# Linear Algebra¶

## Standard Functions¶

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

*(A, B)

Matrix multiplication.

\(A, B)

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

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

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

dot(x, y)
⋅(x, y)

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

vecdot(x, y)

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

cross(x, y)
×(x, y)

Compute the cross product of two 3-vectors.

factorize(A)

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

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

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

Example:

A = diagm(rand(5)) + diagm(rand(4),1); #A is really bidiagonal
factorize(A) #factorize will check to see that A is already factorized


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

full(F)

Reconstruct the matrix A from the factorization F=factorize(A).

Diagonal(A::AbstractMatrix)

Constructs a matrix from the diagonal of A.

Diagonal(V::AbstractVector)

Constructs a matrix with V as its diagonal.

Bidiagonal(dv, ev, isupper::Bool)

Constructs an upper (isupper=true) or lower (isupper=false) bidiagonal matrix using the given diagonal (dv) and off-diagonal (ev) vectors. The result is of type Bidiagonal and provides efficient specialized linear solvers, but may be converted into a regular matrix with full(). ev‘s length must be one less than the length of dv.

Example

dv = rand(5)
ev = rand(4)
Bu = Bidiagonal(dv, ev, true) #e is on the first superdiagonal
Bl = Bidiagonal(dv, ev, false) #e is on the first subdiagonal

Bidiagonal(dv, ev, uplo::Char)

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

Example

dv = rand(5)
ev = rand(4)
Bu = Bidiagonal(dv, ev, 'U') #e is on the first superdiagonal
Bl = Bidiagonal(dv, ev, 'L') #e is on the first subdiagonal

Bidiagonal(A, isupper::Bool)

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

Example

A = rand(5,5)
Bu = Bidiagonal(A, true) #contains the main diagonal and first superdiagonal of A
Bl = Bidiagonal(A, false) #contains the main diagonal and first subdiagonal of A

SymTridiagonal(dv, ev)

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

Tridiagonal(dl, d, du)

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

Symmetric(A, uplo=:U)

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

Example

A = randn(10,10)
Supper = Symmetric(A)
Slower = Symmetric(A,:L)
eigfact(Supper)


eigfact will use a method specialized for matrices known to be symmetric. Note that Supper will not be equal to Slower unless A is itself symmetric (e.g. if A == A.').

Hermitian(A, uplo=:U)

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

Example

A = randn(10,10)
Hupper = Hermitian(A)
Hlower = Hermitian(A,:L)
eigfact(Hupper)


eigfact will use a method specialized for matrices known to be Hermitian. Note that Hupper will not be equal to Hlower unless A is itself Hermitian (e.g. if A == A').

lu(A) → L, U, p

Compute the LU factorization of A, such that A[p,:] = L*U.

lufact(A[, pivot=Val{true}]) → F::LU

Compute the LU factorization of A.

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

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

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

The relationship between F and A is

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

F further supports the following functions:

