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NAG Toolbox

NAG Toolbox: nag_lapack_zpbtrf (f07hr)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zpbtrf (f07hr) computes the Cholesky factorization of a complex Hermitian positive definite band matrix.

Syntax

[ab, info] = f07hr(uplo, kd, ab, 'n', n)
[ab, info] = nag_lapack_zpbtrf(uplo, kd, ab, 'n', n)

Description

nag_lapack_zpbtrf (f07hr) forms the Cholesky factorization of a complex Hermitian positive definite band matrix A either as A=UHU if uplo='U' or A=LLH if uplo='L', where U (or L) is an upper (or lower) triangular band matrix with the same number of superdiagonals (or subdiagonals) as A.

References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville http://www.netlib.org/lapack/lawnspdf/lawn14.pdf
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 whether the upper or lower triangular part of A is stored and how A is to be factorized.
uplo='U'
The upper triangular part of A is stored and A is factorized as UHU, where U is upper triangular.
uplo='L'
The lower triangular part of A is stored and A is factorized as LLH, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2:     kd int64int32nag_int scalar
kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: kd0.
3:     abldab: – complex array
The first dimension of the array ab must be at least kd+1.
The second dimension of the array ab must be at least max1,n.
The n by n Hermitian positive definite band matrix A.
The matrix is stored in rows 1 to kd+1, more precisely,
  • if uplo='U', the elements of the upper triangle of A within the band must be stored with element Aij in abkd+1+i-jj​ for ​max1,j-kdij;
  • if uplo='L', the elements of the lower triangle of A within the band must be stored with element Aij in ab1+i-jj​ for ​jiminn,j+kd.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the second dimension of the array ab.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     abldab: – complex array
The first dimension of the array ab will be kd+1.
The second dimension of the array ab will be max1,n.
The upper or lower triangle of A stores the Cholesky factor U or L as specified by uplo, using the same storage format as described above.
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.
   info>0
The leading minor of order _ is not positive definite and the factorization could not be completed. Hence A itself is not positive definite. This may indicate an error in forming the matrix A. There is no function specifically designed to factorize a Hermitian band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling nag_lapack_zgbtrf (f07br) or as a full Hermitian matrix, by calling nag_lapack_zhetrf (f07mr).

Accuracy

If uplo='U', the computed factor U is the exact factor of a perturbed matrix A+E, where
Eck+1εUHU ,  
ck+1 is a modest linear function of k+1, and ε is the machine precision.
If uplo='L', a similar statement holds for the computed factor L. It follows that eijck+1εaiiajj.

Further Comments

The total number of real floating-point operations is approximately 4n k+1 2, assuming nk.
A call to nag_lapack_zpbtrf (f07hr) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dpbtrf (f07hd).

Example

This example computes the Cholesky factorization of the matrix A, where
A= 9.39+0.00i 1.08-1.73i 0.00+0.00i 0.00+0.00i 1.08+1.73i 1.69+0.00i -0.04+0.29i 0.00+0.00i 0.00+0.00i -0.04-0.29i 2.65+0.00i -0.33+2.24i 0.00+0.00i 0.00+0.00i -0.33-2.24i 2.17+0.00i .  
function f07hr_example


fprintf('f07hr example results\n\n');

uplo = 'L';
kd = int64(1);
n  = int64(4);
ab = [ 9.39 + 0i,     1.69 + 0i,      2.65 + 0i,      2.17 + 0i;
       1.08 - 1.73i, -0.04 + 0.29i,  -0.33 + 2.24i    0    + 0i];

% Factorize
[abf, info] = f07hr( ...
                     uplo, kd, ab);

ku = int64(0);
[ifail] = x04de( ...
                 n, n, kd, ku, abf, 'Cholesky factor');


f07hr example results

 Cholesky factor
             1          2          3          4
 1      3.0643
        0.0000

 2      0.3524     1.1167
       -0.5646     0.0000

 3                -0.0358     1.6066
                   0.2597     0.0000

 4                           -0.2054     0.4289
                              1.3942     0.0000

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Chapter Introduction
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