latsqr(3)

NAME
latsqr - latsqr: tall-skinny QR factor

SYNOPSIS

  Functions
subroutine clatsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
CLATSQR
subroutine dlatsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
DLATSQR
subroutine slatsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
SLATSQR
subroutine zlatsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info)
ZLATSQR

Detailed Description

Function Documentation

  subroutine clatsqr (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)

CLATSQR

Purpose:

     CLATSQR computes a blocked Tall-Skinny QR factorization of
     a complex M-by-N matrix A for M >= N:
        A = Q * ( R ),
                ( 0 )
     where:
        Q is a M-by-M orthogonal matrix, stored on exit in an implicit
        form in the elements below the diagonal of the array A and in
        the elements of the array T;
        R is an upper-triangular N-by-N matrix, stored on exit in
        the elements on and above the diagonal of the array A.
        0 is a (M-N)-by-N zero matrix, 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. M >= N >= 0.

    MB

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

    NB

              NB is INTEGER
              The column block size to be used in the blocked QR.
              N >= NB >= 1.

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the elements on and above the diagonal
              of the array contain the N-by-N upper triangular matrix R;
              the elements below the diagonal represent Q by the columns
              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((M-N)/(MB-N))
              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 >= NB.

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= NB*N.
              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:

     Tall-Skinny QR (TSQR) performs QR 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 subdiagonal entries of a block of MB rows of A:
       Q(1) zeros out the subdiagonal entries of rows 1:MB of A
       Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
       Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
       . . .
     Q(1) is computed by GEQRT, 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 GEQRT.
     Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
     stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
     block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
     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 167 of file clatsqr.f.

  subroutine dlatsqr (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)

DLATSQR

Purpose:

     DLATSQR computes a blocked Tall-Skinny QR factorization of
     a real M-by-N matrix A for M >= N:
        A = Q * ( R ),
                ( 0 )
     where:
        Q is a M-by-M orthogonal matrix, stored on exit in an implicit
        form in the elements below the diagonal of the array A and in
        the elements of the array T;
        R is an upper-triangular N-by-N matrix, stored on exit in
        the elements on and above the diagonal of the array A.
        0 is a (M-N)-by-N zero matrix, 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. M >= N >= 0.

    MB

              MB is INTEGER
              The row block size to be used in the blocked QR.
              MB > 0.

    NB

              NB is INTEGER
              The column block size to be used in the blocked QR.
              N >= NB >= 1.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the elements on and above the diagonal
              of the array contain the N-by-N upper triangular matrix R;
              the elements below the diagonal represent Q by the columns
              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((M-N)/(MB-N))
              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 >= NB.

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= NB*N.
              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:

     Tall-Skinny QR (TSQR) performs QR 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 subdiagonal entries of a block of MB rows of A:
       Q(1) zeros out the subdiagonal entries of rows 1:MB of A
       Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
       Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
       . . .
     Q(1) is computed by GEQRT, 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 GEQRT.
     Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
     stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
     block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
     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 167 of file dlatsqr.f.

  subroutine slatsqr (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)

SLATSQR

Purpose:

     SLATSQR computes a blocked Tall-Skinny QR factorization of
     a real M-by-N matrix A for M >= N:
        A = Q * ( R ),
                ( 0 )
     where:
        Q is a M-by-M orthogonal matrix, stored on exit in an implicit
        form in the elements below the diagonal of the array A and in
        the elements of the array T;
        R is an upper-triangular N-by-N matrix, stored on exit in
        the elements on and above the diagonal of the array A.
        0 is a (M-N)-by-N zero matrix, 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. M >= N >= 0.

    MB

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

    NB

              NB is INTEGER
              The column block size to be used in the blocked QR.
              N >= NB >= 1.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the elements on and above the diagonal
              of the array contain the N-by-N upper triangular matrix R;
              the elements below the diagonal represent Q by the columns
              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((M-N)/(MB-N))
              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 >= NB.

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= NB*N.
              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:

     Tall-Skinny QR (TSQR) performs QR 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 subdiagonal entries of a block of MB rows of A:
       Q(1) zeros out the subdiagonal entries of rows 1:MB of A
       Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
       Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
       . . .
     Q(1) is computed by GEQRT, 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 GEQRT.
     Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
     stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
     block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
     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 167 of file slatsqr.f.

  subroutine zlatsqr (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)

ZLATSQR

Purpose:

     ZLATSQR computes a blocked Tall-Skinny QR factorization of
     a complex M-by-N matrix A for M >= N:
        A = Q * ( R ),
                ( 0 )
     where:
        Q is a M-by-M orthogonal matrix, stored on exit in an implicit
        form in the elements below the diagonal of the array A and in
        the elements of the array T;
        R is an upper-triangular N-by-N matrix, stored on exit in
        the elements on and above the diagonal of the array A.
        0 is a (M-N)-by-N zero matrix, 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. M >= N >= 0.

    MB

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

    NB

              NB is INTEGER
              The column block size to be used in the blocked QR.
              N >= NB >= 1.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the elements on and above the diagonal
              of the array contain the N-by-N upper triangular matrix R;
              the elements below the diagonal represent Q by the columns
              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((M-N)/(MB-N))
              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 >= NB.

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= NB*N.
              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:

     Tall-Skinny QR (TSQR) performs QR 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 subdiagonal entries of a block of MB rows of A:
       Q(1) zeros out the subdiagonal entries of rows 1:MB of A
       Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
       Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
       . . .
     Q(1) is computed by GEQRT, 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 GEQRT.
     Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
     stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
     block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
     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 167 of file zlatsqr.f.

Author

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