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NAG Toolbox: nag_lapack_zpftrf (f07wr)
Purpose
nag_lapack_zpftrf (f07wr) computes the Cholesky factorization of a complex Hermitian positive definite matrix stored in Rectangular Full Packed (RFP) format.
Syntax
Description
nag_lapack_zpftrf (f07wr) forms the Cholesky factorization of a complex Hermitian positive definite matrix
either as
if
or
if
, where
is an upper triangular matrix and
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:
– string (length ≥ 1)
-
Specifies whether the normal RFP representation of
or its conjugate transpose is stored.
- The matrix is stored in normal RFP format.
- The conjugate transpose of the RFP representation of the matrix is stored.
Constraint:
or .
- 2:
– string (length ≥ 1)
-
Specifies whether the upper or lower triangular part of
is stored.
- The upper triangular part of is stored, and is factorized as , where is upper triangular.
- The lower triangular part of is stored, and is factorized as , where is lower triangular.
Constraint:
or .
- 3:
– int64int32nag_int scalar
-
, the order of the matrix .
Constraint:
.
- 4:
– complex array
-
The upper or lower triangular part (as specified by
uplo) of the
by
Hermitian matrix
, 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:
– complex array
-
If , the factor or from the Cholesky factorization or , in the same storage format as .
- 2:
– int64int32nag_int scalar
unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
-
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
-
-
The leading minor of order
is not positive definite and
the factorization could not be completed. Hence
itself is not positive
definite. This may indicate an error in forming the matrix
. 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
If
, the computed factor
is the exact factor of a perturbed matrix
, where
is a modest linear function of
, and
is the
machine precision.
If , a similar statement holds for the computed factor . It follows that .
Further Comments
The total number of real floating-point operations is approximately .
A call to
nag_lapack_zpftrf (f07wr) may be followed by calls to the functions:
The real analogue of this function is
nag_lapack_dpftrf (f07wd).
Example
This example computes the Cholesky factorization of the matrix
, where
and is stored using RFP format.
Open in the MATLAB editor:
f07wr_example
function f07wr_example
fprintf('f07wr example results\n\n');
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]);
[ar, info] = f07wr(transr, uplo, n, ar);
if info == 0
[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)
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