f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

NAG Library Function Documentnag_zhptrf (f07prc)

1  Purpose

nag_zhptrf (f07prc) computes the Bunch–Kaufman factorization of a complex Hermitian indefinite matrix, using packed storage.

2  Specification

 #include #include
 void nag_zhptrf (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex ap[], Integer ipiv[], NagError *fail)

3  Description

nag_zhptrf (f07prc) factorizes a complex Hermitian matrix $A$, using the Bunch–Kaufman diagonal pivoting method and packed storage. $A$ is factorized as either $A=PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $A=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, 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:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:     uploNag_UploTypeInput
On entry: specifies whether the upper or lower triangular part of $A$ is stored and how $A$ is to be factorized.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
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.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
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: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     ap[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the $n$ by $n$ Hermitian matrix $A$, packed by rows or columns.
The storage of elements ${A}_{ij}$ depends on the order and uplo arguments as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(j-1\right)×j/2+i-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-j\right)×\left(j-1\right)/2+i-1\right]$, for $i\ge j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-i\right)×\left(i-1\right)/2+j-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(i-1\right)×i/2+j-1\right]$, for $i\ge j$.
On exit: $A$ 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.
5:     ipiv[n]IntegerOutput
On exit: details of the interchanges and the block structure of $D$. More precisely,
• if ${\mathbf{ipiv}}\left[i-1\right]=k>0$, ${d}_{ii}$ 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 ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ and ${\mathbf{ipiv}}\left[i-2\right]={\mathbf{ipiv}}\left[i-1\right]=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\stackrel{-}{d}}_{i,i-1}\\ {\stackrel{-}{d}}_{i,i-1}& {d}_{ii}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i-1\right)$th row and column of $A$ were interchanged with the $l$th row and column;
• if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$ and ${\mathbf{ipiv}}\left[i-1\right]={\mathbf{ipiv}}\left[i\right]=-m<0$, $\left(\begin{array}{cc}{d}_{ii}& {d}_{i+1,i}\\ {d}_{i+1,i}& {d}_{i+1,i+1}\end{array}\right)$ 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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_SINGULAR
$D\left(⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right)$ 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 ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the computed factors $U$ and $D$ are the exact factors of a perturbed matrix $A+E$, where
 $E≤cnεPUDUHPT ,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, a similar statement holds for the computed factors $L$ and $D$.

8  Parallelism and Performance

nag_zhptrf (f07prc) is not threaded by NAG in any implementation.
nag_zhptrf (f07prc) 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.

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 $U$ and $L$ are stored in the corresponding columns of the array ap, but additional row interchanges must be applied to recover $U$ or $L$ explicitly (this is seldom necessary). If ${\mathbf{ipiv}}\left[\mathit{i}-1\right]=\mathit{i}$, for $\mathit{i}=1,2,\dots ,n$ (as is the case when $A$ is positive definite), then $U$ or $L$ are stored explicitly in packed form (except for their unit diagonal elements which are equal to $1$).
The total number of real floating-point operations is approximately $\frac{4}{3}{n}^{3}$.
A call to nag_zhptrf (f07prc) may be followed by calls to the functions:
The real analogue of this function is nag_dsptrf (f07pdc).

10  Example

This example computes the Bunch–Kaufman factorization of the matrix $A$, where
 $A= -1.36+0.00i 1.58+0.90i 2.21-0.21i 3.91+1.50i 1.58-0.90i -8.87+0.00i -1.84-0.03i -1.78+1.18i 2.21+0.21i -1.84+0.03i -4.63+0.00i 0.11+0.11i 3.91-1.50i -1.78-1.18i 0.11-0.11i -1.84+0.00i ,$
using packed storage.

10.1  Program Text

Program Text (f07prce.c)

10.2  Program Data

Program Data (f07prce.d)

10.3  Program Results

Program Results (f07prce.r)