Linear Algebra

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

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

julia> tr(A)
3

julia> det(A)
104.0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Special matrices

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

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

Type	Description
`Symmetric`	Symmetric matrix
`Hermitian`	Hermitian matrix
`UpperTriangular`	Upper triangular matrix
`UnitUpperTriangular`	Upper triangular matrix with unit diagonal
`LowerTriangular`	Lower triangular matrix
`UnitLowerTriangular`	Lower triangular matrix with unit diagonal
`UpperHessenberg`	Upper Hessenberg matrix
`Tridiagonal`	Tridiagonal matrix
`SymTridiagonal`	Symmetric tridiagonal matrix
`Bidiagonal`	Upper/lower bidiagonal matrix
`Diagonal`	Diagonal matrix
`UniformScaling`	Uniform scaling operator

Elementary operations

Matrix type	`+`	`-`	`*`	`\`	Other functions with optimized methods
`Symmetric`				MV	`inv`, `sqrt`, `cbrt`, `exp`
`Hermitian`				MV	`inv`, `sqrt`, `cbrt`, `exp`
`UpperTriangular`			MV	MV	`inv`, `det`, `logdet`
`UnitUpperTriangular`			MV	MV	`inv`, `det`, `logdet`
`LowerTriangular`			MV	MV	`inv`, `det`, `logdet`
`UnitLowerTriangular`			MV	MV	`inv`, `det`, `logdet`
`UpperHessenberg`				MM	`inv`, `det`
`SymTridiagonal`	M	M	MS	MV	`eigmax`, `eigmin`
`Tridiagonal`	M	M	MS	MV
`Bidiagonal`	M	M	MS	MV
`Diagonal`	M	M	MV	MV	`inv`, `det`, `logdet`, `/`
`UniformScaling`	M	M	MVS	MVS	`/`

Legend:

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

Matrix factorizations

Matrix type	LAPACK	`eigen`	`eigvals`	`eigvecs`	`svd`	`svdvals`
`Symmetric`	SY		ARI
`Hermitian`	HE		ARI
`UpperTriangular`	TR	A	A	A
`UnitUpperTriangular`	TR	A	A	A
`LowerTriangular`	TR	A	A	A
`UnitLowerTriangular`	TR	A	A	A
`SymTridiagonal`	ST	A	ARI	AV
`Tridiagonal`	GT
`Bidiagonal`	BD				A	A
`Diagonal`	DI		A

Legend:

Key	Description	Example
A (all)	An optimized method to find all the characteristic values and/or vectors is available	e.g. `eigvals(M)`
R (range)	An optimized method to find the `il`th through the `ih`th characteristic values are available	`eigvals(M, il, ih)`
I (interval)	An optimized method to find the characteristic values in the interval [`vl`, `vh`] is available	`eigvals(M, vl, vh)`
V (vectors)	An optimized method to find the characteristic vectors corresponding to the characteristic values `x=[x1, x2,...]` is available	`eigvecs(M, x)`

The uniform scaling operator

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

To see the UniformScaling operator in action:

julia> U = UniformScaling(2);

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

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

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

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

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

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

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

Matrix factorizations

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

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

Type	Description
`BunchKaufman`	Bunch-Kaufman factorization
`Cholesky`	Cholesky factorization
`CholeskyPivoted`	Pivoted Cholesky factorization
`LDLt`	LDL(T) factorization
`LU`	LU factorization
`QR`	QR factorization
`QRCompactWY`	Compact WY form of the QR factorization
`QRPivoted`	Pivoted QR factorization
`LQ`	QR factorization of `transpose(A)`
`Hessenberg`	Hessenberg decomposition
`Eigen`	Spectral decomposition
`GeneralizedEigen`	Generalized spectral decomposition
`SVD`	Singular value decomposition
`GeneralizedSVD`	Generalized SVD
`Schur`	Schur decomposition
`GeneralizedSchur`	Generalized Schur decomposition

Adjoints and transposes of Factorization objects are lazily wrapped in AdjointFactorization and TransposeFactorization objects, respectively. Generically, transpose of real Factorizations are wrapped as AdjointFactorization.

Orthogonal matrices (`AbstractQ`)

Some matrix factorizations generate orthogonal/unitary "matrix" factors. These factorizations include QR-related factorizations obtained from calls to qr, i.e., QR, QRCompactWY and QRPivoted, the Hessenberg factorization obtained from calls to hessenberg, and the LQ factorization obtained from lq. While these orthogonal/unitary factors admit a matrix representation, their internal representation is, for performance and memory reasons, different. Hence, they should be rather viewed as matrix-backed, function-based linear operators. In particular, reading, for instance, a column of its matrix representation requires running "matrix"-vector multiplication code, rather than simply reading out data from memory (possibly filling parts of the vector with structural zeros). Another clear distinction from other, non-triangular matrix types is that the underlying multiplication code allows for in-place modification during multiplication. Furthermore, objects of specific AbstractQ subtypes as those created via qr, hessenberg and lq can behave like a square or a rectangular matrix depending on context:

julia> using LinearAlgebra

julia> Q = qr(rand(3,2)).Q
3×3 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}

julia> Matrix(Q)
3×2 Matrix{Float64}:
 -0.320597   0.865734
 -0.765834  -0.475694
 -0.557419   0.155628

julia> Q*I
3×3 Matrix{Float64}:
 -0.320597   0.865734  -0.384346
 -0.765834  -0.475694  -0.432683
 -0.557419   0.155628   0.815514

julia> Q*ones(2)
3-element Vector{Float64}:
  0.5451367118802273
 -1.241527373086654
 -0.40179067589600226

julia> Q*ones(3)
3-element Vector{Float64}:
  0.16079054743832022
 -1.674209978965636
  0.41372375588835797

