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 nontriangular square matrices, an LU factorization is used.
For rectangular A
the result is the minimumnorm 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 3vectors.
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 shortcircuit 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 

Positivedefinite  Cholesky (see cholfact() ) 
Dense Symmetric/Hermitian  BunchKaufman (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 nonsquare  QR (see qrfact() ) 
If factorize
is called on a Hermitian positivedefinite 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 offdiagonal (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 offdiagonal (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 subdiagonal (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/superdiagonal, 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 nonnegative 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 semidefinite 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 fillreducing 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 fillreducing 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 normalizeOutputs:
w
 A unit vector in the direction of v
r
 The norm of v
See also:
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 normalizeOutputs:
w
 A unit vector in the direction of v
(This is a mutation of v
).r
 The norm of v
See also:
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:
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
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), s2s13. doi:10.1137/0908009 
[Schreiber1989]  R Schreiber and C Van Loan, “A storageefficient WY representation for products of Householder transformations”, SIAM J Sci Stat Comput 10 (1989), 5357. doi:10.1137/0910005 
qrfact(A) → SPQR.Factorization
Compute the QR factorization of a sparse matrix A
. A fillreducing 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 BunchKaufman [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), 163179. 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,0.01.0im], Complex{Float64}[0.01.0im 0.0+1.0im; 1.00.0im 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 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 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 01im 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 01im 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 k
th 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 k
th 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 k
th 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 MbyN matrix A
and PbyN matrix B
,
F[:U]
is a MbyM orthogonal matrix,F[:V]
is a PbyP orthogonal matrix,F[:Q]
is a NbyN orthogonal matrix,F[:R0]
is a (K+L)byN matrix whose rightmost (K+L)by(K+L) block is nonsingular upper block triangular,F[:D1]
is a Mby(K+L) diagonal matrix with 1s in the first K entries,F[:D2]
is a Pby(K+L) matrix whose top right LbyL 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 k
th 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 k
th 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 k
th 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 k
th superdiagonal, overwriting M
in the process.
diagind(M[, k])
A Range
giving the indices of the k
th diagonal of the matrix M
.
diag(M[, k])
The k
th 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 k
th diagonal.
scale!(A, b)
scale!(b, A)
Scale an array A
by a scalar b
overwriting A
inplace.
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 inplace 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
inplace with respect to the p
norm.
Inputs:
v::AbstractVector
 vector to be normalizedp::Real
 The p
norm to normalize with respect to. Default: 2Output:
v
 A unit vector being the input vector, rescaled to have norm 1. The input vector is modified inplace.See also:
normalize
, qr
normalize(v[, p=2])
Normalize the vector v
with respect to the p
norm.
Inputs:
v::AbstractVector
 vector to be normalizedp::Real
 The p
norm to normalize with respect to. Default: 2Output:
v
 A unit vector being a copy of the input vector, scaled to have norm 1See also:
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])
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 MoorePenrose 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 illconditioned matrices in a leastsquares sense, tol = sqrt(eps(real(float(one(eltype(M))))))
is recommended.
For more information, see [issue8859], [B96], [S84], [KY88].
[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] 

[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, 757763. 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 ith element of inner
specifies the number of times that the individual entries of the ith dimension of A
should be repeated. The ith element of outer
specifies the number of times that a slice along the ith dimension of A
should be repeated. If inner
or outer
are omitted, no repetition is performed.
julia> repeat(1:2, inner=2) 4element Array{Int64,1}: 1 1 2 2 julia> repeat(1:2, outer=2) 4element 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 blockdiagonally. 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
See also:
\
, cov
, std
, mean
expm(A)
Compute the matrix exponential of A
, defined by
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, 11791193. 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. AlMohy and Nicholas J. Higham, “Improved inverse scaling and squaring algorithms for the matrix logarithm”, SIAM Journal on Scientific Computing, 34(4), 2012, C153C169. doi:10.1137/110852553 
[AHR13]  Awad H. AlMohy, 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, C394C410. 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 BjörckHammarling 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, 5253, 1983, 127140. doi:10.1016/00243795(83)80010X 
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 inplace 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 inplace 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 eigenvaluesncv
: 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 iterationseigs
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:
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 eigenvaluesncv
: 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 iterationseigs
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 = 1e3)
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.
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 inplace versions of matrix operations that allow you to supply a preallocated output vector or matrix. This is useful when optimizing critical code in order to avoid the overhead of repeated allocations. These inplace 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
inplace 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 inplace via, e.g., lufact!()
), and performancecritical situations requiring A_ldiv_B!
usually also require finegrained 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 matrixmatrix or matrixvector 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 inplace 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 inplace 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 \(Aᵀ / Bᵀ\).
