NAG FL Interface
f08knf (zgelss)
1
Purpose
f08knf computes the minimum norm solution to a complex linear least squares problem
2
Specification
Fortran Interface
Subroutine f08knf ( |
m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, info) |
Integer, Intent (In) |
:: |
m, n, nrhs, lda, ldb, lwork |
Integer, Intent (Out) |
:: |
rank, info |
Real (Kind=nag_wp), Intent (In) |
:: |
rcond |
Real (Kind=nag_wp), Intent (Inout) |
:: |
s(*), rwork(*) |
Complex (Kind=nag_wp), Intent (Inout) |
:: |
a(lda,*), b(ldb,*) |
Complex (Kind=nag_wp), Intent (Out) |
:: |
work(max(1,lwork)) |
|
C Header Interface
#include <nag.h>
void |
f08knf_ (const Integer *m, const Integer *n, const Integer *nrhs, Complex a[], const Integer *lda, Complex b[], const Integer *ldb, double s[], const double *rcond, Integer *rank, Complex work[], const Integer *lwork, double rwork[], Integer *info) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08knf_ (const Integer &m, const Integer &n, const Integer &nrhs, Complex a[], const Integer &lda, Complex b[], const Integer &ldb, double s[], const double &rcond, Integer &rank, Complex work[], const Integer &lwork, double rwork[], Integer &info) |
}
|
The routine may be called by the names f08knf, nagf_lapackeig_zgelss or its LAPACK name zgelss.
3
Description
f08knf uses the singular value decomposition (SVD) of , where is an by matrix which may be rank-deficient.
Several right-hand side vectors and solution vectors can be handled in a single call; they are stored as the columns of the by right-hand side matrix and the by solution matrix .
The effective rank of
is determined by treating as zero those singular values which are less than
rcond times the largest singular value.
4
References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
-
1:
– Integer
Input
-
On entry: , the number of rows of the matrix .
Constraint:
.
-
2:
– Integer
Input
-
On entry: , the number of columns of the matrix .
Constraint:
.
-
3:
– Integer
Input
-
On entry: , the number of right-hand sides, i.e., the number of columns of the matrices and .
Constraint:
.
-
4:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
a
must be at least
.
On entry: the by matrix .
On exit: the first rows of are overwritten with its right singular vectors, stored row-wise.
-
5:
– Integer
Input
-
On entry: the first dimension of the array
a as declared in the (sub)program from which
f08knf is called.
Constraint:
.
-
6:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
b
must be at least
.
On entry: the by right-hand side matrix .
On exit:
b is overwritten by the
by
solution matrix
. If
and
, the residual sum of squares for the solution in the
th column is given by the sum of squares of the modulus of elements
in that column.
-
7:
– Integer
Input
-
On entry: the first dimension of the array
b as declared in the (sub)program from which
f08knf is called.
Constraint:
.
-
8:
– Real (Kind=nag_wp) array
Output
-
Note: the dimension of the array
s
must be at least
.
On exit: the singular values of in decreasing order.
-
9:
– Real (Kind=nag_wp)
Input
-
On entry: used to determine the effective rank of . Singular values are treated as zero. If , machine precision is used instead.
-
10:
– Integer
Output
-
On exit: the effective rank of , i.e., the number of singular values which are greater than .
-
11:
– Complex (Kind=nag_wp) array
Workspace
-
On exit: if
, the real part of
contains the minimum value of
lwork required for optimal performance.
-
12:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08knf is called.
If
, 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.
Suggested value:
for optimal performance,
lwork should generally be larger. Consider increasing
lwork by at least
, where
is the optimal
block size.
Constraint:
and .
-
13:
– Real (Kind=nag_wp) array
Workspace
-
Note: the dimension of the array
rwork
must be at least
.
-
14:
– Integer
Output
On exit:
unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
-
The algorithm for computing the SVD failed to converge; off-diagonal elements of an intermediate bidiagonal form did not converge to zero.
7
Accuracy
See Section 4.5 of
Anderson et al. (1999) for details.
8
Parallelism and Performance
f08knf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08knf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The real analogue of this routine is
f08kaf.
10
Example
This example solves the linear least squares problem
for the solution,
, of minimum norm, where
and
A tolerance of is used to determine the effective rank of .
Note that the block size (NB) of assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
10.1
Program Text
10.2
Program Data
10.3
Program Results