NAG FL Interface
f07fcf (dsposv)
1
Purpose
f07fcf uses the Cholesky factorization
to compute the solution to a real system of linear equations
where
$A$ is an
$n$ by
$n$ symmetric positive definite matrix and
$X$ and
$B$ are
$n$ by
$r$ matrices.
2
Specification
Fortran Interface
Subroutine f07fcf ( 
uplo, n, nrhs, a, lda, b, ldb, x, ldx, work, swork, iter, info) 
Integer, Intent (In) 
:: 
n, nrhs, lda, ldb, ldx 
Integer, Intent (Out) 
:: 
iter, info 
Real (Kind=nag_wp), Intent (In) 
:: 
b(ldb,*) 
Real (Kind=nag_wp), Intent (Inout) 
:: 
a(lda,*), x(ldx,*) 
Real (Kind=nag_wp), Intent (Out) 
:: 
work(n,nrhs) 
Real (Kind=nag_rp), Intent (Out) 
:: 
swork(n*(n+nrhs)) 
Character (1), Intent (In) 
:: 
uplo 

C Header Interface
#include <nag.h>
void 
f07fcf_ (const char *uplo, const Integer *n, const Integer *nrhs, double a[], const Integer *lda, const double b[], const Integer *ldb, double x[], const Integer *ldx, double work[], float swork[], Integer *iter, Integer *info, const Charlen length_uplo) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f07fcf_ (const char *uplo, const Integer &n, const Integer &nrhs, double a[], const Integer &lda, const double b[], const Integer &ldb, double x[], const Integer &ldx, double work[], float swork[], Integer &iter, Integer &info, const Charlen length_uplo) 
}

The routine may be called by the names f07fcf, nagf_lapacklin_dsposv or its LAPACK name dsposv.
3
Description
f07fcf first attempts to factorize the matrix in reduced precision and use this factorization within an iterative refinement procedure to produce a solution with full precision normwise backward error quality (see below). If the approach fails the method switches to a full precision factorization and solve.
The iterative refinement can be more efficient than the corresponding direct full precision algorithm. Since the strategy implemented by
f07fcf must perform iterative refinement on each righthand side, any efficiency gains will reduce as the number of righthand sides increases. Conversely, as the matrix size increases the cost of these iterative refinements become less significant relative to the cost of factorization. Thus, any efficiency gains will be greatest for a very small number of righthand sides and for large matrix sizes. The cutoff values for the number of righthand sides and matrix size, for which the iterative refinement strategy performs better, depends on the relative performance of the reduced and full precision factorization and backsubstitution.
f07fcf always attempts the iterative refinement strategy first; you are advised to compare the performance of
f07fcf with that of its full precision counterpart
f07faf to determine whether this strategy is worthwhile for your particular problem dimensions.
The iterative refinement process is stopped if
${\mathbf{iter}}>30$ where
iter is the number of iterations carried out thus far. The process is also stopped if for all righthand sides we have
where
$\Vert \mathit{resid}\Vert $ is the
$\infty $norm of the residual,
$\Vert x\Vert $ is the
$\infty $norm of the solution,
$\Vert A\Vert $ is the
$\infty $norm of the matrix
$A$ and
$\epsilon $ is the
machine precision returned by
x02ajf.
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: specifies whether the upper or lower triangular part of
$A$ is stored.
 ${\mathbf{uplo}}=\text{'U'}$
 The upper triangular part of $A$ is stored.
 ${\mathbf{uplo}}=\text{'L'}$
 The lower triangular part 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:
$\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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 positive definite matrix
$A$.
 If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
 If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: if iterative refinement has been successfully used (
${\mathbf{info}}={\mathbf{0}}$ and
${\mathbf{iter}}\ge 0$, see description below), then
a is unchanged. If full precision factorization has been used (
${\mathbf{info}}={\mathbf{0}}$ and
${\mathbf{iter}}<0$, see description below), then the array
$A$ contains 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
f07fcf 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

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 righthand side matrix $B$.

7:
$\mathbf{ldb}$ – Integer
Input

On entry: the first dimension of the array
b as declared in the (sub)program from which
f07fcf is called.
Constraint:
${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

8:
$\mathbf{x}\left({\mathbf{ldx}},*\right)$ – Real (Kind=nag_wp) array
Output

Note: the second dimension of the array
x
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.

9:
$\mathbf{ldx}$ – Integer
Input

On entry: the first dimension of the array
x as declared in the (sub)program from which
f07fcf is called.
Constraint:
${\mathbf{ldx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

10:
$\mathbf{work}\left({\mathbf{n}},{\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) array
Workspace


11:
$\mathbf{swork}\left({\mathbf{n}}\times \left({\mathbf{n}}+{\mathbf{nrhs}}\right)\right)$ – Real (Kind=nag_rp) array
Workspace
Note: this array is utilized in the reduced precision computation, consequently its type nag_rp reflects this usage.

12:
$\mathbf{iter}$ – Integer
Output

On exit: information on the progress of the interative refinement process.
 ${\mathbf{iter}}<0$
 Iterative refinement has failed for one of the reasons given below, full precision factorization has been performed instead.
$1$ 
The routine fell back to full precision for implementation or machinespecific reasons. 
$2$ 
Narrowing the precision induced an overflow, the routine fell back to full precision. 
$3$ 
An intermediate reduced precision factorization failed. 
$31$ 
The maximum permitted number of iterations was exceeded. 
 ${\mathbf{iter}}>0$
 Iterative refinement has been sucessfully used. iter returns the number of iterations.

13:
$\mathbf{info}$ – 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.
 ${\mathbf{info}}>0\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{info}}\le {\mathbf{n}}$

The leading minor of order $\u2329\mathit{\text{value}}\u232a$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed.
7
Accuracy
For each righthand side vector
$b$, the computed solution
$x$ is the exact solution of a perturbed system of equations
$\left(A+E\right)x=b$, where
 if ${\mathbf{uplo}}=\text{'U'}$, $\leftE\right\le c\left(n\right)\epsilon \left{U}^{\mathrm{T}}\right\leftU\right$;
 if ${\mathbf{uplo}}=\text{'L'}$, $\leftE\right\le c\left(n\right)\epsilon \leftL\right\left{L}^{\mathrm{T}}\right$,
$c\left(n\right)$ is a modest linear function of
$n$, and
$\epsilon $ is the
machine precision. See Section 10.1 of
Higham (2002) for further details.
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) for further details.
8
Parallelism and Performance
f07fcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07fcf 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 complex analogue of this routine is
f07fqf.
10
Example
This example solves the equations
where
$A$ is the symmetric positive definite matrix
and
10.1
Program Text
10.2
Program Data
10.3
Program Results