?ggsvd3

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?ggsvd3

Computes generalized SVD.

Syntax

call sggsvd3(jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, iwork, info)

call dggsvd3(jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, iwork, info)

call cggsvd3(jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, rwork, iwork, info)

call zggsvd3(jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, rwork, iwork, info)

Include Files

mkl.fi

Description

?ggsvd3 computes the generalized singular value decomposition (GSVD) of an m-by-n real or complex matrix A and p-by-n real or complex matrix B:

U^T*A*Q = D₁*( 0 R ), V^T*B*Q = D₂*( 0 R ) for real flavors

U^H*A*Q = D₁*( 0 R ), V^H*B*Q = D₂*( 0 R ) for complex flavors

where U, V and Q are orthogonal/unitary matrices.

Let k+l = the effective numerical rank of the matrix (A^TB^T)^T for real flavors or the matrix (A^H,B^H)^H for complex flavors, then R is a (k + l)-by-(k + l) nonsingular upper triangular matrix, D₁ and D₂ are m-by-(k + l) and p-by-(k + l) "diagonal" matrices and of the following structures, respectively:

If m-k-l≥ 0,

where

C = diag( alpha(k+1), ... , alpha(k+l) ),

S = diag( beta(k+1), ... , beta(k+l) ),

C² + S² = I.

If m - k - l < 0,

where

C = diag(alpha(k + 1), ... , alpha(m)),

S = diag(beta(k + 1), ... , beta(m)),

C² + S² = I.

The routine computes C, S, R, and optionally the orthogonal/unitary transformation matrices U, V and Q.

In particular, if B is an n-by-n nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of A*inv(B):

A*inv(B) = U*(D₁*inv(D₂))*V^T for real flavors

A*inv(B) = U*(D₁*inv(D₂))*V^H for complex flavors.

If (A^T,B^T)^T for real flavors or (A^H,B^H)^H for complex flavors has orthonormal columns, then the GSVD of A and B is also equal to the CS decomposition of A and B. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem:

A^T*AX = λ* B^T*BX for real flavors

A^H*AX = λ* B^H*BX for complex flavors

In some literature, the GSVD of A and B is presented in the form

U^T*A*X = ( 0 D₁ ), V^T*B*X = ( 0 D₂ ) for real (A, B)

U^H*A*X = ( 0 D₁ ), V^H*B*X = ( 0 D₂ ) for complex (A, B)

where U and V are orthogonal and X is nonsingular, D₁ and D₂ are "diagonal''. The former GSVD form can be converted to the latter form by taking the nonsingular matrix X as

Input Parameters

jobu

CHARACTER*1. = 'U': Orthogonal/unitary matrix U is computed;

= 'N': U is not computed.

jobv

CHARACTER*1. = 'V': Orthogonal/unitary matrix V is computed;

= 'N': V is not computed.

jobq

CHARACTER*1. = 'Q': Orthogonal/unitary matrix Q is computed;

= 'N': Q is not computed.

m

INTEGER. The number of rows of the matrix A.

m≥ 0.

n

INTEGER. The number of columns of the matrices A and B.

n≥ 0.

p

INTEGER. The number of rows of the matrix B.

p≥ 0.

a

REAL for sggsvd3

DOUBLE PRECISION for dggsvd3

COMPLEX for cggsvd3

DOUBLE COMPLEX for zggsvd3

Array, size (lda, n).

On entry, the m-by-n matrix A.

lda

INTEGER. The leading dimension of the array a.

lda≥ max(1,m).

b

REAL for sggsvd3

DOUBLE PRECISION for dggsvd3

COMPLEX for cggsvd3

DOUBLE COMPLEX for zggsvd3

Array, size (ldb, n).