julia> ones(1,2) * Q'
1×3 Matrix{Float64}:
 0.545137  -1.24153  -0.401791

julia> ones(1,3) * Q'
1×3 Matrix{Float64}:
 0.160791  -1.67421  0.413724

Due to this distinction from dense or structured matrices, the abstract AbstractQ type does not subtype AbstractMatrix, but instead has its own type hierarchy. Custom types that subtype AbstractQ can rely on generic fallbacks if the following interface is satisfied. For example, for

struct MyQ{T} <: LinearAlgebra.AbstractQ{T}
    # required fields
end

provide overloads for

Base.size(Q::MyQ) # size of corresponding square matrix representation
Base.convert(::Type{AbstractQ{T}}, Q::MyQ) # eltype promotion [optional]
LinearAlgebra.lmul!(Q::MyQ, x::AbstractVecOrMat) # left-multiplication
LinearAlgebra.rmul!(A::AbstractMatrix, Q::MyQ) # right-multiplication

If eltype promotion is not of interest, the convert method is unnecessary, since by default convert(::Type{AbstractQ{T}}, Q::AbstractQ{T}) returns Q itself. Adjoints of AbstractQ-typed objects are lazily wrapped in an AdjointQ wrapper type, which requires its own LinearAlgebra.lmul! and LinearAlgebra.rmul! methods. Given this set of methods, any Q::MyQ can be used like a matrix, preferably in a multiplicative context: multiplication via * with scalars, vectors and matrices from left and right, obtaining a matrix representation of Q via Matrix(Q) (or Q*I) and indexing into the matrix representation all work. In contrast, addition and subtraction as well as more generally broadcasting over elements in the matrix representation fail because that would be highly inefficient. For such use cases, consider computing the matrix representation up front and cache it for future reuse.

Pivoting Strategies

Several of Julia's matrix factorizations support pivoting, which can be used to improve their numerical stability. In fact, some matrix factorizations, such as the LU factorization, may fail without pivoting.

In pivoting, first, a pivot element with good numerical properties is chosen based on a pivoting strategy. Next, the rows and columns of the original matrix are permuted to bring the chosen element in place for subsequent computation. Furthermore, the process is repeated for each stage of the factorization.

Consequently, besides the conventional matrix factors, the outputs of pivoted factorization schemes also include permutation matrices.

In the following, the pivoting strategies implemented in Julia are briefly described. Note that not all matrix factorizations may support them. Consult the documentation of the respective matrix factorization for details on the supported pivoting strategies.

LinearAlgebra.NoPivot — Type

NoPivot

Pivoting is not performed. Matrix factorizations such as the LU factorization may fail without pivoting, and may also be numerically unstable for floating-point matrices in the face of roundoff error. This pivot strategy is mainly useful for pedagogical purposes.

LinearAlgebra.RowNonZero — Type

RowNonZero

First non-zero element in the remaining rows is chosen as the pivot element.

Beware that for floating-point matrices, the resulting LU algorithm is numerically unstable — this strategy is mainly useful for comparison to hand calculations (which typically use this strategy) or for other algebraic types (e.g. rational numbers) not susceptible to roundoff errors. Otherwise, the default RowMaximum pivoting strategy should be generally preferred in Gaussian elimination.

Note that the element type of the matrix must admit an iszero method.

LinearAlgebra.RowMaximum — Type

RowMaximum

The maximum-magnitude element in the remaining rows is chosen as the pivot element. This is the default strategy for LU factorization of floating-point matrices, and is sometimes referred to as the "partial pivoting" algorithm.

Note that the element type of the matrix must admit an abs method, whose result type must admit a < method.

LinearAlgebra.ColumnNorm — Type

ColumnNorm

The column with the maximum norm is used for subsequent computation. This is used for pivoted QR factorization.

Note that the element type of the matrix must admit norm and abs methods, whose respective result types must admit a < method.

Standard functions

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

Base.:* — Method

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

Matrix multiplication.

Examples

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

Base.:* — Method

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

Chained multiplication of 3 or 4 matrices is done in the most efficient sequence, based on the sizes of the arrays. That is, the number of scalar multiplications needed for (A * B) * C (with 3 dense matrices) is compared to that for A * (B * C) to choose which of these to execute.

If the last factor is a vector, or the first a transposed vector, then it is efficient to deal with these first. In particular x' * B * y means (x' * B) * y for an ordinary column-major B::Matrix. Unlike dot(x, B, y), this allocates an intermediate array.

If the first or last factor is a number, this will be fused with the matrix multiplication, using 5-arg mul!.

Properties of `A`	type of factorization
Positive-definite	Cholesky (see `cholesky`)
Dense Symmetric/Hermitian	Bunch-Kaufman (see `bunchkaufman`)
Sparse Symmetric/Hermitian	LDLt (see `ldlt`)
Triangular	Triangular
Diagonal	Diagonal
Bidiagonal	Bidiagonal
Tridiagonal	LU (see `lu`)
Symmetric real tridiagonal	LDLt (see `ldlt`)
General square	LU (see `lu`)
General non-square	QR (see `qr`)

Component	Description
`F.L`	`L` (unit lower triangular) part of `LU`
`F.U`	`U` (upper triangular) part of `LU`
`F.p`	(right) permutation `Vector`
`F.P`	(right) permutation `Matrix`

Component	Description
`L`	`L` (lower triangular) part of `LU`
`U`	`U` (upper triangular) part of `LU`
`p`	right permutation `Vector`
`q`	left permutation `Vector`
`Rs`	`Vector` of scaling factors
`:`	`(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}}`
`/`	✓
`\`	✓	✓
`inv`	✓	✓
`det`	✓	✓
`logdet`	✓	✓
`logabsdet`	✓	✓
`size`	✓	✓

Component	Description
`F.L`	`L` (unit lower triangular) part of `LDLt`
`F.D`	`D` (diagonal) part of `LDLt`
`F.Lt`	`Lt` (unit upper triangular) part of `LDLt`
`F.d`	diagonal values of `D` as a `Vector`

See also ldlt.

