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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zpftrs (f07ws)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zpftrs (f07ws) solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
AX=B ,  
using the Cholesky factorization computed by nag_lapack_zpftrf (f07wr) stored in Rectangular Full Packed (RFP) format.

Syntax

[b, info] = f07ws(transr, uplo, ar, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zpftrs(transr, uplo, ar, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zpftrs (f07ws) is used to solve a complex Hermitian positive definite system of linear equations AX=B, the function must be preceded by a call to nag_lapack_zpftrf (f07wr) which computes the Cholesky factorization of A, stored in RFP format. The RFP storage format is described in Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction. The solution X is computed by forward and backward substitution.
If uplo='U', A=UHU, where U is upper triangular; the solution X is computed by solving UHY=B and then UX=Y.
If uplo='L', A=LLH, where L is lower triangular; the solution X is computed by solving LY=B and then LHX=Y.

References

Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

Parameters

Compulsory Input Parameters

1:     transr – string (length ≥ 1)
Specifies whether the normal RFP representation of A or its conjugate transpose is stored.
transr='N'
The matrix A is stored in normal RFP format.
transr='C'
The conjugate transpose of the RFP representation of the matrix A is stored.
Constraint: transr='N' or 'C'.
2:     uplo – string (length ≥ 1)
Specifies how A has been factorized.
uplo='U'
A=UHU, where U is upper triangular.
uplo='L'
A=LLH, where L is lower triangular.
Constraint: uplo='U' or 'L'.
3:     arn×n+1/2 – complex array
The Cholesky factorization of A stored in RFP format, as returned by nag_lapack_zpftrf (f07wr).
4:     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.
The n by r right-hand side matrix B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array ar.
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: – 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.
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 and κA is the condition number when using the -norm.
Note that condA,x can be much smaller than condA.

Further Comments

The total number of real floating-point operations is approximately 8n2r.
The real analogue of this function is nag_lapack_dpftrs (f07we).

Example

This example solves the system of equations AX=B, where
A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i  
and
B= 3.93-06.14i 1.48+06.58i 6.17+09.42i 4.65-04.75i -7.17-21.83i -4.91+02.29i 1.99-14.38i 7.64-10.79i .  
Here A is Hermitian positive definite, stored in RFP format, and must first be factorized by nag_lapack_zpftrf (f07wr).
function f07ws_example


fprintf('f07ws example results\n\n');

% Symmetric matrix in RFP format
transr = 'n';
uplo   = 'l';
ar = [ 4.09 + 0.00i  2.33 - 0.14i;
       3.23 + 0.00i  4.29 + 0.00i;
       1.51 + 1.92i  3.58 + 0.00i;
       1.90 - 0.84i -0.23 - 1.11i;
       0.42 - 2.50i -1.18 - 1.37i];
n  = int64(4);
n2 = (n*(n+1))/2;
ar  = reshape(ar,[n2,1]);

% RHS
b = [ 3.93 -  6.14i,  1.48 +  6.58i;
      6.17 +  9.42i,  4.65 -  4.75i;
     -7.17 - 21.83i, -4.91 +  2.29i;
      1.99 - 14.38i,  7.64 - 10.79i];

% Factorize a
[ar, info] = f07wr(transr, uplo, n, ar);

if info == 0
  % Compute solution
  [b, info] = f07ws( ...
                     transr, uplo, ar, b);
  fprintf('\n');
  ncols  = int64(80);
  indent = int64(0);
  form   = 'f7.4';
  title  = 'Solutions';

  [ifail] = x04db( ...
                   'g', ' ', b, 'bracket', form, title, ...
                   'int', 'int', ncols, indent);
else
  fprintf('\na is not positive definite.\n');
end


f07ws example results


 Solutions
                    1                 2
 1  ( 1.0000,-1.0000) (-1.0000, 2.0000)
 2  (-0.0000, 3.0000) ( 3.0000,-4.0000)
 3  (-4.0000,-5.0000) (-2.0000, 3.0000)
 4  ( 2.0000, 1.0000) ( 4.0000,-5.0000)

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Chapter Introduction
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