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    WELCOME LETTER

    Dear Students,
    Welcome to the Numerical Methods (CSE 235) course. I,  Zannatul Mawa Koli will be your guide throughout this journey. The numerical methods are used for deeper understanding to predict the anomalies which are not possible in the analytical methods because the analytical method can solve only two or three unknown variables but numerical methods can do much more than it very accurately. I care about your success in these courses. I'm glad you are here.
    COURSE TEACHER                                             

    Zannatul Mawa Koli(ZMK)

    Lecturer, 

    Department of CSE,

    Ab-04, Room:727

    Daffodil International University, Ashulia, Dhaka.

    Course Rationals

    Numerical analysis is an area of mathematics and computer science that creates, analyzes, and implements algorithms for obtaining numerical solutions to problems involving continuous variables. Such problems arise throughout the natural sciences, social sciences, engineering, medicine, and business. Since the mid-20th century, the growth in power and availability of digital computers has led to the increasing use of realistic mathematical models in science and engineering, and numerical analysis of increasing sophistication is needed to solve these more detailed models of the world. The formal academic area of numerical analysis ranges from quite theoretical mathematical studies to computer science issues. With the increasing availability of computers, the new discipline of scientific computing, or computational science, emerged during the 1980s and 1990s. The discipline combines numerical analysis, symbolic mathematical computations, computer graphics, and other areas of computer science to make it easier to set up, solve, and interpret complicated mathematical models of the real world.

    Course Objectives

    O1
    Computing integrals and derivatives
    O2
    Solving differential equations
    O3
    Building models based on data, be it through interpolation, Least Square, or other methods
    O4
    Root finding and numerical optimization
    O5
    Estimating the solution to a set of linear and nonlinear equations
    O6
    Computational geometry

    Course Outcomes

    CO1
    Demonstrate an understanding of common numerical methods and how they are used to obtain approximate solutions to otherwise intractable mathematical problems.
    CO2
    Analyze and apply numerical methods to obtain approximate solutions to mathematical problems and estimate errors in the calculation of various methods.
    CO3
    Evaluate the accuracy of numerical methods for various mathematical operations and tasks, such as interpolation, differentiation, integration, the solution of linear and nonlinear equations, and the solution of differential equations.

    TEXT/REFERENCE BOOKS

    01
    Introductory methods of numerical analysis
    by S.S. Sastry
    02
    Numerical Analysis
    by Burden & Faires, current edition
    03
    Numerical Methods in Engineering
    by J. Kiusalaas,Cambridge University Press

    Assessment Plan

    Final Exam
    40
    Mid-term Exam
    25
    Class Test
    15
    Attendance
    07
    Assignment
    05
    Presentation
    08
    Total
    100

  • Topic 1



      Introduction to Numerical Methods



    Topics of Discussion:
        • Introduction to Numerical Methods
        • Discussion on Course Rationales, Objectives, Outcomes, Syllabus, Text Books, etc.

    Expected Learning Outcomes:

        • Recognize the importance of the course and course outcomes.
        • Get the idea of the course 

    Resources of Learning:

    Numerical methods Methods are designed for the constructive solution of mathematical problems requiring particular numerical results, usually on a computer. A numerical method is a complete and unambiguous set of procedures for the solution of a problem, together with computable error estimates (see error analysis). The study and implementation of such methods is the province of numerical analysis.



  • Topic 2



      Introduction to  Error Discussion 



    Topics of Discussion:

        • Introduction and error analysis
        • Solution of Linear Equation by Matrix 

    Expected Learning Outcomes:

        • Perform an error analysis for a given numerical method

    Resources of Learning:

    Error in solving an engineering or science problem can arise due to several factors. First, the error may be in the modeling technique. A mathematical model may be based on using assumptions that are not acceptable. For example, there are two kinds of numbers such as exact and approximate numbers


      

  • Topic 3

    • Students must
      Mark as done



            Bisection and  False Position Method



      Topics of Discussion:
          • Bisection method to solve  algebraic and transcendental equations  with an algorithm
          • False position method to solve algebraic and transcendental equations with Algorithm.

      Expected Learning Outcomes:

          • Prove results for various numerical root-finding methods
          • Perform an error analysis for a given numerical method

      Resources of Learning:

      This article is about searching zeros of continuous functions. For searching a finite sorted array, see binary search algorithm. For the method of determining what software change caused a change in behavior, see Bisection (software engineering). A few steps of the bisection method were applied over the starting range [a1;b1]. The bigger red dot is the root of the function. In mathematics, the bisection method is a root-finding method that applies to any continuous functions for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and then selecting the subinterval in which the function changes sign, and therefore must contain a root. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods.[1] The method is also called the interval halving method,[2] the binary search method,[3] or the dichotomy method.[4] For polynomials, more elaborated methods exist for testing the existence of a root in an interval (Descartes' rule of signs, Sturm's theorem, Budan's theorem). They allow extending the bisection method into efficient algorithms for finding all real roots of a polynomial; see Real-root isolation.

                                 

    • url icon
      Bisection and False position method resources URL
      Students must
      Mark as done

  • Topic 4


          Fix Point Iteration, Newton-Raphson Method and Secant Method


    Topics of Discussion:

        • Fix point iteration method to solve algebraic and transcendental equations with Algorithm.
        • Newton Raphson method to solve algebraic and transcendental equations with Algorithm.
        • Secant method to solve algebraic and transcendental equations with Algorithm.