On entry, the p-by-n matrix B.

ldb

INTEGER. The leading dimension of the array b.

ldb≥ max(1,p).

ldu

INTEGER. The leading dimension of the array u.

ldu≥ max(1,m) if jobu = 'U'; ldu≥ 1 otherwise.

ldv

INTEGER. The leading dimension of the array v.

ldv≥ max(1,p) if jobv = 'V'; ldv≥ 1 otherwise.

ldq

INTEGER. The leading dimension of the array q.

ldq≥ max(1,n) if jobq = 'Q'; ldq≥ 1 otherwise.

work

REAL for sggsvd3

DOUBLE PRECISION for dggsvd3

COMPLEX for cggsvd3

DOUBLE COMPLEX for zggsvd3

Array, size (max(1,lwork)).

lwork

INTEGER. The dimension of the array work.

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla.

rwork

for sggsvd3

for dggsvd3

REAL for cggsvd3

DOUBLE PRECISION for zggsvd3

Array, size (2*n).

iwork

INTEGER. Array, size (n).

Output Parameters

k, l	INTEGER. On exit, k and l specify the dimension of the subblocks described in the Description section. k + l = effective numerical rank of (`A`^T,`B`^T)^T for real flavors or (`A`^H,`B`^H)^H for complex flavors.
a	On exit, a contains the triangular matrix `R`, or part of `R`. If m-k-l≥ 0, `R` is stored in `a(1: k + l,n - k - l + 1:n)`. If m - k - l < 0, is stored in `a(1:m, n - k - l + 1:n)`, and `R33` is stored in b`(m - k + 1:l,n + m - k - l + 1:n)` on exit.
b	On exit, b contains part of the triangular matrix `R` if m - k - l < 0. See Description for details.
alpha	REAL for sggsvd3 DOUBLE PRECISION for dggsvd3 REAL for cggsvd3 DOUBLE PRECISION for zggsvd3 Array, size (n)
beta	REAL for sggsvd3 DOUBLE PRECISION for dggsvd3 REAL for cggsvd3 DOUBLE PRECISION for zggsvd3 Array, size (n) On exit, alpha and beta contain the generalized singular value pairs of a and b; alpha`(1: k)` = 1, beta`(1: k)` = 0, and if m - k - l≥ 0, alpha`(k + 1:k + l)` = `C`, beta`(k + 1:k + l)` = `S`, or if m - k - l < 0, alpha`(k + 1:m)` = `C`, alpha`(m + 1:k + l)` = 0 beta`(k + 1: m)` =`S`, beta`(m + 1: k + l)` = 1 and alpha`(k + l + 1: n)` = 0 beta`(k + l + 1: n)` = 0
u	REAL for sggsvd3 DOUBLE PRECISION for dggsvd3 COMPLEX for cggsvd3 DOUBLE COMPLEX for zggsvd3 Array, size (ldu, m). If jobu = 'U', u contains the m-by-m orthogonal/unitary matrix `U`. If jobu = 'N', u is not referenced.
v	REAL for sggsvd3 DOUBLE PRECISION for dggsvd3 COMPLEX for cggsvd3 DOUBLE COMPLEX for zggsvd3 Array, size (ldv, p). If jobv = 'V', v contains the p-by-p orthogonal/unitary matrix `V`. If jobv = 'N', v is not referenced.
q	REAL for sggsvd3 DOUBLE PRECISION for dggsvd3 COMPLEX for cggsvd3 DOUBLE COMPLEX for zggsvd3 Array, size (ldq, n). If jobq = 'Q', q contains the n-by-n orthogonal/unitary matrix `Q`. If jobq = 'N', q is not referenced.
work	On exit, if info = 0, work(1) returns the optimal lwork.
iwork	On exit, iwork stores the sorting information. More precisely, the following loop uses iwork to sort alpha: for I = k+1, min(m,k + l) swap alpha(I) and alpha(iwork(I)) endfor such that alpha(1) ≥alpha(2) ≥ ... ≥alpha(n).
info	INTEGER. = 0: successful exit. < 0: if info = -`i`, the `i`-th argument had an illegal value. > 0: if info = 1, the Jacobi-type procedure failed to converge. For further details, see subroutine ?tgsja.

Application Notes

?ggsvd3 replaces the deprecated subroutine ?ggsvd.

Parent topic: Generalized Singular Value Decomposition: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?ggsvd3