lamtsqr(3)

NAME
lamtsqr - lamtsqr: multiply by Q from latsqr

SYNOPSIS

  Functions

subroutine clamtsqr (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info)
CLAMTSQR
subroutine dlamtsqr (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info)
DLAMTSQR
subroutine slamtsqr (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info)
SLAMTSQR
subroutine zlamtsqr (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info)
ZLAMTSQR

Detailed Description

Function Documentation

  subroutine clamtsqr (character side, character trans, integer m, integer n, integer k, integer mb, integer nb, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( ldc, * ) c, integer ldc, complex, dimension( * ) work, integer lwork, integer info)

CLAMTSQR

Purpose:

    !>
    !>      CLAMTSQR overwrites the general complex M-by-N matrix C with
    !>
    !>
    !>                 SIDE = 'L'     SIDE = 'R'
    !> TRANS = 'N':      Q * C          C * Q
    !> TRANS = 'C':      Q**H * C       C * Q**H
    !>      where Q is a complex unitary matrix defined as the product
    !>      of blocked elementary reflectors computed by tall skinny
    !>      QR factorization (CLATSQR)
    !>

Parameters
    SIDE

    !>          SIDE is CHARACTER*1
    !>          = 'L': apply Q or Q**H from the Left;
    !>          = 'R': apply Q or Q**H from the Right.
    !>

    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N':  No transpose, apply Q;
    !>          = 'C':  Conjugate Transpose, apply Q**H.
    !>

    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 C. N >= 0.
    !>

    K

    !>          K is INTEGER
    !>          The number of elementary reflectors whose product defines
    !>          the matrix Q. M >= K >= 0;
    !>
    !>

    MB

    !>          MB is INTEGER
    !>          The block size to be used in the blocked QR.
    !>          MB > N. (must be the same as CLATSQR)
    !>

    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,K)
    !>          The i-th column must contain the vector which defines the
    !>          blockedelementary reflector H(i), for i = 1,2,...,k, as
    !>          returned by CLATSQR in the first k columns of
    !>          its array argument A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.
    !>          If SIDE = 'L', LDA >= max(1,M);
    !>          if SIDE = 'R', LDA >= max(1,N).
    !>

    T

    !>          T is COMPLEX array, dimension
    !>          ( N * Number of blocks(CEIL(M-K/MB-K)),
    !>          The blocked upper triangular block reflectors stored in compact form
    !>          as a sequence of upper triangular blocks.  See below
    !>          for further details.
    !>

    LDT

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

    C

    !>          C is COMPLEX array, dimension (LDC,N)
    !>          On entry, the M-by-N matrix C.
    !>          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.
    !>

    LDC

    !>          LDC is INTEGER
    !>          The leading dimension of the array C. LDC >= max(1,M).
    !>

    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.
    !>          If MIN(M,N,K) = 0, LWORK >= 1.
    !>          If SIDE = 'L', LWORK >= max(1,N*NB).
    !>          If SIDE = 'R', LWORK >= max(1,MB*NB).
    !>
    !>          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:

    !> Tall-Skinny QR (TSQR) performs QR by a sequence of unitary transformations,
    !> representing Q as a product of other unitary 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
    !>

  subroutine dlamtsqr (character side, character trans, integer m, integer n, integer k, integer mb, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer lwork, integer info)

DLAMTSQR

Purpose:

    !>
    !>      DLAMTSQR overwrites the general real M-by-N matrix C with
    !>
    !>
    !>                 SIDE = 'L'     SIDE = 'R'
    !> TRANS = 'N':      Q * C          C * Q
    !> TRANS = 'T':      Q**T * C       C * Q**T
    !>      where Q is a real orthogonal matrix defined as the product
    !>      of blocked elementary reflectors computed by tall skinny
    !>      QR factorization (DLATSQR)
    !>

Parameters
    SIDE

    !>          SIDE is CHARACTER*1
    !>          = 'L': apply Q or Q**T from the Left;
    !>          = 'R': apply Q or Q**T from the Right.
    !>

    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N':  No transpose, apply Q;
    !>          = 'T':  Transpose, apply Q**T.
    !>

    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 C. N >= 0.
    !>

    K

    !>          K is INTEGER
    !>          The number of elementary reflectors whose product defines
    !>          the matrix Q. M >= K >= 0;
    !>
    !>

    MB

    !>          MB is INTEGER
    !>          The block size to be used in the blocked QR.
    !>          MB > N. (must be the same as DLATSQR)
    !>

    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,K)
    !>          The i-th column must contain the vector which defines the
    !>          blockedelementary reflector H(i), for i = 1,2,...,k, as
    !>          returned by DLATSQR in the first k columns of
    !>          its array argument A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.
    !>          If SIDE = 'L', LDA >= max(1,M);
    !>          if SIDE = 'R', LDA >= max(1,N).
    !>

    T

    !>          T is DOUBLE PRECISION array, dimension
    !>          ( N * Number of blocks(CEIL(M-K/MB-K)),
    !>          The blocked upper triangular block reflectors stored in compact form
    !>          as a sequence of upper triangular blocks.  See below
    !>          for further details.
    !>

    LDT

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

    C

    !>          C is DOUBLE PRECISION array, dimension (LDC,N)
    !>          On entry, the M-by-N matrix C.
    !>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
    !>

    LDC

    !>          LDC is INTEGER
    !>          The leading dimension of the array C. LDC >= max(1,M).
    !>

