nag_lapack_zpftrf (f07wr) computes the Cholesky factorization of a complex Hermitian positive definite matrix stored in Rectangular Full Packed (RFP) format.

Syntax

[ar, info] = f07wr(transr, uplo, n, ar)

[ar, info] = nag_lapack_zpftrf(transr, uplo, n, ar)

Description

nag_lapack_zpftrf (f07wr) forms the Cholesky factorization of a complex Hermitian positive definite matrix

A

either as

A = U^{H} U

uplo ='U'

A = L L^{H}

uplo ='L'

, where

U

is an upper triangular matrix and

L

is a lower triangular, stored in RFP format. The RFP storage format is described in Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction.

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

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'

'C'

2: $uplo$ – string (length ≥ 1)

Specifies whether the upper or lower triangular part of

A

is stored.

$uplo ='U'$: The upper triangular part of $A$ is stored, and $A$ is factorized as $U^{H} U$ , where $U$ is upper triangular.
$uplo ='L'$: The lower triangular part of $A$ is stored, and $A$ is factorized as $L L^{H}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

3: $n$ – int64int32nag_int scalar

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $ar (n \times (n + 1) / 2)$ – complex array

The upper or lower triangular part (as specified by uplo) of the

n

n

Hermitian matrix

A

, in either normal or transposed RFP format (as specified by transr). The storage format is described in detail in Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction.

Optional Input Parameters

None.

Output Parameters

1: $ar (n \times (n + 1) / 2)$ – complex array: If $info = 0$ , the factor $U$ or $L$ from the Cholesky factorization $A = U^{H} U$ or $A = L L^{H}$ , in the same storage format as $A$ .
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 matrix stored in RFP format which is not positive definite; the matrix must be treated as a full Hermitian matrix, by calling nag_lapack_zhetrf (f07mr).

Accuracy

uplo ='U'

, the computed factor

U

is the exact factor of a perturbed matrix

A + E

, where

|E| \leq c (n) ε |U^{H}| |U|,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

uplo ='L'

, a similar statement holds for the computed factor

L

. It follows that

|e_{i j}| \leq c (n) ε \sqrt{a_{i i} a_{j j}}

Further Comments

The total number of real floating-point operations is approximately

\frac{4}{3} n^{3}

A call to nag_lapack_zpftrf (f07wr) may be followed by calls to the functions:

nag_lapack_zpftrs (f07ws) to solve $A X = B$ ;
nag_lapack_zpftri (f07ww) to compute the inverse of $A$ .

The real analogue of this function is nag_lapack_dpftrf (f07wd).

Example

This example computes the Cholesky factorization of the matrix

A

, where

A = (\begin{array}{r} 3.23 + 0.00 i & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 + 0.00 i & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 + 0.00 i & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 + 0.00 i \end{array}) .

and is stored using RFP format.

Open in the MATLAB editor: f07wr_example

function f07wr_example


fprintf('f07wr 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]);

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

if info == 0
  % Convert factor to full array form for display
  [a, info] = f01vh(transr, uplo, n, ar);
  fprintf('\n');
  ncols  = int64(80);
  indent = int64(0);
  form   = 'f7.4';
  title  = 'Factor L:';
  diag   = 'n';
  [ifail] = x04db( ...
                   uplo, diag, a, 'brackets', form, title, ...
                   'int', 'int', ncols, indent);
else
  fprintf('\na is not positive definite.\n');
end

f07wr example results


 Factor L:
                    1                 2                 3                 4
 1  ( 1.7972, 0.0000)
 2  ( 0.8402, 1.0683) ( 1.3164, 0.0000)
 3  ( 1.0572,-0.4674) (-0.4702, 0.3131) ( 1.5604,-0.0000)
 4  ( 0.2337,-1.3910) ( 0.0834, 0.0368) ( 0.9360, 0.8105) ( 0.8713,-0.0000)

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox