Integer, Intent (In)	::	n, nrhs, lda, ldb, ldx
Integer, Intent (Out)	::	iter, info
Real (Kind=nag_wp), Intent (In)	::	b(ldb,*)
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), x(ldx,)
Real (Kind=nag_wp), Intent (Out)	::	work(n,nrhs)
Real (Kind=nag_rp), Intent (Out)	::	swork(n*(n+nrhs))
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nag.h>

void	f07fcf_ (const char uplo, const Integer n, const Integer nrhs, double a[], const Integer lda, const double b[], const Integer ldb, double x[], const Integer ldx, double work[], float swork[], Integer iter, Integer info, const Charlen length_uplo)

The routine may be called by the names f07fcf, nagf_lapacklin_dsposv or its LAPACK name dsposv.

3 Description

f07fcf first attempts to factorize the matrix in reduced precision and use this factorization within an iterative refinement procedure to produce a solution with full precision normwise backward error quality (see below). If the approach fails the method switches to a full precision factorization and solve.

The iterative refinement can be more efficient than the corresponding direct full precision algorithm. Since the strategy implemented by f07fcf 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. f07fcf always attempts the iterative refinement strategy first; you are advised to compare the performance of f07fcf with that of its full precision counterpart f07faf to determine whether this strategy is worthwhile for your particular problem dimensions.

The iterative refinement process is stopped if

iter > 30

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

-norm of the matrix

A

and

ε

is the machine precision returned by x02ajf.

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

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5 Arguments

1: $uplo$ – Character(1) Input

On entry: specifies whether the upper or lower triangular part of

A

is stored.

$uplo ='U'$: The upper triangular part of $A$ is stored.
$uplo ='L'$: The lower triangular part of $A$ is stored.

Constraint:

uplo ='U'

'L'

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: $a (lda *)$ – 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

n

n

symmetric positive definite matrix

A

If $uplo ='U'$ , the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If $uplo ='L'$ , the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: if iterative refinement has been successfully used (

info = 0

and

iter \geq 0

, see description below), then a is unchanged. If full precision factorization has been used (

info = 0

and

iter < 0

, see description below), then the array

A

contains the factor

U

L

from the Cholesky factorization

A = U^{T} U

A = L L^{T}

5: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f07fcf is called.

Constraint:

lda \geq \max (1, n)

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

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

\max (1, nrhs)

On entry: the right-hand side matrix

B

7: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f07fcf is called.

Constraint:

ldb \geq \max (1, n)

8: $x (ldx *)$ – Real (Kind=nag_wp) array Output

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

\max (1, nrhs)

On exit: if

info = 0

, the

n

r

solution matrix

X

9: $ldx$ – Integer Input

On entry: the first dimension of the array x as declared in the (sub)program from which f07fcf is called.

Constraint:

ldx \geq \max (1, n)

10: $work (n, nrhs)$ – Real (Kind=nag_wp) array Workspace

11: $swork (n \times (n + nrhs))$ – Real (Kind=nag_rp) array Workspace

Note: this array is utilized in the reduced precision computation, consequently its type nag_rp reflects this usage.

12: $iter$ – Integer Output

On exit: information on the progress of the interative refinement process.

$iter < 0$

Iterative refinement has failed for one of the reasons given below, full precision factorization has been performed instead.

$- 1$	The routine fell back to full precision for implementation- or machine-specific reasons.
$- 2$	Narrowing the precision induced an overflow, the routine fell back to full precision.
$- 3$	An intermediate reduced precision factorization failed.
$- 31$	The maximum permitted number of iterations was exceeded.

$iter > 0$

Iterative refinement has been sucessfully used. iter returns the number of iterations.

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.

$info > 0 and info \leq n$: The leading minor of order $〈value〉$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed.

7 Accuracy

For each right-hand side vector

b

, the computed solution

x

is the exact solution of a perturbed system of equations

(A + E) x = b

, where

if $uplo ='U'$ , $|E| \leq c (n) ε |U^{T}| |U|$ ;
if $uplo ='L'$ , $|E| \leq c (n) ε |L| |L^{T}|$ ,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision. See Section 10.1 of Higham (2002) for further details.

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

f07fcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07fcf 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 f07fqf.

10 Example

This example solves the equations

A X = B,

where

A

is the symmetric positive definite matrix

A = (\begin{matrix} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.18 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.18 & 0.34 & 1.18 \end{matrix})

and

B = (\begin{matrix} 8.70 \\ - 13.35 \\ 1.89 \\ - 4.14 \end{matrix}) .

Interfaces: FL CL AD

NAG FL Interface Introduction

F07 (Lapacklin) Chapter Contents

F07 (Lapacklin) Chapter Introduction

f07fc: FL CL AD

NAG FL Interfacef07fcf (dsposv)

▸▿ 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
f07fcf (dsposv)