NAG CL Interface
f07pdc (dsptrf)
1
Purpose
f07pdc computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix, using packed storage.
2
Specification
void 
f07pdc (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
double ap[],
Integer ipiv[],
NagError *fail) 

The function may be called by the names: f07pdc, nag_lapacklin_dsptrf or nag_dsptrf.
3
Description
f07pdc 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 ${\mathbf{uplo}}=\mathrm{Nag\_Upper}$ or $A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ 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 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:
$\mathbf{order}$ – Nag_OrderType
Input

On entry: the
order argument specifies the twodimensional storage scheme being used, i.e., rowmajor ordering or columnmajor ordering. C language defined storage is specified by
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$. See
Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

2:
$\mathbf{uplo}$ – Nag_UploType
Input

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 $PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
 ${\mathbf{uplo}}=\mathrm{Nag\_Lower}$
 The lower triangular part of $A$ is stored and $A$ is factorized as $PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint:
${\mathbf{uplo}}=\mathrm{Nag\_Upper}$ or $\mathrm{Nag\_Lower}$.

3:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

4:
$\mathbf{ap}\left[\mathit{dim}\right]$ – double
Input/Output

Note: the dimension,
dim, 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 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 ${\mathbf{ap}}\left[\left(j1\right)\times j/2+i1\right]$, for $i\le j$;
 if ${\mathbf{order}}=\mathrm{Nag\_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag\_Lower}$,
 ${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2nj\right)\times \left(j1\right)/2+i1\right]$, for $i\ge j$;
 if ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag\_Upper}$,
 ${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2ni\right)\times \left(i1\right)/2+j1\right]$, for $i\le j$;
 if ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag\_Lower}$,
 ${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(i1\right)\times i/2+j1\right]$, 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:
$\mathbf{ipiv}\left[{\mathbf{n}}\right]$ – Integer
Output

On exit: details of the interchanges and the block structure of
$D$. More precisely,
 if ${\mathbf{ipiv}}\left[i1\right]=k>0$, ${d}_{ii}$ 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 ${\mathbf{ipiv}}\left[i2\right]={\mathbf{ipiv}}\left[i1\right]=l<0$, $\left(\begin{array}{cc}{d}_{i1,i1}& {\overline{d}}_{i,i1}\\ {\overline{d}}_{i,i1}& {d}_{ii}\end{array}\right)$ 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 ${\mathbf{uplo}}=\mathrm{Nag\_Lower}$ and ${\mathbf{ipiv}}\left[i1\right]={\mathbf{ipiv}}\left[i\right]=m<0$, $\left(\begin{array}{cc}{d}_{ii}& {d}_{i+1,i}\\ {d}_{i+1,i}& {d}_{i+1,i+1}\end{array}\right)$ 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:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
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.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_SINGULAR

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
${\mathbf{uplo}}=\mathrm{Nag\_Upper}$, 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 ${\mathbf{uplo}}=\mathrm{Nag\_Lower}$, a similar statement holds for the computed factors $L$ and $D$.
8
Parallelism and Performance
f07pdc 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 function. 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 ${\mathbf{ipiv}}\left[\mathit{i}1\right]=\mathit{i}$, 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 floatingpoint operations is approximately $\frac{1}{3}{n}^{3}$.
A call to
f07pdc may be followed by calls to the functions:
 f07pec to solve $AX=B$;
 f07pgc to estimate the condition number of $A$;
 f07pjc to compute the inverse of $A$.
The complex analogues of this function are
f07prc for Hermitian matrices and
f07qrc 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