F08YSF (ZTGSJA) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08YSF (ZTGSJA)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08YSF (ZTGSJA) computes the generalized singular value decomposition (GSVD) of two complex upper trapezoidal matrices A and B, where A is an m by n matrix and B is a p by n matrix.
A and B are assumed to be in the form returned by F08VSF (ZGGSVP) or F08VUF (ZGGSVP3).

2  Specification

SUBROUTINE F08YSF ( 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  M, P, N, K, L, LDA, LDB, LDU, LDV, LDQ, NCYCLE, INFO
REAL (KIND=nag_wp)  TOLA, TOLB, ALPHA(N), BETA(N)
COMPLEX (KIND=nag_wp)  A(LDA,*), B(LDB,*), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(2*N)
CHARACTER(1)  JOBU, JOBV, JOBQ
The routine may be called by its LAPACK name ztgsja.

3  Description

F08YSF (ZTGSJA) computes the GSVD of the matrices A and B which are assumed to have the form as returned by F08VSF (ZGGSVP) or F08VUF (ZGGSVP3)
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 by k matrix A12 and the l by l matrix B13 are nonsingular upper triangular, A23 is l by l upper triangular if m-k-l0 and is m-k by l upper trapezoidal otherwise.
F08YSF (ZTGSJA) computes unitary matrices Q, U and V, diagonal matrices D1 and D2, and an upper triangular matrix R such that
UHAQ = D1 0 R ,   VHBQ = 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-l0 then D1, D2 and R have the form
D1= klkI0l0Cm-k-l00() ,  
D2= kll0Sp-l00() ,  
R = klkR11R12l0R22() ,  
where C=diagαk+1,,,,,,αk+l,  S=diagβk+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=diagαk+1,,,,,,αm,  S=diagβk+1,,,,,,βm.
In both cases the diagonal matrix C has real non-negative diagonal elements, the diagonal matrix S has real 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 http://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 a unitary matrix U1 on entry, and the product U1U is returned.
If JOBU='I', U is initialized to the unit matrix, and the unitary 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 a unitary matrix V1 on entry, and the product V1V is returned.
If JOBV='I', V is initialized to the unit matrix, and the unitary 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 a unitary matrix Q1 on entry, and the product Q1Q is returned.
If JOBQ='I', Q is initialized to the unit matrix, and the unitary matrix Q is returned.
If JOBQ='N', Q is not computed.
Constraint: JOBQ='Q', 'I' or 'N'.
4:     M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint: M0.
5:     P – INTEGERInput
On entry: p, the number of rows of the matrix B.
Constraint: P0.
6:     N – INTEGERInput
On entry: n, the number of columns of the matrices A and B.
Constraint: N0.
7:     K – INTEGERInput
8:     L – INTEGERInput
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 F08YSF (ZTGSJA).
9:     ALDA* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
On entry: the m by n matrix A.
On exit: if m-k-l0, A1:k+ln-k-l+1:n  contains the k+l by k+l upper triangular matrix R.
If m-k-l<0, A1:mn-k-l+1:n  contains the first m rows of the k+l by k+l upper triangular matrix R, and the submatrix R33 is returned in Bm-k+1:ln+m-k-l+1:n .
10:   LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08YSF (ZTGSJA) is called.
Constraint: LDAmax1,M.
11:   BLDB* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least max1,N.
On entry: the p by n matrix B.
On exit: if m-k-l<0 , Bm-k+1:ln+m-k-l+1:n  contains the submatrix R33 of R.
12:   LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08YSF (ZTGSJA) is called.
Constraint: LDBmax1,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 F08VSF (ZGGSVP) or F08VUF (ZGGSVP3), say
TOLA=maxM,NAε, TOLB=maxP,NBε,  
where ε  is the machine precision.
15:   ALPHAN – REAL (KIND=nag_wp) arrayOutput
On exit: see the description of BETA.
16:   BETAN – REAL (KIND=nag_wp) arrayOutput
On exit: ALPHA and BETA contain the generalized singular value pairs of A and B;
  • ALPHAi=1 , BETAi=0 , for i=1,2,,k, and
  • if m-k-l0 , ALPHAi=αi , BETAi=βi , for i=k+1,,k+l, or
  • if m-k-l<0 , ALPHAi=αi , BETAi=βi , for i=k+1,,m and ALPHAi=0 , BETAi=1 , for i=m+1,,k+l.
Furthermore, if k+l<n, ALPHAi= BETAi=0 , for i=k+l+1,,n.
17:   ULDU* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array U must be at least max1,M if JOBU='U' or 'I', and at least 1 otherwise.
On entry: if JOBU='U', U must contain an m by m matrix U1 (usually the unitary matrix returned by F08VSF (ZGGSVP) or F08VUF (ZGGSVP3)).
On exit: if JOBU='U', U contains the product U1U.
If JOBU='I', U contains the unitary matrix U.
If JOBU='N', U is not referenced.
18:   LDU – INTEGERInput
On entry: the first dimension of the array U as declared in the (sub)program from which F08YSF (ZTGSJA) is called.
Constraints:
  • if JOBU='U' or 'I', LDU max1,M ;
  • otherwise LDU1.
19:   VLDV* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array V must be at least max1,P if JOBV='V' or 'I', and at least 1 otherwise.
On entry: if JOBV='V', V must contain an p by p matrix V1 (usually the unitary matrix returned by F08VSF (ZGGSVP) or F08VUF (ZGGSVP3)).
On exit: if JOBV='I', V contains the unitary matrix V.
If JOBV='V', V contains the product V1V.
If JOBV='N', V is not referenced.
20:   LDV – INTEGERInput
On entry: the first dimension of the array V as declared in the (sub)program from which F08YSF (ZTGSJA) is called.
Constraints:
  • if JOBV='V' or 'I', LDV max1,P ;
  • otherwise LDV1.
21:   QLDQ* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array Q must be at least max1,N if JOBQ='Q' or 'I', and at least 1 otherwise.
On entry: if JOBQ='Q', Q must contain an n by n matrix Q1 (usually the unitary matrix returned by F08VSF (ZGGSVP) or F08VUF (ZGGSVP3)).
On exit: if JOBQ='I', Q contains the unitary matrix Q.
If JOBQ='Q', Q contains the product Q1Q.
If JOBQ='N', Q is not referenced.
22:   LDQ – INTEGERInput
On entry: the first dimension of the array Q as declared in the (sub)program from which F08YSF (ZTGSJA) is called.
Constraints:
  • if JOBQ='Q' or 'I', LDQ max1,N ;
  • otherwise LDQ1.
23:   WORK2×N – COMPLEX (KIND=nag_wp) arrayWorkspace
24:   NCYCLE – INTEGEROutput
On exit: the number of cycles required for convergence.
25:   INFO – INTEGEROutput
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

F08YSF (ZTGSJA) 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 real analogue of this routine is F08YEF (DTGSJA).

10  Example

This example finds the generalized singular value decomposition
A = UΣ1 0 R QH ,   B= VΣ2 0 R QH ,  
of the matrix pair A,B, where
A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i 0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i  
and
B = 1 0 -1 0 0 1 0 -1 .  

10.1  Program Text

Program Text (f08ysfe.f90)

10.2  Program Data

Program Data (f08ysfe.d)

10.3  Program Results

Program Results (f08ysfe.r)


F08YSF (ZTGSJA) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016