NAG FL Interface
f08yef (dtgsja)

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1 Purpose

f08yef computes the generalized singular value decomposition (GSVD) of two real upper trapezoidal matrices A and B, where A is an m×n matrix and B is a p×n matrix.
A and B are assumed to be in the form returned by f08vgf.

2 Specification

Fortran Interface
Subroutine f08yef ( 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)
Integer, Intent (In) :: m, p, n, k, l, lda, ldb, ldu, ldv, ldq
Integer, Intent (Out) :: ncycle, info
Real (Kind=nag_wp), Intent (In) :: tola, tolb
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), u(ldu,*), v(ldv,*), q(ldq,*)
Real (Kind=nag_wp), Intent (Out) :: alpha(n), beta(n), work(2*n)
Character (1), Intent (In) :: jobu, jobv, jobq
C Header Interface
#include <nag.h>
void  f08yef_ (const char *jobu, const char *jobv, const char *jobq, const Integer *m, const Integer *p, const Integer *n, const Integer *k, const Integer *l, double a[], const Integer *lda, double b[], const Integer *ldb, const double *tola, const double *tolb, double alpha[], double beta[], double u[], const Integer *ldu, double v[], const Integer *ldv, double q[], const Integer *ldq, double work[], Integer *ncycle, Integer *info, const Charlen length_jobu, const Charlen length_jobv, const Charlen length_jobq)
The routine may be called by the names f08yef, nagf_lapackeig_dtgsja or its LAPACK name dtgsja.

3 Description

f08yef computes the GSVD of the matrices A and B which are assumed to have the form as returned by f08vgf
A= { n-k-lklk0A12A13l00A23m-k-l000() ,   if ​ m-k-l 0; n-k-lklk0A12A13m-k00A23() ,   if ​ m-k-l < 0 ; B= n-k-lkll00B13p-l000() ,  
where the k×k matrix A12 and the l×l matrix B13 are nonsingular upper triangular, A23 is l×l upper triangular if m-k-l0 and is (m-k)×l upper trapezoidal otherwise.
f08yef computes orthogonal matrices Q, U and V, diagonal matrices D1 and D2, and an upper triangular matrix R such that
UTAQ = D1 ( 0 R ) ,   VTBQ = D2 ( 0 R ) .  
Optionally Q, U and V may or may not be computed, or they may be premultiplied by matrices Q1, U1 and V1 respectively.
If (m-k-l)0 then D1, D2 and R have the form
D1= klkI0l0Cm-k-l00() ,  
D2= kll0Sp-l00() ,  
R = klkR11R12l0R22() ,  
where C=diag(αk+1,,,,,,αk+l),  S=diag(βk+1,,,,,,βk+l).
If (m-k-l)<0 then D1, D2 and R have the form
D1= km-kk+l-mkI00m-k0C0() ,  
D2= km-kk+l-mm-k0S0k+l-m00Ip-l000() ,  
R = km-kk+l-mkR11R12R13m-k0R22R23k+l-m00R33() ,  
where C=diag(αk+1,,,,,,αm),  S=diag(βk+1,,,,,,βm).
In both cases the diagonal matrix C has non-negative diagonal elements, the diagonal matrix S has positive diagonal elements, so that S is nonsingular, and C2+S2=1. See Section 2.3.5.3 of Anderson et al. (1999) for further information.