Supported function LU LU{T,Tridiagonal{T}}
/()
$$) cond() det() logdet() logabsdet() size() lufact(A::SparseMatrixCSC) → F::UmfpackLU Compute the LU factorization of a sparse matrix A. For sparse A with real or complex element type, the return type of F is UmfpackLU{Tv, Ti}, with Tv = Float64 or Complex128 respectively and Ti is an integer type (Int32 or Int64). The individual components of the factorization F can be accessed by indexing: Component Description F[:L] L (lower triangular) part of LU F[:U] U (upper triangular) part of LU F[:p] right permutation Vector F[:q] left permutation Vector F[:Rs] Vector of scaling factors F[:(:)] (L,U,p,q,Rs) components The relation between F and A is F[:L]*F[:U] == (F[:Rs] .* A)[F[:p], F[:q]] F further supports the following functions: ** Implementation note ** lufact(A::SparseMatrixCSC) uses the UMFPACK library that is part of SuiteSparse. As this library only supports sparse matrices with Float64 or Complex128 elements, lufact converts A into a copy that is of type SparseMatrixCSC{Float64} or SparseMatrixCSC{Complex128} as appropriate. lufact!(A) → LU lufact! is the same as lufact(), but saves space by overwriting the input A, instead of creating a copy. An InexactError exception is thrown if the factorisation produces a number not representable by the element type of A, e.g. for integer types. chol(A) → U Compute the Cholesky factorization of a positive definite matrix A and return the UpperTriangular matrix U such that A = U'U. chol(x::Number) → y Compute the square root of a non-negative number x. cholfact(A, [uplo::Symbol, ]Val{false}) → Cholesky Compute the Cholesky factorization of a dense symmetric positive definite matrix A and return a Cholesky factorization. The matrix A can either be a Symmetric or Hermitian StridedMatrix or a perfectly symmetric or Hermitian StridedMatrix. In the latter case, the optional argument uplo may be :L for using the lower part or :U for the upper part of A. The default is to use :U. The triangular Cholesky factor can be obtained from the factorization F with: F[:L] and F[:U]. The following functions are available for Cholesky objects: size, \, inv, det. A PosDefException exception is thrown in case the matrix is not positive definite. cholfact(A, [uplo::Symbol, ]Val{true}; tol = 0.0) → CholeskyPivoted Compute the pivoted Cholesky factorization of a dense symmetric positive semi-definite matrix A and return a CholeskyPivoted factorization. The matrix A can either be a Symmetric or Hermitian StridedMatrix or a perfectly symmetric or Hermitian StridedMatrix. In the latter case, the optional argument uplo may be :L for using the lower part or :U for the upper part of A. The default is to use :U. The triangular Cholesky factor can be obtained from the factorization F with: F[:L] and F[:U]. The following functions are available for PivotedCholesky objects: size, \, inv, det, and rank. The argument tol determines the tolerance for determining the rank. For negative values, the tolerance is the machine precision. cholfact(A; shift = 0.0, perm = Int[]) → CHOLMOD.Factor Compute the Cholesky factorization of a sparse positive definite matrix A. A must be a SparseMatrixCSC, Symmetric{SparseMatrixCSC}, or Hermitian{SparseMatrixCSC}. Note that even if A doesn’t have the type tag, it must still be symmetric or Hermitian. A fill-reducing permutation is used. F = cholfact(A) is most frequently used to solve systems of equations with F\b, but also the methods diag, det, logdet are defined for F. You can also extract individual factors from F, using F[:L]. However, since pivoting is on by default, the factorization is internally represented as A == P'*L*L'*P with a permutation matrix P; using just L without accounting for P will give incorrect answers. To include the effects of permutation, it’s typically preferable to extact “combined” factors like PtL = F[:PtL] (the equivalent of P'*L) and LtP = F[:UP] (the equivalent of L'*P). Setting optional shift keyword argument computes the factorization of A+shift*I instead of A. If the perm argument is nonempty, it should be a permutation of 1:size(A,1) giving the ordering to use (instead of CHOLMOD’s default AMD ordering). Note This method uses the CHOLMOD library from SuiteSparse, which only supports doubles or complex doubles. Input matrices not of those element types will be converted to SparseMatrixCSC{Float64} or SparseMatrixCSC{Complex128} as appropriate. Many other functions from CHOLMOD are wrapped but not exported from the Base.SparseArrays.CHOLMOD module. cholfact!(F::Factor, A; shift = 0.0) → CHOLMOD.Factor Compute the Cholesky (\(LL'$$) factorization of A, reusing the symbolic factorization F. A must be a SparseMatrixCSC, Symmetric{SparseMatrixCSC}, or Hermitian{SparseMatrixCSC}. Note that even if A doesn’t have the type tag, it must still be symmetric or Hermitian.

Note

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

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

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

cholfact!(A, [uplo::Symbol, ]Val{true}; tol = 0.0) → CholeskyPivoted

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

lowrankupdate(C::Cholesky, v::StridedVector) → CC::Cholesky

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

lowrankdowndate(C::Cholesky, v::StridedVector) → CC::Cholesky

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

lowrankupdate!(C::Cholesky, v::StridedVector) → CC::Cholesky

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

lowrankdowndate!(C::Cholesky, v::StridedVector) → CC::Cholesky

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

ldltfact(::SymTridiagonal) → LDLt

Compute an LDLt factorization of a real symmetric tridiagonal matrix such that A = L*Diagonal(d)*L' where L is a unit lower triangular matrix and d is a vector. The main use of an LDLt factorization F = ldltfact(A) is to solve the linear system of equations Ax = b with F\b.

ldltfact(A; shift = 0.0, perm=Int[]) → CHOLMOD.Factor

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

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

Note

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

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

ldltfact!(F::Factor, A; shift = 0.0) → CHOLMOD.Factor

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

Note

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

ldltfact!(::SymTridiagonal) → LDLt

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

qr(v::AbstractVector)

Computes the polar decomposition of a vector.

Input:

• v::AbstractVector - vector to normalize

Outputs:

• w - A unit vector in the direction of v
• r - The norm of v

normalize, normalize!, LinAlg.qr!

LinAlg.qr!(v::AbstractVector)

Computes the polar decomposition of a vector. Instead of returning a new vector as qr(v::AbstractVector), this function mutates the input vector v in place.

Input:

• v::AbstractVector - vector to normalize

Outputs:

• w - A unit vector in the direction of v (This is a mutation of v).
• r - The norm of v

normalize, normalize!, qr

qr(A[, pivot=Val{false}][;thin=true]) → Q, R, [p]

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

qrfact(A[, pivot=Val{false}]) → F

Computes the QR factorization of A. The return type of F depends on the element type of A and whether pivoting is specified (with pivot==Val{true}).

