latsqr - Man Page

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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