Numerical Methods

Unit I: Solution of Nonlinear Equations ———————————— 10 hours

1. 1. Review of calculus and Taylor’s theorem
   2. Errors in numerical calculations
   3. Trial and error method
   4. Bisection method
   5. Newton’s method
   6. Secant method and their convergence
   7. Fixed point iteration and is convergence

Unit II: Solution of Linear Algebraic Equations ————————– 10 hours

1. 1. Review of the existence of solutions and properties of matrices
   2. Gaussian Climination method
   3. Pivoting
   4. Ill conditioning
   5. Gauss-Jordan method
   6. Inverse of matrix using Gauss elimination method
   7. Method of factorization
   8. Dolittle algorithm
   9. Cholesky’s factorization
   10. Iterative solutions
   11. Solving eigen value problems using power method

Unit III: NumericalDifferentiation and Integration ———————- 6 hours

1. 1. Newton’s differentiation formulas
   2. Maxima and minima of tabulated function
   3. Netwon’s quadrature formulas
   4. Gaussian integration algorithm
   5. Romberg integration formulas

Unit IV: Interpolation and Approximation ——————————— 8 hours

1. 1. Lagrange’s polynomials
   2. Newton’s interpolation using difference and divided differences
   3. Cubic spline interpolation
   4. Least squares method for linear and nonlinear data

Unit V: Solution of Ordinary Differential Equations ——————— 8 hours

1. 1. Review of differential equations
   2. Initial value problem
   3. Taylor series method picard’s method
   4. Euler’s method and its accuracy
   5. Henu’s method
   6. Runge-Kutta methods
   7. Solution of the higher order equations
   8. Boundary value problems
   9. Shooting method and its algorithm

Unit VII: Solution of Partial Differential Equations ———————- 6 hours

1. 1. Review of partial differential equations
   2. Deriving difference equations
   3. Laplacian equation
   4. Poisson’s equation
   5. Engineering examples

Text Book:

1. 1. C.F. Gerald and P.O. Wheatly: Applied Numerical Analysis, 4th Edition, addison Wesley publishing Company, New York.

Reference Book:

1. 1. W. Chency and D. Kinciad: Numerical Mathematics and Computing, 2nd Edition, Brooks Cole Publishing Co, 1985.
   2. W.H. Press, B.P. Flannery et.al.: Numerical Recipes in C, 1st Edition, Cambridge press, 1988.
   3. S. Yakwitz and F. Szidarovszky: An Introduction to Numerical Computations, 2nd Edition, Macmillan Publishing Co., New York.