NAG FL Interface
f07mrf (zhetrf)
1
Purpose
f07mrf computes the Bunch–Kaufman factorization of a complex Hermitian indefinite matrix.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
n, lda, lwork 
Integer, Intent (Out) 
:: 
ipiv(*), info 
Complex (Kind=nag_wp), Intent (Inout) 
:: 
a(lda,*) 
Complex (Kind=nag_wp), Intent (Out) 
:: 
work(max(1,lwork)) 
Character (1), Intent (In) 
:: 
uplo 

C++ Header Interface
#include <nag.h> extern "C" {
}

The routine may be called by the names f07mrf, nagf_lapacklin_zhetrf or its LAPACK name zhetrf.
3
Description
f07mrf factorizes a complex Hermitian matrix $A$, using the Bunch–Kaufman diagonal pivoting method. $A$ is factorized either as $A=PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'L'}$, where $P$ is a permutation matrix, $U$ (or $L$) is a unit upper (or lower) triangular matrix and $D$ is an Hermitian block diagonal matrix with $1$ by $1$ and $2$ by $2$ diagonal blocks; $U$ (or $L$) has $2$ by $2$ unit diagonal blocks corresponding to the $2$ by $2$ blocks of $D$. Row and column interchanges are performed to ensure numerical stability while keeping the matrix Hermitian.
This method is suitable for Hermitian matrices which are not known to be positive definite. If $A$ is in fact positive definite, no interchanges are performed and no $2$ by $2$ blocks occur in $D$.
4
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments

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 $PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
 ${\mathbf{uplo}}=\text{'L'}$
 The lower triangular part of $A$ is stored and $A$ is factorized as $PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$, 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
$n$ by
$n$ Hermitian indefinite 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 details of the block diagonal matrix
$D$ and the multipliers used to obtain the 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
f07mrf is called.
Constraint:
${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

5:
$\mathbf{ipiv}\left(*\right)$ – Integer array
Output

Note: the dimension of the array
ipiv
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: details of the interchanges and the block structure of
$D$. More precisely,
 if ${\mathbf{ipiv}}\left(i\right)=k>0$, ${d}_{ii}$ is a $1$ by $1$ pivot block and the $i$th row and column of $A$ were interchanged with the $k$th row and column;
 if ${\mathbf{uplo}}=\text{'U'}$ and ${\mathbf{ipiv}}\left(i1\right)={\mathbf{ipiv}}\left(i\right)=l<0$, $\left(\begin{array}{cc}{d}_{i1,i1}& {\overline{d}}_{i,i1}\\ {\overline{d}}_{i,i1}& {d}_{ii}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i1\right)$th row and column of $A$ were interchanged with the $l$th row and column;
 if ${\mathbf{uplo}}=\text{'L'}$ and ${\mathbf{ipiv}}\left(i\right)={\mathbf{ipiv}}\left(i+1\right)=m<0$, $\left(\begin{array}{cc}{d}_{ii}& {d}_{i+1,i}\\ {d}_{i+1,i}& {d}_{i+1,i+1}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i+1\right)$th row and column of $A$ were interchanged with the $m$th row and column.

6:
$\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Complex (Kind=nag_wp) array
Workspace

On exit: if
${\mathbf{info}}=0$,
${\mathbf{work}}\left(1\right)$ contains the minimum value of
lwork required for optimum performance.

7:
$\mathbf{lwork}$ – Integer
Input

On entry: the dimension of the array
work as declared in the (sub)program from which
f07mrf is called, unless
${\mathbf{lwork}}=1$, in which case a workspace query is assumed and the routine only calculates the optimal dimension of
work (using the formula given below).
Suggested value:
for optimum performance
lwork should be at least
${\mathbf{n}}\times \mathit{nb}$, where
$\mathit{nb}$ is the
block size.
Constraint:
${\mathbf{lwork}}\ge 1$ or ${\mathbf{lwork}}=1$.

8:
$\mathbf{info}$ – Integer
Output

On exit:
${\mathbf{info}}=0$ unless the routine detects an error (see
Section 6).
6
Error 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.
 ${\mathbf{info}}>0$

Element $\u2329\mathit{\text{value}}\u232a$ of the diagonal is exactly zero.
The factorization has been completed, but the block diagonal matrix $D$
is exactly singular, and division by zero will occur if it is
used to solve a system of equations.
7
Accuracy
If
${\mathbf{uplo}}=\text{'U'}$, the computed factors
$U$ and
$D$ are the exact factors of a perturbed matrix
$A+E$, where
$c\left(n\right)$ 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 factors $L$ and $D$.
8
Parallelism and Performance
f07mrf 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 implementationspecific information.
The elements of
$D$ overwrite the corresponding elements of
$A$; if
$D$ has
$2$ by
$2$ blocks, only the upper or lower triangle is stored, as specified by
uplo.
The unit diagonal elements of
$U$ or
$L$ and the
$2$ by
$2$ unit diagonal blocks are not stored. The remaining elements of
$U$ or
$L$ are stored in the corresponding columns of the array
a, but additional row interchanges must be applied to recover
$U$ or
$L$ explicitly (this is seldom necessary). If
${\mathbf{ipiv}}\left(\mathit{i}\right)=\mathit{i}$, for
$\mathit{i}=1,2,\dots ,n$ (as is the case when
$A$ is positive definite), then
$U$ or
$L$ is stored explicitly (except for its unit diagonal elements which are equal to
$1$).
The total number of real floatingpoint operations is approximately $\frac{4}{3}{n}^{3}$.
A call to
f07mrf may be followed by calls to the routines:
 f07msf to solve $AX=B$;
 f07muf to estimate the condition number of $A$;
 f07mwf to compute the inverse of $A$.
The real analogue of this routine is
f07mdf.
10
Example
This example computes the Bunch–Kaufman factorization of the matrix
$A$, where
10.1
Program Text
10.2
Program Data
10.3
Program Results