F04JGF (PDF version)
F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F04JGF finds the solution of a linear least squares problem, Ax=b, where A is a real m by nmn matrix and b is an m element vector. If the matrix of observations is not of full rank, then the minimal least squares solution is returned.

2  Specification


3  Description

The minimal least squares solution of the problem Ax=b is the vector x of minimum (Euclidean) length which minimizes the length of the residual vector r=b-Ax.
The real m by nmn matrix A is factorized as
A=Q U 0
where Q is an m by m orthogonal matrix and U is an n by n upper triangular matrix. If U is of full rank, then the least squares solution is given by
If U is not of full rank, then the singular value decomposition of U is obtained so that U is factorized as
where R and P are n by n orthogonal matrices and D is the n by n diagonal matrix
with σ1σ2σn0, these being the singular values of A. If the singular values σk+1,,σn are negligible, but σk is not negligible, relative to the data errors in A, then the rank of A is taken to be k and the minimal least squares solution is given by
x=P S-1 0 0 0 RT 0 0 I QTb,
where S=diagσ1,σ2,,σk.
This routine obtains the factorizations by a call to F02WDF.
The routine also returns the value of the standard error
σ = rTr m-k , if ​m>k, = 0, if ​m=k,
rTr being the residual sum of squares and k the rank of A.

4  References

Lawson C L and Hanson R J (1974) Solving Least-squares Problems Prentice–Hall

5  Parameters

1:     M – INTEGERInput
On entry: m, the number of rows of A.
Constraint: MN.
2:     N – INTEGERInput
On entry: n, the number of columns of A.
Constraint: 1NM.
3:     A(LDA,N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the m by n matrix A.
On exit: if SVD is returned as .FALSE., A is overwritten by details of the QU factorization of A (see F02WDF for further details).
If SVD is returned as .TRUE., the first n rows of A are overwritten by the right-hand singular vectors, stored by rows; and the remaining rows of the array are used as workspace.
4:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F04JGF is called.
Constraint: LDAM.
5:     B(M) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the right-hand side vector b.
On exit: the first n elements of B contain the minimal least squares solution vector x. The remaining m-n elements are used for workspace.
6:     TOL – REAL (KIND=nag_wp)Input
On entry: a relative tolerance to be used to determine the rank of A. TOL should be chosen as approximately the largest relative error in the elements of A. For example, if the elements of A are correct to about 4 significant figures then TOL should be set to about 5×10-4. See Section 8 for a description of how TOL is used to determine rank. If TOL is outside the range ε,1.0, where ε is the machine precision, then the value ε is used in place of TOL. For most problems this is unreasonably small.
7:     SVD – LOGICALOutput
On exit: is returned as .FALSE. if the least squares solution has been obtained from the QU factorization of A. In this case A is of full rank. SVD is returned as .TRUE. if the least squares solution has been obtained from the singular value decomposition of A.
8:     SIGMA – REAL (KIND=nag_wp)Output
On exit: the standard error, i.e., the value rTr/m-k when m>k, and the value zero when m=k. Here r is the residual vector b-Ax and k is the rank of A.
9:     IRANK – INTEGEROutput
On exit: k, the rank of the matrix A. It should be noted that it is possible for IRANK to be returned as n and SVD to be returned as .TRUE.. This means that the matrix U only just failed the test for nonsingularity.
10:   WORK(LWORK) – REAL (KIND=nag_wp) arrayOutput
On exit: if SVD is returned as .FALSE., then the first n elements of WORK contain information on the QU factorization of A (see parameter A above and F02WDF), and WORKn+1 contains the condition number UEU-1E of the upper triangular matrix U.
If SVD is returned as .TRUE., then the first n elements of WORK contain the singular values of A arranged in descending order and WORKn+1 contains the total number of iterations taken by the QR algorithm. The rest of WORK is used as workspace.
11:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F04JGF is called.
Constraint: LWORK4×N.
12:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
On entry,N<1,
The QR algorithm has failed to converge to the singular values in 50×N iterations. This failure can only happen when the singular value decomposition is employed, but even then it is not likely to occur.

7  Accuracy

The computed factors Q, U, R, D and PT satisfy the relations
Q U 0 =A+E,Q R 0 0 I D 0 PT=A+F,
E2c1ε A2,
ε being the machine precision, and c1 and c2 being modest functions of m and n. Note that A2=σ1.
For a fuller discussion, covering the accuracy of the solution x see Lawson and Hanson (1974), especially pages 50 and 95.

8  Further Comments

If the least squares solution is obtained from the QU factorization then the time taken by the routine is approximately proportional to n23m-n. If the least squares solution is obtained from the singular value decomposition then the time taken is approximately proportional to n23m+19n. The approximate proportionality factor is the same in each case.
This routine is column biased and so is suitable for use in paged environments.
Following the QU factorization of A the condition number
cU=UE U-1E
is determined and if cU is such that
then U is regarded as singular and the singular values of A are computed. If this test is not satisfied, U is regarded as nonsingular and the rank of A is set to n. When the singular values are computed the rank of A, say k, is returned as the largest integer such that
unless σ1=0 in which case k is returned as zero. That is, singular values which satisfy σiTOL×σ1 are regarded as negligible because relative perturbations of order TOL can make such singular values zero.

9  Example

This example obtains a least squares solution for r=b-Ax, where
A= 0.05 0.05 0.25 -0.25 0.25 0.25 0.05 -0.05 0.35 0.35 1.75 -1.75 1.75 1.75 0.35 -0.35 0.30 -0.30 0.30 0.30 0.40 -0.40 0.40 0.40 ,  b= 1 2 3 4 5 6
and the value TOL is to be taken as 5×10-4.

9.1  Program Text

Program Text (f04jgfe.f90)

9.2  Program Data

Program Data (f04jgfe.d)

9.3  Program Results

Program Results (f04jgfe.r)

F04JGF (PDF version)
F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

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