void	nag_dgelsy (Nag_OrderType order, Integer m, Integer n, Integer nrhs, double a[], Integer pda, double b[], Integer pdb, Integer jpvt[], double rcond, Integer rank, NagError fail)

3 Description

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

m

r

matrix

B

and the solution vectors in the

n

r

matrix

X

nag_dgelsy (f08bac) 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 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 Arguments

1: $order$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

4: $nrhs$ – IntegerInput

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

5: $a [\dim]$ – doubleInput/Output

Note: the dimension, dim, of the array a must be at least

$\max (1, pda \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pda)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

m

n

matrix

A

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

6: $pda$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraints:

if $order = Nag_ColMajor$ , $pda \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

7: $b [\dim]$ – doubleInput/Output

Note: the dimension, dim, of the array b must be at least

$\max (1, pdb \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, \max (1, m, n) \times pdb)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

m

r

right-hand side matrix

B

On exit: the

n

r

solution matrix

X

8: $pdb$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, m, n)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

9: $jpvt [\dim]$ – IntegerInput/Output

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

\max (1, n)

On entry: if

jpvt [i - 1] \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 - 1] = k

, then the

i

th column of

A P

was the

k

th column of

A

10: $rcond$ – doubleInput

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 nag_machine_precision (X02AJC)) 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.

11: $rank$ – 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

12: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $nrhs = 〈value〉$ .
Constraint: $nrhs \geq 0$ .

On entry, $pda = 〈value〉$ .
Constraint: $pda > 0$ .

On entry, $pdb = 〈value〉$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pda \geq \max (1, m)$ .

On entry, $pda = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pda \geq \max (1, n)$ .

On entry, $pdb = 〈value〉$ and $nrhs = 〈value〉$ .
Constraint: $pdb \geq \max (1, nrhs)$ .
NE_INT_3: On entry, $pdb = 〈value〉$ , $m = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq \max (1, m, n)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7 Accuracy

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

8 Parallelism and Performance

nag_dgelsy (f08bac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dgelsy (f08bac) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The complex analogue of this function is nag_zgelsy (f08bnc).

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

NAG Library Function Documentnag_dgelsy (f08bac)

▸▿ 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 Library Function Document

nag_dgelsy (f08bac)