SUBROUTINE 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 )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
$ NCYCLE, P
REAL TOLA, TOLB
* ..
* .. Array Arguments ..
REAL A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), Q( LDQ, * ), U( LDU, * ),
$ V( LDV, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* STGSJA computes the generalized singular value decomposition (GSVD)
* of two real 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 SGGSVP
* 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'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ),
*
* where U, V and Q are orthogonal matrices, Z' denotes the transpose
* of Z, 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 orthogonal 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.
*
* Arguments
* =========
*
* JOBU (input) CHARACTER*1
* = 'U': U must contain an orthogonal matrix U1 on entry, and
* the product U1*U is returned;
* = 'I': U is initialized to the unit matrix, and the
* orthogonal matrix U is returned;
* = 'N': U is not computed.
*
* JOBV (input) CHARACTER*1
* = 'V': V must contain an orthogonal matrix V1 on entry, and
* the product V1*V is returned;
* = 'I': V is initialized to the unit matrix, and the
* orthogonal matrix V is returned;
* = 'N': V is not computed.
*
* JOBQ (input) CHARACTER*1
* = 'Q': Q must contain an orthogonal matrix Q1 on entry, and
* the product Q1*Q is returned;
* = 'I': Q is initialized to the unit matrix, and the
* orthogonal matrix Q is returned;
* = 'N': Q is not computed.
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* P (input) INTEGER
* The number of rows of the matrix B. P >= 0.
*
* N (input) INTEGER
* The number of columns of the matrices A and B. N >= 0.
*
* K (input) INTEGER
* L (input) 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 STGSJA.
* See Further details.
*
* A (input/output) REAL 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 (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) REAL 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 (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,P).
*
* TOLA (input) REAL
* TOLB (input) 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 (output) REAL array, dimension (N)
* BETA (output) 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 and
* BETA(K+L+1:N) = 0.
*
* U (input/output) REAL array, dimension (LDU,M)
* On entry, if JOBU = 'U', U must contain a matrix U1 (usually
* the orthogonal matrix returned by SGGSVP).
* On exit,
* if JOBU = 'I', U contains the orthogonal matrix U;
* if JOBU = 'U', U contains the product U1*U.
* If JOBU = 'N', U is not referenced.
*
* LDU (input) INTEGER
* The leading dimension of the array U. LDU >= max(1,M) if
* JOBU = 'U'; LDU >= 1 otherwise.
*
* V (input/output) REAL array, dimension (LDV,P)
* On entry, if JOBV = 'V', V must contain a matrix V1 (usually
* the orthogonal matrix returned by SGGSVP).
* On exit,
* if JOBV = 'I', V contains the orthogonal matrix V;
* if JOBV = 'V', V contains the product V1*V.
* If JOBV = 'N', V is not referenced.
*
* LDV (input) INTEGER
* The leading dimension of the array V. LDV >= max(1,P) if
* JOBV = 'V'; LDV >= 1 otherwise.
*
* Q (input/output) REAL array, dimension (LDQ,N)
* On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
* the orthogonal matrix returned by SGGSVP).
* On exit,
* if JOBQ = 'I', Q contains the orthogonal matrix Q;
* if JOBQ = 'Q', Q contains the product Q1*Q.
* If JOBQ = 'N', Q is not referenced.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q. LDQ >= max(1,N) if
* JOBQ = 'Q'; LDQ >= 1 otherwise.
*
* WORK (workspace) REAL array, dimension (2*N)
*
* NCYCLE (output) INTEGER
* The number of cycles required for convergence.
*
* INFO (output) 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.
*
* Further Details
* ===============
*
* STGSJA 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'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,
*
* where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose
* of Z. C1 and S1 are diagonal matrices satisfying
*
* C1**2 + S1**2 = I,
*
* and R1 is an L-by-L nonsingular upper triangular matrix.
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXIT
PARAMETER ( MAXIT = 40 )
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
*
LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
INTEGER I, J, KCYCLE
REAL A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, ERROR,
$ GAMMA, RWK, SNQ, SNU, SNV, SSMIN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SLAGS2, SLAPLL, SLARTG, SLASET, SROT,
$ SSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
INITU = LSAME( JOBU, 'I' )
WANTU = INITU .OR. LSAME( JOBU, 'U' )
*
INITV = LSAME( JOBV, 'I' )
WANTV = INITV .OR. LSAME( JOBV, 'V' )
*
INITQ = LSAME( JOBQ, 'I' )
WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' )
*
INFO = 0
IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( P.LT.0 ) THEN
INFO = -5
ELSE IF( N.LT.0 ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
INFO = -12
ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
INFO = -18
ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
INFO = -20
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
INFO = -22
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'STGSJA', -INFO )
RETURN
END IF
*
* Initialize U, V and Q, if necessary
*
IF( INITU )
$ CALL SLASET( 'Full', M, M, ZERO, ONE, U, LDU )
IF( INITV )
$ CALL SLASET( 'Full', P, P, ZERO, ONE, V, LDV )
IF( INITQ )
$ CALL SLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
*
* Loop until convergence
*
UPPER = .FALSE.