Note

LinearAlgebra.QR — Type

QR <: Factorization

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

\[A = Q R\]

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

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

Iterating the decomposition produces the components Q and R.

The object has two fields:

factors is an m×n matrix.
- The upper triangular part contains the elements of $R$, that is R = triu(F.factors) for a QR object F.
- The subdiagonal part contains the reflectors $v_i$ stored in a packed format where $v_i$ is the $i$th column of the matrix V = I + tril(F.factors, -1).
τ is a vector of length min(m,n) containing the coefficients $au_i$.

LinearAlgebra.QRCompactWY — Type

QRCompactWY <: Factorization

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

\[A = Q R\]

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

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

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

Iterating the decomposition produces the components Q and R.

The object has two fields:

factors, as in the QR type, is an m×n matrix.
- The upper triangular part contains the elements of $R$, that is R = triu(F.factors) for a QR object F.
- The subdiagonal part contains the reflectors $v_i$ stored in a packed format such that V = I + tril(F.factors, -1).
T is a $n_b$-by-$\min(m,n)$ matrix as described above. The subdiagonal elements for each triangular matrix $T_j$ are ignored.

Note

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

LinearAlgebra.QRPivoted — Type

QRPivoted <: Factorization

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

\[A P = Q R\]

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

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

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

The object has three fields:

factors is an m×n matrix.
- The upper triangular part contains the elements of $R$, that is R = triu(F.factors) for a QR object F.
- The subdiagonal part contains the reflectors $v_i$ stored in a packed format where $v_i$ is the $i$th column of the matrix V = I + tril(F.factors, -1).
τ is a vector of length min(m,n) containing the coefficients $au_i$.
jpvt is an integer vector of length n corresponding to the permutation $P$.

LinearAlgebra.qr — Function

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

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

Note

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

Examples

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

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

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

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

\[A = Q R\]

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

if pivot == ColumnNorm() then F is a QRPivoted object,
otherwise if the element type of A is a BLAS type (Float32, Float64, ComplexF32 or ComplexF64), then F is a QRCompactWY object,
otherwise F is a QR object.

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

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

Note

Each reference to the upper triangular factor via F.R allocates a new array. It is therefore advisable to cache that array, say, by R = F.R and continue working with R.

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

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

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

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

Julia 1.4

The blocksize keyword argument requires Julia 1.4 or later.

Examples

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

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

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

Note

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

LinearAlgebra.qr! — Function

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

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

Julia 1.4

The blocksize keyword argument requires Julia 1.4 or later.

Examples

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

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

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

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

LinearAlgebra.LQ — Type

LQ <: Factorization

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

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

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

Examples

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

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

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

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

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

LinearAlgebra.lq — Function

lq(A) -> S::LQ

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

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

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

Examples

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

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

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

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

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

LinearAlgebra.lq! — Function

lq!(A) -> LQ

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

LinearAlgebra.BunchKaufman — Type

BunchKaufman <: Factorization

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

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

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

Examples

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

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

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

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

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

LinearAlgebra.bunchkaufman — Function

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

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

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

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

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

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

Examples

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

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

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

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

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

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

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

LinearAlgebra.bunchkaufman! — Function

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

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

LinearAlgebra.Eigen — Type

Eigen <: Factorization

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

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

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

Examples

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

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

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

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

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

LinearAlgebra.GeneralizedEigen — Type

GeneralizedEigen <: Factorization

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

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

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

Examples

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

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

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

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

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

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

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

LinearAlgebra.eigvals — Function

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

Return the eigenvalues of A.

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

Examples

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

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

For a scalar input, eigvals will return a scalar.

Examples

julia> eigvals(-2)
-2

eigvals(A, B) -> values

Compute the generalized eigenvalues of A and B.

Examples

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

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

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

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

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

Examples

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

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

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

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

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

Examples

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

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

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

LinearAlgebra.eigvals! — Function

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

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

Note

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

Examples

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

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

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

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

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

Note

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

Examples

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

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

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

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

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

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

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

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

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

LinearAlgebra.eigmax — Function

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

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

Examples

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

julia> eigmax(A)
1.0

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

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

LinearAlgebra.eigmin — Function

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

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

Examples

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

julia> eigmin(A)
-1.0

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

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

LinearAlgebra.eigvecs — Function

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

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

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

Examples

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

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

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

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

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

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

Examples

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

eigvecs(A, B) -> Matrix

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

Examples

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

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

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

LinearAlgebra.eigen — Function

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

Compute the eigenvalue decomposition of A, returning an Eigen factorization object F which contains the eigenvalues in F.values and the eigenvectors in the columns of the matrix F.vectors. This corresponds to solving an eigenvalue problem of the form Ax = λx, where A is a matrix, x is an eigenvector, and λ is an eigenvalue. (The kth eigenvector can be obtained from the slice F.vectors[:, k].)

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

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

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

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

Examples

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

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

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

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

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

eigen(A, B; sortby) -> GeneralizedEigen

Compute the generalized eigenvalue decomposition of A and B, returning a GeneralizedEigen factorization object F which contains the generalized eigenvalues in F.values and the generalized eigenvectors in the columns of the matrix F.vectors. This corresponds to solving a generalized eigenvalue problem of the form Ax = λBx, where A, B are matrices, x is an eigenvector, and λ is an eigenvalue. (The kth generalized eigenvector can be obtained from the slice F.vectors[:, k].)

