?tgsja

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 3/22/2024

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?tgsja

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

Syntax

call stgsja(jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb, tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq, work, ncycle, info)

call dtgsja(jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb, tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq, work, ncycle, info)

call ctgsja(jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb, tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq, work, ncycle, info)

call ztgsja(jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb, tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq, work, ncycle, info)

call tgsja(a, b, tola, tolb, k, l [,u] [,v] [,q] [,jobu] [,jobv] [,jobq] [,alpha] [,beta] [,ncycle] [,info])

Include Files

mkl.fi, lapack.f90

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+1),...,alpha(k+l))

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

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+1),...,alpha(m)),

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

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

jobu

CHARACTER*1. 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

CHARACTER*1. 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

CHARACTER*1. 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

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

p

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

n

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

k, l

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

a, b, u, v, q, work

REAL for stgsja

DOUBLE PRECISION for dtgsja

COMPLEX for ctgsja

DOUBLE COMPLEX for ztgsja.

Arrays:

a(lda,*) contains the m-by-n matrix A.

The second dimension of a must be at least max(1, n).

b(ldb,*) contains the p-by-n matrix B.

The second dimension of b must be at least max(1, n).

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

The second dimension of u must be at least max(1, m).

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

The second dimension of v must be at least max(1, p).

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

The second dimension of q must be at least max(1, n).

work(*) is a workspace array.

The dimension of work must be at least max(1, 2n).

lda

INTEGER. The leading dimension of a; at least max(1, m).

ldb

INTEGER. The leading dimension of b; at least 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.

tola, tolb

REAL for single-precision flavors

DOUBLE PRECISION for double-precision flavors.

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

REAL for single-precision flavors

DOUBLE PRECISION for double-precision flavors.

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

INTEGER. The number of cycles required for convergence.

info

INTEGER.

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.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine tgsja interface are the following:

a

Holds the matrix A of size (m,n).

b

Holds the matrix B of size (p,n).

u

Holds the matrix U of size (m,m).

v

Holds the matrix V of size (p,p).

q

Holds the matrix Q of size (n,n).

alpha

Holds the vector of length n.

beta

Holds the vector of length n.

jobu

If omitted, this argument is restored based on the presence of argument u as follows:

jobu = 'U', if u is present,

jobu = 'N', if u is omitted.

If present, jobu must be equal to 'I' or 'U' and the argument u must also be present.

Note that there will be an error condition if jobu is present and u omitted.

jobv

If omitted, this argument is restored based on the presence of argument v as follows:

jobv = 'V', if v is present,

jobv = 'N', if v is omitted.

If present, jobv must be equal to 'I' or 'V' and the argument v must also be present.

Note that there will be an error condition if jobv is present and v omitted.

jobq

If omitted, this argument is restored based on the presence of argument q as follows:

jobq = 'Q', if q is present,

jobq = 'N', if q is omitted.

If present, jobq must be equal to 'I' or 'Q' and the argument q must also be present.

Note that there will be an error condition if jobq is present and q omitted.

Parent topic: Generalized Singular Value Decomposition: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?tgsja

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes