Wilkinson’s iterative refinement of solution with automatic step-size control for linear system of equations
Introduction
Throughout the presentation of this study we concerned with the numerical solution for linear system of algebraic equationswhere det(A) ≠ 0. It is well known that the strategy of iteration refinement is to generate an iterative sequence of the approximate solution that converges to the solution of (1). A pioneer work of this topic is Wilkinson’s iterative refinement of solution [2], [3] and has remained active in this area (for example, see [5], [6], [7], [8], [9]). The condition number of a nonsingular square matrix A with respect to a given matrix norm ∥·∥ is defined to beLinear system (1) of algebraic equations is called ill-conditioned if the condition number cond(A) of the coefficient matrix A is very large. It is a difficult challenge for the numerical computation to solve an ill-conditioned linear system of algebraic equations, especially for the system of algebraic equations with Hilbert matrices. This is mainly because small errors in the data may cause large errors in the solution for an ill-conditioned linear system of algebraic equations. Therefore, its numerical solution requires quite more care. Usually, the computed solution z0 of ill-conditioned system (1) may not be sufficient accurate if direct methods are employed, even may be little or no interest if matrix A is seriously ill-conditioned. For this reason the strategy of iteration refinement was employed to produce an iterative sequence of the approximate solution that converges to the solution of (1). It was showed in [1] that, Wilkinson’s iterative improvement of solution can be viewed as an iterative process with starting vector x0 = 0:or equivalently written aswith x0 = 0, in which computed solution z0 is given by a certain direct method. It is revealed from (4) that, Wilkinson’s iterative improvement of solution can be regarded as an explicit Euler method with fixed step size hk ≡ 1 for solving the system of linear ordinary differential equationswhere is a unique globally asymptotically stable equilibrium point of dynamic system (5). So that if the solution of systems (5) is expressed in the formthen we haveHowever, in general speaking, explicit Euler method with a fixed step size hn ≡ 1 may not be suitable for the system, from the standpoint of numerical solution of ordinary differential equations, because the method is not competent for generating numerical approximate sequence {zn} along the orbit defined by (5). For achieving this practical error estimates and step size selection are necessary. On the one hand, in order to yield required precision of the computed results the step sizes hn should be chosen sufficiently small, but on the other hand, in order to avoid redundant computational work the step sizes should be chosen large enough. Consequently, in the following text we will consider an embedded pair of Euler method and trapezoid rule adapted to the numerical integration of (5).
Section snippets
The necessity of Wilkinson’s iterative refinement of solution with automatic step-size control
From the analysis in last section, for the strategy of Wilkinson’s iteration refinement, we observe that Wilkinson’s iterative improvement of solution can be viewed as an explicit Euler method with fixed step size hn ≡ 1 for solving the system of linear ordinary differential equations. With this consideration, thus, it is natural to ask why we should use the step size hn ≡ 1, for it is not convinced that the step size hn ≡ 1 should be the best choice for the numerical solution of differential
An embedded pair of Wilkinson’s methods adapted to the numerical integration of (5)
This section describes an embedded pair of Wilkinson’s methods adapted to the numerical integration of (5). Here we consider only one–step methods of the formfor approximating the solution x(t) to the initial-value problem (5). A perfect numerical method should have the property that, the minimal number of mesh points will be used to ensure that the global error x(tn) − xn does not exceed a tolerance ϵ prescribed in advance. Usually, we cannot determine the
The analysis of convergence for iterative formula (9)
If hn ≡ h is a fixed constant then (9) is in fact a linear stationary iterative formula of the formwith the iterative matrix and although the practicable computational versionwill be used. Here I means identity matrix.
Explicitly, iterative methods (12) is completely consistent with the system (1). Theorem 4.1 If hn ≡ h > 0 is a fixed constant, then iterative formula (12) is unconditionally convergent. Proof It is easy to see that for any fixed constant h > 0
Numerical tests
In this stage we are concerned with the numerical tests of the embedded pair of Euler method and trapezoid rule for ill-conditioned linear systems of algebraic equations. We employ both Wilkinson’s iteration improvement and the embedded pair of Euler method and trapezoid rule for solving ill-conditioned linear system of algebraic equations. Consider the typical ill-conditioned linear system of algebraic equations with coefficient matrices of m × m Hilbert matrices A = (aij)12×12where
Conclusion
In this paper, we present an iterative improvement of solution with automatic step-size control for linear system of algebraic equations. We consider dynamic system (5) relating to the ill-conditioned system of algebraic equations (1), such that the asymptotically stable equilibrium point of the dynamic system is just the solution of system (1) and then is a limit point of motion of the dynamic system with any initial value . We solve numerically it using an embedded pair of
Acknowledgement
This Project Supported by the Natural Science Foundation of China (60573157).
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