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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dpotrs (f07fe)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dpotrs (f07fe) solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
AX=B ,  
where A has been factorized by nag_lapack_dpotrf (f07fd).

Syntax

[b, info] = f07fe(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dpotrs(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dpotrs (f07fe) is used to solve a real symmetric positive definite system of linear equations AX=B, this function must be preceded by a call to nag_lapack_dpotrf (f07fd) which computes the Cholesky factorization of A. The solution X is computed by forward and backward substitution.
If uplo='U', A=UTU, where U is upper triangular; the solution X is computed by solving UTY=B and then UX=Y.
If uplo='L', A=LLT, where L is lower triangular; the solution X is computed by solving LY=B and then LTX=Y.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies how A has been factorized.
uplo='U'
A=UTU, where U is upper triangular.
uplo='L'
A=LLT, where L is lower triangular.
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 Cholesky factor of A, as returned by nag_lapack_dpotrf (f07fd).
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 right-hand side matrix B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the arrays a, b and the second dimension of the array a.
n, 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.
Constraint: nrhs_p0.

Output Parameters

1:     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.
The n by r solution matrix X.
2:     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.

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
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x cncondA,xε  
where condA,x=A-1Ax/xcondA=A-1AκA.
Note that condA,x can be much smaller than condA.
Forward and backward error bounds can be computed by calling nag_lapack_dporfs (f07fh), and an estimate for κA (=κ1A) can be obtained by calling nag_lapack_dpocon (f07fg).

Further Comments

The total number of floating-point operations is approximately 2n2r.
This function may be followed by a call to nag_lapack_dporfs (f07fh) to refine the solution and return an error estimate.
The complex analogue of this function is nag_lapack_zpotrs (f07fs).

Example

This example solves the system of equations AX=B, where
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 .  
Here A is symmetric positive definite and must first be factorized by nag_lapack_dpotrf (f07fd).
function f07fe_example


fprintf('f07fe example results\n\n');

% Lower triangular part of symmetric matrix A
uplo = 'Lower';
a = [ 4.16,  0,    0,    0;
     -3.12,  5.03, 0,    0;
      0.56, -0.83, 0.76, 0;
     -0.10,  1.18, 0.34, 1.18];

% Factorize
[L, info] = f07fd( ...
                   uplo, a);

% Rhs
b = [  8.70, 8.30;
     -13.35, 2.13;
       1.89, 1.61;
      -4.14, 5.00];

% Solve
[x, info] = f07fe( ...
                   uplo, L, b);

disp('Solution(s)');
disp(x);


f07fe example results

Solution(s)
    1.0000    4.0000
   -1.0000    3.0000
    2.0000    2.0000
   -3.0000    1.0000


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Chapter Contents
Chapter Introduction
NAG Toolbox

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