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ctgsja-f(3)

ctgsja.f –

Synopsis

Functions/Subroutines

subroutine 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)
CTGSJA

Function/Subroutine Documentation

subroutine ctgsja (characterJOBU, characterJOBV, characterJOBQ, integerM, integerP, integerN, integerK, integerL, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, * )B, integerLDB, realTOLA, realTOLB, real, dimension( * )ALPHA, real, dimension( * )BETA, complex, dimension( ldu, * )U, integerLDU, complex, dimension( ldv, * )V, integerLDV, complex, dimension( ldq, * )Q, integerLDQ, complex, dimension( * )WORK, integerNCYCLE, integerINFO)

CTGSJA

Purpose:

 CTGSJA computes the generalized singular value decomposition (GSVD)
 of two 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 CGGSVP
 from a general M-by-N matrix A and P-by-N matrix B:
 N-K-L K L
 A = K ( 0 A12 A13 ) if M-K-L >= 0;
 L ( 0 0 A23 )
 M-K-L ( 0 0 0 )
 N-K-L K L
 A = K ( 0 A12 A13 ) if M-K-L < 0;
 M-K ( 0 0 A23 )
 N-K-L K L
 B = L ( 0 0 B13 )
 P-L ( 0 0 0 )
 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 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,
 K L
 D1 = K ( I 0 )
 L ( 0 C )
 M-K-L ( 0 0 )
 K L
 D2 = L ( 0 S )
 P-L ( 0 0 )
 N-K-L K L
 ( 0 R ) = K ( 0 R11 R12 ) K
 L ( 0 0 R22 ) L
 where
 C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
 S = diag( BETA(K+1), ... , BETA(K+L) ),
 C**2 + S**2 = I.
 R is stored in A(1:K+L,N-K-L+1:N) on exit.
 If M-K-L < 0,
 K M-K K+L-M
 D1 = K ( I 0 0 )
 M-K ( 0 C 0 )
 K M-K K+L-M
 D2 = M-K ( 0 S 0 )
 K+L-M ( 0 0 I )
 P-L ( 0 0 0 )
 N-K-L K M-K K+L-M
 ( 0 R ) = K ( 0 R11 R12 R13 )
 M-K ( 0 0 R22 R23 )
 K+L-M ( 0 0 0 R33 )
 where
 C = diag( ALPHA(K+1), ... , ALPHA(M) ),
 S = diag( BETA(K+1), ... , BETA(M) ),
 C**2 + S**2 = I.
 R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
 ( 0 R22 R23 )
 in B(M-K+1:L,N+M-K-L+1:N) on exit.
 The computation of the 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.

 

Parameters:

JOBU

 JOBU is CHARACTER*1
 = 'U': U must contain a unitary matrix U1 on entry, and
 the product U1*U is returned;
 = 'I': U is initialized to the unit matrix, and the
 unitary matrix U is returned;
 = 'N': U is not computed.

JOBV

 JOBV is CHARACTER*1
 = 'V': V must contain a unitary matrix V1 on entry, and
 the product V1*V is returned;
 = 'I': V is initialized to the unit matrix, and the
 unitary matrix V is returned;
 = 'N': V is not computed.

JOBQ

 JOBQ is CHARACTER*1
 = 'Q': Q must contain a unitary matrix Q1 on entry, and
 the product Q1*Q is returned;
 = 'I': Q is initialized to the unit matrix, and the
 unitary matrix Q is returned;
 = 'N': Q is not computed.

M

 M is INTEGER
 The number of rows of the matrix A. M >= 0.

P

 P is INTEGER
 The number of rows of the matrix B. P >= 0.

N

 N is INTEGER
 The number of columns of the matrices A and B. N >= 0.

K

 K is INTEGER

L

 L is INTEGER
 K and L specify the subblocks in the input matrices A and B:
 A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N)
 of A and B, whose GSVD is going to be computed by CTGSJA.
 See Further Details.

A

 A is COMPLEX array, dimension (LDA,N)
 On entry, the M-by-N matrix A.
 On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
 matrix R or part of R. See Purpose for details.

LDA

 LDA is INTEGER
 The leading dimension of the array A. LDA >= max(1,M).

B

 B is COMPLEX array, dimension (LDB,N)
 On entry, the P-by-N matrix B.
 On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
 a part of R. See Purpose for details.

LDB

 LDB is INTEGER
 The leading dimension of the array B. LDB >= max(1,P).

TOLA

 TOLA is REAL

TOLB

 TOLB is REAL
 TOLA and TOLB are the convergence criteria for the Jacobi-
 Kogbetliantz iteration procedure. Generally, they are the
 same as used in the preprocessing step, say
 TOLA = MAX(M,N)*norm(A)*MACHEPS,
 TOLB = MAX(P,N)*norm(B)*MACHEPS.

ALPHA

 ALPHA is REAL array, dimension (N)

BETA

 BETA is REAL array, dimension (N)
 On exit, ALPHA and BETA 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)= C, ALPHA(M+1:K+L)= 0
 BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
 Furthermore, if K+L < N,
 ALPHA(K+L+1:N) = 0
 BETA(K+L+1:N) = 0.

U

 U is COMPLEX array, dimension (LDU,M)
 On entry, if JOBU = 'U', U must contain a matrix U1 (usually
 the unitary matrix returned by CGGSVP).
 On exit,
 if JOBU = 'I', U contains the unitary matrix U;
 if JOBU = 'U', U contains the product U1*U.
 If JOBU = 'N', U is not referenced.

LDU

 LDU is INTEGER
 The leading dimension of the array U. LDU >= max(1,M) if
 JOBU = 'U'; LDU >= 1 otherwise.

V

 V is COMPLEX array, dimension (LDV,P)
 On entry, if JOBV = 'V', V must contain a matrix V1 (usually
 the unitary matrix returned by CGGSVP).
 On exit,
 if JOBV = 'I', V contains the unitary matrix V;
 if JOBV = 'V', V contains the product V1*V.
 If JOBV = 'N', V is not referenced.

LDV

 LDV is INTEGER
 The leading dimension of the array V. LDV >= max(1,P) if
 JOBV = 'V'; LDV >= 1 otherwise.

Q

 Q is COMPLEX array, dimension (LDQ,N)
 On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
 the unitary matrix returned by CGGSVP).
 On exit,
 if JOBQ = 'I', Q contains the unitary matrix Q;
 if JOBQ = 'Q', Q contains the product Q1*Q.
 If JOBQ = 'N', Q is not referenced.

LDQ

 LDQ is INTEGER
 The leading dimension of the array Q. LDQ >= max(1,N) if
 JOBQ = 'Q'; LDQ >= 1 otherwise.

WORK

 WORK is COMPLEX array, dimension (2*N)

NCYCLE

 NCYCLE is INTEGER
 The number of cycles required for convergence.

INFO

 INFO is INTEGER
 = 0: successful exit
 < 0: if INFO = -i, the i-th argument had an illegal value.
 = 1: the procedure does not converge after MAXIT cycles.

 

Internal Parameters:

 MAXIT INTEGER
 MAXIT specifies the total loops that the iterative procedure
 may take. If after MAXIT cycles, the routine fails to
 converge, we return INFO = 1.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Further Details:

 CTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
 min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
 matrix B13 to the form:
 U1**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1,
 where U1, V1 and Q1 are unitary matrix.
 C1 and S1 are diagonal matrices satisfying
 C1**2 + S1**2 = I,
 and R1 is an L-by-L nonsingular upper triangular matrix.

 

Definition at line 378 of file ctgsja.f.

Author

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