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

Examples

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

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

julia> F = eigen(A, B);

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

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

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

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

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

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

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

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

Note

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

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

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

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

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

Note

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

LinearAlgebra.eigen! — Function

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

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

LinearAlgebra.Hessenberg — Type

Hessenberg <: Factorization

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

LinearAlgebra.hessenberg — Function

hessenberg(A) -> Hessenberg

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

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

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

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

Examples

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

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

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

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

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

LinearAlgebra.hessenberg! — Function

hessenberg!(A) -> Hessenberg

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

LinearAlgebra.Schur — Type

Schur <: Factorization

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

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

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

Examples

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

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

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

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

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

LinearAlgebra.GeneralizedSchur — Type

GeneralizedSchur <: Factorization

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

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

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

LinearAlgebra.schur — Function

schur(A) -> F::Schur

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

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

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

Examples

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

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

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

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

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

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

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

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

LinearAlgebra.schur! — Function

schur!(A) -> F::Schur

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

Examples

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

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

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

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

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

LinearAlgebra.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.

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

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

LinearAlgebra.ordschur! — Function

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

Same as ordschur but overwrites the factorization F.

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

Same as ordschur but overwrites the factorization F.

LinearAlgebra.SVD — Type

SVD <: Factorization

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

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

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

Examples

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

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

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

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

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

LinearAlgebra.GeneralizedSVD — Type

GeneralizedSVD <: Factorization

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

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

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

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

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

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

Examples

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

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

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

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

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

LinearAlgebra.svd — Function

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

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

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

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

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

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

Julia 1.3

The alg keyword argument requires Julia 1.3 or later.

Examples

julia> A = rand(4,3);

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

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

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

julia> A ≈ U * Diagonal(S) * V'
true

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

julia> Uonly == U
true

svd(A, B) -> GeneralizedSVD

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

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

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

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

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

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

Examples

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

julia> F = svd(A, B);

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

julia> H = R*Q';

julia> [A; B] ≈ [U*C; V*S]*H
true

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

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

julia> U == Uonly
true

LinearAlgebra.svd! — Function

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

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

svd!(A, B) -> GeneralizedSVD

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

LinearAlgebra.svdvals — Function

svdvals(A)

Return the singular values of A in descending order.

Examples

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

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

svdvals(A, B)

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

Examples

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

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

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

LinearAlgebra.svdvals! — Function

svdvals!(A)

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

svdvals!(A, B)

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

LinearAlgebra.Givens — Type

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

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

`tM`	Destination	Source
`'N'`	`B[ir_dest, jr_dest]`	`transpose(M)[jr_src, ir_src]`
`'T'`	`B[ir_dest, jr_dest]`	`M[jr_src, ir_src]`
`'C'`	`B[ir_dest, jr_dest]`	`conj(M)[jr_src, ir_src]`

Low-level matrix operations

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

LinearAlgebra.mul! — Function

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

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

Examples

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

julia> mul!(Y, A, B) === Y
true

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

julia> Y == A * B
true

Implementation

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

mul!(C, A, B, α, β) -> C

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

Julia 1.3

Five-argument mul! requires at least Julia 1.3.

Examples

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

julia> α, β = 100.0, 10.0;

julia> mul!(C, A, B, α, β) === C
true

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

julia> C_original = [1.0 2.0; 3.0 4.0]; # A copy of the original value of C

julia> C == A * B * α + C_original * β
true

LinearAlgebra.lmul! — Function

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

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

Julia 1.1

Prior to Julia 1.1, NaN and ±Inf entries in B were treated inconsistently.

Examples

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

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

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

lmul!(A, B)

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

Examples

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

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

julia> lmul!(A, B);

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

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

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

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

LinearAlgebra.rmul! — Function

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

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

Julia 1.1

Prior to Julia 1.1, NaN and ±Inf entries in A were treated inconsistently.

Examples

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

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

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

rmul!(A, B)

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

Examples

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

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

julia> rmul!(A, B);

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

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

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

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

LinearAlgebra.ldiv! — Function

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

Compute A \ B in-place and store the result in Y, returning the result.

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

Note

Certain structured matrix types, such as Diagonal and UpperTriangular, are permitted, as these are already in a factorized form

Examples

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

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

julia> Y = zero(X);

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

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

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

ldiv!(A, B)

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

Note

Certain structured matrix types, such as Diagonal and UpperTriangular, are permitted, as these are already in a factorized form

Examples

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

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

julia> Y = copy(X);

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

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

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

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

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

Examples

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

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

ldiv!(A::Tridiagonal, B::AbstractVecOrMat) -> B

Compute A \ B in-place by Gaussian elimination with partial pivoting and store the result in B, returning the result. In the process, the diagonals of A are overwritten as well.

Julia 1.11

ldiv! for Tridiagonal left-hand sides requires at least Julia 1.11.

LinearAlgebra.rdiv! — Function

rdiv!(A, B)

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

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

Note

Certain structured matrix types, such as Diagonal and UpperTriangular, are permitted, as these are already in a factorized form

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

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

Examples

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

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

BLAS functions

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

LinearAlgebra.BLAS provides wrappers for some of the BLAS functions. Those BLAS functions that overwrite one of the input arrays have names ending in '!'. Usually, a BLAS function has four methods defined, for Float32, Float64, ComplexF32, and ComplexF64 arrays.

BLAS character arguments

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

Multiplication order

`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.

LinearAlgebra.BLAS — Module

Interface to BLAS subroutines.

LinearAlgebra.BLAS.set_num_threads — Function

set_num_threads(n::Integer)
set_num_threads(::Nothing)

Set the number of threads the BLAS library should use equal to n::Integer.

Also accepts nothing, in which case julia tries to guess the default number of threads. Passing nothing is discouraged and mainly exists for historical reasons.

LinearAlgebra.BLAS.get_num_threads — Function

get_num_threads()

Get the number of threads the BLAS library is using.

Julia 1.6

get_num_threads requires at least Julia 1.6.

BLAS functions can be divided into three groups, also called three levels, depending on when they were first proposed, the type of input parameters, and the complexity of the operation.

