# Numerical Methods: EEE 423

EEE 423: Level 4, Term 2. (Semester: November 2013; April 2012.)

Overview of term final syllabus:

Matrix Inversion and LU decomposition, Matrix analysis, Special matrices, Sets of linear equations, Numerical Integration, Boundary value problems, Finite difference method for ODE and PDE, Eigen value problems, Interpolation.

Curve fitting, Roots of nonlinear equations, Numerical Differentiation, Initial value problems, Finite Element Method.

Study Materials:

Clear understanding of the meaning of mathematical symbols are necessary for most of the contents of this course. The following link provides a good review of common symbols used in mathematical theorems:

Mathematical Symbols and their meaning (PDF file) (External link)

Gaussian quadrature is an efficient way of performing numerical integration. The following slides describes the basics of Gaussian Quadrature:

Gershgorin Circle Theorem (sometimes spelled Gerschgorin) can approximate the regions in a complex plane where the Eigen values of a matrix exist. The method can be used to approximate the position of the Eigen values with very little computational effort. The following article describes the Gershgorin Circule Theorem:

Gershgorin Circle Theorem (PDF file) (External link)

Householder's method converts any matrix to a tridiagonal form without changing the Eigen values of the matrix. It is easier to extract the Eigen values from a tridiagonal matrix than from the original matrix. For this reason, Householder's transformation is often used in Eigen value related problems. The following reading material describes the procedure of Householder's method with examples and exercise.

Householder's Method (PDF file)

Class Test 1:

Date: Jan. 28, 2014 (Tuesday)

Time: 10:00 am

Location: Room No.: 906, ECE building

Duration: 25 minutes

Full marks: 20

Syllabus:

Roots of nonlinear equations: Open methods and Bracketing methods. Bisection method, absolute error in bisection method and a priori calculations of errors, false position method, secant method, Newton-Raphson method, Muller's method, fixed-point iteration method, polynomial division, Bairstow's method, convergence analysis, termination criteria, pseudocodes of various methods, convergence/divergence characteristic of various methods, errors and error analysis of various methods.

Sets of Linear Equations: Gauss elimination, pivoting, Gauss-Jordan elimination, operation counts, scaling vector, order vector.

LU Decomposition: LU decomposition using Gaussian elimination, efficient storing of L an U in the original matrix, memory requirement analysis.

Any other topics covered in class are also included in the syllabus.

Class Test 2:

Date: March 3, 2014. (Monday)

Time: 8 am

Location: Room No.: 906, ECE building

Duration: 23 minutes

Full marks: 20

Syllabus:

Matrix Inversion and LU decomposition: Matrix norms, condition number, ill-conditioned matrices, LU decomposition using Thomas algorithm, Cholesky decomposition, matrix inversion using LU decomposition.

Curve Fitting: Least squares regression, linear curve fitting, polynomial curve fitting, conversion of non-linear curve fitting to linear curve fitting, nonlinear regression and generalized nonlinear curve fitting.

Interpolation: Lagrange interpolation polynomial, Divided difference interpolation, spline curves, cubic spline interpolation, quadratic spline interpolation, Bezier curves.

Numerical Differentiation: Forward finite-divided-difference formulas, backward finite-divided-difference formulas, centered finite-divided-difference formulas, high accuracy differentiation formulas, Richardson extrapolation, differentiation using Lagrange polynomial.

Any other topics covered in class will also be included in the syllabus.

Question breakdown: 4 questions, 20 marks Duration: 25 minutes.

Question 1: Numerical differentiation from tabulated data (4 marks)

Question 2: Interpolation from tabulated data (5 marks)

Question 3: LU decomposition of a matrix (5 marks)

Question 4: Curve fitting problem from tabulated data (6 marks)

Class Test 3:

Date: March 12, 2014. (Monday)

Time: 11 am

Location: Room No.: 906, ECE building

Duration: 18 minutes

Full marks: 20

Syllabus:

Numerical Integration: the trapezoidal rule, composite trapezoidal rule, Simpson's 1/3 rule, composite Simpson's 1/3 rule, Simpson's 3/8 rule, composite Simpson's 3/8 rule, integration using matrix multiplications, double integrals, improper integrals, Romberg integration, Gaussian quadrature.

Class Test 4:

Date: April 1, 2014

Location: Room No.: 906, ECE building

Duration: 18 minutes

Full marks: 20

Archived Items (From April 2012 Semester)

Overview of term final syllabus:

Matrix Inversion and LU decomposition, Matrix analysis, Special matrices, Sets of linear equations, Numerical Integration, Boundary value problems, Finite difference method, Finite element method, Eigen value problems and Householder's method.

