MATH/COSC 3416 Course Outline
Numerical Methods I
January 2010

Course Description

The course has a programming component using MATLAB. Maple may also be used in a few places.

Course Organization

Textbook: Numerical Mathematics and Computing, 6th Edition
Authors: Ward Cheney and David Kincaid
Publisher: Thomson (Brooks/Cole)

Prerequisite: MATH 1037, MATH 1057, MATH 2066, COSC 1046 or ENGR 1077

Instructor: Dr. B.G. Adams
Office: FA-378
Phone: 675-1151 x 2323
Email: badams 'at' laurentian 'dot' ca
Note: Only send messages using your university email address

Lecture Times: Tue, Thur: 8:30 am - 9:45 am
Room: C-301

Grading scheme

Best 4 of 5 assignments	30%	Jan 22, Feb 5, Feb 26, Mar 12, Mar 26
2 Term tests (15% each)	30%	Feb 9, Mar 23
Final exam (3 hours)	40%	Scheduled by registrar

Course Outline

Introduction (2 weeks)

• Machine representation of floating point numbers.
• Absolute and relative roundoff error
• loss of significance theorem and Matlab examples
• Taylor series approximations to functions. Convergence of Taylor series.
• Numerical examples of series summation and convergence using MATLAB.
• The derivative problem using limit definition
• Recurrence relations. Stable and unstable recurrence relations. Backward iteration. MATLAB examples.
• Convergence rates for iterative methods. Examples of slow and fast convergence. MATLAB examples.

Root Finding Methods (2 weeks)

• Bisection method and fixed point iteration and onvergence analysis.
• Newton and secant methods and convergence analysis.
• Writing MATLAB functions for root finding methods
• The MATLAB fzero function.
• Applied examples using MATLAB

Interpolation and Extrapolation (2 weeks)

• Efficient evaluation of polynomials. Lagrange form of interpolating polynomials.
• Newton form of the interpolating polynomials.
• Algorithms for calculating the Newton form based on a divided difference table.
• Writing MATLAB functions for interpolation
• Errors in polynomial interpolation.
• The Runge phenomenon
• Using Chebyshev polynomials for better approximations.
• Estimating derivatives numerically using Richardson extrapolation.
• Obtaining derivative formulas from interpolating polynomials.
• Built-in MATLAB functions for interpolation.

Numerical Integration (2 weeks)

• Riemann integral definition using upper and lower sums.
• Trapezoidal and Simpson rules, composite rule, algorithm, Matlab functions, error analysis.
• Recursive trapezoidal formula, the Romberg extrapolation method, algorithm, MATLAB function.
• Gaussian integration (quadrature)
• Applied examples using MATLAB

Numerical Solution of Linear Systems (2 weeks)

• Brief review of linear algebra, matrix and vector operations, determinants, and the solution of linear systems of equations using row reduction methods. Matrix inverse using row reduction methods.
• Solving Ax = b when A is an upper triangular or lower triangular matrix, algorithms.
• The general case Ax = b using gaussian elimination to upper triangular form and back substitution.
• Naive Gaussian elimination, algorithms, complexity. Eliminating zero pivot elements.
• Roundoff error examples in solution of  Ax = b.
• Reducing roundoff error using maximal column pivoting and scaled column pivoting, algorithms, MATLAB fnuctions. Using pivot vectors (permutation vectors) to avoid explicit row exchanges.
• Separating the Gaussian elimination part from the back substitution part, algorithms and MATLAB functions.
• LU factorization (A = LU) using naive Gaussian elimination. Solving  Ax = b using LU factorization, forward substitution and backward substitution, algorithms and MATLAB functions.
• LU factorization (PA = LU) using maximal column pivoting and the permutation matrix P (defined by pivot vector) to obtain the LU decomposition of PA, algorithms and Matlab functions.
• Efficient calculation of the matrix inverse and determinant using the LU factorization.
• Ill-conditioned matrices and the condition number. Examples using Hilbert and Vandermonde matrices.

Numerical Solution of ODE's (2 weeks)

• Brief review of first order DE's  x'(t) = f(t,x) and systems of DE's. Geometric interpretation of a first order DE.
• Using DFIELD and PPLAT to solve and graph DE's
• Euler's method for the initial value problem x'(t) = f(t,x), x(t0) = x0. Local and global truncation errors, algorithms and Maple procedures.
• Higher order Taylor series methods, algorithms and MATLAB functions.
• Runge-Kutta methods. Derivation of 2nd order and the classical 4th order methods, algorithms, MATLAB functions.
• The 4th order Runge-Kutta method for a system of first order differential equations, algorithm and Maple procedure for the case of a system of two equations x'(t) = f(t,x,y) and y'(t) = g(t,x,y).
• Case study: The "rabbits and foxes" problem as a dynamical system of two first order DE's, closed solution curves and population curves as functions of time, obtained using the 4-th order Runge-Kutta method for systems.
• Built-in MATLAB functions for solving systems of first order DE's.

Assignments and Tests

assign1.pdf 

assign2.pdf 

assign3.pdf 

assign4.pdf 

assign5.pdf 

Test 1 study guide
Term test 1 solutions 

Test 2 study guide
Term test 2 solutions 

Matlab tutorial, m files, and Other Resources

Matlab Tutorial

Scripts from tutorial: fact.m      falling_object.m      quad_script.m      sq_root1.m      sq_root2.m      sq_root3.m

Introductory Notes

Scripts from Introductory Notes

Scripts for root finding

Scripts for interpolation and extrapolation

Numerical Integration

Linear Systems

Differential equations