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Numerical and Computational Methods

This course build on the Introductory Mathematical methods course and consist of three main topics: initial value problems, solving large systems, and optimization. The goal of the course is to provide a good start into each of these fields, focusing more on fundamental ideas than on involved details. Focus will be given on the mathematical understanding as well as on applying the presented concepts. Practical examples and computer programs will be covered.

Topics covered includes: (i) Initial value problems: Linear initial value problems such as the wave equation and the heat equation admit closed form solutions in simple geometries. In a more complex setup they have to be solved numerically. (ii)Solving large systems: The discretization of partial differential equations by finite difference or finite element methods leads to large sparse linear systems, either directly for linear problems or as an auxiliary subproblem for many nonlinear problems. Gaussian elimination destroys the sparse structure, so solvers are required which make use of the specific sparse matrix structure. (iii) Optimization and minimum principles: Optimization problems search for the minimizer of some quantity (cost function), possibly given constraints. Quadratic cost functions lead to linear systems using Lagrange multipliers and Kuhn-Tucker conditions. Saddle point problems, regularization and calculus of variations will be presented as fundamental concepts. A different world in encountered in the case of linear cost functions. Applications are operations research and network problems. Solution algorithms are the simplex method or interior point methods. The underlying principle in all approaches is the concept of duality.

Tthe detailed course structure:

UNIT-1:  Numerical Solution of Algebraic and Transcendental Equations: Bisection method, Secant method, Newton Raphson method, Method of Successive Approximations.

UNIT-2: Numerical Solution of Systems of Linear Algebraic Equations: Gauss elimination method, Gauss Jordan method, Gauss-Seidel iteration method, Jacobi iteration method. Numerical solution of a system of nonlinear algebraic equations: Newton’s method using Jacobian.

UNIT-3: Interpolation: Lagrange interpolation, Difference table, Divided difference interpolation, Newton’s Forward and backward difference interpolation, Piecewise interpolation, Spline interpolation.

UNIT-4: Differentiation and Integration: Numerical differentiation using Lagrange interpolation, using difference operators and using the method of undetermined coefficients. Numerical Integration: Trapezoidal rule, Simpson’s rule and Gaussian quadrature formulae.

UNIT-5: Numerical Solution of Differential Equations: Introduction, Euler’s method, Midpoint rule, Second-order Runge-Kutta method, Fourth-order Runge-Kutta method, Applications to boundary value problems. MATLAB for the solution of ordinary differential equations.

Text and Reference Books:

  1. An Introduction to Numerical Methods and Analysis by James F. Epperson, Wiley & Sons.
  2. Elementary Numerical Analysis by K. Atkinson and W. Han, Wiley & Sons.
  3. Numerical Methods for Scientific and Engineering Computation by M. K. Jain, S. R. K. Iyengar and R. K. Jain, New Age International Publishers.
  4. Computer Oriented Numerical Methods by V. Rajaraman, Prentice Hall of India