Integer, Intent (In)	::	m, n, nrhs, lda, ldb, lwork
Integer, Intent (Inout)	::	jpvt(*)
Integer, Intent (Out)	::	rank, info
Real (Kind=nag_wp), Intent (In)	::	rcond
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), b(ldb,)
Real (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))

C Header Interface

#include <nag.h>

void	f08baf_ (const Integer m, const Integer n, const Integer nrhs, double a[], const Integer lda, double b[], const Integer ldb, Integer jpvt[], const double rcond, Integer rank, double work[], const Integer lwork, Integer *info)

The routine may be called by the names f08baf, nagf_lapackeig_dgelsy or its LAPACK name dgelsy.

3 Description

The right-hand side vectors are stored as the columns of the

m \times r

matrix

B

and the solution vectors in the

n \times r

matrix

X

f08baf first computes a

Q R

factorization with column pivoting

A P = Q (\begin{array}{r} R_{11} & R_{12} \\ 0 & R_{22} \end{array}),

with

R_{11}

defined as the largest leading sub-matrix whose estimated condition number is less than

1 / rcond

. The order of

R_{11}

, rank, is the effective rank of

A

Then,

R_{22}

is considered to be negligible, and

R_{12}

is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization

A P = Q (\begin{array}{r} T_{11} & 0 \\ 0 & 0 \end{array}) Z .

The minimum norm solution is then

X = P Z^{T} (\begin{matrix} T_{11}^{−1} Q_{1}^{T} b \\ 0 \end{matrix})

where

Q_{1}

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 https://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: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

2: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

3: $nrhs$ – Integer Input

On entry:

r

, the number of right-hand sides, i.e., the number of columns of the matrices

B

and

X

Constraint:

nrhs \geq 0

4: $a (lda, *)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array a must be at least

\max (1, n)

On entry: the

m \times n

matrix

A

On exit: a has been overwritten by details of its complete orthogonal factorization.

5: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08baf is called.

Constraint:

lda \geq \max (1, m)

6: $b (ldb, *)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array b must be at least

\max (1, nrhs)

On entry: the

m \times r

right-hand side matrix

B

On exit: the

n \times r

solution matrix

X

7: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f08baf is called.

Constraint:

ldb \geq \max (1, m, n)

8: $jpvt (*)$ – Integer array Input/Output

Note: the dimension of the array jpvt must be at least

\max (1, n)

On entry: if

jpvt (i) \neq 0

, the

i

th column of

A

is permuted to the front of

A P

, otherwise column

i

is a free column.

On exit: if

jpvt (i) = k

, the

i

th column of

A P

was the

k

th column of

A

9: $rcond$ – Real (Kind=nag_wp) Input

On entry: used to determine the effective rank of

A

, which is defined as the order of the largest leading triangular sub-matrix

R_{11}

in the

Q R

factorization of

A

, whose estimated condition number is

< 1 / rcond

Suggested value: if the condition number of a is not known then

rcond = \sqrt{(ε) / 2}

(where

ε

is machine precision, see x02ajf) 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)

that could be larger than its actual rank, leading to meaningless results.

10: $rank$ – Integer Output

On exit: the effective rank of

A

, i.e., the order of the sub-matrix

R_{11}

. This is the same as the order of the sub-matrix

T_{11}

in the complete orthogonal factorization of

A

11: $work (\max (1, lwork))$ – Real (Kind=nag_wp) array Workspace

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

12: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08baf is called.

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 \geq \max (k + 2 \times n + nb \times (n + 1), 2 \times k + nb \times nrhs),

where

k = \min (m, n)

and

nb

is the optimal block size.

Constraint:

lwork \geq k + \max (2 \times k, n + 1, k + nrhs), where ​ k = \min (m, n)

lwork = −1

13: $info$ – Integer Output

On exit:

info = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

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

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08baf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08baf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The complex analogue of this routine is f08bnf.

10 Example

This example solves the linear least squares problem

\min_{x} {‖ b - A x ‖}_{2}

for the solution,

x

, of minimum norm, where

A = (\begin{array}{r} - 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 \end{array}) and b = (\begin{array}{r} 7.4 \\ 4.2 \\ - 8.3 \\ 1.8 \\ 8.6 \\ 2.1 \end{array}) .

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.

f08ba: FL CL CPP AD PY MB

NAG FL Interfacef08baf (dgelsy)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f08baf (dgelsy)