Numerical Computing

Syllabus:

i) INTRODUCTION TO OCTAVE (OR APPROPRIATE PACKAGE): Octave as a calculator, Built-in Variables and Functions, Functions and Commands; Creating Matrices, Subscript Notation for Matrix Elements, Colon Notation, Deleting Elements from Vectors and Matrices, Mathematical Operations with Matrices, Reshaping Matrices, Strings, Working with Data from External Files, Plotting.Script. Scripts m-Files; Function m-Files; Input and Output Parameters; Relational Operators, if...else..., Case Selection with switch , forLoops, whileloops, breakCommand, return Command; Vectorization.
ii) Number representations, finite precision arithmetic, errors in computing. Convergence, iteration, Taylor series.
iii) SOLUTION OF A SINGLE NON-LINEAR EQUATION: Bisection method. Fixed point methods. Newtons method. Convergence to a root, rates of convergence.
iv) REVIEW OF APPLIED LINEAR ALGEBRA: Vectors and matrices. Basic operations, linear combinations, basis, range, rank, vector norms, matrix norms. Special matrices. Solving Systems of equations (Direct Methods): Linear systems. Solution of triangular systems. Gaussian elimination with pivoting. LU decomposition, multiple right-hand sides.
v) LEAST SQUARES FITTING OF DATA: Fitting a line to data. Generalized least squares. QR decomposition.
vi) INTERPOLATION: Polynomial interpolation by Lagrange polynomials. Alternate bases: Monomials, Newton, divided differences. Piecewise polynomial interpolation. Cubic Hermite polynomials and splines.
vii) NUMERICAL QUADRATURE: Newton - Cotes Methods: Trapezoid and Simpson quadrature. Gaussian quadrature. Adaptive quadrature.
viii) ORDINARY DIFFERENTIAL EQUATIONS: Eulers Method. Accuracy and Stability. Trapezoid method. Runge - Kutta method. Boundary value problems and finite differences.

https://www.isibang.ac.in/~adean/infsys/database/Bmath/NC.html