nag_dgelss (f08kac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dgelss (f08kac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgelss (f08kac) computes the minimum norm solution to a real linear least squares problem
minx b-Ax2 .

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dgelss (Nag_OrderType order, Integer m, Integer n, Integer nrhs, double a[], Integer pda, double b[], Integer pdb, double s[], double rcond, Integer *rank, NagError *fail)

3  Description

nag_dgelss (f08kac) 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:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3:     nIntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
4:     nrhsIntegerInput
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]doubleInput/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:     pdaIntegerInput
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]doubleInput/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 elements n+1,,m in that column.
8:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdb max1,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:   rconddoubleInput
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:   rankInteger *Output
On exit: the effective rank of A, i.e., the number of singular values which are greater than rcond×s[0].
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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: pdb max1,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.

7  Accuracy

See Section 4.5 of Anderson et al. (1999) for details.

8  Further Comments

The complex analogue of this function is nag_zgelss (f08knc).

9  Example

This example solves the linear least squares problem
minx b-Ax2
for the solution, x, of minimum norm, where
A = -0.09 0.14 -0.46 0.68 1.29 -1.56 0.20 0.29 1.09 0.51 -1.48 -0.43 0.89 -0.71 -0.96 -1.09 0.84 0.77 2.11 -1.27 0.08 0.55 -1.13 0.14 1.74 -1.59 -0.72 1.06 1.24 0.34   and  b= 7.4 4.2 -8.3 1.8 8.6 2.1 .
A tolerance of 0.01 is used to determine the effective rank of A.

9.1  Program Text

Program Text (f08kace.c)

9.2  Program Data

Program Data (f08kace.d)

9.3  Program Results

Program Results (f08kace.r)


nag_dgelss (f08kac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012