void	f07acc (Nag_OrderType order, Integer n, Integer nrhs, double a[], Integer pda, Integer ipiv[], const double b[], Integer pdb, double x[], Integer pdx, Integer iter, NagError fail)

The function may be called by the names: f07acc, nag_lapacklin_dsgesv or nag_dsgesv.

3 Description

f07acc first attempts to factorize the matrix in single precision and use this factorization within an iterative refinement procedure to produce a solution with full double precision accuracy. If the approach fails the method switches to a double precision factorization and solve.

The iterative refinement process is stopped if

iter > itermax,

where iter is the number of iterations carried out thus far and

itermax

is the maximum number of iterations allowed, which is fixed at

30

iterations. The process is also stopped if for all right-hand sides we have

‖ resid ‖ < \sqrt{n} ‖ x ‖ ‖ A ‖ ε,

where

‖ resid ‖

is the

\infty

-norm of the residual,

‖ x ‖

is the

\infty

-norm of the solution,

‖ A ‖

is the

\infty

-operator-norm of the matrix

A

and

ε

is the machine precision returned by X02AJC.

The iterative refinement strategy used by f07acc can be more efficient than the corresponding direct full precision algorithm. Since this strategy must perform iterative refinement on each right-hand side, any efficiency gains will reduce as the number of right-hand sides increases. Conversely, as the matrix size increases the cost of these iterative refinements become less significant relative to the cost of factorization. Thus, any efficiency gains will be greatest for a very small number of right-hand sides and for large matrix sizes. The cut-off values for the number of right-hand sides and matrix size, for which the iterative refinement strategy performs better, depends on the relative performance of the reduced and full precision factorization and back-substitution. For now, f07acc always attempts the iterative refinement strategy first; you are advised to compare the performance of f07acc with that of its full precision counterpart f07aac to determine whether this strategy is worthwhile for your particular problem dimensions.

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

Buttari A, Dongarra J, Langou J, Langou J, Luszczek P and Kurzak J (2007) Mixed precision iterative refinement techniques for the solution of dense linear systems International Journal of High Performance Computing Applications

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: $n$ – Integer Input

On entry:

n

, the number of linear equations, i.e., the order 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 matrix

B

Constraint:

nrhs \geq 0

4: $a [\dim]$ – double Input/Output

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

\max (1, pda \times n)

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

n \times n

coefficient matrix

A

On exit: if iterative refinement has been successfully used (i.e., if

fail . code =

NE_NOERROR and

iter \geq 0

), then

A

is unchanged. If double precision factorization has been used (when

fail . code =

NE_NOERROR and

iter < 0

A

contains the factors

L

and

U

from the factorization

A = P L U

; the unit diagonal elements of

L

are not stored.

5: $pda$ – Integer Input

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

Constraint:

pda \geq \max (1, n)

6: $ipiv [n]$ – Integer Output

On exit: if no constraints are violated, the pivot indices that define the permutation matrix

P

; at the

i

th step row

i

of the matrix was interchanged with row

ipiv [i - 1]

ipiv [i - 1] = i

indicates a row interchange was not required.

ipiv

corresponds either to the single precision factorization (if

fail . code =

NE_NOERROR and

iter \geq 0

) or to the double precision factorization (if

fail . code =

NE_NOERROR and

iter < 0

7: $b [\dim]$ – const double Input

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

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

n \times r

right-hand side matrix

B

8: $pdb$ – Integer Input

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

Constraints:

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

9: $x [\dim]$ – double Output

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

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

The

(i, j)

th element of the matrix

X

is stored in

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

On exit: if

fail . code =

NE_NOERROR, the

n \times r

solution matrix

X

10: $pdx$ – Integer Input

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

Constraints:

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

11: $iter$ – Integer * Output

On exit: if

iter > 0

, iterative refinement has been successfully used and iter is the number of iterations carried out.

iter < 0

, iterative refinement has failed for one of the reasons given below and double precision factorization has been carried out instead.

$iter = −1$: Taking into account machine parameters, and the values of n and nrhs, it is not worth working in single precision.
$iter = −2$: Overflow of an entry occurred when moving from double to single precision.
$iter = −3$: An intermediate single precision factorization failed.
$iter = −31$: The maximum permitted number of iterations was exceeded.

12: $fail$ – 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_INT: 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: $pdb > 0$ .

On entry, $pdx = ⟨ value ⟩$ .
Constraint: $pdx > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .

On entry, $pdb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, n)$ .

On entry, $pdb = ⟨ value ⟩$ and $nrhs = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, nrhs)$ .

On entry, $pdx = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdx \geq \max (1, n)$ .

On entry, $pdx = ⟨ value ⟩$ and $nrhs = ⟨ value ⟩$ .
Constraint: $pdx \geq \max (1, nrhs)$ .
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.
NE_SINGULAR: Element $⟨ value ⟩$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution could not be computed.

7 Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies the equation of the form

(A + E) \hat{x} = b,

where

{‖ E ‖}_{1} = O (ε) {‖ A ‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖ \hat{x} - x ‖}_{1}}{{‖ x ‖}_{1}} \leq κ (A) \frac{{‖ E ‖}_{1}}{{‖ A ‖}_{1}}

where

κ (A) = {‖ A^{- 1} ‖}_{1} {‖ A ‖}_{1}

, the condition number of

A

with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

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

f07acc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07acc 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 f07aqc.

10 Example

This example solves the equations

A x = b,

where

A

is the general matrix

A = (\begin{matrix} 1.80 & 2.88 & 2.05 & - 0.89 \\ 5.25 & - 2.95 & - 0.95 & - 3.80 \\ 1.58 & - 2.69 & - 2.90 & - 1.04 \\ - 1.11 & - 0.66 & - 0.59 & 0.80 \end{matrix}) and b = (\begin{matrix} 9.52 \\ 24.35 \\ 0.77 \\ - 6.22 \end{matrix}) .

f07ac: FL CL CPP AD PY MB

NAG CL Interfacef07acc (dsgesv)

▸▿ 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
f07acc (dsgesv)