void	f08aac (Nag_OrderType order, Nag_TransType trans, Integer m, Integer n, Integer nrhs, double a[], Integer pda, double b[], Integer pdb, NagError *fail)

The function may be called by the names: f08aac, nag_lapackeig_dgels or nag_dgels.

3 Description

The following options are provided:

1.If $trans = Nag_NoTrans$ and $m \geq n$ : find the least squares solution of an overdetermined system, i.e., solve the least squares problem

$\min_{x} {‖ b - A x ‖}_{2} .$
2.If $trans = Nag_NoTrans$ and $m < n$ : find the minimum norm solution of an underdetermined system $A x = b$ .
3.If $trans = Nag_Trans$ and $m \geq n$ : find the minimum norm solution of an undetermined system $A^{T} x = b$ .
4.If $trans = Nag_Trans$ 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 \times r

right-hand side matrix

B

and the

n \times 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 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: $order$ – Nag_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $trans$ – Nag_TransType Input

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: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

4: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

5: $nrhs$ – 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

6: $a [\dim]$ – double Input/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 \times n

matrix

A

On exit: if

m \geq n

, a is overwritten by details of its

Q R

factorization, as returned by f08aec.

m < n

, a is overwritten by details of its

L Q

factorization, as returned by f08ahc.

7: $pda$ – Integer Input

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: $b [\dim]$ – double Input/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 \times r

trans = Nag_NoTrans

, or

n \times 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: $pdb$ – Integer Input

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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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.

f08aac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

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 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.

f08aa: FL CL CPP AD PY MB

NAG CL Interfacef08aac (dgels)

▸▿ 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 CL Interface
f08aac (dgels)