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getsls - getsls: least squares using tall-skinny QR/LQ
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Detailed Description
Function Documentation
subroutinecgetsls(charactertrans,integerm,integern,integernrhs,complex,dimension(lda,*)a,integerlda,complex,dimension(ldb,*)b,integerldb,complex,dimension(*)work,integerlwork,integerinfo)CGETSLSPurpose:
CGETSLS solves overdetermined or underdetermined complex linear systems
involving an M-by-N matrix A, using a tall skinny QR or short wide LQ
factorization of A.
It is assumed that A has full rank, and only a rudimentary protection
against rank-deficient matrices is provided. This subroutine only detects
exact rank-deficiency, where a diagonal element of the triangular factor
of A is exactly zero.
It is conceivable for one (or more) of the diagonal elements of the triangular
factor of A to be subnormally tiny numbers without this subroutine signalling
an error. The solutions computed for such almost-rank-deficient matrices may
be less accurate due to a loss of numerical precision.
The following options are provided:
1. If TRANS = 'N' and m >= n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A*X ||.
2. If TRANS = 'N' and m < n: find the minimum norm solution of
an underdetermined system A * X = B.
3. If TRANS = 'C' and m >= n: find the minimum norm solution of
an undetermined system A**T * X = B.
4. If TRANS = 'C' and m < n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A**T * X ||.
Several right hand side vectors b and solution vectors x can be
handled in a single call; they are stored as the columns of the
M-by-NRHS right hand side matrix B and the N-by-NRHS solution
matrix X.
ParametersTRANS
TRANS is CHARACTER*1
= 'N': the linear system involves A;
= 'C': the linear system involves A**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 A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of
columns of the matrices B and X. NRHS >=0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
A is overwritten by details of its QR or LQ
factorization as returned by CGEQR or CGELQ.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B
B is COMPLEX array, dimension (LDB,NRHS)
On entry, the matrix B of right hand side vectors, stored
columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
if TRANS = 'C'.
On exit, if INFO = 0, B is overwritten by the solution
vectors, stored columnwise:
if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
squares solution vectors.
if TRANS = 'N' and m < n, rows 1 to N of B contain the
minimum norm solution vectors;
if TRANS = 'C' and m >= n, rows 1 to M of B contain the
minimum norm solution vectors;
if TRANS = 'C' and m < n, rows 1 to M of B contain the
least squares solution vectors.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= MAX(1,M,N).
WORK
(workspace) COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
or optimal, if query was assumed) LWORK.
See LWORK for details.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= 1.
If LWORK = -1 or -2, then a workspace query is assumed.
If LWORK = -1, the routine calculates optimal size of WORK for the
optimal performance and returns this value in WORK(1).
If LWORK = -2, the routine calculates minimal size of WORK and
returns this value in WORK(1).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of the
triangular factor of A is exactly zero, so that A does not have
full rank; the least squares solution could not be
computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutinedgetsls(charactertrans,integerm,integern,integernrhs,doubleprecision,dimension(lda,*)a,integerlda,doubleprecision,dimension(ldb,*)b,integerldb,doubleprecision,dimension(*)work,integerlwork,integerinfo)DGETSLSPurpose:
DGETSLS solves overdetermined or underdetermined real linear systems
involving an M-by-N matrix A, using a tall skinny QR or short wide LQ
factorization of A.
It is assumed that A has full rank, and only a rudimentary protection
against rank-deficient matrices is provided. This subroutine only detects
exact rank-deficiency, where a diagonal element of the triangular factor
of A is exactly zero.
It is conceivable for one (or more) of the diagonal elements of the triangular
factor of A to be subnormally tiny numbers without this subroutine signalling
an error. The solutions computed for such almost-rank-deficient matrices may
be less accurate due to a loss of numerical precision.
The following options are provided:
1. If TRANS = 'N' and m >= n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A*X ||.
2. If TRANS = 'N' and m < n: find the minimum norm solution of
an underdetermined system A * X = B.
3. If TRANS = 'T' and m >= n: find the minimum norm solution of
an undetermined system A**T * X = B.
4. If TRANS = 'T' and m < n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A**T * X ||.
Several right hand side vectors b and solution vectors x can be
handled in a single call; they are stored as the columns of the
M-by-NRHS right hand side matrix B and the N-by-NRHS solution
matrix X.
ParametersTRANS
TRANS is CHARACTER*1
= 'N': the linear system involves A;
= 'T': the linear system involves A**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 A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of
columns of the matrices B and X. NRHS >=0.
A
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
A is overwritten by details of its QR or LQ
factorization as returned by DGEQR or DGELQ.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B
B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the matrix B of right hand side vectors, stored
columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
if TRANS = 'T'.
On exit, if INFO = 0, B is overwritten by the solution
vectors, stored columnwise:
if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
squares solution vectors.
if TRANS = 'N' and m < n, rows 1 to N of B contain the
minimum norm solution vectors;
if TRANS = 'T' and m >= n, rows 1 to M of B contain the
minimum norm solution vectors;
if TRANS = 'T' and m < n, rows 1 to M of B contain the
least squares solution vectors.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= MAX(1,M,N).
WORK
(workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
or optimal, if query was assumed) LWORK.
See LWORK for details.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= 1.
If LWORK = -1 or -2, then a workspace query is assumed.
If LWORK = -1, the routine calculates optimal size of WORK for the
optimal performance and returns this value in WORK(1).
If LWORK = -2, the routine calculates minimal size of WORK and
returns this value in WORK(1).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of the
triangular factor of A is exactly zero, so that A does not have
full rank; the least squares solution could not be
computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutinesgetsls(charactertrans,integerm,integern,integernrhs,real,dimension(lda,*)a,integerlda,real,dimension(ldb,*)b,integerldb,real,dimension(*)work,integerlwork,integerinfo)SGETSLSPurpose:
SGETSLS solves overdetermined or underdetermined real linear systems
involving an M-by-N matrix A, using a tall skinny QR or short wide LQ
factorization of A.
