Topic outline

  • WELCOME LETTER

    Dear Students,
    Welcome to the Numerical Methods (CSE312) course. I,  Tirtha Roy will be your co-pilot in this online journey of learning. 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.



    Instructor

    Tirtha Roy

    Lecturer

    Lecturer, 
    Department of CSE
    Daffodil International University


    Office: AB4
    Email: tirtha.cse0403.c@diu.edu.bd
    Phone: +8801515217840

        

    Course Rationals

    Numerical analysis, 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
    Develop Derive and 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

    Quick Access Link

    gbd     ide    board   board



  •   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.



    •                      



  •   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


      

    •                      



    •       Bisection and  Newton Raphson Method



      Topics of Discussion:

          • Solution of Linear Equation by Matrix 
          • Bisection method to solve  algebraic and transcendental equations  with an algorithm
          • Newton Raphson method to solve algebraic and transcendental equations with Algorithm.

      Expected Learning Outcomes:

          • Ability to solve linear equations by matrix Visualize the applications 
          • 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.

                                 


    •                      


  •       False Position and  Sacant Method


    Topics of Discussion:

        • False position method to solve algebraic and transcendental equations with Algorithm.
        • Sacant method to solve algebraic and transcendental equations with Algorithm.

    Expected Learning Outcomes:

        • Method of False Position:- An algorithm for finding roots which retain that prior estimate for which the function value has the opposite sign from the function value at the current best estimate of the root. In this way, the method of false position keeps the root bracketed
        • The existence of a fixed point is therefore of paramount importance in several areas of mathematics and other sciences. Fixed point results provide conditions under which maps have solutions. The theory itself is a beautiful mixture of analysis (pure and applied), topology, and geometry.

    Resources of Learning:



    •                      

  • Online Makeup Class


  • Lagrange  and  Spline interpolation


    Topics of Discussion:

        • Interpolation: Newton’s Forward Difference Method.
        • Interpolation: Newton’s Backward Difference Method
        • Solving Linear Equation (SLE)
        •  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

        •  In order to reduce the numerical computations associated to the repeated application of the     
              existing interpolation formula in computing a large number of interpolated values, a formula  
              has been derived from Newton’s forward/backward interpolation formula for representing the
              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:



    •                      


  • Newton Forward Backward and  Divided difference table


    Topics of Discussion:

        • Interpolation: Newton’s Forward Difference Method.
        • Interpolation: Newton’s Backward Difference Method
        • Solving Linear Equation (SLE)
        •  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

        •  In order to reduce the numerical computations associated to the repeated application of the     
              existing interpolation formula in computing a large number of interpolated values, a formula  
              has been derived from Newton’s forward/backward interpolation formula for representing the
              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:



    •                      



  •   Welcome to Mid-Term Exam



       



    • Jacobe Gauss Seidel and Gauss Elimination


      Topics of Discussion:

          • Gauss Elimination Iteration method
          • Gauss-Seidal Iteration method

      Expected Learning Outcomes:

          • Gauss-Seidel Method is used to solve the linear system Equations. This method is named after the German Scientist Carl Friedrich Gauss and Philipp Ludwig Siedel. It is a method of iteration for solving  n linear equation   with the unknown variables. This method is very simple and uses in digital computers for computing.

          • Jacobi iterative method is considered as an iterative algorithm which is used for determining the solutions for the system of linear equations in numerical linear algebra, which is diagonally dominant. In this method, an approximate value is filled in for each diagonal element.

      Resources of Learning:



      •                      


    • Gauss Elimination by pivoting and Triangular systems and back substitution


      Topics of Discussion:

          • Gauss Elimination Iteration method
          • Triangular systems and back substitution Iteration method

      Expected Learning Outcomes:

          • Gauss Elimination Method is used to solve the linear system Equations. This method is named after the German Scientist Carl Friedrich Gauss and Philipp Ludwig Siedel. It is a method of iteration for solving  n linear equation   with the unknown variables. This method is very simple and uses in digital computers for computing.

          • Triangular systems and back substitution Iteration method is considered as an iterative algorithm which is used for determining the solutions for the system of linear equations in numerical linear algebra, which is diagonally dominant. In this method, an approximate value is filled in for each diagonal element.

      Resources of Learning:



      •                      



    •   LU decomposition and Cholesky’s method 


                


      Topics of Discussion:

          • Lower-Upper Factorization (LU Factorization)

      Expected Learning Outcomes:

          • LU decomposition is a better way to implement Gauss elimination, especially for repeated solving a number of equations with the same left-hand side. This provides the motivation for LU decomposition where a matrix A is written as a product of a lower triangular matrix L and an upper triangular matrix U. 

      Resources of Learning:   



      •                      

    • Assignment


        Welcome to Assignment 



         



      • Gauss Jordan method and Linear and polynomial regression


        Topics of Discussion:

            • Numerical solution of ordinary differential equations: Runge-kutta method of fourth order
            • Curve fitting: Least square method for linear and non-linear case

        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.  

            • 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:



        •                      


      • Exponential and Trigonometric functions and  Chebyshev polynomial 


        Topics of Discussion:

            • Numerical solution of ordinary differential equations: Runge-kutta method of fourth order
            • Curve fitting: Least square method for linear and non-linear case

        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.  

            • 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:



        •                      



      •   Numerical Integration & Differentiation 


                  


        Topics of Discussion:

            • Numerical solution of ordinary differential equations: Runge-kutta method of fourth order
            • Curve fitting: Least square method for linear and non-linear case

        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.  

            • 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:   



        •                      

      • Presentation


          Welcome to Presentation



        • Quiz


            Welcome to Quiz



             


          • Topic 19

            • Topic 20

              • Topic 21

                • Topic 22

                  • Topic 23

                    • Topic 24

                      • Topic 25

                        • Topic 26

                          • Topic 27

                            • Topic 28