Return type eltype(A) pivot Relationship between F and A
QR not BlasFloat either A==F[:Q]*F[:R]
QRCompactWY BlasFloat Val{false} A==F[:Q]*F[:R]
QRPivoted BlasFloat Val{true} A[:,F[:p]]==F[:Q]*F[:R]

BlasFloat refers to any of: Float32, Float64, Complex64 or Complex128.

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

Component Description QR QRCompactWY QRPivoted
F[:Q] Q (orthogonal/unitary) part of QR ✓ (QRPackedQ) ✓ (QRCompactWYQ) ✓ (QRPackedQ)
F[:R] R (upper right triangular) part of QR
F[:p] pivot Vector
F[:P] (pivot) permutation Matrix

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

Multiplication with respect to either thin or full Q is allowed, i.e. both F[:Q]*F[:R] and F[:Q]*A are supported. A Q matrix can be converted into a regular matrix with full() which has a named argument thin.

Note

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

The data contained in QR or QRPivoted can be used to construct the QRPackedQ type, which is a compact representation of the rotation matrix:

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

where $$\tau_i$$ is the scale factor and $$v_i$$ is the projection vector associated with the $$i^{th}$$ Householder elementary reflector.

The data contained in QRCompactWY can be used to construct the QRCompactWYQ type, which is a compact representation of the rotation matrix

$Q = I + Y T Y^T$

where Y is $$m \times r$$ lower trapezoidal and T is $$r \times r$$ upper triangular. The compact WY representation [Schreiber1989] is not to be confused with the older, WY representation [Bischof1987]. (The LAPACK documentation uses V in lieu of Y.)

 [Bischof1987] (1, 2) C Bischof and C Van Loan, “The WY representation for products of Householder matrices”, SIAM J Sci Stat Comput 8 (1987), s2-s13. doi:10.1137/0908009
 [Schreiber1989] R Schreiber and C Van Loan, “A storage-efficient WY representation for products of Householder transformations”, SIAM J Sci Stat Comput 10 (1989), 53-57. doi:10.1137/0910005
qrfact(A) → SPQR.Factorization

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

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

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

full(QRCompactWYQ[, thin=true]) → Matrix

Converts an orthogonal or unitary matrix stored as a QRCompactWYQ object, i.e. in the compact WY format [Bischof1987], to a dense matrix.

Optionally takes a thin Boolean argument, which if true omits the columns that span the rows of R in the QR factorization that are zero. The resulting matrix is the Q in a thin QR factorization (sometimes called the reduced QR factorization). If false, returns a Q that spans all rows of R in its corresponding QR factorization.

lqfact!(A) → LQ

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

lqfact(A) → LQ

Compute the LQ factorization of A. See also lq().

lq(A; [thin=true]) → L, Q

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

bkfact(A) → BunchKaufman

Compute the Bunch-Kaufman [Bunch1977] factorization of a real symmetric or complex Hermitian matrix A and return a BunchKaufman object. The following functions are available for BunchKaufman objects: size, \, inv, issymmetric, ishermitian.

 [Bunch1977] J R Bunch and L Kaufman, Some stable methods for calculating inertia and solving symmetric linear systems, Mathematics of Computation 31:137 (1977), 163-179. url.
bkfact!(A) → BunchKaufman

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

eig(A,[irange,][vl,][vu,][permute=true,][scale=true]) → D, V

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

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


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

eig(A, B) → D, V

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

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

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

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

julia> eig(A, B)
(Complex{Float64}[0.0+1.0im,0.0-1.0im],
Complex{Float64}[0.0-1.0im 0.0+1.0im; -1.0-0.0im -1.0+0.0im])

eigvals(A,[irange,][vl,][vu]) → values

Returns the eigenvalues of A. If A is Symmetric, Hermitian or SymTridiagonal, it is possible to calculate only a subset of the eigenvalues by specifying either a UnitRange irange covering indices of the sorted eigenvalues, or a pair vl and vu for the lower and upper boundaries of the eigenvalues.

For general non-symmetric 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 moreequal in norm. The default is true for both options.

eigvals!(A,[irange,][vl,][vu]) → values

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

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

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

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

julia> eigmax(A)
1.0

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

julia> eigmax(A)
ERROR: DomainError:
in #eigmax#30(::Bool, ::Bool, ::Function, ::Array{Complex{Int64},2}) at ./linalg/eigen.jl:186
in eigmax(::Array{Complex{Int64},2}) at ./linalg/eigen.jl:184
...

