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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dgelsy (f08ba)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dgelsy (f08ba) 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.

Syntax

[a, b, jpvt, rank, info] = f08ba(a, b, jpvt, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p)
[a, b, jpvt, rank, info] = nag_lapack_dgelsy(a, b, jpvt, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p)

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_lapack_dgelsy (f08ba) 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.

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

Parameters

Compulsory Input Parameters

1:     alda: – double array
The first dimension of the array a must be at least max1,m.
The second dimension of the array a must be at least max1,n.
The m by n matrix A.
2:     bldb: – double array
The first dimension of the array b must be at least max1,m,n.
The second dimension of the array b must be at least max1,nrhs_p.
The m by r right-hand side matrix B.
3:     jpvt: int64int32nag_int array
The dimension of the array jpvt must be at least max1,n
If jpvti0, the ith column of A is permuted to the front of AP, otherwise column i is a free column.
4:     rcond – double scalar
Suggested value: if the condition number of a is not known then rcond=ε/2 (where ε is machine precision, see nag_machine_precision (x02aj)) 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.
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.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the first dimension of the array a.
m, the number of rows of the matrix A.
Constraint: m0.
2:     n int64int32nag_int scalar
Default: the second dimension of the array a.
n, the number of columns of the matrix A.
Constraint: n0.
3:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the array b.
r, the number of right-hand sides, i.e., the number of columns of the matrices B and X.
Constraint: nrhs_p0.

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,m.
The second dimension of the array a will be max1,n.
a stores details of its complete orthogonal factorization.
2:     bldb: – double array
The first dimension of the array b will be max1,m,n.
The second dimension of the array b will be max1,nrhs_p.
The n by r solution matrix X.
3:     jpvt: int64int32nag_int array
The dimension of the array jpvt will be max1,n
If jpvti=k, then the ith column of AP was the kth column of A.
4:     rank int64int32nag_int scalar
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.
5:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   info=-i
If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: m, 2: n, 3: nrhs_p, 4: a, 5: lda, 6: b, 7: ldb, 8: jpvt, 9: rcond, 10: rank, 11: work, 12: lwork, 13: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

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

Further Comments

The complex analogue of this function is nag_lapack_zgelsy (f08bn).

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.
function f08ba_example


fprintf('f08ba example results\n\n');

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];
b = [ 7.4;
      4.2;
     -8.3;
      1.8;
      8.6;
      2.1];
[m,n] = size(a);

jpvt = zeros(n,1,'int64');
rcond = 0.01;

[af, x, jpvt, rank, info] = f08ba( ...
				   a, b, jpvt, rcond);

disp('Least squares solution');
disp(x');
disp('Tolerance used to estimate the rank of A');
disp(rcond);
disp('Estimated rank of A');
disp(rank);


f08ba example results

Least squares solution
    0.6344    0.9699   -1.4402    3.3678    3.3992   -0.0035

Tolerance used to estimate the rank of A
    0.0100

Estimated rank of A
                    4


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