

source

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

source

Base.LinAlg.eigfact — Function.

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

source

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].)

source

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

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.

source

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

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

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

Note

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

source

Base.LinAlg.eigfact! — Function.

eigfact!(A, [B])

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

source

Base.LinAlg.hessfact — Function.

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

source

Base.LinAlg.hessfact! — Function.

hessfact!(A) -> Hessenberg

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

source

Base.LinAlg.schurfact — Function.

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

source

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].

source

Base.LinAlg.schurfact! — Function.

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

Same as schurfact but uses the input argument as workspace.

source

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

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

source

Base.LinAlg.schur — Function.

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.

See schurfact.

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

See schurfact.

source

Base.LinAlg.ordschur — Function.

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.

source

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.

source

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].

source

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.

source

Base.LinAlg.ordschur! — Function.

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

Same as ordschur but overwrites the factorization F.

source

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

Same as ordschur but overwrites the input arguments.

source

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

Same as ordschur but overwrites the factorization F.

source

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.

source

Base.LinAlg.svdfact — Function.

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

source

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).

source

Base.LinAlg.svdfact! — Function.

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.

source

svdfact!(A, B) -> GeneralizedSVD

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

source

Base.LinAlg.svd — Function.

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

source

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'.

source

Base.LinAlg.svdvals — Function.

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

source

svdvals(A, B)

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

source

Base.LinAlg.Givens — Type.

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'.

`which`	type of eigenvalues
`:LM`	eigenvalues of largest magnitude (default)
`:SM`	eigenvalues of smallest magnitude
`:LR`	eigenvalues of largest real part
`:SR`	eigenvalues of smallest real part
`:LI`	eigenvalues of largest imaginary part (nonsymmetric or complex `A` only)
`:SI`	eigenvalues of smallest imaginary part (nonsymmetric or complex `A` only)
`:BE`	compute half of the eigenvalues from each end of the spectrum, biased in favor of the high end. (real symmetric `A` only)

`sigma`	iteration mode	`which` refers to eigenvalues of
`nothing`	ordinary (forward)	$A$
real or complex	inverse with level shift `sigma`	$(A - \sigma I )^{-1}$

`which`	type of eigenvalues
`:LM`	eigenvalues of largest magnitude (default)
`:SM`	eigenvalues of smallest magnitude
`:LR`	eigenvalues of largest real part
`:SR`	eigenvalues of smallest real part
`:LI`	eigenvalues of largest imaginary part (nonsymmetric or complex `A` only)
`:SI`	eigenvalues of smallest imaginary part (nonsymmetric or complex `A` only)
`:BE`	compute half of the eigenvalues from each end of the spectrum, biased in favor of the high end. (real symmetric `A` only)

`sigma`	iteration mode	`which` refers to the problem
`nothing`	ordinary (forward)	$Av = Bv\lambda$
real or complex	inverse with level shift `sigma`	$(A - \sigma B )^{-1}B = v\nu$

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.

Base.LinAlg.A_ldiv_B! — Function.

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

Base.A_ldiv_Bc — Function.

A_ldiv_Bc(A, B)

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

source

Base.A_ldiv_Bt — Function.

A_ldiv_Bt(A, B)

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

source

Base.LinAlg.A_mul_B! — Function.

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

Base.A_mul_Bc — Function.

A_mul_Bc(A, B)

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

source

Base.A_mul_Bt — Function.

A_mul_Bt(A, B)

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

source

Base.A_rdiv_Bc — Function.

A_rdiv_Bc(A, B)

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

source

Base.A_rdiv_Bt — Function.

A_rdiv_Bt(A, B)

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

source

Base.Ac_ldiv_B — Function.

Ac_ldiv_B(A, B)

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

source

Base.LinAlg.Ac_ldiv_B! — Function.

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

Base.Ac_ldiv_Bc — Function.

Ac_ldiv_Bc(A, B)

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

source

Base.Ac_mul_B — Function.

Ac_mul_B(A, B)

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

source

Base.Ac_mul_Bc — Function.

Ac_mul_Bc(A, B)

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

source

Base.Ac_rdiv_B — Function.

Ac_rdiv_B(A, B)

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

source

Base.Ac_rdiv_Bc — Function.

Ac_rdiv_Bc(A, B)

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

source

Base.At_ldiv_B — Function.

At_ldiv_B(A, B)

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

source

Base.LinAlg.At_ldiv_B! — Function.

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).

source

Base.At_ldiv_Bt — Function.

At_ldiv_Bt(A, B)

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

source

Base.At_mul_B — Function.

At_mul_B(A, B)

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

source

Base.At_mul_Bt — Function.

At_mul_Bt(A, B)

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

source

Base.At_rdiv_B — Function.