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

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

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

julia> eigmin(A)
-1.0

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

julia> eigmin(A)
ERROR: DomainError:
in #eigmin#31(::Bool, ::Bool, ::Function, ::Array{Complex{Int64},2}) at ./linalg/eigen.jl:226
in eigmin(::Array{Complex{Int64},2}) at ./linalg/eigen.jl:224
...

eigvecs(A, [eigvals,][permute=true,][scale=true]) → Matrix

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

For SymTridiagonal matrices, if the optional vector of eigenvalues eigvals is specified, returns the specific corresponding eigenvectors.

eigfact(A,[irange,][vl,][vu,][permute=true,][scale=true]) → Eigen

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

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

If A is Symmetric, Hermitian or SymTridiagonal, it is possible to calculate only a subset of the eigenvalues by specifying either a UnitRange irange covering indices of the sorted eigenvalues or a pair vl and vu for the lower and upper boundaries of the eigenvalues.

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.

eigfact(A, B) → GeneralizedEigen

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

eigfact!(A[, B])

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

hessfact(A)

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

hessfact!(A)

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

schurfact(A::StridedMatrix) → F::Schur

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

schurfact!(A::StridedMatrix) → F::Schur

Same as schurfact but uses the input argument as workspace.

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

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

See schurfact.

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::Schur, select::Union{Vector{Bool}, BitVector}) → F::Schur

Same as ordschur but overwrites the factorization F.

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

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

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

Same as ordschur but overwrites the input arguments.

schurfact(A::StridedMatrix, B::StridedMatrix) → F::GeneralizedSchur

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

schurfact!(A::StridedMatrix, B::StridedMatrix) → F::GeneralizedSchur

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

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

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

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

Same as ordschur but overwrites the factorization F.

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

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

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

Same as ordschur but overwrites the factorization the input arguments.

schur(A::StridedMatrix, B::StridedMatrix) → S::StridedMatrix, T::StridedMatrix, Q::StridedMatrix, Z::StridedMatrix, α::Vector, β::Vector

See schurfact.

svdfact(A[, thin=true]) → SVD

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

U, S, V and Vt can be obtained from the factorization F with F[:U], F[:S], F[:V] and F[:Vt], such that A = U*diagm(S)*Vt. The algorithm produces Vt and hence Vt is more efficient to extract than V.

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

svdfact!(A[, thin=true]) → SVD

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

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

svd(A[, thin=true]) → U, S, V

Computes the SVD of A, returning U, vector S, and V such that A == U*diagm(S)*V'.

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

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

svdvals(A)

Returns the singular values of A.

svdvals!(A)

Returns the singular values of A, saving space by overwriting the input.

svdfact(A, B) → GeneralizedSVD

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

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

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

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

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

svd(A, B) → U, V, Q, D1, D2, R0

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

svdvals(A, B)

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

LinAlg.Givens(i1, i2, c, s) → G

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

See also: givens()

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

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

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


the result of the multiplication

y = G*x


has the property that

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


See also: LinAlg.Givens

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

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

B = G*x


has the property that

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


See also: LinAlg.Givens

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

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

B = G*A


has the property that

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


See also: LinAlg.Givens

triu(M)

Upper triangle of a matrix.

triu(M, k)

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

triu!(M)

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

triu!(M, k)

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

tril(M)

Lower triangle of a matrix.

tril(M, k)

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

tril!(M)

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

tril!(M, k)

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

diagind(M[, k])

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

diag(M[, k])

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

diagm(v[, k])

Construct a diagonal matrix and place v on the kth diagonal.

scale!(A, b)
scale!(b, A)

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

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

Tridiagonal(dl, d, du)

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

rank(M)

Compute the rank of a matrix.

norm(A[, p])

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

For vectors, p can assume any numeric value (even though not all values produce a mathematically valid vector norm). In particular, norm(A, Inf) returns the largest value in abs(A), whereas norm(A, -Inf) returns the smallest.

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

vecnorm(A[, p])

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

For example, if A is a matrix and p=2, then this is equivalent to the Frobenius norm.

normalize!(v[, p=2])

Normalize the vector v in-place with respect to the p-norm.

Inputs:

• v::AbstractVector - vector to be normalized
• p::Real - The p-norm to normalize with respect to. Default: 2

Output:

• v - A unit vector being the input vector, rescaled to have norm 1. The input vector is modified in-place.

normalize, qr

normalize(v[, p=2])

Normalize the vector v with respect to the p-norm.

Inputs:

• v::AbstractVector - vector to be normalized
• p::Real - The p-norm to normalize with respect to. Default: 2

Output:

• v - A unit vector being a copy of the input vector, scaled to have norm 1

normalize!, qr

cond(M[, p])

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

condskeel(M[, x, p])
$\begin{split}\kappa_S(M, p) & = \left\Vert \left\vert M \right\vert \left\vert M^{-1} \right\vert \right\Vert_p \\ \kappa_S(M, x, p) & = \left\Vert \left\vert M \right\vert \left\vert M^{-1} \right\vert \left\vert x \right\vert \right\Vert_p\end{split}$

Skeel condition number $$\kappa_S$$ of the matrix M, optionally with respect to the vector x, as computed using the operator p-norm. p is Inf by default, if not provided. Valid values for p are 1, 2, or Inf.