It is assumed that A has full rank, and only a rudimentary protection
against rank-deficient matrices is provided. This subroutine only detects
exact rank-deficiency, where a diagonal element of the triangular factor
of A is exactly zero.
It is conceivable for one (or more) of the diagonal elements of the triangular
factor of A to be subnormally tiny numbers without this subroutine signalling
an error. The solutions computed for such almost-rank-deficient matrices may
be less accurate due to a loss of numerical precision.
The following options are provided:
1. If TRANS = 'N' and m >= n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A*X ||.
2. If TRANS = 'N' and m < n: find the minimum norm solution of
an underdetermined system A * X = B.
3. If TRANS = 'T' and m >= n: find the minimum norm solution of
an undetermined system A**T * X = B.
4. If TRANS = 'T' and m < n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A**T * X ||.
Several right hand side vectors b and solution vectors x can be
handled in a single call; they are stored as the columns of the
M-by-NRHS right hand side matrix B and the N-by-NRHS solution
matrix X.
ParametersTRANS
TRANS is CHARACTER*1
= 'N': the linear system involves A;
= 'T': the linear system involves A**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 A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of
columns of the matrices B and X. NRHS >=0.
A
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
A is overwritten by details of its QR or LQ
factorization as returned by SGEQR or SGELQ.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B
B is REAL array, dimension (LDB,NRHS)
On entry, the matrix B of right hand side vectors, stored
columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
if TRANS = 'T'.
On exit, if INFO = 0, B is overwritten by the solution
vectors, stored columnwise:
if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
squares solution vectors.
if TRANS = 'N' and m < n, rows 1 to N of B contain the
minimum norm solution vectors;
if TRANS = 'T' and m >= n, rows 1 to M of B contain the
minimum norm solution vectors;
if TRANS = 'T' and m < n, rows 1 to M of B contain the
least squares solution vectors.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= MAX(1,M,N).
WORK
(workspace) REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
or optimal, if query was assumed) LWORK.
See LWORK for details.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= 1.
If LWORK = -1 or -2, then a workspace query is assumed.
If LWORK = -1, the routine calculates optimal size of WORK for the
optimal performance and returns this value in WORK(1).
If LWORK = -2, the routine calculates minimal size of WORK and
returns this value in WORK(1).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of the
triangular factor of A is exactly zero, so that A does not have
full rank; the least squares solution could not be
computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutinezgetsls(charactertrans,integerm,integern,integernrhs,complex*16,dimension(lda,*)a,integerlda,complex*16,dimension(ldb,*)b,integerldb,complex*16,dimension(*)work,integerlwork,integerinfo)ZGETSLSPurpose:
ZGETSLS solves overdetermined or underdetermined complex linear systems
involving an M-by-N matrix A, using a tall skinny QR or short wide LQ
factorization of A.
It is assumed that A has full rank, and only a rudimentary protection
against rank-deficient matrices is provided. This subroutine only detects
exact rank-deficiency, where a diagonal element of the triangular factor
of A is exactly zero.
It is conceivable for one (or more) of the diagonal elements of the triangular
factor of A to be subnormally tiny numbers without this subroutine signalling
an error. The solutions computed for such almost-rank-deficient matrices may
be less accurate due to a loss of numerical precision.
The following options are provided:
1. If TRANS = 'N' and m >= n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A*X ||.
2. If TRANS = 'N' and m < n: find the minimum norm solution of
an underdetermined system A * X = B.
3. If TRANS = 'C' and m >= n: find the minimum norm solution of
an undetermined system A**T * X = B.
4. If TRANS = 'C' and m < n: find the least squares solution of
an overdetermined system, i.e., solve the least squares problem
minimize || B - A**T * X ||.
Several right hand side vectors b and solution vectors x can be
handled in a single call; they are stored as the columns of the
M-by-NRHS right hand side matrix B and the N-by-NRHS solution
matrix X.
ParametersTRANS
TRANS is CHARACTER*1
= 'N': the linear system involves A;
= 'C': the linear system involves A**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 A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of
columns of the matrices B and X. NRHS >=0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
A is overwritten by details of its QR or LQ
factorization as returned by ZGEQR or ZGELQ.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B
B is COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the matrix B of right hand side vectors, stored
columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
if TRANS = 'C'.
On exit, if INFO = 0, B is overwritten by the solution
vectors, stored columnwise:
if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
squares solution vectors.
if TRANS = 'N' and m < n, rows 1 to N of B contain the
minimum norm solution vectors;
if TRANS = 'C' and m >= n, rows 1 to M of B contain the
minimum norm solution vectors;
if TRANS = 'C' and m < n, rows 1 to M of B contain the
least squares solution vectors.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= MAX(1,M,N).
WORK
(workspace) COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
or optimal, if query was assumed) LWORK.
See LWORK for details.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= 1.
If LWORK = -1 or -2, then a workspace query is assumed.
If LWORK = -1, the routine calculates optimal size of WORK for the
optimal performance and returns this value in WORK(1).
If LWORK = -2, the routine calculates minimal size of WORK and
returns this value in WORK(1).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element of the
triangular factor of A is exactly zero, so that A does not have
full rank; the least squares solution could not be
computed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Name
getsls - getsls: least squares using tall-skinny QR/LQ
Synopsis
Functions
subroutine cgetsls (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
CGETSLS
subroutine dgetsls (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
DGETSLS
subroutine sgetsls (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
SGETSLS
subroutine zgetsls (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
ZGETSLS
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