NAG FL Interface
f07pdf (dsptrf)
1
Purpose
f07pdf computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix, using packed storage.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
n 
Integer, Intent (Out) 
:: 
ipiv(n), info 
Real (Kind=nag_wp), Intent (Inout) 
:: 
ap(*) 
Character (1), Intent (In) 
:: 
uplo 

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

The routine may be called by the names f07pdf, nagf_lapacklin_dsptrf or its LAPACK name dsptrf.
3
Description
f07pdf factorizes a real symmetric matrix $A$, using the Bunch–Kaufman diagonal pivoting method and packed storage. $A$ is factorized as either $A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ 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 a symmetric 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 preserving symmetry.
This method is suitable for symmetric 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 $PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
 ${\mathbf{uplo}}=\text{'L'}$
 The lower triangular part of $A$ is stored and $A$ is factorized as $PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, 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:
$\mathbf{ap}\left(*\right)$ – Real (Kind=nag_wp) array
Input/Output

Note: the dimension of the array
ap
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\times \left({\mathbf{n}}+1\right)/2\right)$.
On entry: the
$n$ by
$n$ symmetric matrix
$A$, packed by columns.
More precisely,
 if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j\left(j1\right)/2\right)$ for $i\le j$;
 if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2nj\right)\left(j1\right)/2\right)$ 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.

4:
$\mathbf{ipiv}\left({\mathbf{n}}\right)$ – Integer array
Output

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.

5:
$\mathbf{info}$ – 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
f07pdf 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 $L$ overwrite elements in the corresponding columns of $A$, 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 $L$ are stored explicitly in packed form (except for their unit diagonal elements which are equal to $1$).
The total number of floatingpoint operations is approximately $\frac{1}{3}{n}^{3}$.
A call to
f07pdf may be followed by calls to the routines:
 f07pef to solve $AX=B$;
 f07pgf to estimate the condition number of $A$;
 f07pjf to compute the inverse of $A$.
The complex analogues of this routine are
f07prf for Hermitian matrices and
f07qrf for symmetric matrices.
10
Example
This example computes the Bunch–Kaufman factorization of the matrix
$A$, where
using packed storage.
10.1
Program Text
10.2
Program Data
10.3
Program Results