Computational Methods

 Course Objectives :

1. To understand numerical methods to find roots of functions and first order unconstrained minimization of functions.
2. To introduce concept of interpolation methods and numerical integration.
3. To understand numerical methods to solve systems of algebraic equations and curve fitting by splines.
4. To understand numerical methods for the solution of Ordinary and partial differential equations.

Course Outcomes (CO)

• CO1 Ability to develop mathematical models of low level engineering problems
• CO2 Ability to apply interpolation methods and numerical integration.
• CO3 Ability to solve simultaneous linear equations and curve fitting by splines
• CO4 Ability to numerically solve ordinary differential equations that are initial value or boundary value problems

UNIT-I

Review of Taylor Series, Rolle ’s Theorem and Mean Value Theorem, Approximations and Errors in numerical computations, Data representation and computer arithmetic, Loss of significance in computation

Location of roots of equation: Bisection method (convergence analysis and implementation), Newton Method (convergence analysis and implementation), Secant Method (convergence analysis and implementation).

Unconstrained one variable function minimization by Fibonacci search, Golden Section Search and Newton’s method. Multivariate function minimization by the method of steepest descent, Nelder- Mead Algorithm.

UNIT-II

Interpolation: Assumptions for interpolation, errors in polynomial interpolation, Finite differences, GregoryNewton’s Forward Interpolation, Gregory-Newton’s backward Interpolation, Lagrange’s Interpolation, Newton’s divided difference interpolation

Numerical Integration: Definite Integral, Newton-Cote’s Quadrature formula, Trapezoidal Rule, Simpson’s onethird rule, simpson’s three-eight rule, Errors in quadrature formulae, Romberg’s Algorithm, Gaussian Quadrature formula.

UNIT-III

System of Linear Algebraic Equations: Existence of solution, Gauss elimination method and its computational effort, concept of Pivoting, Gauss Jordan method and its computational effort, Triangular Matrix factorization methods: Dolittle algorithm, Crout’s Algorithm, Cholesky method, Eigen value problem: Power method

Approximation by Spline Function: First-Degree and second degree Splines, Natural Cubic Splines, B Splines, Interpolation and Approximation

UNIT - IV

Numerical solution of ordinary Differential Equations: Picard’s method, Taylor series method, Euler’s and Runge-Kutta’s methods, Predictor-corrector methods: Euler’s method, Adams-Bashforth method, Milne’s method.

Numerical Solution of Partial Differential equations: Parabolic, Hyperbolic, and elliptic equations Implementation to be done in C/C++

Textbook(s):

1. E. Ward Cheney & David R. Kincaid , “Numerical Mathematics and Computing” Cengage; 7th ed (2013).

References:

1. R. L. Burden and J. D. Faires, “Numerical Analysis”, CENGAGE Learning Custom Publishing; 10th Edition (2015).
2. S. D. Conte and C. de Boor, “Elementary Numerical Analysis: An Algorithmic Approach”, McGraw Hill, 3rd ed. (2005).
3. H. M. Antia, “Numerical Methods for Scientists & Engineers”, Hindustan Book Agency, (2002).
4. E Balagurusamy “Numerical Methods” McGraw Hill Education (2017).