# NAG Library Routine Document

## 1Purpose

f07krf (zpstrf) computes the Cholesky factorization with complete pivoting of a complex Hermitian positive semidefinite matrix.

## 2Specification

Fortran Interface
 Subroutine f07krf ( uplo, n, a, lda, piv, rank, tol, work, info)
 Integer, Intent (In) :: n, lda Integer, Intent (Out) :: piv(n), rank, info Real (Kind=nag_wp), Intent (In) :: tol Real (Kind=nag_wp), Intent (Out) :: work(2*n) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*) Character (1), Intent (In) :: uplo
C Header Interface
#include <nagmk26.h>
 void f07krf_ (const char *uplo, const Integer *n, Complex a[], const Integer *lda, Integer piv[], Integer *rank, const double *tol, double work[], Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name zpstrf.

## 3Description

f07krf (zpstrf) forms the Cholesky factorization of a complex Hermitian positive semidefinite matrix $A$ either as ${P}^{\mathrm{T}}AP={U}^{\mathrm{H}}U$ if ${\mathbf{uplo}}=\text{'U'}$ or ${P}^{\mathrm{T}}AP=L{L}^{\mathrm{H}}$ if ${\mathbf{uplo}}=\text{'L'}$, where $P$ is a permutation matrix, $U$ is an upper triangular matrix and $L$ is lower triangular.
This algorithm does not attempt to check that $A$ is positive semidefinite.
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
Lucas C (2004) LAPACK-style codes for Level 2 and 3 pivoted Cholesky factorizations LAPACK Working Note No. 161. Technical Report CS-04-522 Department of Computer Science, University of Tennessee, 107 Ayres Hall, Knoxville, TN 37996-1301, USA http://www.netlib.org/lapack/lawnspdf/lawn161.pdf

## 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:     $\mathbf{n}$ – IntegerInput
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) arrayInput/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 $n$ by $n$ Hermitian positive semidefinite 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: if ${\mathbf{uplo}}=\text{'U'}$, the first rank rows of the upper triangle of $A$ are overwritten with the nonzero elements of the Cholesky factor $U$, and the remaining rows of the triangle are destroyed.
If ${\mathbf{uplo}}=\text{'L'}$, the first rank columns of the lower triangle of $A$ are overwritten with the nonzero elements of the Cholesky factor $L$, and the remaining columns of the triangle are destroyed.
4:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07krf (zpstrf) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5:     $\mathbf{piv}\left({\mathbf{n}}\right)$ – Integer arrayOutput
On exit: piv is such that the nonzero entries of $P$ are $P\left({\mathbf{piv}}\left(\mathit{k}\right),\mathit{k}\right)=1$, for $\mathit{k}=1,2,\dots ,n$.
6:     $\mathbf{rank}$ – IntegerOutput
On exit: the computed rank of $A$ given by the number of steps the algorithm completed.
7:     $\mathbf{tol}$ – Real (Kind=nag_wp)Input
On entry: user defined tolerance. If ${\mathbf{tol}}<0$,  will be used. The algorithm terminates at the $r$th step if the $\left(r+1\right)$th step pivot $\text{}<{\mathbf{tol}}$.
8:     $\mathbf{work}\left(2*{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
9:     $\mathbf{info}$ – IntegerOutput
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.
${\mathbf{info}}=1$
The matrix $A$ is not positive definite. It is either positive semidefinite with computed rank as returned in rank and less than $n$, or it may be indefinite, see Section 9.

## 7Accuracy

If ${\mathbf{uplo}}=\text{'L'}$ and ${\mathbf{rank}}=r$, the computed Cholesky factor $L$ and permutation matrix $P$ satisfy the following upper bound
 $A - PLLHPT 2 A2 ≤ 2r cr ε W 2 + 1 2 + Oε2 ,$
where
 $W = L 11 -1 L12 , L = L11 0 L12 0 , L11 ∈ ℂr×r ,$
$c\left(r\right)$ is a modest linear function of $r$, $\epsilon$ is machine precision, and
 $W2 ≤ 13 n-r 4r-1 .$
So there is no guarantee of stability of the algorithm for large $n$ and $r$, although ${‖W‖}_{2}$ is generally small in practice.

## 8Parallelism and Performance

f07krf (zpstrf) 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.

## 9Further Comments

The total number of real floating-point operations is approximately $4n{r}^{2}-8/3{r}^{3}$, where $r$ is the computed rank of $A$.
This algorithm does not attempt to check that $A$ is positive semidefinite, and in particular the rank detection criterion in the algorithm is based on $A$ being positive semidefinite. If there is doubt over semidefiniteness then you should use the indefinite factorization f07mrf (zhetrf). See Lucas (2004) for further information.
The real analogue of this routine is f07kdf (dpstrf).

## 10Example

This example computes the Cholesky factorization of the matrix $A$, where
 $A= 12.40+0.00i 2.39+0.00i 5.50+0.05i 4.47+0.00i 11.89+0.00i 2.39+0.00i 1.63+0.00i 1.04+0.10i 1.14+0.00i 1.81+0.00i 5.50+0.05i 1.04+0.10i 2.45+0.00i 1.98-0.03i 5.28-0.02i 4.47+0.00i 1.14+0.00i 1.98-0.03i 1.71+0.00i 4.14+0.00i 11.89+0.00i 1.81+0.00i 5.28-0.02i 4.14+0.00i 11.63+0.00i .$

### 10.1Program Text

Program Text (f07krfe.f90)

### 10.2Program Data

Program Data (f07krfe.d)

### 10.3Program Results

Program Results (f07krfe.r)