laswlq(3)

NAME
laswlq - laswlq: short-wide LQ factor

SYNOPSIS

  Functions
subroutine claswlq (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
CLASWLQ
subroutine dlaswlq (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
DLASWLQ
subroutine slaswlq (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
SLASWLQ
subroutine zlaswlq (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
ZLASWLQ

Detailed Description

Function Documentation

  subroutine claswlq (integer m, integer n, integer mb, integer nb, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldt, *) t, integer ldt, complex, dimension( * ) work, integer lwork, integer info)

CLASWLQ

Purpose:

     CLASWLQ computes a blocked Tall-Skinny LQ factorization of
     a complex M-by-N matrix A for M <= N:
        A = ( L 0 ) *  Q,
     where:
        Q is a n-by-N orthogonal matrix, stored on exit in an implicit
        form in the elements above the diagonal of the array A and in
        the elements of the array T;
        L is a lower-triangular M-by-M matrix stored on exit in
        the elements on and below the diagonal of the array A.
        0 is a M-by-(N-M) zero matrix, if M < N, and is not stored.

Parameters
    M

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

    N

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

    MB

              MB is INTEGER
              The row block size to be used in the blocked QR.
              M >= MB >= 1

    NB

              NB is INTEGER
              The column block size to be used in the blocked QR.
              NB > 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the elements on and below the diagonal
              of the array contain the N-by-N lower triangular matrix L;
              the elements above the diagonal represent Q by the rows
              of blocked V (see Further Details).

    LDA

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

    T

              T is COMPLEX array,
              dimension (LDT, N * Number_of_row_blocks)
              where Number_of_row_blocks = CEIL((N-M)/(NB-M))
              The blocked upper triangular block reflectors stored in compact form
              as a sequence of upper triangular blocks.
              See Further Details below.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= MB.

    WORK

             (workspace) COMPLEX array, dimension (MAX(1,LWORK))

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= MB*M.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

     Short-Wide LQ (SWLQ) performs LQ by a sequence of orthogonal transformations,
     representing Q as a product of other orthogonal matrices
       Q = Q(1) * Q(2) * . . . * Q(k)
     where each Q(i) zeros out upper diagonal entries of a block of NB rows of A:
       Q(1) zeros out the upper diagonal entries of rows 1:NB of A
       Q(2) zeros out the bottom MB-N rows of rows [1:M,NB+1:2*NB-M] of A
       Q(3) zeros out the bottom MB-N rows of rows [1:M,2*NB-M+1:3*NB-2*M] of A
       . . .
     Q(1) is computed by GELQT, which represents Q(1) by Householder vectors
     stored under the diagonal of rows 1:MB of A, and by upper triangular
     block reflectors, stored in array T(1:LDT,1:N).
     For more information see Further Details in GELQT.
     Q(i) for i>1 is computed by TPLQT, which represents Q(i) by Householder vectors
     stored in columns [(i-1)*(NB-M)+M+1:i*(NB-M)+M] of A, and by upper triangular
     block reflectors, stored in array T(1:LDT,(i-1)*M+1:i*M).
     The last Q(k) may use fewer rows.
     For more information see Further Details in TPQRT.
     For more details of the overall algorithm, see the description of
     Sequential TSQR in Section 2.2 of [1].
     [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
         J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
         SIAM J. Sci. Comput, vol. 34, no. 1, 2012

Definition at line 165 of file claswlq.f.

  subroutine dlaswlq (integer m, integer n, integer mb, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldt, *) t, integer ldt, double precision, dimension( * ) work, integer lwork, integer info)

DLASWLQ

Purpose:

     DLASWLQ computes a blocked Tall-Skinny LQ factorization of
     a real M-by-N matrix A for M <= N:
        A = ( L 0 ) *  Q,
     where:
        Q is a n-by-N orthogonal matrix, stored on exit in an implicit
        form in the elements above the diagonal of the array A and in
        the elements of the array T;
        L is a lower-triangular M-by-M matrix stored on exit in
        the elements on and below the diagonal of the array A.
        0 is a M-by-(N-M) zero matrix, if M < N, and is not stored.

