Numerical Methods

1. Introduction 
2. Importance of numerical methods

  • Review of calculus
  • Taylor’s theorem
  • Errors in numerical computations
  • Use of computer programming in numerical methods

3. Solution of Nonlinear Equations

  • Nonlinear equations and their solutions
  • Trial and error method
  • Graphical method
  • Iterative methods: Bisection method
  • False position method
  • Secant method
  • Newton’s method
  • Rate of convergence of iterative methods
  • Newton’s method for polynomials and Horner’s rule

4. Solution of Set of Algebraic Equations

  • Existence of solutions for linear set of equations
  • Gaussian elimination method
  • pivoting
  • ill- conditioning
  • Gauss-Jordan method
  • Matrix inversion
  • Matrix factorization: Doolittle algorithm
  • Cholesky’s factorization
  • Iterative solution using Gauss Seidel method
  • Eigen value and eigen vector using power method

5. Interpolation and Approximation

  •  Lagrange interpolation
  • Newton’s interpolation using divided differences and difference table
  • Cubic spline interpolation
  • Least squares method of fitting linear and nonlinear function for given data

6. Numerical Differentiation and Integration\

  •  Numerical differentiation formulas
  • Maxima and minima of a tabulated function
  • Newton- Cote’s quadrature formulas: Trapezoidal
  • Simpson’s 1/3 and 3/8 rule
  • Romberg integration
  • Gaussian integration

7. Solution of Ordinary Differential Equations 

  • Review of differential equations
  • Initial value problem
  • Taylor series method
  • Euler’s method and its accuracy
  • Henu’s method
  • Runge-Kutta methods
  • Solution of higher order equations
  • Solution of boundary value problems using finite difference and Shooting method

8. Solution of Partial Differential Equations 

  • Review of partial differential equations
  • Deriving difference equations
  • Solution of Laplacian equation and Poisson’s equation