DO 40 KCYCLE = 1, MAXIT
*
UPPER = .NOT.UPPER
*
DO 20 I = 1, L - 1
DO 10 J = I + 1, L
*
A1 = ZERO
A2 = ZERO
A3 = ZERO
IF( K+I.LE.M )
$ A1 = A( K+I, N-L+I )
IF( K+J.LE.M )
$ A3 = A( K+J, N-L+J )
*
B1 = B( I, N-L+I )
B3 = B( J, N-L+J )
*
IF( UPPER ) THEN
IF( K+I.LE.M )
$ A2 = A( K+I, N-L+J )
B2 = B( I, N-L+J )
ELSE
IF( K+J.LE.M )
$ A2 = A( K+J, N-L+I )
B2 = B( J, N-L+I )
END IF
*
CALL SLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
$ CSV, SNV, CSQ, SNQ )
*
* Update (K+I)-th and (K+J)-th rows of matrix A: U'*A
*
IF( K+J.LE.M )
$ CALL SROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ),
$ LDA, CSU, SNU )
*
* Update I-th and J-th rows of matrix B: V'*B
*
CALL SROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
$ CSV, SNV )
*
* Update (N-L+I)-th and (N-L+J)-th columns of matrices
* A and B: A*Q and B*Q
*
CALL SROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
$ A( 1, N-L+I ), 1, CSQ, SNQ )
*
CALL SROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ,
$ SNQ )
*
IF( UPPER ) THEN
IF( K+I.LE.M )
$ A( K+I, N-L+J ) = ZERO
B( I, N-L+J ) = ZERO
ELSE
IF( K+J.LE.M )
$ A( K+J, N-L+I ) = ZERO
B( J, N-L+I ) = ZERO
END IF
*
* Update orthogonal matrices U, V, Q, if desired.
*
IF( WANTU .AND. K+J.LE.M )
$ CALL SROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
$ SNU )
*
IF( WANTV )
$ CALL SROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV )
*
IF( WANTQ )
$ CALL SROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1, CSQ,
$ SNQ )
*
10 CONTINUE
20 CONTINUE
*
IF( .NOT.UPPER ) THEN
*
* The matrices A13 and B13 were lower triangular at the start
* of the cycle, and are now upper triangular.
*
* Convergence test: test the parallelism of the corresponding
* rows of A and B.
*
ERROR = ZERO
DO 30 I = 1, MIN( L, M-K )
CALL SCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
CALL SCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 )
CALL SLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN )
ERROR = MAX( ERROR, SSMIN )
30 CONTINUE
*
IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) )
$ GO TO 50
END IF
*
* End of cycle loop
*
40 CONTINUE
*
* The algorithm has not converged after MAXIT cycles.
*
INFO = 1
GO TO 100
*
50 CONTINUE
*
* If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
* Compute the generalized singular value pairs (ALPHA, BETA), and
* set the triangular matrix R to array A.
*
DO 60 I = 1, K
ALPHA( I ) = ONE
BETA( I ) = ZERO
60 CONTINUE
*
DO 70 I = 1, MIN( L, M-K )
*
A1 = A( K+I, N-L+I )
B1 = B( I, N-L+I )
*
IF( A1.NE.ZERO ) THEN
GAMMA = B1 / A1
*
* change sign if necessary
*
IF( GAMMA.LT.ZERO ) THEN
CALL SSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
IF( WANTV )
$ CALL SSCAL( P, -ONE, V( 1, I ), 1 )
END IF
*
CALL SLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ),
$ RWK )
*
IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
CALL SSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ),
$ LDA )
ELSE
CALL SSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
$ LDB )
CALL SCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
$ LDA )
END IF
*
ELSE
*
ALPHA( K+I ) = ZERO
BETA( K+I ) = ONE
CALL SCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
$ LDA )
*
END IF
*
70 CONTINUE
*
* Post-assignment
*
DO 80 I = M + 1, K + L
ALPHA( I ) = ZERO
BETA( I ) = ONE
80 CONTINUE
*
IF( K+L.LT.N ) THEN
DO 90 I = K + L + 1, N
ALPHA( I ) = ZERO
BETA( I ) = ZERO
90 CONTINUE
END IF
*
100 CONTINUE
NCYCLE = KCYCLE
RETURN
*
* End of STGSJA
*
END