    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.
    !>          If MIN(M,N,K) = 0, LWORK >= 1.
    !>          If SIDE = 'L', LWORK >= max(1,N*NB).
    !>          If SIDE = 'R', LWORK >= max(1,MB*NB).
    !>
    !>          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:

    !> 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
    !>

  subroutine slamtsqr (character side, character trans, integer m, integer n, integer k, integer mb, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( ldt, * ) t, integer ldt, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer lwork, integer info)

SLAMTSQR

Purpose:

    !>
    !>      SLAMTSQR overwrites the general real M-by-N matrix C with
    !>
    !>
    !>                 SIDE = 'L'     SIDE = 'R'
    !> TRANS = 'N':      Q * C          C * Q
    !> TRANS = 'T':      Q**T * C       C * Q**T
    !>      where Q is a real orthogonal matrix defined as the product
    !>      of blocked elementary reflectors computed by tall skinny
    !>      QR factorization (SLATSQR)
    !>

Parameters
    SIDE

    !>          SIDE is CHARACTER*1
    !>          = 'L': apply Q or Q**T from the Left;
    !>          = 'R': apply Q or Q**T from the Right.
    !>

    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N':  No transpose, apply Q;
    !>          = 'T':  Transpose, apply Q**T.
    !>

    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 C. N >= 0.
    !>

    K

    !>          K is INTEGER
    !>          The number of elementary reflectors whose product defines
    !>          the matrix Q. M >= K >= 0;
    !>
    !>

    MB

    !>          MB is INTEGER
    !>          The block size to be used in the blocked QR.
    !>          MB > N. (must be the same as SLATSQR)
    !>

    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,K)
    !>          The i-th column must contain the vector which defines the
    !>          blockedelementary reflector H(i), for i = 1,2,...,k, as
    !>          returned by SLATSQR in the first k columns of
    !>          its array argument A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.
    !>          If SIDE = 'L', LDA >= max(1,M);
    !>          if SIDE = 'R', LDA >= max(1,N).
    !>

    T

    !>          T is REAL array, dimension
    !>          ( N * Number of blocks(CEIL(M-K/MB-K)),
    !>          The blocked upper triangular block reflectors stored in compact form
    !>          as a sequence of upper triangular blocks.  See below
    !>          for further details.
    !>

    LDT

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

    C

    !>          C is REAL array, dimension (LDC,N)
    !>          On entry, the M-by-N matrix C.
    !>          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
    !>

    LDC

    !>          LDC is INTEGER
    !>          The leading dimension of the array C. LDC >= max(1,M).
    !>

    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.
    !>          If MIN(M,N,K) = 0, LWORK >= 1.
    !>          If SIDE = 'L', LWORK >= max(1,N*NB).
    !>          If SIDE = 'R', LWORK >= max(1,MB*NB).
    !>
    !>          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:

    !> 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
    !>

  subroutine zlamtsqr (character side, character trans, integer m, integer n, integer k, integer mb, integer nb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( ldc, * ) c, integer ldc, complex*16, dimension( * ) work, integer lwork, integer info)

ZLAMTSQR

Purpose:

    !>
    !>      ZLAMTSQR overwrites the general complex M-by-N matrix C with
    !>
    !>
    !>                 SIDE = 'L'     SIDE = 'R'
    !> TRANS = 'N':      Q * C          C * Q
    !> TRANS = 'C':      Q**H * C       C * Q**H
    !>      where Q is a complex unitary matrix defined as the product
    !>      of blocked elementary reflectors computed by tall skinny
    !>      QR factorization (ZLATSQR)
    !>

Parameters
    SIDE

    !>          SIDE is CHARACTER*1
    !>          = 'L': apply Q or Q**H from the Left;
    !>          = 'R': apply Q or Q**H from the Right.
    !>

    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N':  No transpose, apply Q;
    !>          = 'C':  Conjugate Transpose, apply Q**H.
    !>

    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 C. N >= 0.
    !>

    K

    !>          K is INTEGER
    !>          The number of elementary reflectors whose product defines
    !>          the matrix Q. M >= K >= 0;
    !>
    !>

    MB

    !>          MB is INTEGER
    !>          The block size to be used in the blocked QR.
    !>          MB > N. (must be the same as ZLATSQR)
    !>

    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,K)
    !>          The i-th column must contain the vector which defines the
    !>          blockedelementary reflector H(i), for i = 1,2,...,k, as
    !>          returned by ZLATSQR in the first k columns of
    !>          its array argument A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.
    !>          If SIDE = 'L', LDA >= max(1,M);
    !>          if SIDE = 'R', LDA >= max(1,N).
    !>

    T

    !>          T is COMPLEX*16 array, dimension
    !>          ( N * Number of blocks(CEIL(M-K/MB-K)),
    !>          The blocked upper triangular block reflectors stored in compact form
    !>          as a sequence of upper triangular blocks.  See below
    !>          for further details.
    !>

    LDT

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

    C

    !>          C is COMPLEX*16 array, dimension (LDC,N)
    !>          On entry, the M-by-N matrix C.
    !>          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.
    !>

    LDC

    !>          LDC is INTEGER
    !>          The leading dimension of the array C. LDC >= max(1,M).
    !>

    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.
    !>          If MIN(M,N,K) = 0, LWORK >= 1.
    !>          If SIDE = 'L', LWORK >= max(1,N*NB).
    !>          If SIDE = 'R', LWORK >= max(1,MB*NB).
    !>
    !>          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:

    !> Tall-Skinny QR (TSQR) performs QR by a sequence of unitary transformations,
    !> representing Q as a product of other unitary 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
    !>

Author

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