Level 1 BLAS functions

The level 1 BLAS functions were first proposed in [(Lawson, 1979)][Lawson-1979] and define operations between scalars and vectors.

[Lawson-1979]: https://dl.acm.org/doi/10.1145/355841.355847

LinearAlgebra.BLAS.rot! — Function

rot!(n, X, incx, Y, incy, c, s)

Overwrite X with c*X + s*Y and Y with -conj(s)*X + c*Y for the first n elements of array X with stride incx and first n elements of array Y with stride incy. Returns X and Y.

Julia 1.5

rot! requires at least Julia 1.5.

LinearAlgebra.BLAS.scal! — Function

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

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

If n and incx are not provided, length(X) and stride(X,1) are used.

LinearAlgebra.BLAS.scal — Function

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

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

If n and incx are not provided, length(X) and stride(X,1) are used.

LinearAlgebra.BLAS.blascopy! — Function

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

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

LinearAlgebra.BLAS.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.

Examples

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

LinearAlgebra.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.

Examples

julia> BLAS.dotu(10, fill(1.0im, 10), 1, fill(1.0+im, 20), 2)
-10.0 + 10.0im

LinearAlgebra.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.

Examples

julia> BLAS.dotc(10, fill(1.0im, 10), 1, fill(1.0+im, 20), 2)
10.0 - 10.0im

LinearAlgebra.BLAS.nrm2 — Function

nrm2(n, X, incx)

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

Examples

julia> BLAS.nrm2(4, fill(1.0, 8), 2)
2.0

julia> BLAS.nrm2(1, fill(1.0, 8), 2)
1.0

LinearAlgebra.BLAS.asum — Function

asum(n, X, incx)

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

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

Examples

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

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

LinearAlgebra.BLAS.iamax — Function

iamax(n, dx, incx)
iamax(dx)

Find the index of the element of dx with the maximum absolute value. n is the length of dx, and incx is the stride. If n and incx are not provided, they assume default values of n=length(dx) and incx=stride1(dx).

Level 2 BLAS functions

The level 2 BLAS functions were published in [(Dongarra, 1988)][Dongarra-1988], and define matrix-vector operations.

[Dongarra-1988]: https://dl.acm.org/doi/10.1145/42288.42291

return a vector

LinearAlgebra.BLAS.gemv! — Function

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

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

LinearAlgebra.BLAS.gemv — Method

gemv(tA, alpha, A, x)

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

LinearAlgebra.BLAS.gemv — Method

gemv(tA, A, x)

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

LinearAlgebra.BLAS.gbmv! — Function

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

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

LinearAlgebra.BLAS.gbmv — Function

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

Return alpha*A*x or alpha*A'*x according to trans. The matrix A is a general band matrix of dimension m by size(A,2) with kl sub-diagonals and ku super-diagonals, and alpha is a scalar.

LinearAlgebra.BLAS.hemv! — Function

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

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

LinearAlgebra.BLAS.hemv — Method

hemv(ul, alpha, A, x)

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

LinearAlgebra.BLAS.hemv — Method

hemv(ul, A, x)

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

LinearAlgebra.BLAS.hpmv! — Function

hpmv!(uplo, α, AP, x, β, y)

Update vector y as α*A*x + β*y, where A is a Hermitian matrix provided in packed format AP.

With uplo = 'U', the array AP must contain the upper triangular part of the Hermitian matrix packed sequentially, column by column, so that AP[1] contains A[1, 1], AP[2] and AP[3] contain A[1, 2] and A[2, 2] respectively, and so on.

With uplo = 'L', the array AP must contain the lower triangular part of the Hermitian matrix packed sequentially, column by column, so that AP[1] contains A[1, 1], AP[2] and AP[3] contain A[2, 1] and A[3, 1] respectively, and so on.

The scalar inputs α and β must be complex or real numbers.

The array inputs x, y and AP must all be of ComplexF32 or ComplexF64 type.

Return the updated y.

Julia 1.5

hpmv! requires at least Julia 1.5.

LinearAlgebra.BLAS.symv! — Function

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

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

LinearAlgebra.BLAS.symv — Method

symv(ul, alpha, A, x)

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

LinearAlgebra.BLAS.symv — Method

symv(ul, A, x)

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

LinearAlgebra.BLAS.sbmv! — Function

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

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

Return the updated y.

LinearAlgebra.BLAS.sbmv — Method

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

Return alpha*A*x where A is a symmetric band matrix of order size(A,2) with k super-diagonals stored in the argument A. Only the uplo triangle of A is used.

LinearAlgebra.BLAS.sbmv — Method

sbmv(uplo, k, A, x)

Return A*x where A is a symmetric band matrix of order size(A,2) with k super-diagonals stored in the argument A. Only the uplo triangle of A is used.

LinearAlgebra.BLAS.spmv! — Function

spmv!(uplo, α, AP, x, β, y)

Update vector y as α*A*x + β*y, where A is a symmetric matrix provided in packed format AP.

With uplo = 'U', the array AP must contain the upper triangular part of the symmetric matrix packed sequentially, column by column, so that AP[1] contains A[1, 1], AP[2] and AP[3] contain A[1, 2] and A[2, 2] respectively, and so on.

With uplo = 'L', the array AP must contain the lower triangular part of the symmetric matrix packed sequentially, column by column, so that AP[1] contains A[1, 1], AP[2] and AP[3] contain A[2, 1] and A[3, 1] respectively, and so on.

The scalar inputs α and β must be real.

The array inputs x, y and AP must all be of Float32 or Float64 type.

Return the updated y.

Julia 1.5

spmv! requires at least Julia 1.5.

LinearAlgebra.BLAS.trmv! — Function

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

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

LinearAlgebra.BLAS.trmv — Function

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

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

LinearAlgebra.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. Return the updated b.