Parameters
    M

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

    N

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

    MB

              MB is INTEGER
              The row block size to be used in the blocked QR.
              M >= MB >= 1

    NB

              NB is INTEGER
              The column block size to be used in the blocked QR.
              NB > 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the elements on and below the diagonal
              of the array contain the N-by-N lower triangular matrix L;
              the elements above the diagonal represent Q by the rows
              of blocked V (see Further Details).

    LDA

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

    T

              T is DOUBLE PRECISION array,
              dimension (LDT, N * Number_of_row_blocks)
              where Number_of_row_blocks = CEIL((N-M)/(NB-M))
              The blocked upper triangular block reflectors stored in compact form
              as a sequence of upper triangular blocks.
              See Further Details below.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= MB.

    WORK

             (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= MB*M.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

     Short-Wide LQ (SWLQ) performs LQ by a sequence of orthogonal transformations,
     representing Q as a product of other orthogonal matrices
       Q = Q(1) * Q(2) * . . . * Q(k)
     where each Q(i) zeros out upper diagonal entries of a block of NB rows of A:
       Q(1) zeros out the upper diagonal entries of rows 1:NB of A
       Q(2) zeros out the bottom MB-N rows of rows [1:M,NB+1:2*NB-M] of A
       Q(3) zeros out the bottom MB-N rows of rows [1:M,2*NB-M+1:3*NB-2*M] of A
       . . .
     Q(1) is computed by GELQT, which represents Q(1) by Householder vectors
     stored under the diagonal of rows 1:MB of A, and by upper triangular
     block reflectors, stored in array T(1:LDT,1:N).
     For more information see Further Details in GELQT.
     Q(i) for i>1 is computed by TPLQT, which represents Q(i) by Householder vectors
     stored in columns [(i-1)*(NB-M)+M+1:i*(NB-M)+M] of A, and by upper triangular
     block reflectors, stored in array T(1:LDT,(i-1)*M+1:i*M).
     The last Q(k) may use fewer rows.
     For more information see Further Details in TPQRT.
     For more details of the overall algorithm, see the description of
     Sequential TSQR in Section 2.2 of [1].
     [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
         J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
         SIAM J. Sci. Comput, vol. 34, no. 1, 2012

Definition at line 165 of file dlaswlq.f.

  subroutine slaswlq (integer m, integer n, integer mb, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( ldt, *) t, integer ldt, real, dimension( * ) work, integer lwork, integer info)

SLASWLQ

Purpose:

     SLASWLQ computes a blocked Tall-Skinny LQ factorization of
     a real M-by-N matrix A for M <= N:
        A = ( L 0 ) *  Q,
     where:
        Q is a n-by-N orthogonal matrix, stored on exit in an implicit
        form in the elements above the diagonal of the array A and in
        the elements of the array T;
        L is a lower-triangular M-by-M matrix stored on exit in
        the elements on and below the diagonal of the array A.
        0 is a M-by-(N-M) zero matrix, if M < N, and is not stored.

Parameters
    M

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

    N

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

    MB

              MB is INTEGER
              The row block size to be used in the blocked QR.
              M >= MB >= 1

    NB

              NB is INTEGER
              The column block size to be used in the blocked QR.
              NB > 0.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the elements on and below the diagonal
              of the array contain the N-by-N lower triangular matrix L;
              the elements above the diagonal represent Q by the rows
              of blocked V (see Further Details).

    LDA

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

    T

              T is REAL array,
              dimension (LDT, N * Number_of_row_blocks)
              where Number_of_row_blocks = CEIL((N-M)/(NB-M))
              The blocked upper triangular block reflectors stored in compact form
              as a sequence of upper triangular blocks.
              See Further Details below.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= MB.

