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gelst - gelst: least squares using QR/LQ with T matrix
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Detailed Description
Function Documentation
subroutinecgelst(charactertrans,integerm,integern,integernrhs,complex,dimension(lda,*)a,integerlda,complex,dimension(ldb,*)b,integerldb,complex,dimension(*)work,integerlwork,integerinfo)CGELSTsolvesoverdeterminedorunderdeterminedsystemsforGEmatricesusingQRorLQfactorizationwithcompactWYrepresentationofQ.Purpose:
CGELST solves overdetermined or underdetermined real linear systems
involving an M-by-N matrix A, or its conjugate-transpose, using a QR
or LQ factorization of A with compact WY representation of Q.
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 underdetermined 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,
if M >= N, A is overwritten by details of its QR
factorization as returned by CGEQRT;
if M < N, A is overwritten by details of its LQ
factorization as returned by CGELQT.
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; the residual sum of squares for the
solution in each column is given by the sum of squares of
modulus of elements N+1 to M in that column;
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; the residual sum of squares
for the solution in each column is given by the sum of
squares of the modulus of elements M+1 to N in that column.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= MAX(1,M,N).
WORK
WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
LWORK >= max( 1, MN + max( MN, NRHS ) ).
For optimal performance,
LWORK >= max( 1, (MN + max( MN, NRHS ))*NB ).
where MN = min(M,N) and NB is the optimum block size.
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
> 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.
Contributors:
November 2022, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
subroutinedgelst(charactertrans,integerm,integern,integernrhs,doubleprecision,dimension(lda,*)a,integerlda,doubleprecision,dimension(ldb,*)b,integerldb,doubleprecision,dimension(*)work,integerlwork,integerinfo)DGELSTsolvesoverdeterminedorunderdeterminedsystemsforGEmatricesusingQRorLQfactorizationwithcompactWYrepresentationofQ.Purpose:
DGELST solves overdetermined or underdetermined real linear systems
involving an M-by-N matrix A, or its transpose, using a QR or LQ
factorization of A with compact WY representation of Q.
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 underdetermined 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,
if M >= N, A is overwritten by details of its QR
factorization as returned by DGEQRT;
if M < N, A is overwritten by details of its LQ
factorization as returned by DGELQT.
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; the residual sum of squares for the
solution in each column is given by the sum of squares of
elements N+1 to M in that column;
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; the residual sum of squares
for the solution in each column is given by the sum of
squares of elements M+1 to N in that column.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= MAX(1,M,N).
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
LWORK >= max( 1, MN + max( MN, NRHS ) ).
For optimal performance,
LWORK >= max( 1, (MN + max( MN, NRHS ))*NB ).
where MN = min(M,N) and NB is the optimum block size.
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
> 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.
Contributors:
November 2022, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
subroutinesgelst(charactertrans,integerm,integern,integernrhs,real,dimension(lda,*)a,integerlda,real,dimension(ldb,*)b,integerldb,real,dimension(*)work,integerlwork,integerinfo)SGELSTsolvesoverdeterminedorunderdeterminedsystemsforGEmatricesusingQRorLQfactorizationwithcompactWYrepresentationofQ.Purpose:
SGELST solves overdetermined or underdetermined real linear systems
involving an M-by-N matrix A, or its transpose, using a QR or LQ
factorization of A with compact WY representation of Q.
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 underdetermined 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,
if M >= N, A is overwritten by details of its QR
factorization as returned by SGEQRT;
if M < N, A is overwritten by details of its LQ
factorization as returned by SGELQT.
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; the residual sum of squares for the
solution in each column is given by the sum of squares of
elements N+1 to M in that column;
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; the residual sum of squares
for the solution in each column is given by the sum of
squares of elements M+1 to N in that column.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= MAX(1,M,N).
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
LWORK >= max( 1, MN + max( MN, NRHS ) ).
For optimal performance,
LWORK >= max( 1, (MN + max( MN, NRHS ))*NB ).
where MN = min(M,N) and NB is the optimum block size.
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
> 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.
Contributors:
November 2022, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
subroutinezgelst(charactertrans,integerm,integern,integernrhs,complex*16,dimension(lda,*)a,integerlda,complex*16,dimension(ldb,*)b,integerldb,complex*16,dimension(*)work,integerlwork,integerinfo)ZGELSTsolvesoverdeterminedorunderdeterminedsystemsforGEmatricesusingQRorLQfactorizationwithcompactWYrepresentationofQ.Purpose:
ZGELST solves overdetermined or underdetermined real linear systems
involving an M-by-N matrix A, or its conjugate-transpose, using a QR
or LQ factorization of A with compact WY representation of Q.
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 underdetermined 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,
if M >= N, A is overwritten by details of its QR
factorization as returned by ZGEQRT;
if M < N, A is overwritten by details of its LQ
factorization as returned by ZGELQT.
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; the residual sum of squares for the
solution in each column is given by the sum of squares of
modulus of elements N+1 to M in that column;
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; the residual sum of squares
for the solution in each column is given by the sum of
squares of the modulus of elements M+1 to N in that column.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= MAX(1,M,N).
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK.
LWORK >= max( 1, MN + max( MN, NRHS ) ).
For optimal performance,
LWORK >= max( 1, (MN + max( MN, NRHS ))*NB ).
where MN = min(M,N) and NB is the optimum block size.
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
> 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.
Contributors:
November 2022, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
Name
gelst - gelst: least squares using QR/LQ with T matrix
Synopsis
Functions
subroutine cgelst (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
CGELSTsolvesoverdeterminedorunderdeterminedsystemsforGEmatricesusingQRorLQfactorizationwithcompactWYrepresentationofQ.
subroutine dgelst (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
DGELSTsolvesoverdeterminedorunderdeterminedsystemsforGEmatricesusingQRorLQfactorizationwithcompactWYrepresentationofQ.
subroutine sgelst (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
SGELSTsolvesoverdeterminedorunderdeterminedsystemsforGEmatricesusingQRorLQfactorizationwithcompactWYrepresentationofQ.
subroutine zgelst (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
ZGELSTsolvesoverdeterminedorunderdeterminedsystemsforGEmatricesusingQRorLQfactorizationwithcompactWYrepresentationofQ.
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