Man page - latsqr(3)

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Manual

latsqr

NAME
SYNOPSIS
Functions
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)
subroutine dlatsqr (integer m, integer n, integer mb, integer nb, doubleprecision, dimension( lda, * ) a, integer lda, double precision,dimension( ldt, * ) t, integer ldt, double precision, dimension( * )work, integer lwork, integer info)
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)
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, integerlwork, integer info)
Author

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))
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 >= NB*N, 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:

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, doubleprecision, 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))
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 >= NB*N, 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:

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))
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 >= NB*N, 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:

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, integerlwork, 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))
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 >= NB*N, 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:

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

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