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

trace(M)

Matrix trace.

det(M)

Matrix determinant.

logdet(M)

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

logabsdet(M)

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

inv(M)

Matrix inverse.

pinv(M[, tol])

Computes the Moore-Penrose pseudoinverse.

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

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

 [issue8859] Issue 8859, “Fix least squares”, https://github.com/JuliaLang/julia/pull/8859
 [B96] Åke Björck, “Numerical Methods for Least Squares Problems”, SIAM Press, Philadelphia, 1996, “Other Titles in Applied Mathematics”, Vol. 51. doi:10.1137/1.9781611971484
 [S84] Stewart, “Rank Degeneracy”, SIAM Journal on Scientific and Statistical Computing, 5(2), 1984, 403-413. doi:10.1137/0905030
 [KY88] Konstantinos Konstantinides and Kung Yao, “Statistical analysis of effective singular values in matrix rank determination”, IEEE Transactions on Acoustics, Speech and Signal Processing, 36(5), 1988, 757-763. doi:10.1109/29.1585
nullspace(M)

Basis for nullspace of M.

repmat(A, n, m)

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

repeat(A::AbstractArray; inner=ntuple(x->1, ndims(A)), outer=ntuple(x->1, ndims(A)))

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

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

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

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

kron(A, B)

Kronecker tensor product of two vectors or two matrices.

blkdiag(A...)

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

linreg(x, y)

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

Examples:

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


\, cov, std, mean

expm(A)

Compute the matrix exponential of A, defined by

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

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

 [H05] Nicholas J. Higham, “The squaring and scaling method for the matrix exponential revisited”, SIAM Journal on Matrix Analysis and Applications, 26(4), 2005, 1179-1193. doi:10.1137/090768539
logm(A::StridedMatrix)

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

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

 [AH12] Awad H. Al-Mohy and Nicholas J. Higham, “Improved inverse scaling and squaring algorithms for the matrix logarithm”, SIAM Journal on Scientific Computing, 34(4), 2012, C153-C169. doi:10.1137/110852553
 [AHR13] Awad H. Al-Mohy, Nicholas J. Higham and Samuel D. Relton, “Computing the Fréchet derivative of the matrix logarithm and estimating the condition number”, SIAM Journal on Scientific Computing, 35(4), 2013, C394-C410. doi:10.1137/120885991
sqrtm(A)

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

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

 [BH83] Åke Björck and Sven Hammarling, “A Schur method for the square root of a matrix”, Linear Algebra and its Applications, 52-53, 1983, 127-140. doi:10.1016/0024-3795(83)80010-X
lyap(A, C)

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

sylvester(A, B, C)

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

issymmetric(A) → Bool

Test whether a matrix is symmetric.

isposdef(A) → Bool

Test whether a matrix is positive definite.

isposdef!(A) → Bool

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

istril(A) → Bool

Test whether a matrix is lower triangular.

istriu(A) → Bool

Test whether a matrix is upper triangular.

isdiag(A) → Bool

Test whether a matrix is diagonal.

ishermitian(A) → Bool

Test whether a matrix is Hermitian.

transpose(A)

The transposition operator (.').

transpose!(dest, src)

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

ctranspose(A)

The conjugate transposition operator (').

ctranspose!(dest, src)

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

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

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

The following keyword arguments are supported:

• nev: Number of eigenvalues
• ncv: Number of Krylov vectors used in the computation; should satisfy nev+1 <= ncv <= n for real symmetric problems and nev+2 <= ncv <= n for other problems, where n is the size of the input matrix A. The default is ncv = max(20,2*nev+1). Note that these restrictions limit the input matrix A to be of dimension at least 2.
• which: type of eigenvalues to compute. See the note below.
which type of eigenvalues
:LM eigenvalues of largest magnitude (default)
:SM eigenvalues of smallest magnitude
:LR eigenvalues of largest real part
:SR eigenvalues of smallest real part
:LI eigenvalues of largest imaginary part (nonsymmetric or complex A only)
:SI eigenvalues of smallest imaginary part (nonsymmetric or complex A only)
:BE compute half of the eigenvalues from each end of the spectrum, biased in favor of the high end. (real symmetric A only)
• tol: parameter defining the relative tolerance for convergence of Ritz values (eigenvalue estimates). A Ritz value $$θ$$ is considered converged when its associated residual is less than or equal to the product of tol and $$max(ɛ^{2/3}, |θ|)$$, where ɛ = eps(real(eltype(A)))/2 is LAPACK’s machine epsilon. The residual associated with $$θ$$ and its corresponding Ritz vector $$v$$ is defined as the norm $$||Av - vθ||$$. The specified value of tol should be positive; otherwise, it is ignored and $$ɛ$$ is used instead. Default: $$ɛ$$.
• maxiter: Maximum number of iterations (default = 300)
• sigma: Specifies the level shift used in inverse iteration. If nothing (default), defaults to ordinary (forward) iterations. Otherwise, find eigenvalues close to sigma using shift and invert iterations.
• ritzvec: Returns the Ritz vectors v (eigenvectors) if true
• v0: starting vector from which to start the iterations