Curve fitting, Interpolation, Roots of nonlinear equations, Numerical Differentiation, Numerical Integration, Initial value problems.

For complete syllabus description, students are advised to check the syllabus of the class tests. They should also refer to their class notes.

Class Test 1:

Date: 22nd May, 2012. (Tuesday)

Time: 1:05 pm

Location: Room No.: 236, ECE building

Duration: 25 minutes

Full marks: 20

Syllabus:

Roots of nonlinear equations: Open methods and Bracketing methods. Bisection method, absolute error in bisection method and a priori calculations of errors, false position method, secant method, Newton-Raphson method, Muller's method, fixed-point iteration method, graphical representation of various methods, convergence analysis, termination criteria, pseudocodes of various methods, convergence/divergence characteristic of various methods, errors and error analysis of various methods.

Any other topics covered in class are also included in the syllabus.

Class Test 2:

Date: 13th June, 2012. (Wednesday)

Time: 11 am

Location: Room No.: 236, ECE building

Duration: 20 minutes

Full marks: 20

Syllabus:

Sets of Linear Equations: Gauss elimination, pivoting, Gauss-Jordan elimination, operation counts, scaling vector, order vector, Gauss-Seidel iterative method, convergence criterion of Gauss-Seidel method.

Matrix Inversion and LU decomposition: Matrix norms, condition number, ill-conditioned matrices, LU decomposition using Gauss elimination, LU decomposition using Thomas algorithm, Cholesky decomposition.

Curve Fitting: Least squares regression, linear curve fitting, polynomial curve fitting, conversion of non-linear curve fitting to linear curve fitting.

Interpolation: Lagrange interpolation polynomial, Divided difference interpolation, spline curves, cubic spline interpolation.

Any other topics covered in class will also be included in the syllabus.

Class Test 3:

Date: 26th September, 2012. (Wednesday)

Time: 11:00 am

Location: Room No.: 236, ECE building

Duration: 25 minutes

Full marks: 20

Syllabus:

Numerical Differentiation: forward finite-divided-difference formulas, backward finite-divided-difference formulas, centered finite-divided-difference formulas, high accuracy differentiation formulas, Richardson extrapolation.

Numerical Integration: the trapezoidal rule, composite trapezoidal rule, Simpson's 1/3 rule, composite Simpson's 1/3 rule, Simpson's 3/8 rule, composite Simpson's 3/8 rule, Romberg integration, Gauss quadrature.

Ordinary Differential Equations:

Initial Value Problems (IVPs): Taylor series method, Euler's method, Modified Euler's method (predictor-corrector/Heun's method), modified Euler's method (mid-point method), Runge-Kutta methods, second order RK method, fourth order RK method, Runge-Kutta-Fehlberg method, solution of simultaneous ODEs, higher order ODEs.

Boundary Value Problems (BVPs): Shooting method, Finite difference method for Dirichlet boundary value problems.

Eigen Value Problems: Fadeev-Leverrier method for finding characteristic polynomial, Solving Eigen value problems, Gershgorin Circle Theorem.

Any other topics covered in class (including the class of 24th September, 2012) will also be included in the syllabus.

Class Test 4:

Date: 17th October, 2012. (Wednesday)

Time: 10:00 am

Location: Room No.: 236, ECE building

Duration: 22 minutes

Full marks: 20

Syllabus:

Finite Difference Method: Solving Neumann and mixed boundary value problems using Finite Difference Method.

Finite Element Method: Solving Dirichlet, Neumann and mixed boundary value problems using Finite Element Method.

Eigen Value Problems: The power method for finding maximum and minimum Eigen values and Eigen vectors, Characteristics of matrices and approximating Eigen values, Householder's method for converting a matrix to tridiagonal form.

Any other topics covered in class will also be included in the syllabus.

Incredible People: Carl Friedrich Gauss (1777 – 1855)

“God does arithmetic.”

A true child prodigy, Carl Gauss was a mathematical genius. Gauss amazed his parents by learning to add numbers and make calculations before he was able to talk. Gauss taught himself to read and, when fourteen years old, received a stipend from Duke Ferdinand of Brunswick to study science. He eventually entered the University of Göttingen in 1795 and obtained his doctorate degree in 1799. Gauss is best known for his mathematical advances. He devised the method of least squares while a teenager; calculated the orbit of the asteroid Ceres , permitting it to be rediscovered after it had been lost; calculated theories of perturbations between the planets, which led to the discovery of Neptune; constructed an equilateral polygon of seventeen sides; and established a non-Euclidean geometry .