NAG Library Function Document
nag_zgelsd (f08kqc)
1
Purpose
nag_zgelsd (f08kqc) computes the minimum norm solution to a complex linear least squares problem
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_zgelsd (Nag_OrderType order,
Integer m,
Integer n,
Integer nrhs,
Complex a[],
Integer pda,
Complex b[],
Integer pdb,
double s[],
double rcond,
Integer *rank,
NagError *fail) |
|
3
Description
nag_zgelsd (f08kqc) uses the singular value decomposition (SVD) of , where is a complex 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 problem is solved in three steps:
1. |
reduce the coefficient matrix to bidiagonal form with Householder transformations, reducing the original problem into a ‘bidiagonal least squares problem’ (BLS); |
2. |
solve the BLS using a divide-and-conquer approach; |
3. |
apply back all the Householder transformations to solve the original least squares problem. |
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
http://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:
– Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– IntegerInput
-
On entry: , the number of rows of the matrix .
Constraint:
.
- 3:
– IntegerInput
-
On entry: , the number of columns of the matrix .
Constraint:
.
- 4:
– IntegerInput
-
On entry: , the number of right-hand sides, i.e., the number of columns of the matrices and .
Constraint:
.
- 5:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
a
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by coefficient matrix .
On exit: the contents of
a are destroyed.
- 6:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraints:
- if ,
;
- if , .
- 7:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
b
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
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.
- 8:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 9:
– doubleOutput
-
Note: the dimension,
dim, of the array
s
must be at least
.
On exit: the singular values of in decreasing order.
- 10:
– doubleInput
-
On entry: used to determine the effective rank of . Singular values are treated as zero. If , machine precision is used instead.
- 11:
– Integer *Output
-
On exit: the effective rank of , i.e., the number of singular values which are greater than .
- 12:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CONVERGENCE
-
The algorithm for computing the SVD failed to converge; off-diagonal elements of an intermediate bidiagonal form did not converge to zero.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INT_3
-
On entry, , and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
See Section 4.5 of
Anderson et al. (1999) for details.
8
Parallelism and Performance
nag_zgelsd (f08kqc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgelsd (f08kqc) 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 function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The real analogue of this function is
nag_dgelsd (f08kcc).
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 .
10.1
Program Text
Program Text (f08kqce.c)
10.2
Program Data
Program Data (f08kqce.d)
10.3
Program Results
Program Results (f08kqce.r)