At_rdiv_B(A, B)

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

source

Base.At_rdiv_Bt — Function.

At_rdiv_Bt(A, B)

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

source

BLAS Functions

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

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

`side`	Meaning
`'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`/`ul`	Meaning
`'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`/`tX`	Meaning
`'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`/`dX`	Meaning
`'N'`	The diagonal values of the matrix `X` will be read.
`'U'`	The diagonal of the matrix `X` is assumed to be all ones.

Base.LinAlg.BLAS.dotu — Function.

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

Base.LinAlg.BLAS.dotc — Function.

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

source

Base.LinAlg.BLAS.blascopy! — Function.

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

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

source

Base.LinAlg.BLAS.nrm2 — Function.

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

source

Base.LinAlg.BLAS.asum — Function.

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

source

Base.LinAlg.axpy! — Function.

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

source

Base.LinAlg.BLAS.scal! — Function.

scal!(n, a, X, incx)

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

source

Base.LinAlg.BLAS.scal — Function.

scal(n, a, X, incx)

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

source

Base.LinAlg.BLAS.ger! — Function.

ger!(alpha, x, y, A)

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

source

Base.LinAlg.BLAS.syr! — Function.

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.

source

Base.LinAlg.BLAS.syrk! — Function.

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.

source

Base.LinAlg.BLAS.syrk — Function.

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.

source

Base.LinAlg.BLAS.her! — Function.

her!(uplo, alpha, x, A)

Methods for complex arrays only. Rank-1 update of the Hermitian matrix A with vector x as alpha*x*x' + A. uplo controls which triangle of A is updated. Returns A.

source

Base.LinAlg.BLAS.herk! — Function.

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

Methods for complex arrays only. Rank-k update of the Hermitian matrix C as alpha*A*A' + beta*C or alpha*A'*A + beta*C according to trans. Only the uplo triangle of C is updated. Returns C.

source

Base.LinAlg.BLAS.herk — Function.

herk(uplo, trans, alpha, A)

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

source

Base.LinAlg.BLAS.gbmv! — Function.

gbmv!(trans, m, kl, ku, alpha, A, x, beta, y)

Update vector y as alpha*A*x + beta*y or alpha*A'*x + beta*y according to trans. The matrix A is a general band matrix of dimension m by size(A,2) with kl sub-diagonals and ku super-diagonals. alpha and beta are scalars. Returns the updated y.

source

Base.LinAlg.BLAS.gbmv — Function.

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.

source

Base.LinAlg.BLAS.sbmv! — Function.

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.

source

Base.LinAlg.BLAS.sbmv — Method.

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.

source

Base.LinAlg.BLAS.sbmv — Method.

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.

source

Base.LinAlg.BLAS.gemm! — Function.

gemm!(tA, tB, alpha, A, B, beta, C)

Update C as alpha*A*B + beta*C or the other three variants according to tA and tB. Returns the updated C.

source

Base.LinAlg.BLAS.gemm — Method.

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

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

source

Base.LinAlg.BLAS.gemm — Method.

gemm(tA, tB, A, B)

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

source

Base.LinAlg.BLAS.gemv! — Function.

gemv!(tA, alpha, A, x, beta, y)

Update the vector y as alpha*A*x + beta*y or alpha*A'x + beta*y according to tA. alpha and beta are scalars. Returns the updated y.

source

Base.LinAlg.BLAS.gemv — Method.

gemv(tA, alpha, A, x)

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

source

Base.LinAlg.BLAS.gemv — Method.

gemv(tA, A, x)

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

source

Base.LinAlg.BLAS.symm! — Function.

symm!(side, ul, alpha, A, B, beta, C)

Update C as alpha*A*B + beta*C or alpha*B*A + beta*C according to side. A is assumed to be symmetric. Only the ul triangle of A is used. Returns the updated C.

source

Base.LinAlg.BLAS.symm — Method.

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.

source

Base.LinAlg.BLAS.symm — Method.

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.

source

Base.LinAlg.BLAS.symv! — Function.

symv!(ul, alpha, A, x, beta, y)

Update the vector y as alpha*A*x + beta*y. A is assumed to be symmetric. Only the ul triangle of A is used. alpha and beta are scalars. Returns the updated y.

source

Base.LinAlg.BLAS.symv — Method.

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.

source

Base.LinAlg.BLAS.symv — Method.

symv(ul, A, x)

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

source

Base.LinAlg.BLAS.trmm! — Function.

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

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

source

Base.LinAlg.BLAS.trmm — Function.

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

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

source

Base.LinAlg.BLAS.trsm! — Function.

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

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

source

Base.LinAlg.BLAS.trsm — Function.

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.

source

Base.LinAlg.BLAS.trmv! — Function.

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.

source

Base.LinAlg.BLAS.trmv — Function.