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

Note

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

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

Note

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

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

1. Parlett, “The Symmetric Eigenvalue Problem”, SIAM: Philadelphia, 2/e (1998), Ch. 13.2, “Accessing Accuracy in Lanczos Problems”, pp. 290-292 ff.
1. Lehoucq and D. C. Sorensen, “Deflation Techniques for an Implicitly Restarted Arnoldi Iteration”, SIAM Journal on Matrix Analysis and Applications (1996), 17(4), 789–821. doi:10.1137/S0895479895281484
eigs(A, B; nev=6, ncv=max(20, 2*nev+1), which="LM", tol=0.0, maxiter=300, sigma=nothing, ritzvec=true, v0=zeros((0, ))) → (d,[v,],nconv,niter,nmult,resid)

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

The following keyword arguments are supported:

• nev: Number of eigenvalues
• ncv: Number of Krylov vectors used in the computation; should satisfy nev+1 <= ncv <= n for real symmetric problems and nev+2 <= ncv <= n for other problems, where n is the size of the input matrices A and B. The default is ncv = max(20,2*nev+1). Note that these restrictions limit the input matrix A to be of dimension at least 2.
• which: type of eigenvalues to compute. See the note below.
which type of eigenvalues
:LM eigenvalues of largest magnitude (default)
:SM eigenvalues of smallest magnitude
:LR eigenvalues of largest real part
:SR eigenvalues of smallest real part
:LI eigenvalues of largest imaginary part (nonsymmetric or complex A only)
:SI eigenvalues of smallest imaginary part (nonsymmetric or complex A only)
:BE compute half of the eigenvalues from each end of the spectrum, biased in favor of the high end. (real symmetric A only)
• tol: relative tolerance used in the convergence criterion for eigenvalues, similar to tol in the eigs() method for the ordinary eigenvalue problem, but effectively for the eigenvalues of $$B^{-1} A$$ instead of $$A$$. See the documentation for the ordinary eigenvalue problem in eigs() and the accompanying note about tol.
• maxiter: Maximum number of iterations (default = 300)
• sigma: Specifies the level shift used in inverse iteration. If nothing (default), defaults to ordinary (forward) iterations. Otherwise, find eigenvalues close to sigma using shift and invert iterations.
• ritzvec: Returns the Ritz vectors v (eigenvectors) if true
• v0: starting vector from which to start the iterations

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

Example

X = sprand(10, 5, 0.2)
eigs(X, nsv = 2, tol = 1e-3)


Note

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

sigma iteration mode which refers to the problem
nothing ordinary (forward) $$Av = Bv\lambda$$
real or complex inverse with level shift sigma $$(A - \sigma B )^{-1}B = v\nu$$
svds(A; nsv=6, ritzvec=true, tol=0.0, maxiter=1000, ncv=2*nsv, u0=zeros((0, )), v0=zeros((0, ))) → (SVD([left_sv,] s, [right_sv,]), nconv, niter, nmult, resid)

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

Inputs

• A: Linear operator whose singular values are desired. A may be represented as a subtype of AbstractArray, e.g., a sparse matrix, or any other type supporting the four methods size(A), eltype(A), A * vector, and A' * vector.
• nsv: Number of singular values. Default: 6.
• ritzvec: If true, return the left and right singular vectors left_sv and right_sv. If false, omit the singular vectors. Default: true.
• tol: tolerance, see eigs().
• maxiter: Maximum number of iterations, see eigs(). Default: 1000.
• ncv: Maximum size of the Krylov subspace, see eigs() (there called nev). Default: 2*nsv.
• u0: Initial guess for the first left Krylov vector. It may have length m (the first dimension of A), or 0.
• v0: Initial guess for the first right Krylov vector. It may have length n (the second dimension of A), or 0.

Outputs

• svd: An SVD object containing the left singular vectors, the requested values, and the right singular vectors. If ritzvec = false, the left and right singular vectors will be empty.
• nconv: Number of converged singular values.
• niter: Number of iterations.
• nmult: Number of matrix–vector products used.
• resid: Final residual vector.

