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Numerical Analysis

Numerical Analysis. 4 Semester Hours.
An introduction to the numerical solution of mathematical problems. Primaryemphasis is upon the development of use of computational algorithms to obtain anaccurate numerical solution as well as methods for establishing error estimatesand bounds for this solution. These algorithms will primarily be implemented onthe computer using the Mathematica system and a programming language such asC/C++, FORTRAN, or Pascal. Some work will also be done by using a scientificgraphing calculator such as the TI-83 or TI-86. Grades will be based onassignments and exams. Prerequisites: MATH 202, MATH 205, COMP 150, andfamiliarity with the scientific graphing calculator. This course is cross-listedas MATH 320. Students may enroll in either COMP 320 or MATH 320, but not both.Mathematical-reasoning intensive.

Thetextbook for this course is NumericalMethods for Engineers, Third Edition, and Raymond P.Canale (McGraw-Hill). Notes and handouts (through hardcopy and file access) will also be used. You will have access to all the files in our COMP 320 course folder(subdirectory) on the Q-Drive during this course. Selected class assignments, sample programs and data, and otherguidelines will be stored in this folder. You will be using several numerical analysis tools during this courseincluding: a (TI) graphing calculator, a high-level language (C++, Fortran, orPascal), and Mathematica (Q:\ClassPrograms\Mathematica 4.bat).

Allthe references below are to the textbook unless otherwise indicated. Since one prerequisite for this course is COMP 150, it is assumed thatyou are familiar with using high-level programming language constructs throughthe topic of 1-D and 2-D arrays. It is also assumed that you are familiar with matrix algebra (MATH 205)as well as differential and integral calculus (MATH 201 and MATH 202). All of these disciplines will be used during this course.

General Syllabus

Part  I:Modeling, Computers, and Error Analysis (Chapters 1-4)
MathematicalModeling and Problem-Solving
     - Analytic vs.Numerical Solutions
     - Computers,Algorithms, and Software
     - Convergence
     -Typesof Errors
Real Number Arithmetic vs. Floating-Point Computer Arithmetic
     - Roundoff Error, Underflow and Overflow
     - Computer Arithmetic and its Sensitivity
TaylorSeries and Truncation Error (Two Forms)
     - ErrorPropagation
     - ProblemCondition and Algorithm Stability
     - Total NumericalError
     - Introduction to Mathematica
Programming Style and Documentation

PartII: Rootsof Nonlinear Equations (Chapters 5-8)
Bracketing Methods: Bisection, False-Position
     - OpenMethods: Fixed-Point Iteration, Secant, Newton-Raphson
     - PolynomialEvaluation and Root-Finding Methods
     - MultipleRoots and Complex Roots
    - ErrorAnalysis


Part III: LinearAlgebraic Systems of Equations (Chapters 9-12)
     - Matrices,Vectors and Linear Equations
     - Vectorand Matrix P-Norms (P=1,2,)
     - DirectSolution Methods: Gaussian Elimination, Pivoting and LU Decomposition
     - Ill-Conditioning
    - System Solution vs. Matrix Inversion
     - IterativeImprovement
     - Complex Linear Systems
     - Indirect(Iterative) Solution Methods: Gauss-Jordan, Gauss-Seidel
     - NonlinearSystems of Equations
     - ErrorAnalysis

Part IV: Optimization(Chapters 13-16)
- UnconstrainedOptimization
     - One-Dimensional Methods: Golden-Section Search,Quadratic Interpolation
     - N-Dimensional Methods: Direct, Gradient
     - Constrained Optimization
     - Linear Programming
     - Nonlinear Programming
     - Error Analysis

Exam 2

Part V: Dataand Function Approximation (Chapters 17-18)
    - Least-Squares Curve Fitting
     - Linear and Multiple Regression (Basis Functions)
     - Interpolation Methods: Lagrange, Newton
     - Spline Fitting
     - Error Analysis

