The routine may be called by the names f07fcf, nagf_lapacklin_dsposv or its LAPACK name dsposv.
3Description
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 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 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 right-hand 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.
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 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
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: $\mathbf{a}({\mathbf{lda}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: the $n\times n$ 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 iter), then a is unchanged. If full precision factorization has been used (${\mathbf{info}}={\mathbf{0}}$ and ${\mathbf{iter}}<0$, see iter), 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 is called.
The leading minor of order $\u27e8\mathit{\text{value}}\u27e9$ 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 $(A+E)x=b$, 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
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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
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 implementation-specific information.