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.

source

Base.LinAlg.BLAS.trsv! — Function.

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.

source

Base.LinAlg.BLAS.trsv — Function.

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.

source

Base.LinAlg.BLAS.set_num_threads — Function.

set_num_threads(n)

Set the number of threads the BLAS library should use.

source

Base.LinAlg.I — Constant.

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.

Base.LinAlg.LAPACK.gbtrf! — Function.

gbtrf!(kl, ku, m, AB) -> (AB, ipiv)

Compute the LU factorization of a banded matrix AB. kl is the first subdiagonal containing a nonzero band, ku is the last superdiagonal containing one, and m is the first dimension of the matrix AB. Returns the LU factorization in-place and ipiv, the vector of pivots used.

source

Base.LinAlg.LAPACK.gbtrs! — Function.

gbtrs!(trans, kl, ku, m, AB, ipiv, B)

Solve the equation AB * X = B. trans determines the orientation of AB. It may be N (no transpose), T (transpose), or C (conjugate transpose). kl is the first subdiagonal containing a nonzero band, ku is the last superdiagonal containing one, and m is the first dimension of the matrix AB. ipiv is the vector of pivots returned from gbtrf!. Returns the vector or matrix X, overwriting B in-place.

source

Base.LinAlg.LAPACK.gebal! — Function.

gebal!(job, A) -> (ilo, ihi, scale)

Balance the matrix A before computing its eigensystem or Schur factorization. job can be one of N (A will not be permuted or scaled), P (A will only be permuted), S (A will only be scaled), or B (A will be both permuted and scaled). Modifies A in-place and returns ilo, ihi, and scale. If permuting was turned on, A[i,j] = 0 if j > i and 1 < j < ilo or j > ihi. scale contains information about the scaling/permutations performed.

source

Base.LinAlg.LAPACK.gebak! — Function.

gebak!(job, side, ilo, ihi, scale, V)

Transform the eigenvectors V of a matrix balanced using gebal! to the unscaled/unpermuted eigenvectors of the original matrix. Modifies V in-place. side can be L (left eigenvectors are transformed) or R (right eigenvectors are transformed).

source

Base.LinAlg.LAPACK.gebrd! — Function.

gebrd!(A) -> (A, d, e, tauq, taup)

Reduce A in-place to bidiagonal form A = QBP'. Returns A, containing the bidiagonal matrix B; d, containing the diagonal elements of B; e, containing the off-diagonal elements of B; tauq, containing the elementary reflectors representing Q; and taup, containing the elementary reflectors representing P.

source

Base.LinAlg.LAPACK.gelqf! — Function.

gelqf!(A, tau)

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

Returns A and tau modified in-place.

source

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

Compute the LQ factorization of A, A = LQ.

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

source

Base.LinAlg.LAPACK.geqlf! — Function.

geqlf!(A, tau)

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

Returns A and tau modified in-place.

source

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.

source

Base.LinAlg.LAPACK.geqrf! — Function.

geqrf!(A, tau)

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

Returns A and tau modified in-place.

source

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.

source

Base.LinAlg.LAPACK.geqp3! — Function.

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.

source

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.

source

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.

source

Base.LinAlg.LAPACK.gerqf! — Function.

gerqf!(A, tau)

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

Returns A and tau modified in-place.

source

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.

source

Base.LinAlg.LAPACK.geqrt! — Function.

geqrt!(A, T)

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

Returns A and T modified in-place.

source

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.

source

Base.LinAlg.LAPACK.geqrt3! — Function.

geqrt3!(A, T)

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

Returns A and T modified in-place.

source

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.

source

Base.LinAlg.LAPACK.getrf! — Function.

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

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

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

source

Base.LinAlg.LAPACK.tzrzf! — Function.

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

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

source

Base.LinAlg.LAPACK.ormrz! — Function.

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

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

source

Base.LinAlg.LAPACK.gels! — Function.

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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Base.LinAlg.LAPACK.gesv! — Function.

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

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

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Base.LinAlg.LAPACK.getrs! — Function.

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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Base.LinAlg.LAPACK.getri! — Function.

getri!(A, ipiv)

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

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Base.LinAlg.LAPACK.gesvx! — Function.

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

Solves the linear equation A * X = B (trans = N), 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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Base.LinAlg.LAPACK.gelsd! — Function.

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

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

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Base.LinAlg.LAPACK.gelsy! — Function.

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

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

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Base.LinAlg.LAPACK.gglse! — Function.

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

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

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Base.LinAlg.LAPACK.geev! — Function.

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

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

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Base.LinAlg.LAPACK.gesdd! — Function.

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

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

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Base.LinAlg.LAPACK.gesvd! — Function.

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

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

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

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Base.LinAlg.LAPACK.ggsvd! — Function.

