hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dsposv (f07fc)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dsposv (f07fc) uses the Cholesky factorization
A=UTU   or   A=LLT  
to compute the solution to a real system of linear equations
AX=B ,  
where A is an n by n symmetric positive definite matrix and X and B are n by r matrices.

Syntax

[a, x, iter, info] = f07fc(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[a, x, iter, info] = nag_lapack_dsposv(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dsposv (f07fc) 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 nag_lapack_dsposv (f07fc) 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. nag_lapack_dsposv (f07fc) always attempts the iterative refinement strategy first; you are advised to compare the performance of nag_lapack_dsposv (f07fc) with that of its full precision counterpart nag_lapack_dposv (f07fa) 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 < n x A ε ,  
where resid is the -norm of the residual, x is the -norm of the solution, A is the -norm of the matrix A and ε is the machine precision returned by nag_machine_precision (x02aj).

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

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
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' or 'L'.
2:     alda: – double array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The n by 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.
3:     bldb: – double array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,nrhs_p.
The right-hand side matrix B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array n.
n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
2:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the array b.
r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs_p0.

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
If iterative refinement has been successfully used (info=0 and iter0, 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 or L from the Cholesky factorization A=UTU or A=LLT.
2:     xldx: – double array
The first dimension of the array x will be max1,n.
The second dimension of the array x will be max1,nrhs_p.
If info=0, the n by r solution matrix X.
3:     iter int64int32nag_int scalar
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.
-1The function fell back to full precision for implementation- or machine-specific reasons.
-2Narrowing the precision induced an overflow, the function fell back to full precision.
-3An intermediate reduced precision factorization failed.
-31The maximum permitted number of iterations was exceeded.
iter>0
Iterative refinement has been sucessfully used. iter returns the number of iterations.
4:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

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>0andinfon
The leading minor of order _ of A is not positive definite, so the factorization could not be completed, and the solution has not been computed.

Accuracy

For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations A+Ex=b, where cn 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
x^ - x 1 x 1 κA E 1 A 1  
where κA = 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.

Further Comments

The complex analogue of this function is nag_lapack_zcposv (f07fq).

Example

This example solves the equations
AX=B ,  
where A  is the symmetric positive definite matrix
A = 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  
and
B = 8.70 -13.35 1.89 -4.14 .  
function f07fc_example


fprintf('f07fc example results\n\n');

a = [ 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,  1.19,  0.34,  1.18];

b = [ 8.70; -13.35; 1.89; -4.14];

% Solve the equations Ax = b for x
[a, x, iter, info] = f07fc( ...
                            'upper', a, b);

% Print Solution
fprintf('Solution:\n');
disp(x');


f07fc example results

Solution:
    1.0000   -1.0000    2.0000   -3.0000


PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015