# NAG Library Routine Document

## 1Purpose

f07fcf (dsposv) uses the Cholesky factorization
 $A=UTU or A=LLT$
to compute 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.

## 2Specification

Fortran Interface
 Subroutine f07fcf ( uplo, n, nrhs, a, lda, b, ldb, x, ldx, work, 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
#include nagmk26.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)
The routine may be called by its LAPACK name dsposv.

## 3Description

f07fcf (dsposv) 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 (dsposv) must perform iterative refinement on each right-hand side, any efficiency gains will reduce as the number of right-hand 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 right-hand sides and for large matrix sizes. The cut-off values for the number of right-hand 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 back-substitution. f07fcf (dsposv) always attempts the iterative refinement strategy first; you are advised to compare the performance of f07fcf (dsposv) with that of its full precision counterpart f07faf (dposv) 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 right-hand sides we have
 $resid < n x A ε ,$
where $‖\mathit{resid}‖$ is the $\infty$-norm of the residual, $‖x‖$ is the $\infty$-norm of the solution, $‖A‖$ is the $\infty$-norm of the matrix $A$ and $\epsilon$ is the machine precision returned by x02ajf.

## 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
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

## 5Arguments

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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand 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) 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 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:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07fcf (dsposv) 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) arrayInput
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 right-hand side matrix $B$.
7:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07fcf (dsposv) 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) arrayOutput
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:     $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07fcf (dsposv) 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) arrayWorkspace
11:   $\mathbf{swork}\left({\mathbf{n}}×\left({\mathbf{n}}+{\mathbf{nrhs}}\right)\right)$ – Real (Kind=nag_rp) arrayWorkspace
Note: this array is utilized in the reduced precision computation, consequently its type nag_rp reflects this usage.
12:   $\mathbf{iter}$ – IntegerOutput
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 machine-specific 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:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error 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.
${\mathbf{info}}>0 \text{and} {\mathbf{info}}\le {\mathbf{n}}$
The leading minor of order $〈\mathit{\text{value}}〉$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed.

## 7Accuracy

For each right-hand 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 ${\mathbf{uplo}}=\text{'U'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|{U}^{\mathrm{T}}\right|\left|U\right|$;
• if ${\mathbf{uplo}}=\text{'L'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|L\right|\left|{L}^{\mathrm{T}}\right|$,
$c\left(n\right)$ 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
 $x^ - x 1 x 1 ≤ κA E 1 A 1$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, 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.

## 8Parallelism and Performance

f07fcf (dsposv) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07fcf (dsposv) 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 complex analogue of this routine is f07fqf (zcposv).

## 10Example

This example solves the equations
 $AX=B ,$
where $A$ is the symmetric positive definite matrix
 $A = 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18$
and
 $B = 8.70 -13.35 1.89 -4.14 .$

### 10.1Program Text

Program Text (f07fcfe.f90)

### 10.2Program Data

Program Data (f07fcfe.d)

### 10.3Program Results

Program Results (f07fcfe.r)

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