1. Introduction to Numerical Methods
Numerical methods are algorithms used to obtain approximate solutions for mathematical problems that are challenging to solve analytically. This chapter discusses the various types of errors that arise during numerical computations, the significance of floating-point representation, and the concepts of conditioning and stability in numerical algorithms. Understanding these foundational concepts is crucial for enhancing the reliability and accuracy of computational techniques in various scientific fields.
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What we have learnt
- Numerical methods help resolve mathematical problems that lack analytical solutions.
- Errors in numerical computations can stem from approximations, rounding, and algorithm selection.
- Floating-point representation, while necessary, introduces limitations such as rounding errors and loss of precision.
- Conditioning measures a problem's sensitivity to changes in input, influencing the accuracy of the solution.
- Stability of numerical algorithms is essential for minimizing the propagation of errors during calculations.
Key Concepts
- -- Absolute Error
- The difference between the exact value and approximate value.
- -- Relative Error
- The absolute error normalized by the exact value.
- -- FloatingPoint Representation
- Method for representing real numbers in a computer, using scientific notation.
- -- WellConditioned Problem
- A problem where small changes to input produce small changes in output.
- -- IllConditioned Problem
- A problem where small changes in input can cause large changes in output.
- -- Stability
- The property of an algorithm to control the growth of numerical errors during computations.
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