# NAG Library Routine Document

## 1Purpose

f07pdf (dsptrf) computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix, using packed storage.

## 2Specification

Fortran Interface
 Subroutine f07pdf ( uplo, n, ap, ipiv, info)
 Integer, Intent (In) :: n Integer, Intent (Out) :: ipiv(n), info Real (Kind=nag_wp), Intent (Inout) :: ap(*) Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f07pdf_ (const char *uplo, const Integer *n, double ap[], Integer ipiv[], Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name dsptrf.

## 3Description

f07pdf (dsptrf) 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 $D$.

## 4References

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 $PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{ap}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension 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 $n$ 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 ${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ 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 arrayOutput
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 ${\mathbf{ipiv}}\left(i-1\right)={\mathbf{ipiv}}\left(i\right)=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\stackrel{-}{d}}_{i,i-1}\\ {\stackrel{-}{d}}_{i,i-1}& {d}_{ii}\end{array}\right)$ 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 ${\mathbf{uplo}}=\text{'L'}$ 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:     $\mathbf{info}$ – 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.
${\mathbf{info}}>0$
Element $〈\mathit{\text{value}}〉$ 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.

## 7Accuracy

If ${\mathbf{uplo}}=\text{'U'}$, the computed factors $U$ and $D$ are the exact factors of a perturbed matrix $A+E$, where
 $E≤cnεPUDUTPT ,$
$c\left(n\right)$ 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 $D$.

## 8Parallelism and Performance

f07pdf (dsptrf) 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 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 floating-point operations is approximately $\frac{1}{3}{n}^{3}$.
A call to f07pdf (dsptrf) may be followed by calls to the routines:
• f07pef (dsptrs) to solve $AX=B$;
• f07pgf (dspcon) to estimate the condition number of $A$;
• f07pjf (dsptri) to compute the inverse of $A$.
The complex analogues of this routine are f07prf (zhptrf) for Hermitian matrices and f07qrf (zsptrf) for symmetric matrices.

## 10Example

This example computes the Bunch–Kaufman factorization of the matrix $A$, where
 $A= 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 ,$
using packed storage.

### 10.1Program Text

Program Text (f07pdfe.f90)

### 10.2Program Data

Program Data (f07pdfe.d)

### 10.3Program Results

Program Results (f07pdfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017