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: jobu Character(1) Input
On entry: if jobu='U', u must contain an orthogonal matrix U1 on entry, and the product U1U is returned.
If jobu='I', u is initialized to the unit matrix, and the orthogonal matrix U is returned.
If jobu='N', U is not computed.
Constraint: jobu='U', 'I' or 'N'.
2: jobv Character(1) Input
On entry: if jobv='V', v must contain an orthogonal matrix V1 on entry, and the product V1V is returned.
If jobv='I', v is initialized to the unit matrix, and the orthogonal matrix V is returned.
If jobv='N', V is not computed.
Constraint: jobv='V', 'I' or 'N'.
3: jobq Character(1) Input
On entry: if jobq='Q', q must contain an orthogonal matrix Q1 on entry, and the product Q1Q is returned.
If jobq='I', q is initialized to the unit matrix, and the orthogonal matrix Q is returned.
If jobq='N', Q is not computed.
Constraint: jobq='Q', 'I' or 'N'.
4: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
5: p Integer Input
On entry: p, the number of rows of the matrix B.
Constraint: p0.
6: n Integer Input
On entry: n, the number of columns of the matrices A and B.
Constraint: n0.
7: k Integer Input
8: l Integer Input
On entry: k and l specify the sizes, k and l, of the subblocks of A and B, whose GSVD is to be computed by f08yef.
9: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the m×n matrix A.
On exit: if m-k-l0, a(1:k+l,n-k-l+1:n) contains the (k+l)×(k+l) upper triangular matrix R.
If m-k-l<0, a(1:m,n-k-l+1:n) contains the first m rows of the (k+l)×(k+l) upper triangular matrix R, and the submatrix R33 is returned in b(m-k+1:l,n+m-k-l+1:n) .
10: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08yef is called.
Constraint: ldamax(1,m).
11: b(ldb,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least max(1,n).
On entry: the p×n matrix B.
On exit: if m-k-l<0 , b(m-k+1:l,n+m-k-l+1:n) contains the submatrix R33 of R.
12: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f08yef is called.
Constraint: ldbmax(1,p).
13: tola Real (Kind=nag_wp) Input
14: tolb Real (Kind=nag_wp) Input
On entry: tola and tolb are the convergence criteria for the Jacobi–Kogbetliantz iteration procedure. Generally, they should be the same as used in the preprocessing step performed by f08vgf, say
tola=max(m,n)Aε, tolb=max(p,n)Bε,  
where ε is the machine precision.
15: alpha(n) Real (Kind=nag_wp) array Output
On exit: see the description of beta.
16: beta(n) Real (Kind=nag_wp) array Output
On exit: alpha and beta contain the generalized singular value pairs of A and B;
  • alpha(i)=1 , beta(i)=0 , for i=1,2,,k, and
  • if m-k-l0 , alpha(i)=αi , beta(i)=βi , for i=k+1,,k+l, or
  • if m-k-l<0 , alpha(i)=αi , beta(i)=βi , for i=k+1,,m and alpha(i)=0 , beta(i)=1 , for i=m+1,,k+l.
Furthermore, if k+l<n, alpha(i)= beta(i)=0 , for i=k+l+1,,n.
17: u(ldu,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array u must be at least max(1,m) if jobu='U' or 'I', and at least 1 otherwise.
On entry: if jobu='U', u must contain an m×m matrix U1 (usually the orthogonal matrix returned by f08vgf).
On exit: if jobu='U', u contains the product U1U.
If jobu='I', u contains the orthogonal matrix U.
If jobu='N', u is not referenced.
18: ldu Integer Input
On entry: the first dimension of the array u as declared in the (sub)program from which f08yef is called.
Constraints:
  • if jobu='U' or 'I', ldu max(1,m) ;
  • otherwise ldu1.
19: v(ldv,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array v must be at least max(1,p) if jobv='V' or 'I', and at least 1 otherwise.
On entry: if jobv='V', v must contain an p×p matrix V1 (usually the orthogonal matrix returned by f08vgf).
On exit: if jobv='I', v contains the orthogonal matrix V.
If jobv='V', v contains the product V1V.
If jobv='N', v is not referenced.
20: ldv Integer Input
On entry: the first dimension of the array v as declared in the (sub)program from which f08yef is called.
Constraints:
  • if jobv='V' or 'I', ldv max(1,p) ;
  • otherwise ldv1.
21: q(ldq,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array q must be at least max(1,n) if jobq='Q' or 'I', and at least 1 otherwise.
On entry: if jobq='Q', q must contain an n×n matrix Q1 (usually the orthogonal matrix returned by f08vgf).
On exit: if jobq='I', q contains the orthogonal matrix Q.
If jobq='Q', q contains the product Q1Q.
If jobq='N', q is not referenced.
22: ldq Integer Input
On entry: the first dimension of the array q as declared in the (sub)program from which f08yef is called.
Constraints:
  • if jobq='Q' or 'I', ldq max(1,n) ;
  • otherwise ldq1.
23: work(2×n) Real (Kind=nag_wp) array Workspace
24: ncycle Integer Output
On exit: the number of cycles required for convergence.
25: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
info=1
The procedure does not converge after 40 cycles.

7 Accuracy

The computed generalized singular value decomposition is nearly the exact generalized singular value decomposition for nearby matrices (A+E) and (B+F), where
E2 = Oε A2   and   F2= Oε B2 ,  
and ε is the machine precision. See Section 4.12 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08yef makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The complex analogue of this routine is f08ysf.

10 Example

This example finds the generalized singular value decomposition
A = UΣ1 ( 0 R ) QT ,   B= VΣ2 ( 0 R ) QT ,  
of the matrix pair (A,B), where
A = ( 1 2 3 3 2 1 4 5 6 7 8 8 )   and   B= ( −2 −3 3 4 6 5 ) .  

10.1 Program Text

Program Text (f08yefe.f90)

10.2 Program Data

Program Data (f08yefe.d)

10.3 Program Results

Program Results (f08yefe.r)