F08KAF (DGELSS) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08KAF (DGELSS)

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

F08KAF (DGELSS) computes the minimum norm solution to a real linear least squares problem
minx b-Ax2 .

2  Specification

SUBROUTINE F08KAF ( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, INFO)
INTEGER  M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
REAL (KIND=nag_wp)  A(LDA,*), B(LDB,*), S(*), RCOND, WORK(max(1,LWORK))
The routine may be called by its LAPACK name dgelss.

3  Description

F08KAF (DGELSS) 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  Parameters

1:     M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint: M0.
2:     N – INTEGERInput
On entry: n, the number of columns of the matrix A.
Constraint: N0.
3:     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.
4:     A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
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.
5:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08KAF (DGELSS) is called.
Constraint: LDAmax1,M.
6:     B(LDB,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least max1,NRHS.
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.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08KAF (DGELSS) is called.
Constraint: LDB max1,maxM,N .
8:     S(*) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array S must be at least max1,minM,N .
On exit: the singular values of A in decreasing order.
9:     RCOND – REAL (KIND=nag_wp)Input
On entry: used to determine the effective rank of A. Singular values SiRCOND×S1 are treated as zero. If RCOND<0, machine precision is used instead.
10:   RANK – INTEGEROutput
On exit: the effective rank of A, i.e., the number of singular values which are greater than RCOND×S1.
11:   WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
12:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08KAF (DGELSS) is called.
If LWORK=-1, a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.
Suggested value: for optimal performance, LWORK should generally be larger. Consider increasing LWORK by at least nb×minM,N , where nb  is the optimal block size.
Constraint: LWORK1, and also
LWORK3×minM,N+max2×minM,N,maxM,N,NRHS.
13:   INFO – INTEGEROutput

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
INFO>0
The algorithm for computing the SVD failed to converge; if INFO=i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero.

7  Accuracy

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

8  Further Comments

The complex analogue of this routine is F08KNF (ZGELSS).

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.
Note that the block size (NB) of 64 assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

9.1  Program Text

Program Text (f08kafe.f90)

9.2  Program Data

Program Data (f08kafe.d)

9.3  Program Results

Program Results (f08kafe.r)


F08KAF (DGELSS) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2011