    Expected Learning Outcomes:

        • Solving non-linear equations and finding root using open end methods.
        • calculating accuracy and tolerance of fix point iteration, Newton-Raphson method and secant method, then differentiating between them to find the best method to apply

    Resources of Learning:





  • Assignment

    Assignment Submission

    Topic:

    You already know about the process of solving non-linear equation and finding their root using Bisection, False-position, Fix point iteration, Newton-Raphson and secant method. Now you are given an algebraic equation \( x^2-4x-10=0 \) . Find the root of the equation using all of the method mentioned above and compare their tolerance percentage.

    Submission Date: 01 march, 2023.

    • assign icon
      Section: PC-A Assignment
      Opened: Monday, 20 February 2023, 12:00 AM
      Due: Tuesday, 7 March 2023, 12:00 AM
      Not available unless: You belong to Section: PC-A
    • assign icon
      Section: PC-B Assignment
      Opened: Monday, 20 February 2023, 12:00 AM
      Due: Tuesday, 7 March 2023, 12:00 AM
      Not available unless: You belong to Section: PC-B
    • assign icon
      Section: PC-C Assignment
      Opened: Monday, 20 February 2023, 12:00 AM
      Due: Tuesday, 7 March 2023, 12:00 AM
      Not available unless: You belong to Section: PC-C
    • assign icon
      Section: A Assignment
      Opened: Monday, 20 February 2023, 12:00 AM
      Due: Tuesday, 7 March 2023, 12:00 AM
      Not available unless: You belong to Section: 59-A

  • Topic 6


    Linear  and  Lagrange interpolation


    Topics of Discussion:

        • Solving Linear Interpolation problems.
        •  Lagrange Interpolation Formula

    Expected Learning Outcomes:

        • Approximate a function using an appropriate numerical method

        • Able to find maximum and minimum value of a tabulated functions., able to use in cryptography, able to forecast missing data.

        • Numerical data on a pair of variables by a polynomial curve.

        • Approximate a function using an appropriate numerical method

        • Code a numerical method in a modern computer language

    Resources of Learning:




  • Topic 7



      Welcome to Mid-Term Exam



       


    • Students must
      Mark as done

      Syllabus :

       
      Numbers and Errors:
      Introduction, Accuracy and errors, Significant digits, Absolute and relative error, Rounding error in functional evaluation, Propagation of error in the arithmetic process, and Truncation errors.
      Solution of Non-linear Equation:
      Method of iteration, Bisection method, Newton–Raphson method, False position method, Secant method, Fixed point method.
      Interpolation:

  • Topic 8


    Newton Interpolation Formula


    Topics of Discussion:

    Expected Learning Outcomes:

    1. Student will learn to solve mathematical problem using newton interpolation formula and find out missing data point from a data set.

    2. They will learn to build divided difference table for solving problems using newton interpolation formula.

    3. They will learn to apply forward and backward difference methods for finding out missing data points.

     

    Resources of Learning:




  • Topic 9


    Gauss Elimination by pivoting, Gauss-Jordan and LU decomposition


    Topics of Discussion:

    Expected Learning Outcomes:

        • These methods are used for solving linear equation problem.
        • Student will learn different aspect and techniques for solving linear equation and find out better method for approaching.

    Resources of Learning:





  • Topic 10



      Numerical Integration


              


    Topics of Discussion:

        • Trapezoidal Rule, Simpson's 1/3 Rule, Simpson's 3/8 Rule, Weddle's Rule and Boole's Rule

    Expected Learning Outcomes:

        • Student will understand different numerical Integration methods and their application to solve various integral problems.

    Resources of Learning:   




  • Topic 11


    Jacobi Iteration and gauss Seidel Iteration method


    Topics of Discussion:

        • Jacobi iteration method
        • Gauss Seidel iteration method

    Expected Learning Outcomes:

    Resources of Learning:


    https://www.youtube.com/watch?v=jPv5pP1kOco

    https://www.youtube.com/watch?v=gxy6VI1hEfs


  • Topic 12


    Curve Fitting: Regression


    Topics of Discussion:

        • Curve fitting: Least square method for linear and non-linear case

    Expected Learning Outcomes:

        • Construct a curve or mathematical function that has the best fit to a series of data points.

        • Curve fitting, also known as regression analysis, is used to find the "best fit" line or curve for a series of data points. Most of the time, the curve fit will produce an equation that can be used to find points anywhere along the curve.  

    Resources of Learning:





  • Topic 13



    Runge Kutta Method 


              


    Topics of Discussion:

        • Numerical solution of ordinary differential equations: Runge-kutta method of fourth order

    Expected Learning Outcomes:

        • Solve a differential equation using an appropriate numerical method
        • Runge–Kutta method is an effective and widely used method for solving the initial-value problems of differential equations. Runge–Kutta method can be used to construct high order accurate numerical method by functions' self without needing the high order derivatives of functions.  

    Resources of Learning:   




  • Final Exam

    Welcome To Final Exam

    Syllabus:

    Interpolation: Difference tables, Newton’s forward and backward interpolation formula. 

    Solution of Linear Equations: Gaussian Elimination, LU Decomposition,  Iteration Method of Jacob, and Gauss-Seidel. 

    Curve fitting: Linear and polynomial regression, Fitting exponential and trigonometric functions, 

    Numerical Integration & Differentiation: Trapezoidal rule, Simpson’s 1/3 rule, Simpson’s 3/8 rule, and Weddle method 

    Solution of Ordinary Differential Equations: Runge-Kutta method