NAG FL Interface
f07pnf (zhpsv)
1
Purpose
f07pnf computes the solution to a complex system of linear equations
where
$A$ is an
$n$ by
$n$ Hermitian matrix stored in packed format and
$X$ and
$B$ are
$n$ by
$r$ matrices.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
n, nrhs, ldb 
Integer, Intent (Out) 
:: 
ipiv(n), info 
Complex (Kind=nag_wp), Intent (Inout) 
:: 
ap(*), b(ldb,*) 
Character (1), Intent (In) 
:: 
uplo 

C Header Interface
#include <nag.h>
void 
f07pnf_ (const char *uplo, const Integer *n, const Integer *nrhs, Complex ap[], Integer ipiv[], Complex b[], const Integer *ldb, Integer *info, const Charlen length_uplo) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f07pnf_ (const char *uplo, const Integer &n, const Integer &nrhs, Complex ap[], Integer ipiv[], Complex b[], const Integer &ldb, Integer &info, const Charlen length_uplo) 
}

The routine may be called by the names f07pnf, nagf_lapacklin_zhpsv or its LAPACK name zhpsv.
3
Description
f07pnf uses the diagonal pivoting method to factor $A$ as $A=UD{U}^{\mathrm{H}}$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=LD{L}^{\mathrm{H}}$ if ${\mathbf{uplo}}=\text{'L'}$, where $U$ (or $L$) is a product of permutation and unit upper (lower) triangular matrices, $D$ is Hermitian and block diagonal with $1$ by $1$ and $2$ by $2$ diagonal blocks. The factored form of $A$ is then used to solve the system of equations $AX=B$.
4
References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5
Arguments

1:
$\mathbf{uplo}$ – Character(1)
Input

On entry: if
${\mathbf{uplo}}=\text{'U'}$, the upper triangle of
$A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ is stored.
Constraint:
${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.

2:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

3:
$\mathbf{nrhs}$ – Integer
Input

On entry: $r$, the number of righthand sides, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{nrhs}}\ge 0$.

4:
$\mathbf{ap}\left(*\right)$ – Complex (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
$n$ Hermitian 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: the block diagonal matrix
$D$ and the multipliers used to obtain the factor
$U$ or
$L$ from the factorization
$A=UD{U}^{\mathrm{H}}$ or
$A=LD{L}^{\mathrm{H}}$ as computed by
f07prf, stored as a packed triangular matrix in the same storage format as
$A$.

5:
$\mathbf{ipiv}\left({\mathbf{n}}\right)$ – 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.

6:
$\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (Kind=nag_wp) array
Input/Output

Note: the second dimension of the array
b
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
to solve the equations
$Ax=b$, where
$b$ is a single righthand side,
b may be supplied as a onedimensional array with length
${\mathbf{ldb}}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $r$ righthand side matrix $B$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.

7:
$\mathbf{ldb}$ – Integer
Input

On entry: the first dimension of the array
b as declared in the (sub)program from which
f07pnf is called.
Constraint:
${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

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, so the solution could not be computed.
7
Accuracy
The computed solution for a single righthand side,
$\hat{x}$, satisfies an equation of the form
where
and
$\epsilon $ is the
machine precision. An approximate error bound for the computed solution is given by
where
$\kappa \left(A\right)={\Vert {A}^{1}\Vert}_{1}{\Vert A\Vert}_{1}$, the condition number of
$A$ with respect to the solution of the linear equations. See Section 4.4 of
Anderson et al. (1999) and Chapter 11 of
Higham (2002) for further details.
f07ppf is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively,
f04cjf solves
$AX=B$ and returns a forward error bound and condition estimate.
f04cjf calls
f07pnf to solve the equations.
8
Parallelism and Performance
f07pnf 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 total number of floatingpoint operations is approximately $\frac{4}{3}{n}^{3}+8{n}^{2}r$, where $r$ is the number of righthand sides.
The real analogue of this routine is
f07paf. The complex symmetric analogue of this routine is
f07qnf.
10
Example
This example solves the equations
where
$A$ is the Hermitian matrix
and
Details of the factorization of $A$ are also output.
10.1
Program Text
10.2
Program Data
10.3
Program Results