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))
    !>          On exit, if INFO = 0, WORK(1) returns the minimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MB*M, otherwise.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the minimal 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
    !>

  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))
    !>          On exit, if INFO = 0, WORK(1) returns the minimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MB*M, otherwise.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the minimal 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
    !>

  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))
    !>          On exit, if INFO = 0, WORK(1) returns the minimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MB*M, otherwise.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the minimal 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
    !>

  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))
    !>          On exit, if INFO = 0, WORK(1) returns the minimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MB*M, otherwise.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the minimal 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
    !>

Author

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