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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zposv (f07fn)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zposv (f07fn) computes the solution to a complex system of linear equations
AX=B ,  
where A is an n by n Hermitian positive definite matrix and X and B are n by r matrices.

Syntax

[a, b, info] = f07fn(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[a, b, info] = nag_lapack_zposv(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zposv (f07fn) uses the Cholesky decomposition to factor A as A=UHU if uplo='U' or A=LLH 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
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)
If uplo='U', the upper triangle of A is stored.
If uplo='L', the lower triangle of A is stored.
Constraint: uplo='U' or 'L'.
2:     alda: – complex 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 Hermitian 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: – complex 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.
Note: to solve the equations Ax=b, where b is a single right-hand side, b may be supplied as a one-dimensional array with length ldb=max1,n.
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 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: – complex array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
If info=0, the factor U or L from the Cholesky factorization A=UHU or A=LLH.
2:     bldb: – complex array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,nrhs_p.
Note: to solve the equations Ax=b, where b is a single right-hand side, b may be supplied as a one-dimensional array with length ldb=max1,n.
If info=0, the n by r solution matrix X.
3:     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>0
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

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. See Section 4.4 of Anderson et al. (1999) for further details.
nag_lapack_zposvx (f07fp) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_complex_posdef_solve (f04cd) solves Ax=b  and returns a forward error bound and condition estimate. nag_linsys_complex_posdef_solve (f04cd) calls nag_lapack_zposv (f07fn) to solve the equations.

Further Comments

The total number of floating-point operations is approximately 43 n3 + 8n2r , where r  is the number of right-hand sides.
The real analogue of this function is nag_lapack_dposv (f07fa).

Example

This example solves the equations
Ax=b ,  
where A  is the symmetric positive definite matrix
A = 3.23i+0.00 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58i+0.00 -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09i+0.00 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29i+0.00  
and
b = 3.93-06.14i 6.17+09.42i -7.17-21.83i 1.99-14.38i .  
Details of the Cholesky factorization of A  are also output.
function f07fn_example


fprintf('f07fn example results\n\n');

% Upper triangular part of Hermitian matrix A
uplo = 'Upper';
a = [ 3.23  + 0i,  1.51 - 1.92i,  1.90 + 0.84i,  0.42 + 2.50i;
      0     + 0i,  3.58 + 0i,    -0.23 + 1.11i, -1.18 + 1.37i;
      0     + 0i,  0    + 0i,     4.09 + 0i,     2.33 - 0.14i;
      0     + 0i,  0    + 0i,     0    + 0i,     4.29 + 0i];

% RHS
b = [ 3.93 -  6.14i;
      6.17 +  9.42i;
     -7.17 - 21.83i;
      1.99 - 14.38i];

% Solve
[af, x, info] = f07fn( ...
                       uplo, a, b);

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

[ifail] = x04da( ...
                 uplo, 'Non-unit', af, 'Cholesky factor U');


f07fn example results

Solution
   1.0000 - 1.0000i
  -0.0000 + 3.0000i
  -4.0000 - 5.0000i
   2.0000 + 1.0000i

 Cholesky factor U
             1          2          3          4
 1      1.7972     0.8402     1.0572     0.2337
        0.0000    -1.0683     0.4674     1.3910

 2                 1.3164    -0.4702     0.0834
                   0.0000    -0.3131    -0.0368

 3                            1.5604     0.9360
                              0.0000    -0.9900

 4                                       0.6603
                                         0.0000

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