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NAG Toolbox: nag_lapack_dgelsy (f08ba)

Purpose

nag_lapack_dgelsy (f08ba) computes the minimum norm solution to a real linear least squares problem
min bAx2
x
minx b-Ax2
using a complete orthogonal factorization of AA. AA is an mm by nn matrix which may be rank-deficient. Several right-hand side vectors bb and solution vectors xx 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 mm by rr matrix BB and the solution vectors in the nn by rr matrix XX.
nag_lapack_dgelsy (f08ba) first computes a QRQR factorization with column pivoting
AP = Q
(R11R12)
0 R22
,
AP= Q R11 R12 0 R22 ,
with R11R11 defined as the largest leading sub-matrix whose estimated condition number is less than 1 / rcond1/rcond. The order of R11R11, rank, is the effective rank of AA.
Then, R22R22 is considered to be negligible, and R12R12 is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization
AP = Q
(T110)
0 0
Z .
AP= Q T11 0 0 0 Z .
The minimum norm solution is then
X = PZT
( T111 Q1T b )
0
X = PZT T11-1 Q1T b 0
where Q1Q1 consists of the first rank columns of QQ.

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:     a(lda, : :) – double array
The first dimension of the array a must be at least max (1,m)max(1,m)
The second dimension of the array must be at least max (1,n)max(1,n)
The mm by nn matrix AA.
2:     b(ldb, : :) – double array
The first dimension of the array b must be at least max (1,m,n)max(1,m,n)
The second dimension of the array must be at least max (1,nrhs)max(1,nrhs)
The mm by rr right-hand side matrix BB.
3:     jpvt( : :) – int64int32nag_int array
Note: the dimension of the array jpvt must be at least max (1,n)max(1,n).
If jpvt(i)0jpvti0, the iith column of AA is permuted to the front of APAP, otherwise column ii is a free column.
4:     rcond – double scalar
Used to determine the effective rank of AA, which is defined as the order of the largest leading triangular sub-matrix R11R11 in the QRQR factorization of AA, whose estimated condition number is < 1 / rcond<1/rcond.
Suggested value: if the condition number of a is not known then rcond = sqrt((ε) / 2)rcond=(ε)/2 (where εε 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 = min (m,n)rank=min(m,n) that could be larger than its actual rank, leading to meaningless results.

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
mm, the number of rows of the matrix AA.
Constraint: m0m0.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array a.
nn, the number of columns of the matrix AA.
Constraint: n0n0.
3:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
rr, the number of right-hand sides, i.e., the number of columns of the matrices BB and XX.
Constraint: nrhs0nrhs0.

Input Parameters Omitted from the MATLAB Interface

lda ldb work lwork

Output Parameters

1:     a(lda, : :) – double array
The first dimension of the array a will be max (1,m)max(1,m)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,m)ldamax(1,m).
a stores details of its complete orthogonal factorization.
2:     b(ldb, : :) – double array
The first dimension of the array b will be max (1,m,n)max(1,m,n)
The second dimension of the array will be max (1,nrhs)max(1,nrhs)
ldbmax (1,m,n)ldbmax(1,m,n).
The nn by rr solution matrix XX.
3:     jpvt( : :) – int64int32nag_int array
Note: the dimension of the array jpvt must be at least max (1,n)max(1,n).
If jpvt(i) = kjpvti=k, then the iith column of APAP was the kkth column of AA.
4:     rank – int64int32nag_int scalar
The effective rank of AA, i.e., the order of the sub-matrix R11R11. This is the same as the order of the sub-matrix T11T11 in the complete orthogonal factorization of AA.
5:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii 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

function nag_lapack_dgelsy_example
a = [-0.09, 0.14, -0.46, 0.68, 1.29;
     -1.56, 0.2, 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];
jpvt = [int64(0);0;0;0;0];
rcond = 0.01;
[aOut, bOut, jpvtOut, rank, info] = nag_lapack_dgelsy(a, b, jpvt, rcond)
 

aOut =

   -2.9366   -0.7273    1.9134   -1.0317   -0.0890
    0.5234    2.8681    0.1721    0.7329   -0.1692
    0.4966   -0.4778   -2.3028   -0.6896    0.1052
    0.3657   -0.5520   -0.3306    1.2344   -0.3173
   -0.0268    0.6259   -0.0197    0.1934   -0.0034
    0.5335   -0.0259    0.0087   -0.3961   -0.6352


bOut =

    0.6344
    0.9699
   -1.4402
    3.3678
    3.3992
   -0.0035


jpvtOut =

                    1
                    5
                    4
                    2
                    3


rank =

                    4


info =

                    0


function f08ba_example
a = [-0.09, 0.14, -0.46, 0.68, 1.29;
     -1.56, 0.2, 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];
jpvt = [int64(0);0;0;0;0];
rcond = 0.01;
[aOut, bOut, jpvtOut, rank, info] = f08ba(a, b, jpvt, rcond)
 

aOut =

   -2.9366   -0.7273    1.9134   -1.0317   -0.0890
    0.5234    2.8681    0.1721    0.7329   -0.1692
    0.4966   -0.4778   -2.3028   -0.6896    0.1052
    0.3657   -0.5520   -0.3306    1.2344   -0.3173
   -0.0268    0.6259   -0.0197    0.1934   -0.0034
    0.5335   -0.0259    0.0087   -0.3961   -0.6352


bOut =

    0.6344
    0.9699
   -1.4402
    3.3678
    3.3992
   -0.0035


jpvtOut =

                    1
                    5
                    4
                    2
                    3


rank =

                    4


info =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
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