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Chapter 5: Problem 128

Suggest some practical ways of reducing the roundoff error.

Short Answer

Expert verified
Answer: Roundoff errors can be minimized in numerical computations by using higher precision arithmetic, performing sensitivity analysis, avoiding subtracting nearly equal numbers, using compensated summation algorithms, scaling variables appropriately, using interval arithmetic, and choosing appropriate algorithms for better numerical stability.

Step by step solution

01

1. Use Higher Precision Arithmetic

One way to reduce roundoff error is to use higher precision arithmetic. By default, many programming languages use single or double precision, which might not be adequate for calculations with small roundoff errors. You can choose higher precision arithmetic libraries or data types in your programming language to minimize these errors.
02

2. Sensitivity Analysis

Perform a sensitivity analysis to determine the numerical stability of your algorithms. By understanding how errors propagate through your calculations, you can identify and focus on the parts of the algorithm that contribute most to the roundoff error.
03

3. Avoid Subtracting Nearly Equal Numbers

Subtracting two numbers that are very close in value can lead to significant roundoff errors. In this case, use algebraic manipulations to reorganize your calculations to avoid such subtraction.
04

4. Use Compensated Summation Algorithms

When summing a large array of numbers, use compensated summation algorithms, such as Kahan summation or pairwise summation. These algorithms track and correct the roundoff errors as the summation proceeds, resulting in a more accurate final result.
05

5. Scale Variables Appropriately

If your problem involves calculations with variables that have vastly different magnitudes, consider rescaling the variables before performing calculations. This helps in reducing the effects of roundoff error when combining the variables in calculations.
06

6. Use Interval Arithmetic

Interval arithmetic represents numbers as intervals rather than exact values. This allows the roundoff errors to be tracked and controlled, providing bounds on the error in the final result.
07

7. Choose Appropriate Algorithms

Some algorithms are more susceptible to roundoff errors than others. Choose an algorithm that is well-suited for the problem at hand and has good numerical stability properties. Research and use numerically stable algorithms for problems where roundoff errors can pose significant issues. By following these practical ways, you can effectively reduce the roundoff error and enhance the accuracy of your numerical computations.

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