NAG FL Interfacef07frf (zpotrf)

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1Purpose

f07frf computes the Cholesky factorization of a complex Hermitian positive definite matrix.

2Specification

Fortran Interface
 Subroutine f07frf ( uplo, n, a, lda, info)
 Integer, Intent (In) :: n, lda Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*) Character (1), Intent (In) :: uplo
#include <nag.h>
 void f07frf_ (const char *uplo, const Integer *n, Complex a[], const Integer *lda, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07frf, nagf_lapacklin_zpotrf or its LAPACK name zpotrf.

3Description

f07frf forms the Cholesky factorization of a complex Hermitian positive definite matrix $A$ either as $A={U}^{\mathrm{H}}U$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=L{L}^{\mathrm{H}}$ if ${\mathbf{uplo}}=\text{'L'}$, where $U$ is an upper triangular matrix and $L$ is lower triangular.

4References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville https://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

5Arguments

1: $\mathbf{uplo}$Character(1) Input
On entry: specifies whether the upper or lower triangular part of $A$ is stored and how $A$ is to be factorized.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ is stored and $A$ is factorized as ${U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored and $A$ is factorized as $L{L}^{\mathrm{H}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n×n$ Hermitian positive definite matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: the upper or lower triangle of $A$ is overwritten by the Cholesky factor $U$ or $L$ as specified by uplo.
4: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f07frf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
The leading minor of order $⟨\mathit{\text{value}}⟩$ 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 $A$. To factorize a Hermitian matrix which is not positive definite, call f07mrf instead.

7Accuracy

If ${\mathbf{uplo}}=\text{'U'}$, the computed factor $U$ is the exact factor of a perturbed matrix $A+E$, where
 $|E|≤c(n)ε|UH||U| ,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. If ${\mathbf{uplo}}=\text{'L'}$, a similar statement holds for the computed factor $L$. It follows that $|{e}_{ij}|\le c\left(n\right)\epsilon \sqrt{{a}_{ii}{a}_{jj}}$.

8Parallelism and Performance

f07frf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07frf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of real floating-point operations is approximately $\frac{4}{3}{n}^{3}$.
A call to f07frf may be followed by calls to the routines:
• f07fsf to solve $AX=B$;
• f07fuf to estimate the condition number of $A$;
• f07fwf to compute the inverse of $A$.
The real analogue of this routine is f07fdf.

10Example

This example computes the Cholesky factorization of the matrix $A$, where
 $A= ( 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i ) .$

10.1Program Text

Program Text (f07frfe.f90)

10.2Program Data

Program Data (f07frfe.d)

10.3Program Results

Program Results (f07frfe.r)