void	nag_dgels (Nag_OrderType order, Nag_TransType trans, Integer m, Integer n, Integer nrhs, double a[], Integer pda, double b[], Integer pdb, NagError *fail)

3 Description

The following options are provided:

trans = Nag_NoTrans

and

m \geq n

: find the least squares solution of an overdetermined system, i.e., solve the least squares problem

\min_{x} {‖b - A x‖}_{2} .

trans = Nag_NoTrans

and

m < n

: find the minimum norm solution of an underdetermined system

A x = b

trans = Nag_Trans

and

m \geq n

: find the minimum norm solution of an undetermined system

A^{T} x = b

trans = Nag_Trans

and

m < n

: find the least squares solution of an overdetermined system, i.e., solve the least squares problem

\min_{x} {‖b - A^{T} x‖}_{2} .

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

r

right-hand side matrix

B

and the

n

r

solution matrix

X

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: $trans$ – Nag_TransTypeInput

On entry: if

trans = Nag_NoTrans

, the linear system involves

A

trans = Nag_Trans

, the linear system involves

A^{T}

Constraint:

trans = Nag_NoTrans

Nag_Trans

3: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

4: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

5: $nrhs$ – 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

6: $a [\dim]$ – 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: if

m \geq n

, a is overwritten by details of its

Q R

factorization, as returned by nag_dgeqrf (f08aec).

m < n

, a is overwritten by details of its

L Q

factorization, as returned by nag_dgelqf (f08ahc).

7: $pda$ – 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 $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

8: $b [\dim]$ – 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 matrix

B

of right-hand side vectors, stored in rows or columns; b is

m

r

trans = Nag_NoTrans

, or

n

r

trans = Nag_Trans

On exit: b is overwritten by the solution vectors,

x

, stored in rows or columns:

if $trans = Nag_NoTrans$ and $m \geq n$ , or $trans = Nag_Trans$ and $m < n$ , elements $1$ to $\min (m, n)$ in each column of b contain the least squares solution vectors; the residual sum of squares for the solution is given by the sum of squares of the modulus of elements $(\min (m, n) + 1)$ to $\max (m, n)$ in that column;
otherwise, elements $1$ to $\max (m, n)$ in each column of b contain the minimum norm solution vectors.

9: $pdb$ – 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 $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

10: $fail$ – 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_FULL_RANK: Diagonal element $〈value〉$ of the triangular factor of $A$ is zero, so that $A$ does not have full rank; the least squares solution could not be computed.
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_dgels (f08aac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dgels (f08aac) 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 total number of floating-point operations required to factorize

A

is approximately

\frac{2}{3} n^{2} (3 m - n)

m \geq n

and

\frac{2}{3} m^{2} (3 n - m)

otherwise. Following the factorization the solution for a single vector

x

requires

O (\min (m^{2}, n^{2}))

operations.

The complex analogue of this function is nag_zgels (f08anc).

10 Example

This example solves the linear least squares problem

\min_{x} {‖b - A x‖}_{2},

where

A = (\begin{array}{r} - 0.57 & - 1.28 & - 0.39 & 0.25 \\ - 1.93 & 1.08 & - 0.31 & - 2.14 \\ 2.30 & 0.24 & 0.40 & - 0.35 \\ - 1.93 & 0.64 & - 0.66 & 0.08 \\ 0.15 & 0.30 & 0.15 & - 2.13 \\ - 0.02 & 1.03 & - 1.43 & 0.50 \end{array}) and b = (\begin{array}{r} - 2.67 \\ - 0.55 \\ 3.34 \\ - 0.77 \\ 0.48 \\ 4.10 \end{array}) .

The square root of the residual sum of squares is also output.

NAG Library Function Documentnag_dgels (f08aac)

▸▿ 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_dgels (f08aac)