NAG Library Routine Document
f04bhf (real_symm_solve)
1
Purpose
f04bhf computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n$ by $n$ symmetric 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.
2
Specification
Fortran Interface
Subroutine f04bhf ( 
uplo, n, nrhs, a, lda, ipiv, b, ldb, rcond, errbnd, ifail) 
Integer, Intent (In)  ::  n, nrhs, lda, ldb  Integer, Intent (Inout)  ::  ifail  Integer, Intent (Out)  ::  ipiv(n)  Real (Kind=nag_wp), Intent (Inout)  ::  a(lda,*), b(ldb,*)  Real (Kind=nag_wp), Intent (Out)  ::  rcond, errbnd  Character (1), Intent (In)  ::  uplo 

C Header Interface
#include nagmk26.h
void 
f04bhf_ (const char *uplo, const Integer *n, const Integer *nrhs, double a[], const Integer *lda, Integer ipiv[], double b[], const Integer *ldb, double *rcond, double *errbnd, Integer *ifail, const Charlen length_uplo) 

3
Description
The diagonal pivoting method is used to factor $A$ as $A=UD{U}^{\mathrm{T}}$, if ${\mathbf{uplo}}=\text{'U'}$, or $A=LD{L}^{\mathrm{T}}$, if ${\mathbf{uplo}}=\text{'L'}$, where $U$ (or $L$) is a product of permutation and unit upper (lower) triangular matrices, and $D$ is symmetric 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
http://www.netlib.org/lapack/lug
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 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 righthand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{nrhs}}\ge 0$.
 4: $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (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
$n$ symmetric 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
$A=UD{U}^{\mathrm{T}}$ or
$A=LD{L}^{\mathrm{T}}$ as computed by
f07mdf (dsytrf).
 5: $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array
a as declared in the (sub)program from which
f04bhf 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
f07mdf (dsytrf).
 ${\mathbf{ipiv}}\left(k\right)>0$
 Rows and columns $k$ and ${\mathbf{ipiv}}\left(k\right)$ were interchanged, and ${d}_{kk}$ is a $1$ by $1$ diagonal block.
 ${\mathbf{uplo}}=\text{'U'}$ and ${\mathbf{ipiv}}\left(k\right)={\mathbf{ipiv}}\left(k1\right)<0$
 Rows and columns $k1$ and ${\mathbf{ipiv}}\left(k\right)$ were interchanged and ${d}_{k1:k,k1:k}$ is a $2$ by $2$ diagonal block.
 ${\mathbf{uplo}}=\text{'L'}$ and ${\mathbf{ipiv}}\left(k\right)={\mathbf{ipiv}}\left(k+1\right)<0$
 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: $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (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 righthand 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
f04bhf 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 ${\mathbf{rcond}}=1/\left({\Vert A\Vert}_{1}{\Vert {A}^{1}\Vert}_{1}\right)$.
 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
$\hat{x}$, such that
${\Vert \hat{x}x\Vert}_{1}/{\Vert x\Vert}_{1}\le {\mathbf{errbnd}}$, where
$\hat{x}$ 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).
6
Error 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:
 ${\mathbf{ifail}}>0\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{ifail}}\le {\mathbf{n}}$

Diagonal block $\u2329\mathit{\text{value}}\u232a$ 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':
${\mathbf{uplo}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 ${\mathbf{ifail}}=3$

On entry, ${\mathbf{nrhs}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
 ${\mathbf{ifail}}=5$

On entry, ${\mathbf{lda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
 ${\mathbf{ifail}}=8$

On entry, ${\mathbf{ldb}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
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 integer allocatable memory required is n, and the real allocatable memory required is $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2\times {\mathbf{n}},{\mathbf{lwork}}\right)$, where lwork is the optimum workspace required by f07maf (dsysv). If this failure occurs it may be possible to solve the equations by calling the packed storage version of f04bhf, f04bjf, or by calling f07maf (dsysv) 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.
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.
f04bhf uses the approximation
${\Vert E\Vert}_{1}=\epsilon {\Vert A\Vert}_{1}$ to estimate
errbnd. See Section 4.4 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
f04bhf 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 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.
The complex analogues of
f04bhf are
f04chf for complex Hermitian matrices, and
f04dhf for complex symmetric matrices.
10
Example
This example solves the equations
where
$A$ is the symmetric indefinite matrix
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.
10.1
Program Text
Program Text (f04bhfe.f90)
10.2
Program Data
Program Data (f04bhfe.d)
10.3
Program Results
Program Results (f04bhfe.r)