# 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