FS-727: Numerical Methods Credits: 4 (2-1-2) Objective: The Objective of this course is to help student to understand basic concepts in numerical methods and able to solve engineering as well as social problems. COURSE DESCRIPTION: UNIT I: Introduction to Numerical solutions of algebraic and transcendental equations Newton-Raphson and Regula-Falsi methods. Solution of linear simultaneous equations: Gauss elimination and Gauss Jordon methods. Iterative methods: Gauss Jacobi and Gauss-Seidel methods. Definition of Eigen values and Eigen vectors of a square matrix. Computation of largest Eigen value and the corresponding Eigen vector by Rayleigh’s power method. UNIT II: Interpolation and Approximation Finite differences (Forward and Backward differences) Interpolation, Newton’s forward and backward interpolation formulae. Newton’s divided difference formula. Lagrange’s interpolation and inverse interpolation formulae. UNIT III: Numerical differentiation and Numerical Integration Numerical differentiation using Newton’s forward and backward interpolation formulae. Numerical Integration: Simpson’s one third and three eighth’s value, Boole’s rule, Weddle’s rule, Romberg’s method, two and three point Gaussian quadrature formulae, Double integrals using trapezoidal and Simpson’s rules. UNIT IV: Ordinary Differential Equations Single-step explicit methods: Taylor series method, Euler and modified Euler methods, Solution of first order and simultaneous equations by Euler’s and Picard’s method, Fourth order Runge Kutta method for solving first and second order equations, single-step implicit methods, multistep linear methods, Milne’s and Adam’s predictor and corrector methods. Text Books: 1. Balagurusamy, E., “Numerical Methods”, Tata McGraw-Hill Pub.Co.Ltd, New Delhi, 1999. 2. S.S. Sastry, “Introductory Methods of Numerical Analysis”, PHI learning Pvt Ltd. 3. Curtis F. Gerald and Patrick O.Wheatley, “Applied Numerical Analysis”, Pearson Education Ltd 4. Gerald, C.F, and Wheatley, P.O, “Applied Numerical Analysis”, Sixth Edition, Pearson Education Asia,

New Delhi, 2002. 5. Higher Engineering Mathematics by Dr. B.S. Grewal, 36th Edition, Khanna Publishers. 6. S.C. Chapra and R.P. Canale, “Numerical Methods for Engineers with Programming and Software

Applications”, McGraw-Hill, Newyork – 1998

7. Kandasamy, P., Thilagavathy, K. and Gunavathy, K., “Numerical Methods”, S.Chand Co. Ltd., New Delhi, 2003.

8. Burden, R.L and Faires, T.D., “Numerical Analysis”, Seventh Edition, Thomson Asia Pvt. Ltd., Singapore, 2002.

Course Plan: Week Unit Topics Hours

Lecture Tutorial Practical 1 I Introduction to Numerical solutions of algebraic and

transcendental equations: Newton-Raphson and Regula-Falsi methods.

2 1 2

2 I Solution of linear simultaneous equations: Gauss elimination and Gauss Jordon methods.

2 1 2

3 I Iterative methods: Gauss Jacobi and Gauss-Seidel methods.

2 1 2

4 I Definition of Eigen values and Eigen vectors of a square matrix. Computation of largest Eigen value and the corresponding Eigen vector by Rayleigh’s power method.

2 1 2

5 II Finite differences (Forward and Backward differences) Interpolation, Newton’s forward and backward interpolation formulae.

2 1 2

6 II Divided differences: Newton’s divided difference formula. Lagrange’s interpolation and inverse interpolation formulae.

2 1 2

7 III Numerical differentiation using Newton’s forward and backward interpolation formulae.

2 1 2

8 III Numerical Integration: Simpson’s one third and three eighth’s value, Boole’s rule, Weddle’s rule

2 1 2

9 III Romberg’s method, two and Three point Gaussian quadrature formulas.

2 1 2

10 III Double integrals using trapezoidal and Simpson’s rules.

2 1 2

11 IV Single-step explicit methods: Taylor series method, Euler and modified Euler methods,

2 1 2

12 IV simultaneous equations by Euler’s and Picard’s method

2 1 2

13 IV Fourth order Runge Kutta method for solving first and second order equations.

2 1 2

14 IV Single-step implicit methods, multistep linear methods,

2 1 2

15 IV Milne’s and Adam’s predictor and corrector methods.

2 1 2

16 Revision 2 1 2 TOTAL 32 16 32