NAG Library Function Document

nag_zgelss (f08knc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_zgelss (f08knc) computes the minimum norm solution to a complex linear least squares problem
minx b-Ax2 .  

2
Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zgelss (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_zgelss (f08knc) uses the singular value decomposition (SVD) of A, where 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 r right-hand side matrix B and the n by r solution matrix X.
The effective rank of A 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:     order 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 order=Nag_RowMajor. 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: order=Nag_RowMajor or Nag_ColMajor.
2:     m IntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3:     n IntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
4:     nrhs IntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrices B and X.
Constraint: nrhs0.
5:     a[dim] ComplexInput/Output
Note: the dimension, dim, of the array a must be at least
  • max1,pda×n when order=Nag_ColMajor;
  • max1,m×pda when order=Nag_RowMajor.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the m by n matrix A.
On exit: the first minm,n rows of A are overwritten with its right singular vectors, stored row-wise.
6:     pda IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if order=Nag_ColMajor, pdamax1,m;
  • if order=Nag_RowMajor, pdamax1,n.
7:     b[dim] ComplexInput/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×nrhs when order=Nag_ColMajor;
  • max1,max1,m,n×pdb when order=Nag_RowMajor.
The i,jth element of the matrix B is stored in
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the m by r right-hand side matrix B.
On exit: b is overwritten by the n by r solution matrix X. If mn and rank=n, the residual sum of squares for the solution in the ith column is given by the sum of squares of the modulus of elements n+1,,m in that column.
8:     pdb IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,m,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
9:     s[dim] doubleOutput
Note: the dimension, dim, of the array s must be at least max1,minm,n .
On exit: the singular values of A in decreasing order.
10:   rcond doubleInput
On entry: used to determine the effective rank of A. Singular values s[i-1]rcond×s[0] are treated as zero. If rcond<0, machine precision is used instead.
11:   rank Integer *Output
On exit: the effective rank of A, i.e., the number of singular values which are greater than rcond×s[0].
12:   fail 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 value had an illegal value.
NE_CONVERGENCE
The algorithm for computing the SVD failed to converge; value off-diagonal elements of an intermediate bidiagonal form did not converge to zero.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pda=value and m=value.
Constraint: pdamax1,m.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
NE_INT_3
On entry, pdb=value, m=value and n=value.
Constraint: pdbmax1,m,n.
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_zgelss (f08knc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgelss (f08knc) 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.

9
Further Comments

The real analogue of this function is nag_dgelss (f08kac).

10
Example

This example solves the linear least squares problem
minx b-Ax2  
for the solution, x, of minimum norm, where
A = 0.47-0.34i -0.40+0.54i 0.60+0.01i 0.80-1.02i -0.32-0.23i -0.05+0.20i -0.26-0.44i -0.43+0.17i 0.35-0.60i -0.52-0.34i 0.87-0.11i -0.34-0.09i 0.89+0.71i -0.45-0.45i -0.02-0.57i 1.14-0.78i -0.19+0.06i 0.11-0.85i 1.44+0.80i 0.07+1.14i  
and
b = -1.08-2.59i -2.61-1.49i 3.13-3.61i 7.33-8.01i 9.12+7.63i .  
A tolerance of 0.01 is used to determine the effective rank of A.

10.1
Program Text

Program Text (f08knce.c)

10.2
Program Data

Program Data (f08knce.d)

10.3
Program Results

Program Results (f08knce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017