NAG Library Routine Document

1Purpose

f04chf computes the solution to a complex system of linear equations $AX=B$, where $A$ is an $n$ by $n$ Hermitian matrix and $X$ and $B$ are $n$ by $r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.

2Specification

Fortran Interface
 Subroutine f04chf ( uplo, n, nrhs, a, lda, ipiv, b, ldb,
 Integer, Intent (In) :: n, nrhs, lda, ldb Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ipiv(n) Real (Kind=nag_wp), Intent (Out) :: rcond, errbnd Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*) Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f04chf_ (const char *uplo, const Integer *n, const Integer *nrhs, Complex a[], const Integer *lda, Integer ipiv[], Complex b[], const Integer *ldb, double *rcond, double *errbnd, Integer *ifail, const Charlen length_uplo)

3Description

The diagonal pivoting method is used 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, and $D$ 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 $AX=B$.

4References

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 http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5Arguments

1:     $\mathbf{uplo}$ – Character(1)Input
On entry: if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of the matrix $A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of the matrix $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{nrhs}$ – IntegerInput
On entry: the number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
4:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ Hermitian matrix $A$.
If ${\mathbf{uplo}}=\text{'U'}$, the leading n by n upper triangular part of the array a contains the upper triangular part of the matrix $A$, and the strictly lower triangular part of a is not referenced.
If ${\mathbf{uplo}}=\text{'L'}$, the leading n by n lower triangular part of the array a contains the lower triangular part of the matrix $A$, and the strictly upper triangular part of a is not referenced.
On exit: if ${\mathbf{ifail}}\ge {\mathbf{0}}$, 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 f07mrf (zhetrf).
5:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f04chf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6:     $\mathbf{ipiv}\left({\mathbf{n}}\right)$ – Integer arrayOutput
On exit: if ${\mathbf{ifail}}\ge {\mathbf{0}}$, details of the interchanges and the block structure of $D$, as determined by f07mrf (zhetrf).
• If ${\mathbf{ipiv}}\left(k\right)>0$, then rows and columns $k$ and ${\mathbf{ipiv}}\left(k\right)$ were interchanged, and ${d}_{kk}$ is a $1$ by $1$ diagonal block;
• if ${\mathbf{uplo}}=\text{'U'}$ and ${\mathbf{ipiv}}\left(k\right)={\mathbf{ipiv}}\left(k-1\right)<0$, then rows and columns $k-1$ and $-{\mathbf{ipiv}}\left(k\right)$ were interchanged and ${d}_{k-1:k,k-1:k}$ is a $2$ by $2$ diagonal block;
• if ${\mathbf{uplo}}=\text{'L'}$ and ${\mathbf{ipiv}}\left(k\right)={\mathbf{ipiv}}\left(k+1\right)<0$, then rows and columns $k+1$ and $-{\mathbf{ipiv}}\left(k\right)$ were interchanged and ${d}_{k:k+1,k:k+1}$ is a $2$ by $2$ diagonal block.
7:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n$ by $r$ matrix of right-hand sides $B$.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, the $n$ by $r$ solution matrix $X$.
8:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f04chf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:     $\mathbf{rcond}$ – Real (Kind=nag_wp)Output
On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
10:   $\mathbf{errbnd}$ – Real (Kind=nag_wp)Output
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that ${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{errbnd}}$, where $\stackrel{^}{x}$ is a column of the computed solution returned in the array b and $x$ is the corresponding column of the exact solution $X$. If rcond is less than machine precision, errbnd is returned as unity.
11:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}>0 \text{and} {\mathbf{ifail}}\le {\mathbf{n}}$
Diagonal block $〈\mathit{\text{value}}〉$ of the block diagonal matrix is zero. The factorization has been completed, but the solution could not be computed.
${\mathbf{ifail}}={\mathbf{n}}+1$
A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
${\mathbf{ifail}}=-1$
On entry, uplo not one of 'U' or 'u' or 'L' or 'l': ${\mathbf{uplo}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
On entry, ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
${\mathbf{ifail}}=-5$
On entry, ${\mathbf{lda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
${\mathbf{ifail}}=-8$
On entry, ${\mathbf{ldb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
The real allocatable memory required is n, and the complex allocatable memory required is $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2×{\mathbf{n}},{\mathbf{lwork}}\right)$, where lwork is the optimum workspace required by f07mnf (zhesv). If this failure occurs it may be possible to solve the equations by calling the packed storage version of f04chf, f04cjf, or by calling f07mnf (zhesv) directly with less than the optimum workspace (see Chapter F07).
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^ = b ,$
where
 $E1 = Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. f04chf uses the approximation ${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

8Parallelism and Performance

f04chf 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 total number of floating-point operations required to solve the equations $AX=B$ is proportional to $\left(\frac{1}{3}{n}^{3}+2{n}^{2}r\right)$. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
Routine f04dhf is for complex symmetric matrices, and the real analogue of f04chf is f04bhf.

10Example

This example solves the equations
 $AX=B,$
where $A$ is the Hermitian indefinite matrix
 $A= -1.84i+0.00 0.11-0.11i -1.78-1.18i 3.91-1.50i 0.11+0.11i -4.63i+0.00 -1.84+0.03i 2.21+0.21i -1.78+1.18i -1.84-0.03i -8.87i+0.00 1.58-0.90i 3.91+1.50i 2.21-0.21i 1.58+0.90i -1.36i+0.00$
and
 $B= 2.98-10.18i 28.68-39.89i -9.58+03.88i -24.79-08.40i -0.77-16.05i 4.23-70.02i 7.79+05.48i -35.39+18.01i .$
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.

10.1Program Text

Program Text (f04chfe.f90)

10.2Program Data

Program Data (f04chfe.d)

10.3Program Results

Program Results (f04chfe.r)

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