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

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

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Base.LinAlg.LAPACK.ggsvd3! — Function.

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

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

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Base.LinAlg.LAPACK.geevx! — Function.

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

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

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Base.LinAlg.LAPACK.ggev! — Function.

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

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

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Base.LinAlg.LAPACK.gtsv! — Function.

gtsv!(dl, d, du, B)

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

Overwrites B with the solution X and returns it.

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Base.LinAlg.LAPACK.gttrf! — Function.

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

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

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

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Base.LinAlg.LAPACK.gttrs! — Function.

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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Base.LinAlg.LAPACK.orglq! — Function.

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

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

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Base.LinAlg.LAPACK.orgqr! — Function.

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

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

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Base.LinAlg.LAPACK.orgql! — Function.

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

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

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Base.LinAlg.LAPACK.orgrq! — Function.

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

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

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Base.LinAlg.LAPACK.ormlq! — Function.

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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Base.LinAlg.LAPACK.ormqr! — Function.

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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Base.LinAlg.LAPACK.ormql! — Function.

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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Base.LinAlg.LAPACK.ormrq! — Function.

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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Base.LinAlg.LAPACK.gemqrt! — Function.

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

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Base.LinAlg.LAPACK.posv! — Function.

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

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

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Base.LinAlg.LAPACK.potrf! — Function.

potrf!(uplo, A)

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

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Base.LinAlg.LAPACK.potri! — Function.

potri!(uplo, A)

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

A is overwritten by its inverse and returned.

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Base.LinAlg.LAPACK.potrs! — Function.

potrs!(uplo, A, B)

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

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Base.LinAlg.LAPACK.pstrf! — Function.

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

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

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

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Base.LinAlg.LAPACK.ptsv! — Function.

ptsv!(D, E, B)

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

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Base.LinAlg.LAPACK.pttrf! — Function.

pttrf!(D, E)

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

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Base.LinAlg.LAPACK.pttrs! — Function.

pttrs!(D, E, B)

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

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Base.LinAlg.LAPACK.trtri! — Function.

trtri!(uplo, diag, A)

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

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Base.LinAlg.LAPACK.trtrs! — Function.

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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Base.LinAlg.LAPACK.trcon! — Function.

trcon!(norm, uplo, diag, A)

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

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Base.LinAlg.LAPACK.trevc! — Function.

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

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

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Base.LinAlg.LAPACK.trrfs! — Function.

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

Estimates the error in the solution to A * X = B (trans = N), 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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Base.LinAlg.LAPACK.stev! — Function.

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

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

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Base.LinAlg.LAPACK.stebz! — Function.

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

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

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Base.LinAlg.LAPACK.stegr! — Function.

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

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

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Base.LinAlg.LAPACK.stein! — Function.

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

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

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Base.LinAlg.LAPACK.syconv! — Function.

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

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

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Base.LinAlg.LAPACK.sysv! — Function.

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

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

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Base.LinAlg.LAPACK.sytrf! — Function.

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

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

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

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Base.LinAlg.LAPACK.sytri! — Function.

sytri!(uplo, A, ipiv)

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

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Base.LinAlg.LAPACK.sytrs! — Function.

sytrs!(uplo, A, ipiv, B)

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

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Base.LinAlg.LAPACK.hesv! — Function.

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

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

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Base.LinAlg.LAPACK.hetrf! — Function.

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

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

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Base.LinAlg.LAPACK.hetri! — Function.

hetri!(uplo, A, ipiv)

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

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Base.LinAlg.LAPACK.hetrs! — Function.

hetrs!(uplo, A, ipiv, B)

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

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Base.LinAlg.LAPACK.syev! — Function.

syev!(jobz, uplo, A)

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Base.LinAlg.LAPACK.syevr! — Function.

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

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

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

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Base.LinAlg.LAPACK.sygvd! — Function.

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

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

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Base.LinAlg.LAPACK.bdsqr! — Function.

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

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

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

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Base.LinAlg.LAPACK.bdsdc! — Function.

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

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

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

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Base.LinAlg.LAPACK.gecon! — Function.

gecon!(normtype, A, anorm)

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

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Base.LinAlg.LAPACK.gehrd! — Function.

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

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

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Base.LinAlg.LAPACK.orghr! — Function.

orghr!(ilo, ihi, A, tau)

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

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Base.LinAlg.LAPACK.gees! — Function.

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

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

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

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Base.LinAlg.LAPACK.gges! — Function.

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

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

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

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Base.LinAlg.LAPACK.trexc! — Function.

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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Base.LinAlg.LAPACK.trsen! — Function.

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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Base.LinAlg.LAPACK.tgsen! — Function.

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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Base.LinAlg.LAPACK.trsyl! — Function.

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

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

Returns X (overwriting C) and scale.

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