In Julia (as in much of scientific computation), dense linearalgebra operations are based on the LAPACK library, which in turn is built on top of basic linearalgebra buildingblocks known as the BLAS. There are highly optimized implementations of BLAS available for every computer architecture, and sometimes in highperformance 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)
2norm 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)
Rank1 update of the matrix A
with vectors x
and y
as alpha*x*y' + A
.
syr!(uplo, alpha, x, A)
Rank1 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)
Rankk 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. Rank1 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. Rankk 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
subdiagonals and ku
superdiagonals. 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
subdiagonals and ku
superdiagonals.
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
superdiagonals stored in the argument A
. The storage layout for A
is described the reference BLAS module, level2 BLAS at <http://www.netlib.org/lapack/explorehtml/>.
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
superdiagonals 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
superdiagonals 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 unittriangular (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 unittriangular (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 unittriangular (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 unittriangular (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 unittriangular (the diagonal is assumed to be all ones if U
, or nonunit if N
). The multiplication occurs inplace 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 unittriangular (the diagonal is assumed to be all ones if U
, or nonunit 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 unittriangular (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 unittriangular (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.
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 userfacing, 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 inplace 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
inplace.
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
inplace 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
inplace. side
can be L
(left eigenvectors are transformed) or R
(right eigenvectors are transformed).
gebrd!(A) → (A, d, e, tauq, taup)
Reduce A
inplace to bidiagonal form A = QBP'
. Returns A
, containing the bidiagonal matrix B
; d
, containing the diagonal elements of B
; e
, containing the offdiagonal 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 inplace.
gelqf!(A) → (A, tau)
Compute the LQ
factorization of A
, A = LQ
.
Returns A
, modified inplace, 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 inplace.
geqlf!(A) → (A, tau)
Compute the QL
factorization of A
, A = QL
.
Returns A
, modified inplace, 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 inplace.
geqrf!(A) → (A, tau)
Compute the QR
factorization of A
, A = QR
.
Returns A
, modified inplace, 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 inplace.
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 inplace, 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 inplace, 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 inplace.
gerqf!(A) → (A, tau)
Compute the RQ
factorization of A
, A = RQ
.
Returns A
, modified inplace, 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 inplace.
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 inplace, 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 inplace.
geqrt3!(A) → (A, T)
Recursively computes the blocked QR
factorization of A
, A = QR
.
Returns A
, modified inplace, 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 inplace, 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 inplace. 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 leftsided (side = L, Q*C
) or rightsided (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 inplace 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 noequilibration, notranspose 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 dividingandconquering 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 dividingandconquering 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 sumofsquares.
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
inplace 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 rightsided 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 rightsided 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 rightsided 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 rightsided 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 rightsided 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 positivedefinite matrix A
. A
is overwritten and returned with an info code.
potri!(uplo, A)
Computes the inverse of positivedefinite 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 positivedefinite matrix A
with a userset 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 rankdeficient.
ptsv!(D, E, B)
Solves A * X = B
for positivedefinite tridiagonal A
. D
is the diagonal of A
and E
is the offdiagonal. B
is overwritten with the solution X
and returned.
pttrf!(D, E)
Computes the LDLt factorization of a positivedefinite tridiagonal matrix with D
as diagonal and E
as offdiagonal. D
and E
are overwritten and returned.
pttrs!(D, E, B)
Solves A * X = B
for positivedefinite tridiagonal A
with diagonal D
and offdiagonal 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 nonunit 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 nonunit 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 nonunit 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 righthanded 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 nonunit 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 componentwise.
stev!(job, dv, ev) → (dv, Zmat)
Computes the eigensystem for a symmetric tridiagonal matrix with dv
as diagonal and ev
as offdiagonal. 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 offdiagonal. If range = A
, all the eigenvalues are found. If range = V
, the eigenvalues in the halfopen 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 offdiagonal. If range = A
, all the eigenvalues are found. If range = V
, the eigenvalues in the halfopen 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 offdiagonal. 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 BunchKaufman factorization. ipiv
contains pivoting information about the factorization.
sytrf!(uplo, A) → (A, ipiv, info)
Computes the BunchKaufman 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 nonnegative 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 BunchKaufman factorization. ipiv
contains pivoting information about the factorization.
hetrf!(uplo, A) → (A, ipiv, info)
Computes the BunchKaufman 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 nonnegative 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 halfopen 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 positivedefinite 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 offdiagonal. 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 offdiagonal 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 quasiupper 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
.
© 2009–2016 Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and other contributors
Licensed under the MIT License.
http://docs.julialang.org/en/release0.5/stdlib/linalg/