PartVI: NumericalDifferentiation, Integration and ODEs (Chapters 21-23, 25)
- DifferentiationFormulas: Forward, Backward, Central
    -Richardson Extrapolation
     - Newton-Cotes and Gauss Integration Formulas
     - Romberg Integration
     - Differential Equation (IVP) Methods: Euler, Heun,Runge-Kutta
     - Error Analysis                     

Exam 3

Final Exam (Comprehensive)


James L. Noyes
     Office: Room 329B Science.
Hours:Regular hours are posted on office door; also by appointment.
Office Telephone:327-7858
E-Mail: (normallychecked 3-4 times daily on weekdays).
     For more information, see the instructors on the Web.

Course Goals:

  • Learn about mathematical models and understand the various types of errors.
  • Learn the limits of floating-point computation.
  • Learn when numerical algorithms should be used for a given problem.

  • Learn which numerical algorithms should be used for a given problem.

  • Learn how to derive and use some of the more fundamental algorithms.

  • Learn how to do an error analysis to bound or to estimate the approximate solution accuracy.

  • Learn effective and efficient computer science code implementations of algorithms.

  • Learn to effectively use Mathematica in solving and checking solutions.

Assignments (Including Possible Projects):

Therewill be two types of assignments: programming assignments that implement and oruse numerical algorithms and analytical assignments to analyze properties ofalgorithms and errors.  Good programdesign, style and documentation are (still) important in all of the programmingassignments.  Clear and preciselogical steps and writing are important in all of the analytical assignments. However, the main grading criteria is correctness. If you have a question on any of the material you should raise it inclass as soon as possible.  Assignmentswill be accepted in class.  They mayalso be turned in at my office by 5:00pm on the day assigned with no penalty. After that, up to 10% of the total points possible will be DEDUCTEDper day late (including weekends).  Assignmentswill not be accepted after three (3) days unless there is some type ofemergency situation or special arrangements are made ahead of time. Late assignments should be slid under my office door (or under thedepartment door, if it is locked) - be sure my name is on it. 

Tests(Exams, Quizzes, and Final):

Examsand quizzes are typically (although not always) based upon what has been coveredby lecture notes, assignments, handouts, and text material. Exams and quizzes CANNOT betaken later without a legitimately excused absence (e.g., death in the family,personal illness, class field trip, necessary Witt-sponsored activity).  This excuse should be in writing (e.g., e-mail), and theinstructor must be notified as far in ADVANCEas possible.


Thegrade for this course will be based upon assignments and projects (up to 500points), three equally-spaced tests (100 points each), and a comprehensive finalexamination (200 points).  Some ofthese points may also be obtained from unannounced quizzes. Assignments will bedue by 5:00pm on the day assigned with up to 10% of the total points possibleDEDUCTED per day late; assignments will not be accepted after three (3) daysunless special arrangements have been made. Quizzes are typically unannounced and cannot be taken later.  Exams cannot be taken later without advance notice and anofficial excused absence.  Allprogramming assignments MUST initially be submitted with a new 3.5"diskette with a signed listing with the class account, date and time shown. The diskette will contain the program itself as shown in the listing, andany necessary data files.  Classattendance and participation is very important and will have a positive effecton your grade.  The final grade forthis course will be based upon the individual class average relative to the restof the class.  If the scoredistribution starts in the 90%-100% range, then a ten-point spread will probablybe used (e.g., 90% and above would be A-, A, or A+, 80% up to 90% is B-, B, B+,etc.). (Note: All point values are approximate.)


Academic dishonesty of any kind on homework or exams is notacceptable.  This includes, but isnot limited to, plagiarism or collaboration with another student on homework ortests.  At a minimum it willtypically result in a reduced score (typically 0) for all parties involved andit could result in a failing grade for this course. In addition, there may be other University sanctions. See your StudentHandbook for additional details regarding Academic dishonesty.

May 12, 2000.