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Numerical Methods and Programming

Course Code: MA 305; MM 405  |  Credits: 4  |  Prerequisite: NIL

Syllabus Details

  • Unit I: Flow charts and algorithms, sample C-Programmes: compilation and execution of the programmes. C-alphabet: ASCII Character set, basic data types, variables and constants in C. Operators in C: Hierarchy and associativity. Flow control instructions in C: Decision control (if else), loop control (for, while, do-while), case control (switch). The break and continue statements, functions, arrays, structures (user defined data types).
  • Unit II: (At least 10 hands on sessions): MATLAB/Octave, implementation of algorithms which are in Units III - VI.
  • Unit III: (Numerical Computations) Representation of integers and fractions, fixed point and floating point arithmetics, error propagation, loss of significance, condition and instability, computational method of error propagation. Root finding: bisection method, secant method, regula-falsi method, Newton-Raphson method.
  • Unit IV: LU decomposition, Gauss elimination with and without pivoting, Gauss-Jacobi method, Gauss-Seidel method, Power method, Jacobi method to find eigenvalues.
  • Unit V: (Interpolation) Lagrange’s interpolation, Newton’s divided difference interpolation (forward, backward), Newton-Gregory formulae, Sterling’s formula.
  • Unit VI: (Numerical integration) Newton-Cotes (closed type formulae)- trapezoidal rule, Simpson’s $\tfrac{1}{3}$-rd rule, Simpson’s $\tfrac{3}{8}$-th rule.

Course Learning Objective

  1. Compute approximate values of zeros of polynomial equations, transcendental equations.
  2. Compute the interpolating polynomial for a given data and apply various algorithms to find the value of the interpolating polynomial at a given point.
  3. Compute an approximate solution of a system of linear equations and to compute approximate eigenvalues and approximate eigenvectors of a square matrix.
  4. Compute approximate values of definite integrals by applying various forms of Newton-Cotes formulae.
  5. Write programs for the algorithms that they learn in Units III to VI in both C language as well as MATLAB.
  6. Draw 2-D and 3-D plots using MATLAB to get motivation for various theorems, counter examples in Analysis. This can include courses in 2nd, 3rd, and 4th semesters also. For example, draw graphs of $x \sin\left(\tfrac{1}{x}\right)$, $x^{2} \sin\left(\tfrac{1}{x}\right)$ to conclude which one has more total variation.
  1. Yashavant P. Kanetkar, Test Your C Skills, BPB publications, fifth edition, 2009.
  2. Conte, S. D. and deBoor, C., Elementary Numerical Analysis - An Algorithmic Approach, third edition, McGraw Hill, 1981.
  3. Henrici, P., Elements of Numerical Analysis, John Wiley & Sons, 1964.
  4. Froberg, C. E., Numerical Mathematics Theory and Computer Applications, The Benjamin Cummings Pub. Co. 1985.
  5. Quarteroni, A.; Saleri, F. and Gervasio, P., Scientific Computing with MATLAB and Octave, third edition, Springer, 2010.
  6. Rudra Pratap, Getting Started with MATLAB 7: A Quick Introduction for Scientists and Engineers, Oxford University Press, 2005.
  7. Stoer, J. and Bulirsch, R., Introduction to Numerical Analysis, Texts in Applied Mathematics, Springer, 2002.
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