table of contents
| latsqr(3) | LAPACK | 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
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
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
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
Author¶
Generated automatically by Doxygen for LAPACK from the source code.
| Wed Feb 7 2024 11:30:40 | Version 3.12.0 |