ztgsja.f

          SUBROUTINE ZTGSJA( 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
          DOUBLE PRECISION   TOLA, TOLB
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   ALPHA( * ), BETA( * )
          COMPLEX*16         A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
         $                   U( LDU, * ), V( LDV, * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  ZTGSJA 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 ZGGSVP
    *  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 unitary matrices, Z' denotes the conjugate
    *  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 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.
    *
    *  Arguments
    *  =========
    *
    *  JOBU    (input) 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    (input) 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    (input) 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       (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 ZTGSJA.
    *          See Further details.
    *
    *  A       (input/output) COMPLEX*16 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) COMPLEX*16 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) DOUBLE PRECISION
    *  TOLB    (input) DOUBLE PRECISION
    *          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)*MAZHEPS,
    *              TOLB = MAX(P,N)*norm(B)*MAZHEPS.
    *
    *  ALPHA   (output) DOUBLE PRECISION array, dimension (N)
    *  BETA    (output) DOUBLE PRECISION 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       (input/output) COMPLEX*16 array, dimension (LDU,M)
    *          On entry, if JOBU = 'U', U must contain a matrix U1 (usually
    *          the unitary matrix returned by ZGGSVP).
    *          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     (input) INTEGER
    *          The leading dimension of the array U. LDU >= max(1,M) if
    *          JOBU = 'U'; LDU >= 1 otherwise.
    *
    *  V       (input/output) COMPLEX*16 array, dimension (LDV,P)
    *          On entry, if JOBV = 'V', V must contain a matrix V1 (usually
    *          the unitary matrix returned by ZGGSVP).
    *          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     (input) INTEGER
    *          The leading dimension of the array V. LDV >= max(1,P) if
    *          JOBV = 'V'; LDV >= 1 otherwise.
    *
    *  Q       (input/output) COMPLEX*16 array, dimension (LDQ,N)
    *          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
    *          the unitary matrix returned by ZGGSVP).
    *          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     (input) INTEGER
    *          The leading dimension of the array Q. LDQ >= max(1,N) if
    *          JOBQ = 'Q'; LDQ >= 1 otherwise.
    *
    *  WORK    (workspace) COMPLEX*16 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
    *  ===============
    *
    *  ZTGSJA 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 unitary matrix, and Z' is the conjugate
    *  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 )
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
          COMPLEX*16         CZERO, CONE
          PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
         $                   CONE = ( 1.0D+0, 0.0D+0 ) )
    *     ..
    *     .. Local Scalars ..
    *
          LOGICAL            INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
          INTEGER            I, J, KCYCLE
          DOUBLE PRECISION   A1, A3, B1, B3, CSQ, CSU, CSV, ERROR, GAMMA,
         $                   RWK, SSMIN
          COMPLEX*16         A2, B2, SNQ, SNU, SNV
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          EXTERNAL           LSAME
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DLARTG, XERBLA, ZCOPY, ZDSCAL, ZLAGS2, ZLAPLL,
         $                   ZLASET, ZROT
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, DBLE, DCONJG, 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( 'ZTGSJA', -INFO )
             RETURN
          END IF
    *
    *     Initialize U, V and Q, if necessary
    *
          IF( INITU )
         $   CALL ZLASET( 'Full', M, M, CZERO, CONE, U, LDU )
          IF( INITV )
         $   CALL ZLASET( 'Full', P, P, CZERO, CONE, V, LDV )
          IF( INITQ )
         $   CALL ZLASET( 'Full', N, N, CZERO, CONE, 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 = CZERO
                   A3 = ZERO
                   IF( K+I.LE.M )
         $            A1 = DBLE( A( K+I, N-L+I ) )
                   IF( K+J.LE.M )
         $            A3 = DBLE( A( K+J, N-L+J ) )
    *
                   B1 = DBLE( B( I, N-L+I ) )
                   B3 = DBLE( 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 ZLAGS2( 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 ZROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ),
         $                       LDA, CSU, DCONJG( SNU ) )
    *
    *              Update I-th and J-th rows of matrix B: V'*B
    *
                   CALL ZROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
         $                    CSV, DCONJG( SNV ) )
    *
    *              Update (N-L+I)-th and (N-L+J)-th columns of matrices
    *              A and B: A*Q and B*Q
    *
                   CALL ZROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
         $                    A( 1, N-L+I ), 1, CSQ, SNQ )
    *
                   CALL ZROT( 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 ) = CZERO
                      B( I, N-L+J ) = CZERO
                   ELSE
                      IF( K+J.LE.M )
         $               A( K+J, N-L+I ) = CZERO
                      B( J, N-L+I ) = CZERO
                   END IF
    *
    *              Ensure that the diagonal elements of A and B are real.
    *
                   IF( K+I.LE.M )
         $            A( K+I, N-L+I ) = DBLE( A( K+I, N-L+I ) )
                   IF( K+J.LE.M )
         $            A( K+J, N-L+J ) = DBLE( A( K+J, N-L+J ) )
                   B( I, N-L+I ) = DBLE( B( I, N-L+I ) )
                   B( J, N-L+J ) = DBLE( B( J, N-L+J ) )
    *
    *              Update unitary matrices U, V, Q, if desired.
    *
                   IF( WANTU .AND. K+J.LE.M )
         $            CALL ZROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
         $                       SNU )
    *
                   IF( WANTV )
         $            CALL ZROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV )
    *
                   IF( WANTQ )
         $            CALL ZROT( 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 ZCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
                   CALL ZCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 )
                   CALL ZLAPLL( 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 = DBLE( A( K+I, N-L+I ) )
             B1 = DBLE( B( I, N-L+I ) )
    *
             IF( A1.NE.ZERO ) THEN
                GAMMA = B1 / A1
    *
                IF( GAMMA.LT.ZERO ) THEN
                   CALL ZDSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
                   IF( WANTV )
         $            CALL ZDSCAL( P, -ONE, V( 1, I ), 1 )
                END IF
    *
                CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ),
         $                   RWK )
    *
                IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
                   CALL ZDSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ),
         $                      LDA )
                ELSE
                   CALL ZDSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
         $                      LDB )
                   CALL ZCOPY( 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 ZCOPY( 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 ZTGSJA
    *
          END