In this series I will talk about concepts of numerical analysis. To start we’ll look briefly at the definition of numerical analysis, next we’ll talk about types of errors that can occur in calculations and finally we’ll discuss how numerical analysis works on a high level.
Numerical analysis is the study of algorithms that use numerical approximation.Usually you will work with symbols, formulae and other abstract concepts while solving a mathematical problem. At a certain point you will have to use data with your mathematical solution to obtain a result. To do this we introduce numerical methods. These methods are usually in the form an algorithm.
There are three kinds of errors we can distinguish:Inaccurate data: the input data can be to some extent inaccurate, and as a result these errors are propagated throughout the calculations.Rounding errors: a computer has finite memory and uses a finite number of digits to store data and therefore it cannot store all data with the level of precision it might require.Some solutions may require an infinite amount of operations to complete; so to produce exact results you’d need an infinite amount of running time, which is ofcourse not feasable
Consider the following example: the equation “x” to the power of two minus 742 “x” plus 2 equals 0; find the possible values of x.
To solve the equation, we usually make use of this algorithm.
Using the algorithm from the previous slide, we find these solutions.Note that the second solution x2 only has one digit. This is a result of subtracting two numbers that are nearly equal to each other: such kind of operations will reduce the accuracy of the applied method.
The alternative method to finding x2 gives us a result with much more known digits
Previous example illustrates the numerical stability of a method: one method will produce more accurate results than the other method.Another aspect of designing algorithms has to do with efficiency. We will also want a method with very low cost or “time complexity”.
In conclusion:We looked at the definition of numerical analysis, talked about reasons for inaccuracies,Looked at an example and concluded with how we should design algorithms.
Numerical Analysis | 1. Introduction
Numerical method to algorithm
• The study of algorithms that use numerical
• Inaccurate data
• Rounding errors
• Infinite amount of operations