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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_linsys_real_posdef_solve (f04bd)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_linsys_real_posdef_solve (f04bd) computes 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. An estimate of the condition number of A and an error bound for the computed solution are also returned.

Syntax

[a, b, rcond, errbnd, ifail] = f04bd(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[a, b, rcond, errbnd, ifail] = nag_linsys_real_posdef_solve(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

Description

The Cholesky factorization is used to factor A as A=UTU, if uplo='U', or A=LLT, if uplo='L', where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations AX=B.

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

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
If uplo='U', the upper triangle of the matrix A is stored.
If uplo='L', the lower triangle of the matrix 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 matrix A.
If uplo='U', the leading n by n upper triangular part of a contains the upper triangular part of the matrix A, and the strictly lower triangular part of a is not referenced.
If uplo='L', the leading n by n lower triangular part of a contains the lower triangular part of the matrix A, and the strictly upper triangular part of a is 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 n by r matrix of right-hand sides B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array b.
The number of linear equations n, i.e., the order of the matrix A.
Constraint: n0.
2:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the array b.
The number of right-hand sides r, 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 ifail=0 or n+1, the factor U or L from the Cholesky factorization A=UTU or A=LLT.
2:     bldb: – double array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,nrhs_p.
If ifail=0 or n+1, the n by r solution matrix X.
3:     rcond – double scalar
If ifail=0 or n+1, an estimate of the reciprocal of the condition number of the matrix A, computed as rcond=1/A1A-11.
4:     errbnd – double scalar
If ifail=0 or n+1, an estimate of the forward error bound for a computed solution x^, such that x^-x1/x1errbnd, where x^ is a column of the computed solution returned in the array b and x is the corresponding column of the exact solution X. If rcond is less than machine precision, then errbnd is returned as unity.
5:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   ifail>0andifailn
The principal minor of order _ of the matrix A is not positive definite. The factorization has not been completed and the solution could not be computed.
W  ifail=n+1
A solution has been computed, but rcond is less than machine precision so that the matrix A is numerically singular.
   ifail=-1
On entry, uplo'U' or 'L'.
   ifail=-2
Constraint: n0.
   ifail=-3
Constraint: nrhs_p0.
   ifail=-5
Constraint: ldamax1,n.
   ifail=-7
Constraint: ldbmax1,n.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.
The integer allocatable memory required is n, and the double allocatable memory required is 3×n. Allocation failed before the solution could be computed.

Accuracy

The computed solution for a single right-hand side, x^, satisfies an equation of the form
A+E x^=b,  
where
E1 = Oε A1  
and ε is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κA E1 A1 ,  
where κA = A-11 A1 , the condition number of A with respect to the solution of the linear equations. nag_linsys_real_posdef_solve (f04bd) uses the approximation E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The total number of floating-point operations required to solve the equations AX=B is proportional to 13n3+n2r. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The complex analogue of nag_linsys_real_posdef_solve (f04bd) is nag_linsys_complex_posdef_solve (f04cd).

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 8.30 -13.35 2.13 1.89 1.61 -4.14 5.00 .  
An estimate of the condition number of A and an approximate error bound for the computed solutions are also printed.
function f04bd_example


fprintf('f04bd example results\n\n');

% Solve Ax = b for positive definite A with error bound and condition number
uplo = 'Upper';
a = [  4.16, -3.12,  0.56, -0.10;
       0,     5.03, -0.83,  1.18;
       0,     0,     0.76,  0.34;
       0,     0,     0,     1.18];
b = [  8.7,   8.30;
     -13.35,  2.13;
       1.89,  1.61;
      -4.14,  5];

[a, x, rcond, errbnd, ifail] = f04bd(uplo, a, b);

disp('Solution');
disp(x);
disp('Estimate of condition number');
fprintf('%10.1f\n\n',1/rcond);
disp('Estimate of error bound for computed solutions');
fprintf('%10.1e\n\n',errbnd);



f04bd example results

Solution
    1.0000    4.0000
   -1.0000    3.0000
    2.0000    2.0000
   -3.0000    1.0000

Estimate of condition number
      97.3

Estimate of error bound for computed solutions
   1.1e-14


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