Example

X = sprand(10, 5, 0.2)
svds(X, nsv = 2)


Implementation note

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

peakflops(n; parallel=false)

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

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

## Low-level matrix operations¶

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

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

A_ldiv_B!([Y, ]A, B) → Y

Compute A \ B in-place and store the result in Y, returning the result. If only two arguments are passed, then A_ldiv_B!(A, B) overwrites B with the result.

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

A_ldiv_Bc(A, B)

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

A_ldiv_Bt(A, B)

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

A_mul_B!(Y, A, B) → Y

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

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

julia> Y
2×2 Array{Float64,2}:
3.0  3.0
7.0  7.0

A_mul_Bc(A, B)

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

A_mul_Bt(A, B)

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

A_rdiv_Bc(A, B)

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

A_rdiv_Bt(A, B)

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

Ac_ldiv_B(A, B)

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

Ac_ldiv_B!([Y, ]A, B) → Y

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

Ac_ldiv_Bc(A, B)

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

Ac_mul_B(A, B)

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

Ac_mul_Bc(A, B)

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

Ac_rdiv_B(A, B)

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

Ac_rdiv_Bc(A, B)

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

At_ldiv_B(A, B)

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

At_ldiv_B!([Y, ]A, B) → Y

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

At_ldiv_Bt(A, B)

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

At_mul_B(A, B)

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

At_mul_Bt(A, B)

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

At_rdiv_B(A, B)

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

At_rdiv_Bt(A, B)

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

## BLAS Functions¶

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

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

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.

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

Dot function for two complex vectors.

dotc(n, X, incx, U, incy)

Dot function for two complex vectors conjugating the first vector.

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

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

nrm2(n, X, incx)

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

asum(n, X, incx)

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

axpy!(a, X, Y)

Overwrite Y with a*X + Y. Returns Y.

scal!(n, a, X, incx)

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

scal(n, a, X, incx)

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

ger!(alpha, x, y, A)

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

syr!(uplo, alpha, x, A)

Rank-1 update of the symmetric matrix A with vector x as alpha*x*x.' + A. When uplo is ‘U’ the upper triangle of A is updated (‘L’ for lower triangle). Returns A.

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

Rank-k update of the symmetric matrix C as alpha*A*A.' + beta*C or alpha*A.'*A + beta*C according to whether trans is ‘N’ or ‘T’. When uplo is ‘U’ the upper triangle of C is updated (‘L’ for lower triangle). Returns C.

syrk(uplo, trans, alpha, A)

Returns either the upper triangle or the lower triangle, according to uplo (‘U’ or ‘L’), of alpha*A*A.' or alpha*A.'*A, according to trans (‘N’ or ‘T’).

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. When uplo is ‘U’ the upper triangle of A is updated (‘L’ for lower triangle). Returns A.

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

Methods for complex arrays only. Rank-k update of the Hermitian matrix C as alpha*A*A' + beta*C or alpha*A'*A + beta*C according to whether trans is ‘N’ or ‘T’. When uplo is ‘U’ the upper triangle of C is updated (‘L’ for lower triangle). Returns C.

herk(uplo, trans, alpha, A)

Methods for complex arrays only. Returns either the upper triangle or the lower triangle, according to uplo (‘U’ or ‘L’), of alpha*A*A' or alpha*A'*A, according to trans (‘N’ or ‘T’).

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 (‘N’ or ‘T’). The matrix A is a general band matrix of dimension m by size(A,2) with kl sub-diagonals and ku super-diagonals. Returns the updated y.

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

Returns alpha*A*x or alpha*A'*x according to trans (‘N’ or ‘T’). The matrix A is a general band matrix of dimension m by size(A,2) with kl sub-diagonals and ku super-diagonals.

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

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

Returns the updated y.

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

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

sbmv(uplo, k, A, x)

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

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

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

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

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

gemm(tA, tB, A, B)

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

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

Update the vector y as alpha*A*x + beta*y or alpha*A'x + beta*y according to tA (transpose A). Returns the updated y.

gemv(tA, alpha, A, x)

Returns alpha*A*x or alpha*A'x according to tA (transpose A).

gemv(tA, A, x)

Returns A*x or A'x according to tA (transpose A).

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

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

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

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

symm(side, ul, A, B)

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

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

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

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. Returns the updated y.

symv(ul, alpha, A, x)

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

symv(ul, A, x)

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

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

Update B as alpha*A*B or one of the other three variants determined by side (A on left or right) and tA (transpose A). Only the ul triangle of A is used. dA indicates if A is unit-triangular (the diagonal is assumed to be all ones). Returns the updated B.

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

Returns alpha*A*B or one of the other three variants determined by side (A on left or right) and tA (transpose A). Only the ul triangle of A is used. dA indicates if A is unit-triangular (the diagonal is assumed to be all ones).

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

Overwrite B with the solution to A*X = alpha*B or one of the other three variants determined by side (A on left or right of X) and tA (transpose A). Only the ul triangle of A is used. dA indicates if A is unit-triangular (the diagonal is assumed to be all ones). Returns the updated B.

