Introduction

Numerical Methods are techniques to approximate mathematical processes like Integrals, Differential Equations, Nonlinear Equations.

Approximations are needed because we cannot solve the procedure analytically, such as the standard normal cumulative distribution function.

Steps of Solving an Engineering Problem

Quantifying Errors

Lesson 1: True Error

In numerical analysis, errors will arise during the calculations. To be able to deal with the issue of errors we

(A) identify where the error is coming from, followed by

(B) quantify the error, and lastly

(C) minimize the error as per our needs. we'll focus on quantify the error.

True Error

What is True Error?

True error denoted by Et, is the difference between the true value(actual value) and approximate value.

Et=True value−Approximate value

Let's take an example with differential equation -

Here we are given a function value of f(x) and h then we need to find the value of derivative function f'(x). with following conditions -

Sol a) Using value of x=2 and h=0.3 try to determine the approximate value of derivative function -

Sol b) In a problem we got the approximate value, now we are said to calculate true value of the function f(x). Now we'll calculate by using our knowledge of differential calculus -

Sol c) For True error,

We know that, True error = True value - Approximate value

Et=True value−Approximate value=9.5140−10.263=−0.749

Relative True Error

Relative error is nothing but the ratio between the True Error and the True Value

ϵt = True Error / True Value
So, using previous example we've got,

True Value = 9.5410

Approximate value = 10.263

So, Et=True value−Approximate

value=9.5140−10.263=−0.749

Relative true errors are also presented as percentages. For this example,

ϵt=−0.078726×100%=−7.8726%

Approximate Errors

In the above discussion, we talked about True Errors. These errors are calculated only when true values are known. For example, calculating true errors is useful when checking if a program is working correctly and we have some known true values. However, most of the time, we won't know the true values. After all, why would you need approximate values if you already know the true values? So, when solving a problem numerically, we will only have access to approximate values. That's why we need to know how to quantify or calculate these types of errors.

Approximate error is denoted by Ea and is defined as the difference between the current approximation and the previous approximation.

Ea = Present Approximation − Previous Approximation

Let's use an example to better understand what Present Approximation and Previous Approximation are.

We'll use the previous function with some modifications for this example -

Problem:

Conditions :

Sol a) Similarly Calculate approximate value for the function f'(2) using values x=2 and h=0.3

Sol b)

Sol c) We know that, Approximate error = Present Approximation - Previous Approximation

Ea=Present Approximation−Previous Approximation=9.8800−10.263=−0.38300

Relative Approximate Error

Similar to Relative True Error, use this formula:

ϵa = Approximate Error / Present Approximation

That's all for today. In summary, we've learned about quantifying errors: 1) True Error, and 2) Approximate Error. Happy learning!