NAG FL Interface
f04bdf (real_posdef_solve)
1
Purpose
f04bdf computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n$ by $n$ symmetric positive definite 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
Integer, Intent (In) 
:: 
n, nrhs, lda, ldb 
Integer, Intent (Inout) 
:: 
ifail 
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 <nag.h>
void 
f04bdf_ (const char *uplo, const Integer *n, const Integer *nrhs, double a[], const Integer *lda, double b[], const Integer *ldb, double *rcond, double *errbnd, Integer *ifail, const Charlen length_uplo) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f04bdf_ (const char *uplo, const Integer &n, const Integer &nrhs, double a[], const Integer &lda, double b[], const Integer &ldb, double &rcond, double &errbnd, Integer &ifail, const Charlen length_uplo) 
}

The routine may be called by the names f04bdf or nagf_linsys_real_posdef_solve.
3
Description
The Cholesky factorization is used to factor $A$ as $A={U}^{\mathrm{T}}U$, if ${\mathbf{uplo}}=\text{'U'}$, or $A=L{L}^{\mathrm{T}}$, if ${\mathbf{uplo}}=\text{'L'}$, where $U$ is an upper triangular matrix and $L$ is a lower triangular matrix. 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
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:
$\mathbf{n}$ – Integer
Input

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

3:
$\mathbf{nrhs}$ – Integer
Input

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) array
Input/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
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
a contains the lower triangular part of the matrix
$A$, and the strictly upper triangular part of
a is not referenced.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, the factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$.

5:
$\mathbf{lda}$ – Integer
Input

On entry: the first dimension of the array
a as declared in the (sub)program from which
f04bdf is called.
Constraint:
${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

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

Note: the second dimension of the array
b
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
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 $X$.

7:
$\mathbf{ldb}$ – Integer
Input

On entry: the first dimension of the array
b as declared in the (sub)program from which
f04bdf is called.
Constraint:
${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

8:
$\mathbf{rcond}$ – Real (Kind=nag_wp)
Output

On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({\Vert A\Vert}_{1}{\Vert {A}^{1}\Vert}_{1}\right)$.

9:
$\mathbf{errbnd}$ – 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.

10:
$\mathbf{ifail}$ – Integer
Input/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 4 in the Introduction to the NAG Library FL Interface 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:
 ${\mathbf{ifail}}>0\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{ifail}}\le {\mathbf{n}}$

The principal minor of order $\u2329\mathit{\text{value}}\u232a$ of the matrix $A$ is not positive definite. The factorization has not been completed and 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, ${\mathbf{uplo}}\ne \text{'U'}$ or $\text{'L'}$: ${\mathbf{uplo}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=2$

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: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
 ${\mathbf{ifail}}=7$

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 7 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
The integer allocatable memory required is
n, and the real allocatable memory required is
$3\times {\mathbf{n}}$. Allocation failed before the solution could be computed.
See
Section 9 in the Introduction to the NAG Library FL Interface 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.
f04bdf 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
f04bdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f04bdf 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 $\left(\frac{1}{3}{n}^{3}+{n}^{2}r\right)$. 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 analogue of
f04bdf is
f04cdf.
10
Example
This example solves the equations
where
$A$ is the symmetric positive definite 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
10.2
Program Data
10.3
Program Results