| Date |
Week 1
21,23 August 2002 |
- Discussed the importance of an understanding of numerical analysis in computational engineering.
- Gave an example of loss of significance in subtracting two numbers that are close in value, and
cited this phenomenon when doing a numerical derivative.
- Presented the Taylor Theorem & The Mean Value Theorem.
- Demonstrated how poorly converging a Taylor series expansion can be when expanding about a remote point and why the series converges more rapidly when the expansion point is close.
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Week 2
4,6 Sept |
- Discussed rounding and chopping and the maximum error in both.
- Gave some example programming suggestions from Section 1.1.
- Discussed base changes between binary, octal, decimal and hexadecimal.
- Gave additional explanation of Taylor and Mean-Value Theorems.
- Discussed Programming Experiment 1A for the determination of f'(X)= [f(x+h)-f(x)]/h and laid out a systematic procedure for analyzing the errors as h is decreased.
- Displayed a chart of log(error) vs log(h) for the programming experiment and postulated two opposing phenomena: (1) Taylor-series truncation error and (2)loss of significance.
- Derived the maximum error bound for the basic definition of a numerical derivative and explained how to determine the mean value ξ 2 from the error calculation in Prog.Experiment 1A.
- Derived the maxima and minima error bound using the "central difference formula" for the numerical derivation and how to determine the mean value ξ 3 from the error calculation in Prog.Experiment 1B.
- Gave examples of the above.
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Week 3
9,11,13 Sept |
- Presented the loss of (Binary) Precision Theorem (p77), and derived a theorem that describes Loss of
Decimal Precision.
- Discussed the "error" columns in Programming experiments 1A and 1B in light of Taylor-Series truncation error and loss-of-significance error.
- Derived expression for determing the mean values ξ 2 and ξ 3
based on the errors determined in Programming Experiments 1A and 1B:
ξ 2 = sin-1(2|error|/h)
and ξ 3=cos-1(6|error|/h2)
where the errors are taken from the programming experiments, and noted whether x <=ξ2<= x+h and
x-h<=
ξ3<= x+h. - Students should be prepared to explain their programming experiment results based on all of the prior concepts discussed.
- Lectured on the IEEE Floating Point Arithmetic Format, standard.
- Lectured on locating roots of equations, and gave several examples.
- Developed the Bisectional and Newton's Methods of locating roots with examples and gave convergence analyses.
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Week 4
16,18,20 Sept
|
- Derived the Secant Method, gave examples and compared its convergence with those of Newton and Bisectional Methods.
- Discussed the results of student submissions of Programming Experiment 1 and gave a detailed analysis of the resulting computations.
- Gave a summary of Chapters 1-3 and provided guidance for preparation of the Midterm Exam to be held on Monday 23 Sept,2002.
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Week 5
23,25,27 Sept
|
- Presented Newton's Method of finding roots to Nonlinear Equations using tensor notation, showed the parallel between single- and multi-function root finding Newton's Method. Gave the Jacobian gradient tensor and illustrated the method without the need to explicitly invert the Jacobian.
- Returned the exam papers and assigned special problems on an individual basis to cover the missed concepts.
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| Week 6
30Sept,2,4 Oct
|
- Began Ch.4 on interpolation methods in general and polynomial interpolation's specifically. Illustrated the Lagrange and Newton Forms for determining the interpolating polynomial and calculating the coefficients using divided differences.
- Classes were cancelled on 2nd and 4 th of October due to potential
adverse weather.
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Week 7
7,9,11 Oct
|
- Presented the Divided Difference Method of fitting a polynomial to a set of
data and gave an example.
- Lectured on Estimating Derivatives using Richardson Extraploation and gave
examples.
- Derived the second-level Richardson Extrapolation form and demonstrated the
increase in convergence rates using 1st of 2nd level
Richardson Extrapolations in Programming Experiment No.1.
- Began Chapter 5-methods of Numerical Integration.
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Week 8
14,16,18 Oct
|
- Discussed the Lower and Upper- Sums method of numerical integration and the
associated error criterion.
- Derived the Trapezoid-Rule form for numerical integration and discussed the
associated Theorem on Precision.
- Explained the Romberg Algorithm and The Gaussian Quadrature methods of
numerical integration and outlined the solutions to problems using these
methods.
- Reviewed and Summarized Chapters 4 and 5 to prepare standard for the second
Midterm Exam.
|
Week 9
21,23,25 Oct
|
- 2 nd Midterm exam on Chapters 4&5.
- Returned Midterm Exam #2 and gave grade statistics.
- Lectured on The solution to systems of linear equations by the Naive
Gaussian Elimination (NGE) and gave a detailed example of forward elimination
and back substitution.
- Gave examples of the failure of The NGE method
- Lectured on Gaussian Elimination with Scaled Partial Pivoting in general a
gave an example.
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Week 10
28,30 Oct,1 Nov
|
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Week 11
4,6,8 Nov
|
- Summarized the objectives of the remainder of the course.
- Defined terminology to be used in numerical solutions to (single) ordinary and partial differential equations (ODE and PDE).
- Discussed analytical and numerical solutions to ODEs, with initial-value
condtions, with examples using Euler's method from the first-order Taylor
Series.
- Discussed numerical solution methodology with order-four Taylor Series.
- Derived Runge-Kutta method of order 2 demonstrating a Taylor-Series type of
expansion without the need for determining derivatives.
- Described Runge-Kutta methods of order 4.
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| Week 12
11,13,15 Nov |
- Introduced systems of ordinary differential equations (Ch.9) and their
solutions using Taylor Series and Runge-Kutta method in vector notation.
-
Developed the procedure of reducing an nth - order differential
equation to n first-order differential equations with examples.
-
Developed the procedure for reducing systems of higher-order coupled
differential equations to a new system of first-order differential
equations with examples.
- Retruning to Chapter 8, developed "Stability
and Adaptive RK(multistep) methods. Described the Fehiberg method of order
4 and discussed the Adams-Bashforth formula.
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| Week 13
18,20,22 Nov
|
- Presented the Adams-Moulton Predictor-Corrector scheme and gave a
top-level flow diagram for the pseudecodes used in its solutions.
-
Summarized the objectives of Chapters 8 and 9 in light of extending from the
calculation of x(t+h) where h is small to general values using the methods
of AM-PC.
- Student, Scott Sheppard gave his project presentation on Random
Integers
(Programming Project 11.1-15). - Gordan Suljkanovie on Comparisons of
Pseudo-random Number Generators.
- Ricky Crochet on MonteCarlo Estimation;
Computer. Problem 11.2-2.
- Sampie Brown on The Solution to the Heat
Conduction Parabolic Differential Equation; using Mathlab to solve Computer
Problrm 13.1-1.
- Scott Sheppard on Random Numbers; Computer Problem
11.1-15.
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| Week 14
25,27,29 Nov |
- Brad Breaux on Card Shuffeling Simulation;
Computer Problem 11.3-20.
- Wade Salazar on Hyperbolic Functions; Computer Problem 13.2-3.
- Ryan Crow on Adams-Moulton Predictor-Corrector Scheme; Chapter 9.3.
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| Week 15
2,4 Dec |
- Jessie Morgan on Stability and Adaptive Runge-Kutta and Multi-Step
Methods; Computer Problem. 8.3-5.
- Marc Jackson on computer Problem 8.3-8.
-
Emir Suljkanovic on The Method of Least Squares; Computer Problem 10.1-2.
-
Mahdi Mekic on Orthogonal Systems and Chebyshev Polynomials; Computer
Problem 10.2-2.
-
Reviewed the course and highlighted the salient points to remember.
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