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NAG Toolbox: nag_lapack_dpftrf (f07wd)

Purpose

nag_lapack_dpftrf (f07wd) computes the Cholesky factorization of a real symmetric positive definite matrix stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section [Rectangular Full Packed (RFP) Storage] in the F07 Chapter Introduction.

Syntax

[a, info] = f07wd(transr, uplo, n, a)
[a, info] = nag_lapack_dpftrf(transr, uplo, n, a)

Description

nag_lapack_dpftrf (f07wd) forms the Cholesky factorization of a real symmetric positive definite matrix AA either as A = UTUA=UTU if uplo = 'U'uplo='U' or A = LLTA=LLT if uplo = 'L'uplo='L', where UU is an upper triangular matrix and LL is a lower triangular, stored in RFP format.

References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville
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 RFP representation of AA is normal or transposed.
transr = 'N'transr='N'
The matrix AA is stored in normal RFP format.
transr = 'T'transr='T'
The matrix AA is stored in transposed RFP format.
Constraint: transr = 'N'transr='N' or 'T''T'.
2:     uplo – string (length ≥ 1)
Specifies whether the upper or lower triangular part of AA is stored.
uplo = 'U'uplo='U'
The upper triangular part of AA is stored, and AA is factorized as UTUUTU, where UU is upper triangular.
uplo = 'L'uplo='L'
The lower triangular part of AA is stored, and AA is factorized as LLTLLT, where LL is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
3:     n – int64int32nag_int scalar
nn, the order of the matrix AA.
Constraint: n0n0.
4:     a(n × (n + 1) / 2n×(n+1)/2) – double array
The nn by nn symmetric matrix AA, stored in RFP format, as described in Section [Rectangular Full Packed (RFP) Storage] in the F07 Chapter Introduction.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     a(n × (n + 1) / 2n×(n+1)/2) – double array
If INFO = 0INFO=0, the factor UU or LL from the Cholesky factorization A = UTUA=UTU or A = LLTA=LLT, in the same storage format as AA.
2:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W INFO > 0INFO>0
The leading minor of order __ is not positive definite and the factorization could not be completed. Hence AA itself is not positive definite. This may indicate an error in forming the matrix AA. There is no function specifically designed to factorize a symmetric band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling nag_lapack_dgbtrf (f07bd) or as a full symmetric matrix, by calling nag_lapack_dsytrf (f07md).

Accuracy

If uplo = 'U'uplo='U', the computed factor UU is the exact factor of a perturbed matrix A + EA+E, where
|E|c(n)ε|UT||U| ,
|E|c(n)ε|UT||U| ,
c(n)c(n) is a modest linear function of nn, and εε is the machine precision.
If uplo = 'L'uplo='L', a similar statement holds for the computed factor LL. It follows that |eij|c(n)ε×sqrt(aiiajj)|eij|c(n)εaiiajj.

Further Comments

The total number of floating point operations is approximately (1/3)n213n2.
A call to nag_lapack_dpftrf (f07wd) may be followed by calls to the functions:
The complex analogue of this function is nag_lapack_zpftrf (f07wr).

Example

function nag_lapack_dpftrf_example
a = [0.76; 4.16; -3.12; 0.56; -0.10; 0.34; 1.18; 5.03; -0.83; 1.18];
transr = 'n';
uplo   = 'l';
n      = int64(4);

% Factorize a
[aOut, info] = nag_lapack_dpftrf(transr, uplo, n, a);

if info == 0
  % Convert factor to full array form, and print it
  [f, info] = nag_matop_dtfttr(transr, uplo, n, aOut);
  fprintf('\n');
  [ifail] = nag_file_print_matrix_real_gen(uplo, 'n', f, 'Factor');
else
  fprintf('\na is not positive definite.\n');
end
 

 Factor
             1          2          3          4
 1      2.0396
 2     -1.5297     1.6401
 3      0.2746    -0.2500     0.7887
 4     -0.0490     0.6737     0.6617     0.5347

function f07wd_example
a = [0.76; 4.16; -3.12; 0.56; -0.10; 0.34; 1.18; 5.03; -0.83; 1.18];
transr = 'n';
uplo   = 'l';
n      = int64(4);

% Factorize a
[aOut, info] = f07wd(transr, uplo, n, a);

if info == 0
  % Convert factor to full array form, and print it
  [f, info] = f01vg(transr, uplo, n, aOut);
  fprintf('\n');
  [ifail] = x04ca(uplo, 'n', f, 'Factor');
else
  fprintf('\na is not positive definite.\n');
end
 

 Factor
             1          2          3          4
 1      2.0396
 2     -1.5297     1.6401
 3      0.2746    -0.2500     0.7887
 4     -0.0490     0.6737     0.6617     0.5347


PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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