LinearAlgebra.BLAS.trsv — Function

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

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

return a matrix

LinearAlgebra.BLAS.ger! — Function

ger!(alpha, x, y, A)

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

LinearAlgebra.BLAS.her! — Function

her!(uplo, alpha, x, A)

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

LinearAlgebra.BLAS.syr! — Function

syr!(uplo, alpha, x, A)

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

LinearAlgebra.BLAS.spr! — Function

spr!(uplo, α, x, AP)

Update matrix A as A+α*x*x', where A is a symmetric matrix provided in packed format AP and x is a vector.

The scalar input α must be real.

The array inputs x and AP must all be of Float32 or Float64 type. Return the updated AP.

Julia 1.8

spr! requires at least Julia 1.8.

Level 3 BLAS functions

The level 3 BLAS functions were published in [(Dongarra, 1990)][Dongarra-1990], and define matrix-matrix operations.

[Dongarra-1990]: https://dl.acm.org/doi/10.1145/77626.79170

LinearAlgebra.BLAS.gemmt! — Function

gemmt!(uplo, tA, tB, alpha, A, B, beta, C)

Update the lower or upper triangular part specified by uplo of C as alpha*A*B + beta*C or the other variants according to tA and tB. Return the updated C.

Julia 1.11

gemmt! requires at least Julia 1.11.

LinearAlgebra.BLAS.gemmt — Method

gemmt(uplo, tA, tB, alpha, A, B)

Return the lower or upper triangular part specified by uplo of A*B or the other three variants according to tA and tB.

Julia 1.11

gemmt requires at least Julia 1.11.

LinearAlgebra.BLAS.gemmt — Method

gemmt(uplo, tA, tB, A, B)

Return the lower or upper triangular part specified by uplo of A*B or the other three variants according to tA and tB.

Julia 1.11

gemmt requires at least Julia 1.11.

LinearAlgebra.BLAS.gemm! — Function

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

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

LinearAlgebra.BLAS.gemm — Method

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

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

LinearAlgebra.BLAS.gemm — Method

gemm(tA, tB, A, B)

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

LinearAlgebra.BLAS.symm! — Function

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

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

LinearAlgebra.BLAS.symm — Method

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

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

LinearAlgebra.BLAS.symm — Method

symm(side, ul, A, B)

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

LinearAlgebra.BLAS.hemm! — Function

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

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

LinearAlgebra.BLAS.hemm — Method

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

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

LinearAlgebra.BLAS.hemm — Method

hemm(side, ul, A, B)

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

LinearAlgebra.BLAS.syrk! — Function

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

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

LinearAlgebra.BLAS.syrk — Function

syrk(uplo, trans, alpha, A)

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

LinearAlgebra.BLAS.herk! — Function

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

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

LinearAlgebra.BLAS.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.

LinearAlgebra.BLAS.syr2k! — Function

syr2k!(uplo, trans, alpha, A, B, beta, C)

Rank-2k update of the symmetric matrix C as alpha*A*transpose(B) + alpha*B*transpose(A) + beta*C or alpha*transpose(A)*B + alpha*transpose(B)*A + beta*C according to trans. Only the uplo triangle of C is used. Returns C.

LinearAlgebra.BLAS.syr2k — Function

syr2k(uplo, trans, alpha, A, B)

Returns the uplo triangle of alpha*A*transpose(B) + alpha*B*transpose(A) or alpha*transpose(A)*B + alpha*transpose(B)*A, according to trans.

syr2k(uplo, trans, A, B)

Return the uplo triangle of A*transpose(B) + B*transpose(A) or transpose(A)*B + transpose(B)*A, according to trans.

LinearAlgebra.BLAS.her2k! — Function

her2k!(uplo, trans, alpha, A, B, beta, C)

Rank-2k update of the Hermitian matrix C as alpha*A*B' + alpha*B*A' + beta*C or alpha*A'*B + alpha*B'*A + beta*C according to trans. The scalar beta has to be real. Only the uplo triangle of C is used. Return C.

LinearAlgebra.BLAS.her2k — Function

her2k(uplo, trans, alpha, A, B)

Return the uplo triangle of alpha*A*B' + alpha*B*A' or alpha*A'*B + alpha*B'*A, according to trans.

her2k(uplo, trans, A, B)

Return the uplo triangle of A*B' + B*A' or A'*B + B'*A, according to trans.

LinearAlgebra.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. Return the updated B.

LinearAlgebra.BLAS.trmm — Function

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

Return 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.

LinearAlgebra.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.

LinearAlgebra.BLAS.trsm — Function

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

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

LAPACK functions

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

Usually a function has 4 methods defined, one each for Float64, Float32, ComplexF64 and ComplexF32 arrays.

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

LinearAlgebra.LAPACK — Module

Interfaces to LAPACK subroutines.

LinearAlgebra.LAPACK.gbtrf! — Function

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

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

LinearAlgebra.LAPACK.gbtrs! — Function

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

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

LinearAlgebra.LAPACK.gebal! — Function

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

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

LinearAlgebra.LAPACK.gebak! — Function

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

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

LinearAlgebra.LAPACK.gebrd! — Function

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

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

LinearAlgebra.LAPACK.gelqf! — Function

gelqf!(A, tau)

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

Returns A and tau modified in-place.

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

Compute the LQ factorization of A, A = LQ.

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

LinearAlgebra.LAPACK.geqlf! — Function

geqlf!(A, tau)

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

Returns A and tau modified in-place.

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

Compute the QL factorization of A, A = QL.

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

LinearAlgebra.LAPACK.geqrf! — Function

geqrf!(A, tau)

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

Returns A and tau modified in-place.

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

Compute the QR factorization of A, A = QR.