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

Returns the solution to A*X = alpha*B or one of the other three variants determined by side (A on left or right of X) and tA (transpose A). Only the ul triangle of A is used. dA indicates if A is unit-triangular (the diagonal is assumed to be all ones).

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

Returns op(A)*b, where op is determined by tA (N for identity, T for transpose A, and C for conjugate transpose A). Only the ul triangle (U for upper, L for lower) of A is used. dA indicates if A is unit-triangular (the diagonal is assumed to be all ones if U, or non-unit if N). The multiplication occurs in-place on b.

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

Returns op(A)*b, where op is determined by tA (N for identity, T for transpose A, and C for conjugate transpose A). Only the ul triangle (U for upper, L for lower) of A is used. dA indicates if A is unit-triangular (the diagonal is assumed to be all ones if U, or non-unit if N).

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 (transpose A) and ul (triangle of A used). dA indicates if A is unit-triangular (the diagonal is assumed to be all ones). Returns the updated b.

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

Returns the solution to A*x = b or one of the other two variants determined by tA (transpose A) and ul (triangle of A is used.) dA indicates if A is unit-triangular (the diagonal is assumed to be all ones).

set_num_threads(n)

Set the number of threads the BLAS library should use.

I

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

## LAPACK Functions¶

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

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

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

gbtrf!(kl, ku, m, AB) → (AB, ipiv)

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

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

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

gebal!(job, A) → (ilo, ihi, scale)

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

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

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

gebrd!(A) → (A, d, e, tauq, taup)

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

gelqf!(A, tau)

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

Returns A and tau modified in-place.

gelqf!(A) → (A, tau)

Compute the LQ factorization of A, A = LQ.

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

geqlf!(A, tau)

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

Returns A and tau modified in-place.

geqlf!(A) → (A, tau)

Compute the QL factorization of A, A = QL.

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

geqrf!(A, tau)

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

Returns A and tau modified in-place.

geqrf!(A) → (A, tau)

Compute the QR factorization of A, A = QR.

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

geqp3!(A, jpvt, tau)

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

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

geqp3!(A, jpvt) → (A, jpvt, tau)

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

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

geqp3!(A) → (A, jpvt, tau)

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

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

gerqf!(A, tau)

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

Returns A and tau modified in-place.

gerqf!(A) → (A, tau)

Compute the RQ factorization of A, A = RQ.

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

geqrt!(A, T)

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

Returns A and T modified in-place.

geqrt!(A, nb) → (A, T)

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

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

geqrt3!(A, T)

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

Returns A and T modified in-place.

geqrt3!(A) → (A, T)

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

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

getrf!(A) → (A, ipiv, info)

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

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

tzrzf!(A) → (A, tau)

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

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

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

gels!(trans, A, B) → (F, B, ssr)

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

gesv!(A, B) → (B, A, ipiv)

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

getrs!(trans, A, ipiv, B)

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

getri!(A, ipiv)

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

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

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

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

gesvx!(A, B)

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

gelsd!(A, B, rcond) → (B, rnk)

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

gelsy!(A, B, rcond) → (B, rnk)

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

gglse!(A, c, B, d) → (X,res)

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

geev!(jobvl, jobvr, A) → (W, VL, VR)

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

gesdd!(job, A) → (U, S, VT)

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

gesvd!(jobu, jobvt, A) → (U, S, VT)

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

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

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

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

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

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

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

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

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

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

gtsv!(dl, d, du, B)

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

Overwrites B with the solution X and returns it.

gttrf!(dl, d, du) → (dl, d, du, du2, ipiv)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

posv!(uplo, A, B) → (A, B)

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

potrf!(uplo, A)

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

potri!(uplo, A)

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

A is overwritten by its inverse and returned.

potrs!(uplo, A, B)

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

pstrf!(uplo, A, tol) → (A, piv, rank, info)

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

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

ptsv!(D, E, B)

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

pttrf!(D, E)

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

pttrs!(D, E, B)

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

trtri!(uplo, diag, A)

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

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

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

trcon!(norm, uplo, diag, A)

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

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

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

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

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

stev!(job, dv, ev) → (dv, Zmat)

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

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

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

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

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

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.

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.

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.

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.

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.

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.

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.

hetrf!(uplo, A) → (A, ipiv, info)

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

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

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.

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.

syev!(jobz, uplo, A)

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

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.

sygvd!(jobz, range, uplo, A, vl, vu, il, iu, abstol) → (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.

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.

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.

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.

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.

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!.

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.

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.

trexc!(compq, ifst, ilst, T, Q) → (T, Q)

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

trsen!(compq, job, select, T, Q) → (T, Q, w)

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

Returns T, Q, and reordered eigenvalues in w.

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

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

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.