?ggsvd3

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?ggsvd3

Computes generalized SVD.

Syntax

lapack_int LAPACKE_sggsvd3 (int matrix_layout, char jobu, char jobv, char jobq, lapack_int m, lapack_int n, lapack_int p, lapack_int * k, lapack_int * l, float * a, lapack_int lda, float * b, lapack_int ldb, float * alpha, float * beta, float * u, lapack_int ldu, float * v, lapack_int ldv, float * q, lapack_int ldq, lapack_int * iwork);

lapack_int LAPACKE_dggsvd3 (int matrix_layout, char jobu, char jobv, char jobq, lapack_int m, lapack_int n, lapack_int p, lapack_int * k, lapack_int * l, double * a, lapack_int lda, double * b, lapack_int ldb, double * alpha, double * beta, double * u, lapack_int ldu, double * v, lapack_int ldv, double * q, lapack_int ldq, lapack_int * iwork);

lapack_int LAPACKE_cggsvd3 (int matrix_layout, char jobu, char jobv, char jobq, lapack_int m, lapack_int n, lapack_int p, lapack_int * k, lapack_int * l, lapack_complex_float * a, lapack_int lda, lapack_complex_float * b, lapack_int ldb, float * alpha, float * beta, lapack_complex_float * u, lapack_int ldu, lapack_complex_float * v, lapack_int ldv, lapack_complex_float * q, lapack_int ldq, lapack_int * iwork);

lapack_int LAPACKE_zggsvd3 (int matrix_layout, char jobu, char jobv, char jobq, lapack_int m, lapack_int n, lapack_int p, lapack_int * k, lapack_int * l, lapack_complex_double * a, lapack_int lda, lapack_complex_double * b, lapack_int ldb, double * alpha, double * beta, lapack_complex_double * u, lapack_int ldu, lapack_complex_double * v, lapack_int ldv, lapack_complex_double * q, lapack_int ldq, lapack_int * iwork);

Include Files

mkl.h

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

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

jobu

= 'U': Orthogonal/unitary matrix U is computed;

= 'N': U is not computed.

jobv

= 'V': Orthogonal/unitary matrix V is computed;

= 'N': V is not computed.

jobq

= 'Q': Orthogonal/unitary matrix Q is computed;

= 'N': Q is not computed.

m

The number of rows of the matrix A.

m≥ 0.

n

The number of columns of the matrices A and B.

n≥ 0.

p

The number of rows of the matrix B.

p≥ 0.

a

Array, size (lda*n).

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

lda

The leading dimension of the array a.

lda≥ max(1,m).

b

Array, size (ldb*n).

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

ldb

The leading dimension of the array b.

ldb≥ max(1,p).

ldu

The leading dimension of the array u.

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

ldv

The leading dimension of the array v.

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

ldq

The leading dimension of the array q.

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

iwork

Array, size (n).

Output Parameters

k, l	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 the elements of array a corresponding to `A`_{1: k + l,n - k - l + 1:n}. If m - k - l < 0, is stored in the elements of array a corresponding to `A`_{(1:m, n - k - l + 1:n}, and `R33` is stored in bthe elements of array a corresponding to `A`_{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	Array, size (n)
beta	Array, size (n) On exit, alpha and beta contain the generalized singular value pairs of a and b; alpha`[0: k - 1]` = 1, beta`[0: k - 1]` = 0, and if m - k - l≥ 0, alpha`[k:k + l - 1]` = `C`, beta`[k:k + l - 1]` = `S`, or if m - k - l < 0, alpha`[k:m - 1]` = `C`, alpha`[m: k + l - 1]` = 0 beta`[k: m - 1]` =`S`, beta`[m: k + l - 1]` = 1 and alpha`[k + l: n - 1]` = 0 beta`[k + l : n - 1]` = 0
u	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	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	Array, size (ldq*n). If jobq = 'Q', q contains the n-by-n orthogonal/unitary matrix `Q`. If jobq = 'N', q is not referenced.
iwork	On exit, iwork stores the sorting information. More precisely, the following loop uses iwork to sort alpha: for (i = k; i<min(m,k + l); i++) { swap (alpha[i], alpha[iwork[i] - 1]); } such that alpha[0] ≥alpha[1] ≥ ... ≥alpha[n - 1].

Return Values

This function returns a value info.

= 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 C

?ggsvd3