Estimation of Errors in Numerical Solutions

Numerical solutions are not exact; we need some criterion to determine its accuracy
June 26, 2024 by
Estimation of Errors in Numerical Solutions
Hamed Mohammadi
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Introduction

Since numerical solutions for solving science and engineering problems are not exact, we need some criterion to identify to what extent the estimated solution that we just have obtained using numerical method is accurate. We need tools to determine and evaluate errors in this methods. Several measures can be used to estimate the accuracy of an approximate solution. Based on the application of the methods used for solving a specific problem the decision to use one method over another is decided about.

Defining Error in Numerical Methods

Let x T S be the true (exact) solution such that f ( x T S ) = 0 , and let x N S be a numerically approximated solution such that f ( x N S ) = ϵ (where ϵ is a small number). Four measures that can be considered for estimating the error are:


True Error

The true error is the difference between the true solution, x T S , and a numerical solution, x N S :
T r u e E r r o r = x T S - x N S
Unfortunately, however, the true error cannot be calculated because the true solution is generally not known.


Tolerance in f ( x )

Instead of considering the error in the solution, it is possible to consider the deviation of f ( x N S ) from zero. The value of f ( x T S ) at x T S is obviously zero. The tolerance in f ( x ) is defined as the absolute value of the difference between f ( x T S ) and f ( x N S ) :
T o l e r e n c e = | f ( x T S ) - f ( x N S ) | = | 0 - ϵ | = ϵ
The tolerance in f ( x ) then is the absolute value of the function at x N S .

Tolerance in the Solution

A tolerance is the maximum amount by which the true solution can deviate from an approximate numerical solution. A tolerance is useful for estimating the error when bracketing methods are used for calculating the numerical solution. In this case, if it is known that the solution is within the domain [ a , b ] , then the numerical solution can be taken as the midpoint between a and b :
x N S = a + b 2
plus or minus a tolerance that is equal to half the distance between a and b :
T o l e r e n c e = | b - a 2 |

Relative Error

If x N S is an estimated numerical solution, then the True Relative Error is given by:
T r u e R e l a t i v e E r r o r = | x T S - x N S x T S |
This True Relative Error cannot be calculated since the true solution x T S is not known. Instead, it is possible to calculate an Estimated Relative Error when two numerical estimates for the solution are known. This is the case when numerical solutions are calculated iterative, where in each new iteration a more accurate solution is calculated. If x N S ( n ) is the estimated numerical solution in the last iteration and x N S ( n - 1 ) is the estimated numerical solution in the preceding iteration, than Estimated Relative Error can be defined by:
E s t i m a t e d R e l a t i v e E r r o r = | x N S ( n ) - x N S ( n - 1 ) x N S ( n - 1 ) |
When the estimated numerical solutions are close to the true solution, it is anticipated that the difference x N S ( n ) - x N S ( n - 1 ) is small compared to the value of x N S ( n ) , and the Estimated Relative Error is approximately the same as the True Relative Error.
Estimation of Errors in Numerical Solutions
Hamed Mohammadi June 26, 2024
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