1. 1 CSE 231: Numerical Analysis Topics Introduction of Numerical Analysis, Requirements of Numerical Analysis, Different Types of Error Lecture 1 Md. Showrov Hossen Lecturer Dept. of CSE City University, Bangladesh

2. Introduction to Numerical Methods  What are NUMERICAL METHODS ?  Why do we need them?  Text Book recommended: G. Shankar Rao: Numerical Analysis 2

3. 3 Numerical Methods Numerical Methods: Algorithms that are used to obtain numerical solutions of a mathematical problem. Why do we need them? 1. No analytical solution exists, 2. An analytical solution is difficult to obtain or not practical.

4. 4 Numerical Analysis  Numerical analysis is a branch of mathematics that solves continuous problems using numeric approximation.  It involves designing methods that give approximate but accurate numeric solutions, which is useful in cases where the exact solution is impossible or prohibitively expensive to calculate.

5. 5 What do we need? Basic Needs in the Numerical Methods: Practical: Can be computed in a reasonable amount of time. Accurate:  Good approximate to the true value,  Information about the approximation error (Bounds, error order,… ).

6. 6 Solution of Nonlinear Equations  Some simple equations can be solved analytically:  Many other equations have no analytical solution:

7. 7 Errors  One of the most important aspects of numerical analysis is the error analysis. Errors may occur at any stage of the process of solving a problem.  By the error we mean the difference between the true value and the approximate value. Error = True value – Approximate value.

8. 8 Types of Errors o Inherent Errors: These are the errors involved in the statement of a problem. If the data or parameters are in some way determined by physical measurement, they will probably differ from the exact values. Errors inherent in the statement of the problem are called inherent errors.

9. 9 Types of Errors (Cont.) o Analytic Errors: These are the errors introduced due to transforming a physical or mathematical problem into a computational problem. The functions involved in mathematical formulas are frequently specified in the form of infinite sequences or series. For example, If we compute sin (x) by the formula then it leads to an analytical error.

10. 10 Types of Errors (Cont.) o Round-off errors: Roundoff error is the difference between an approximation of a number used in computation and its exact (correct) value. Rounding multiple times can cause error to accumulate. For example, if 9.945309 is rounded to two decimal places (9.95), then rounded again to one decimal place (10.0), the total error is 0.054691.