    WORK

             (workspace) REAL array, dimension (MAX(1,LWORK))

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= MB * M.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

     Short-Wide LQ (SWLQ) performs LQ by a sequence of orthogonal transformations,
     representing Q as a product of other orthogonal matrices
       Q = Q(1) * Q(2) * . . . * Q(k)
     where each Q(i) zeros out upper diagonal entries of a block of NB rows of A:
       Q(1) zeros out the upper diagonal entries of rows 1:NB of A
       Q(2) zeros out the bottom MB-N rows of rows [1:M,NB+1:2*NB-M] of A
       Q(3) zeros out the bottom MB-N rows of rows [1:M,2*NB-M+1:3*NB-2*M] of A
       . . .
     Q(1) is computed by GELQT, which represents Q(1) by Householder vectors
     stored under the diagonal of rows 1:MB of A, and by upper triangular
     block reflectors, stored in array T(1:LDT,1:N).
     For more information see Further Details in GELQT.
     Q(i) for i>1 is computed by TPLQT, which represents Q(i) by Householder vectors
     stored in columns [(i-1)*(NB-M)+M+1:i*(NB-M)+M] of A, and by upper triangular
     block reflectors, stored in array T(1:LDT,(i-1)*M+1:i*M).
     The last Q(k) may use fewer rows.
     For more information see Further Details in TPQRT.
     For more details of the overall algorithm, see the description of
     Sequential TSQR in Section 2.2 of [1].
     [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
         J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
         SIAM J. Sci. Comput, vol. 34, no. 1, 2012

Definition at line 165 of file slaswlq.f.

  subroutine zlaswlq (integer m, integer n, integer mb, integer nb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldt, *) t, integer ldt, complex*16, dimension( * ) work, integer lwork, integer info)

ZLASWLQ

Purpose:

     ZLASWLQ computes a blocked Tall-Skinny LQ factorization of
     a complexx M-by-N matrix A for M <= N:
        A = ( L 0 ) *  Q,
     where:
        Q is a n-by-N orthogonal matrix, stored on exit in an implicit
        form in the elements above the diagonal of the array A and in
        the elements of the array T;
        L is a lower-triangular M-by-M matrix stored on exit in
        the elements on and below the diagonal of the array A.
        0 is a M-by-(N-M) zero matrix, if M < N, and is not stored.

Parameters
    M

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

    N

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

    MB

              MB is INTEGER
              The row block size to be used in the blocked QR.
              M >= MB >= 1

    NB

              NB is INTEGER
              The column block size to be used in the blocked QR.
              NB > 0.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the elements on and below the diagonal
              of the array contain the N-by-N lower triangular matrix L;
              the elements above the diagonal represent Q by the rows
              of blocked V (see Further Details).

    LDA

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

    T

              T is COMPLEX*16 array,
              dimension (LDT, N * Number_of_row_blocks)
              where Number_of_row_blocks = CEIL((N-M)/(NB-M))
              The blocked upper triangular block reflectors stored in compact form
              as a sequence of upper triangular blocks.
              See Further Details below.

    LDT

              LDT is INTEGER
              The leading dimension of the array T.  LDT >= MB.

    WORK

             (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK))

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= MB*M.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

     Short-Wide LQ (SWLQ) performs LQ by a sequence of orthogonal transformations,
     representing Q as a product of other orthogonal matrices
       Q = Q(1) * Q(2) * . . . * Q(k)
     where each Q(i) zeros out upper diagonal entries of a block of NB rows of A:
       Q(1) zeros out the upper diagonal entries of rows 1:NB of A
       Q(2) zeros out the bottom MB-N rows of rows [1:M,NB+1:2*NB-M] of A
       Q(3) zeros out the bottom MB-N rows of rows [1:M,2*NB-M+1:3*NB-2*M] of A
       . . .
     Q(1) is computed by GELQT, which represents Q(1) by Householder vectors
     stored under the diagonal of rows 1:MB of A, and by upper triangular
     block reflectors, stored in array T(1:LDT,1:N).
     For more information see Further Details in GELQT.
     Q(i) for i>1 is computed by TPLQT, which represents Q(i) by Householder vectors
     stored in columns [(i-1)*(NB-M)+M+1:i*(NB-M)+M] of A, and by upper triangular
     block reflectors, stored in array T(1:LDT,(i-1)*M+1:i*M).
     The last Q(k) may use fewer rows.
     For more information see Further Details in TPQRT.
     For more details of the overall algorithm, see the description of
     Sequential TSQR in Section 2.2 of [1].
     [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
         J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
         SIAM J. Sci. Comput, vol. 34, no. 1, 2012

Definition at line 165 of file zlaswlq.f.

Author

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