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

LinearAlgebra.LAPACK.geqp3! — Function

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

Compute the pivoted QR factorization of A, AP = QR using BLAS level 3. P is a pivoting matrix, represented by jpvt. tau stores the elementary reflectors. The arguments jpvt and tau are optional and allow for passing preallocated arrays. When passed, jpvt must have length greater than or equal to n if A is an (m x n) matrix and tau must have length greater than or equal to the smallest dimension of A. On entry, if jpvt[j] does not equal zero then the jth column of A is permuted to the front of AP.

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

LinearAlgebra.LAPACK.gerqf! — Function

gerqf!(A, tau)

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

Returns A and tau modified in-place.

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

Compute the RQ factorization of A, A = RQ.

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

LinearAlgebra.LAPACK.geqrt! — Function

geqrt!(A, T)

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

Returns A and T modified in-place.

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

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

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

LinearAlgebra.LAPACK.geqrt3! — Function

geqrt3!(A, T)

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

Returns A and T modified in-place.

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

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

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

LinearAlgebra.LAPACK.getrf! — Function

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

Compute the pivoted LU factorization of A, A = LU. ipiv contains the pivoting information and info a 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).

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

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

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

LinearAlgebra.LAPACK.tzrzf! — Function

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

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

LinearAlgebra.LAPACK.ormrz! — Function

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

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

LinearAlgebra.LAPACK.gels! — Function

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

Solves the linear equation A * X = B, transpose(A) * X = B, or adjoint(A) * X = B using a QR or LQ factorization. Modifies the matrix/vector B in place with the solution. A is overwritten with its QR or LQ factorization. trans may be one of N (no modification), T (transpose), or C (conjugate transpose). gels! searches for the minimum norm/least squares solution. A may be under or over determined. The solution is returned in B.

LinearAlgebra.LAPACK.gesv! — Function

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

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

LinearAlgebra.LAPACK.getrs! — Function

getrs!(trans, A, ipiv, B)

Solves the linear equation A * X = B, transpose(A) * X = B, or adjoint(A) * X = B for square A. Modifies the matrix/vector B in place with the solution. A is the LU factorization from getrf!, with ipiv the pivoting information. trans may be one of N (no modification), T (transpose), or C (conjugate transpose).

LinearAlgebra.LAPACK.getri! — Function

getri!(A, ipiv)

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

LinearAlgebra.LAPACK.gesvx! — Function

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

Solves the linear equation A * X = B (trans = N), transpose(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) using the LU factorization of A. fact may be E, in which case A will be equilibrated and copied to AF; F, in which case AF and ipiv from a previous LU factorization are inputs; or N, in which case A will be copied to AF and then factored. If fact = F, equed may be N, meaning A has not been equilibrated; R, meaning A was multiplied by Diagonal(R) from the left; C, meaning A was multiplied by Diagonal(C) from the right; or B, meaning A was multiplied by Diagonal(R) from the left and Diagonal(C) from the right. If fact = F and equed = R or B the elements of R must all be positive. If fact = F and equed = C or B the elements of C must all be positive.

Returns the solution X; equed, which is an output if fact is not N, and describes the equilibration that was performed; R, the row equilibration diagonal; C, the column equilibration diagonal; B, which may be overwritten with its equilibrated form Diagonal(R)*B (if trans = N and equed = R,B) or Diagonal(C)*B (if trans = T,C and equed = C,B); rcond, the reciprocal condition number of A after equilbrating; ferr, the forward error bound for each solution vector in X; berr, the forward error bound for each solution vector in X; and work, the reciprocal pivot growth factor.

gesvx!(A, B)

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

LinearAlgebra.LAPACK.gelsd! — Function

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

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

LinearAlgebra.LAPACK.gelsy! — Function

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

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

LinearAlgebra.LAPACK.gglse! — Function

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

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

LinearAlgebra.LAPACK.geev! — Function

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

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

LinearAlgebra.LAPACK.gesdd! — Function

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

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

LinearAlgebra.LAPACK.gesvd! — Function

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

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

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

LinearAlgebra.LAPACK.ggsvd! — Function

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

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

LinearAlgebra.LAPACK.ggsvd3! — Function

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

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

LinearAlgebra.LAPACK.geevx! — Function

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

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

LinearAlgebra.LAPACK.ggev! — Function

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

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

LinearAlgebra.LAPACK.ggev3! — Function

ggev3!(jobvl, jobvr, A, B) -> (alpha, beta, vl, vr)

Finds the generalized eigendecomposition of A and B using a blocked algorithm. 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. This function requires LAPACK 3.6.0.

LinearAlgebra.LAPACK.gtsv! — Function

gtsv!(dl, d, du, B)

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

Overwrites B with the solution X and returns it.

LinearAlgebra.LAPACK.gttrf! — Function

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

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

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

LinearAlgebra.LAPACK.gttrs! — Function

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

Solves the equation A * X = B (trans = N), transpose(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) using the LU factorization computed by gttrf!. B is overwritten with the solution X.

LinearAlgebra.LAPACK.orglq! — Function

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

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

LinearAlgebra.LAPACK.orgqr! — Function

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

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

LinearAlgebra.LAPACK.orgql! — Function

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

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

LinearAlgebra.LAPACK.orgrq! — Function

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

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

LinearAlgebra.LAPACK.ormlq! — Function

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

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a LQ factorization of A computed using gelqf!. C is overwritten.

LinearAlgebra.LAPACK.ormqr! — Function

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

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QR factorization of A computed using geqrf!. C is overwritten.

LinearAlgebra.LAPACK.ormql! — Function

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

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QL factorization of A computed using geqlf!. C is overwritten.

LinearAlgebra.LAPACK.ormrq! — Function

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

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a RQ factorization of A computed using gerqf!. C is overwritten.

LinearAlgebra.LAPACK.gemqrt! — Function

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

Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side = R using Q from a QR factorization of A computed using geqrt!. C is overwritten.

