?tgsja

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?tgsja

Computes the generalized SVD of two upper triangular or trapezoidal matrices.

Syntax

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

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

lapack_int LAPACKE_ctgsja( int matrix_layout, char jobu, char jobv, char jobq, lapack_int m, lapack_int p, lapack_int n, lapack_int k, lapack_int l, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb, float tola, float tolb, 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* ncycle );

lapack_int LAPACKE_ztgsja( int matrix_layout, char jobu, char jobv, char jobq, lapack_int m, lapack_int p, lapack_int n, lapack_int k, lapack_int l, lapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb, double tola, double tolb, 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* ncycle );

Include Files

mkl.h

Description

The routine computes the generalized singular value decomposition (GSVD) of two real/complex upper triangular (or trapezoidal) matrices A and B. On entry, it is assumed that matrices A and B have the following forms, which may be obtained by the preprocessing subroutine ggsvp from a general m-by-n matrix A and p-by-n matrix B:

where the k-by-k matrix A₁₂ and l-by-l matrix B₁₃ are nonsingular upper triangular; A₂₃ is l-by-l upper triangular if m-k-l≥0, otherwise A₂₃ is (m-k)-by-l upper trapezoidal.

On exit,

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

where U, V and Q are orthogonal/unitary matrices, R is a nonsingular upper triangular matrix, and D₁ and D₂ are "diagonal" matrices, which are of the following structures:

If m-k-l≥0,

where

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

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

C² + S² = I

R is stored in a(1:k+l, n-k-l+1:n ) on exit.

If m-k-l < 0,

where

C = diag(alpha[k],...,alpha[m-1]),

S = diag(beta[k],...,beta[m-1]),

C² + S² = I

On exit, Equation 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 ).

The computation of the orthogonal/unitary transformation matrices U, V or Q is optional. These matrices may either be formed explicitly, or they may be postmultiplied into input matrices U₁, V₁, or Q₁.

Input Parameters

matrix_layout

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

jobu

Must be 'U', 'I', or 'N'.

If jobu = 'U', u must contain an orthogonal/unitary matrix U₁ on entry.

If jobu = 'I', u is initialized to the unit matrix.

If jobu = 'N', u is not computed.

jobv

Must be 'V', 'I', or 'N'.

If jobv = 'V', v must contain an orthogonal/unitary matrix V₁ on entry.

If jobv = 'I', v is initialized to the unit matrix.

If jobv = 'N', v is not computed.

jobq

Must be 'Q', 'I', or 'N'.

If jobq = 'Q', q must contain an orthogonal/unitary matrix Q₁ on entry.

If jobq = 'I', q is initialized to the unit matrix.

If jobq = 'N', q is not computed.

m

The number of rows of the matrix A (m≥ 0).

p

The number of rows of the matrix B (p≥ 0).

n

The number of columns of the matrices A and B (n≥ 0).

k, l

Specify the subblocks in the input matrices A and B, whose GSVD is computed.

a, b, u, v, q

Arrays:

a(size at least max(1, lda*n) for column major layout and max(1, lda*m) for row major layout) contains the m-by-n matrix A.

b(size at least max(1, ldb*n) for column major layout and max(1, ldb*p) for row major layout) contains the p-by-n matrix B.

If jobu = 'U', u (size max(1, ldu*m)) must contain a matrix U₁ (usually the orthogonal/unitary matrix returned by ?ggsvp).

If jobv = 'V', v (size at least max(1, ldv*p)) must contain a matrix V₁ (usually the orthogonal/unitary matrix returned by ?ggsvp).

If jobq = 'Q', q (size at least max(1, ldq*n)) must contain a matrix Q₁ (usually the orthogonal/unitary matrix returned by ?ggsvp).

lda

The leading dimension of a; at least max(1, m)for column major layout and max(1, n) for row major layout.

ldb

The leading dimension of b; at least max(1, p) for column major layout and max(1, n) for row major layout.

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.

tola, tolb

tola and tolb are the convergence criteria for the Jacobi-Kogbetliantz iteration procedure. Generally, they are the same as used in ?ggsvp:

tola = max(m, n)*|A|*MACHEPS,

tolb = max(p, n)*|B|*MACHEPS.

Output Parameters

a

On exit, a(n-k+1:n, 1:min(k+l, m)) contains the triangular matrix R or part of R.

b

On exit, if necessary, b(m-k+1: l, n+m-k-l+1: n)) contains a part of R.

alpha, beta

Arrays, size at least max(1, n). 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) = diag(C),

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

or if m-k-l < 0,

alpha(k+1:m)= diag(C), alpha(m+1:k+l)=0

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

beta(m+1:k+l) = 1.

Furthermore, if k+l < n,

alpha(k+l+1:n)= 0 and

beta(k+l+1:n) = 0.

u

If jobu = 'I', u contains the orthogonal/unitary matrix U.

If jobu = 'U', u contains the product U₁*U.

If jobu = 'N', u is not referenced.

v

If jobv = 'I', v contains the orthogonal/unitary matrix U.

If jobv = 'V', v contains the product V₁*V.

If jobv = 'N', v is not referenced.

q

If jobq = 'I', q contains the orthogonal/unitary matrix U.

If jobq = 'Q', q contains the product Q₁*Q.

If jobq = 'N', q is not referenced.

ncycle

The number of cycles required for convergence.

Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

If info = 1, the procedure does not converge after MAXIT cycles.

Parent topic: Generalized Singular Value Decomposition: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?tgsja