SGELSX(3F) SGELSX(3F)
SGELSX - compute the minimum-norm solution to a real linear least squares
problem
SUBROUTINE SGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK,
INFO )
INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
REAL RCOND
INTEGER JPVT( * )
REAL A( LDA, * ), B( LDB, * ), WORK( * )
SGELSX computes the minimum-norm solution to a real linear least squares
problem:
minimize || A * X - B ||
using a complete orthogonal factorization of A. A is an M-by-N matrix
which may be rank-deficient.
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.
The routine first computes a QR factorization with column pivoting:
A * P = Q * [ R11 R12 ]
[ 0 R22 ]
with R11 defined as the largest leading submatrix whose estimated
condition number is less than 1/RCOND. The order of R11, RANK, is the
effective rank of A.
Then, R22 is considered to be negligible, and R12 is annihilated by
orthogonal transformations from the right, arriving at the complete
orthogonal factorization:
A * P = Q * [ T11 0 ] * Z
[ 0 0 ]
The minimum-norm solution is then
X = P * Z' [ inv(T11)*Q1'*B ]
[ 0 ]
where Q1 consists of the first RANK columns of Q.
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
Page 1
SGELSX(3F) SGELSX(3F)
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of
matrices B and X. NRHS >= 0.
A (input/output) REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, A has been overwritten
by details of its complete orthogonal factorization.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (input/output) REAL array, dimension (LDB,NRHS)
On entry, the M-by-NRHS right hand side matrix B. On exit, the
N-by-NRHS solution matrix X. If m >= n and RANK = n, the
residual sum-of-squares for the solution in the i-th column is
given by the sum of squares of elements N+1:M in that column.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,M,N).
JPVT (input/output) INTEGER array, dimension (N)
On entry, if JPVT(i) .ne. 0, the i-th column of A is an initial
column, otherwise it is a free column. Before the QR
factorization of A, all initial columns are permuted to the
leading positions; only the remaining free columns are moved as a
result of column pivoting during the factorization. On exit, if
JPVT(i) = k, then the i-th column of A*P was the k-th column of
A.
RCOND (input) REAL
RCOND is used to determine the effective rank of A, which is
defined as the order of the largest leading triangular submatrix
R11 in the QR factorization with pivoting of A, whose estimated
condition number < 1/RCOND.
RANK (output) INTEGER
The effective rank of A, i.e., the order of the submatrix R11.
This is the same as the order of the submatrix T11 in the
complete orthogonal factorization of A.
WORK (workspace) REAL array, dimension
(max( min(M,N)+3*N, 2*min(M,N)+NRHS )),
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
SGELSX(3F) SGELSX(3F)
SGELSX - compute the minimum-norm solution to a real linear least squares
problem
SUBROUTINE SGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK,
INFO )
INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
REAL RCOND
INTEGER JPVT( * )
REAL A( LDA, * ), B( LDB, * ), WORK( * )
SGELSX computes the minimum-norm solution to a real linear least squares
problem:
minimize || A * X - B ||
using a complete orthogonal factorization of A. A is an M-by-N matrix
which may be rank-deficient.
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.
The routine first computes a QR factorization with column pivoting:
A * P = Q * [ R11 R12 ]
[ 0 R22 ]
with R11 defined as the largest leading submatrix whose estimated
condition number is less than 1/RCOND. The order of R11, RANK, is the
effective rank of A.
Then, R22 is considered to be negligible, and R12 is annihilated by
orthogonal transformations from the right, arriving at the complete
orthogonal factorization:
A * P = Q * [ T11 0 ] * Z
[ 0 0 ]
The minimum-norm solution is then
X = P * Z' [ inv(T11)*Q1'*B ]
[ 0 ]
where Q1 consists of the first RANK columns of Q.
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
Page 1
SGELSX(3F) SGELSX(3F)
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of
matrices B and X. NRHS >= 0.
A (input/output) REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, A has been overwritten
by details of its complete orthogonal factorization.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (input/output) REAL array, dimension (LDB,NRHS)
On entry, the M-by-NRHS right hand side matrix B. On exit, the
N-by-NRHS solution matrix X. If m >= n and RANK = n, the
residual sum-of-squares for the solution in the i-th column is
given by the sum of squares of elements N+1:M in that column.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,M,N).
JPVT (input/output) INTEGER array, dimension (N)
On entry, if JPVT(i) .ne. 0, the i-th column of A is an initial
column, otherwise it is a free column. Before the QR
factorization of A, all initial columns are permuted to the
leading positions; only the remaining free columns are moved as a
result of column pivoting during the factorization. On exit, if
JPVT(i) = k, then the i-th column of A*P was the k-th column of
A.
RCOND (input) REAL
RCOND is used to determine the effective rank of A, which is
defined as the order of the largest leading triangular submatrix
R11 in the QR factorization with pivoting of A, whose estimated
condition number < 1/RCOND.
RANK (output) INTEGER
The effective rank of A, i.e., the order of the submatrix R11.
This is the same as the order of the submatrix T11 in the
complete orthogonal factorization of A.
WORK (workspace) REAL array, dimension
(max( min(M,N)+3*N, 2*min(M,N)+NRHS )),
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
PPPPaaaaggggeeee 2222 [ Back ]
|