LinearAlgebra.LAPACK.posv! — Function

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

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

LinearAlgebra.LAPACK.potrf! — Function

potrf!(uplo, A)

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

LinearAlgebra.LAPACK.potri! — Function

potri!(uplo, A)

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

A is overwritten by its inverse and returned.

LinearAlgebra.LAPACK.potrs! — Function

potrs!(uplo, A, B)

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

LinearAlgebra.LAPACK.pstrf! — Function

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

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

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

LinearAlgebra.LAPACK.ptsv! — Function

ptsv!(D, E, B)

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

LinearAlgebra.LAPACK.pttrf! — Function

pttrf!(D, E)

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

LinearAlgebra.LAPACK.pttrs! — Function

pttrs!(D, E, B)

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

LinearAlgebra.LAPACK.trtri! — Function

trtri!(uplo, diag, A)

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

LinearAlgebra.LAPACK.trtrs! — Function

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

Solves A * X = B (trans = N), transpose(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) for (upper if uplo = U, lower if uplo = L) triangular matrix A. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. B is overwritten with the solution X.

LinearAlgebra.LAPACK.trcon! — Function

trcon!(norm, uplo, diag, A)

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

LinearAlgebra.LAPACK.trevc! — Function

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

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

LinearAlgebra.LAPACK.trrfs! — Function

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

Estimates the error in the solution to A * X = B (trans = N), transpose(A) * X = B (trans = T), adjoint(A) * X = B (trans = C) for side = L, or the equivalent equations a right-handed side = R X * A after computing X using trtrs!. If uplo = U, A is upper triangular. If uplo = L, A is lower triangular. If diag = N, A has non-unit diagonal elements. If diag = U, all diagonal elements of A are one. Ferr and Berr are optional inputs. Ferr is the forward error and Berr is the backward error, each component-wise.

LinearAlgebra.LAPACK.stev! — Function

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

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

LinearAlgebra.LAPACK.stebz! — Function

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

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

LinearAlgebra.LAPACK.stegr! — Function

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

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

LinearAlgebra.LAPACK.stein! — Function

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

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

LinearAlgebra.LAPACK.syconv! — Function

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

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

LinearAlgebra.LAPACK.sysv! — Function

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

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

LinearAlgebra.LAPACK.sytrf! — Function

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

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

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

sytrf!(uplo, A, ipiv) -> (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, the pivot vector ipiv, and the error code info which is a non-negative integer. If info is positive the matrix is singular and the diagonal part of the factorization is exactly zero at position info.

LinearAlgebra.LAPACK.sytri! — Function

sytri!(uplo, A, ipiv)

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

LinearAlgebra.LAPACK.sytrs! — Function

sytrs!(uplo, A, ipiv, B)

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

LinearAlgebra.LAPACK.hesv! — Function

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

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

LinearAlgebra.LAPACK.hetrf! — Function

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

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

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

LinearAlgebra.LAPACK.hetri! — Function

hetri!(uplo, A, ipiv)

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

LinearAlgebra.LAPACK.hetrs! — Function

hetrs!(uplo, A, ipiv, B)

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

LinearAlgebra.LAPACK.syev! — Function

syev!(jobz, uplo, A)

LinearAlgebra.LAPACK.syevr! — Function

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

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

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

LinearAlgebra.LAPACK.syevd! — Function

syevd!(jobz, uplo, A)

Use the divide-and-conquer method, instead of the QR iteration used by syev! or multiple relatively robust representations used by syevr!. See James W. Demmel et al, SIAM J. Sci. Comput. 30, 3, 1508 (2008) for a comparison of the accuracy and performatce of different methods.

LinearAlgebra.LAPACK.sygvd! — Function

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

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

LinearAlgebra.LAPACK.bdsqr! — Function

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

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

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

LinearAlgebra.LAPACK.bdsdc! — Function

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

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

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

LinearAlgebra.LAPACK.gecon! — Function

gecon!(normtype, A, anorm)

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

LinearAlgebra.LAPACK.gehrd! — Function

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

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

LinearAlgebra.LAPACK.orghr! — Function

orghr!(ilo, ihi, A, tau)

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

LinearAlgebra.LAPACK.gees! — Function

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

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

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

LinearAlgebra.LAPACK.gges! — Function

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

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

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

LinearAlgebra.LAPACK.gges3! — Function

gges3!(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 using a blocked algorithm. This function requires LAPACK 3.6.0.

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

LinearAlgebra.LAPACK.trexc! — Function

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

Reorder the Schur factorization T of a matrix, such that the diagonal block of T with row index ifst is moved to row index ilst. If compq = V, the Schur vectors Q are reordered. If compq = N they are not modified. The 4-arg method calls the 5-arg method with compq = V.

LinearAlgebra.LAPACK.trsen! — Function

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

Reorder the Schur factorization of a matrix and optionally finds reciprocal condition numbers. If job = N, no condition numbers are found. If job = E, only the condition number for this cluster of eigenvalues is found. If job = V, only the condition number for the invariant subspace is found. If job = B then the condition numbers for the cluster and subspace are found. If compq = V the Schur vectors Q are updated. If compq = N the Schur vectors are not modified. select determines which eigenvalues are in the cluster. The 3-arg method calls the 5-arg method with job = N and compq = V.

Returns T, Q, reordered eigenvalues in w, the condition number of the cluster of eigenvalues s, and the condition number of the invariant subspace sep.

LinearAlgebra.LAPACK.tgsen! — Function

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

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

LinearAlgebra.LAPACK.trsyl! — Function

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

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

Returns X (overwriting C) and scale.

LinearAlgebra.LAPACK.hseqr! — Function

hseqr!(job, compz, ilo, ihi, H, Z) -> (H, Z, w)

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

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AH16_3Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577
AH16_4Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577
AH16_5Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577
AH16_6Mary Aprahamian and Nicholas J. Higham, "Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms", MIMS EPrint: 2016.4. https://doi.org/10.1137/16M1057577