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author | jason <jason@8a072113-8704-0410-8d35-dd094bca7971> | 2008-10-28 01:38:50 +0000 |
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committer | jason <jason@8a072113-8704-0410-8d35-dd094bca7971> | 2008-10-28 01:38:50 +0000 |
commit | baba851215b44ac3b60b9248eb02bcce7eb76247 (patch) | |
tree | 8c0f5c006875532a30d4409f5e94b0f310ff00a7 /SRC/stgsja.f | |
download | lapack-baba851215b44ac3b60b9248eb02bcce7eb76247.tar.gz lapack-baba851215b44ac3b60b9248eb02bcce7eb76247.tar.bz2 lapack-baba851215b44ac3b60b9248eb02bcce7eb76247.zip |
Move LAPACK trunk into position.
Diffstat (limited to 'SRC/stgsja.f')
-rw-r--r-- | SRC/stgsja.f | 515 |
1 files changed, 515 insertions, 0 deletions
diff --git a/SRC/stgsja.f b/SRC/stgsja.f new file mode 100644 index 00000000..be29a7a4 --- /dev/null +++ b/SRC/stgsja.f @@ -0,0 +1,515 @@ + 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 |