nag_dgelsy (f08bac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dgelsy (f08bac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgelsy (f08bac) computes the minimum norm solution to a real linear least squares problem
minx b-Ax2
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.

2  Specification

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

3  Description

The right-hand side vectors are stored as the columns of the m by r matrix B and the solution vectors in the n by r matrix X.
nag_dgelsy (f08bac) first computes a QR factorization with column pivoting
AP= Q R11 R12 0 R22 ,
with R11 defined as the largest leading sub-matrix 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
AP= Q T11 0 0 0 Z .
The minimum norm solution is then
X = PZT T11-1 Q1T b 0
where Q1 consists of the first rank columns of Q.

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: a has been overwritten by details of its complete orthogonal factorization.
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: the n by r solution matrix X.
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:     jpvt[dim]IntegerInput/Output
Note: the dimension, dim, of the array jpvt must be at least max1,n.
On entry: if jpvt[i-1]0, the ith column of A is permuted to the front of AP, otherwise column i is a free column.
On exit: if jpvt[i-1]=k, then the ith column of AP was the kth column of A.
10:   rconddoubleInput
On entry: used to determine the effective rank of A, which is defined as the order of the largest leading triangular sub-matrix R11 in the QR factorization of A, whose estimated condition number is <1/rcond.
Suggested value: if the condition number of a is not known then rcond=ε/2 (where ε is machine precision, see nag_machine_precision (X02AJC)) is a good choice. Negative values or values less than machine precision should be avoided since this will cause a to have an effective rank=minm,n that could be larger than its actual rank, leading to meaningless results.
11:   rankInteger *Output
On exit: the effective rank of A, i.e., the order of the sub-matrix R11. This is the same as the order of the sub-matrix T11 in the complete orthogonal factorization of A.
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_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 of error bounds.

8  Further Comments

The complex analogue of this function is nag_zgelsy (f08bnc).

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 (f08bace.c)

9.2  Program Data

Program Data (f08bace.d)

9.3  Program Results

Program Results (f08bace.r)


nag_dgelsy (f08bac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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