ELEMENTS OF NUMERICAL ANALYSIS

MA 350

Catalogue Description

Error analysis, finite differences, integrative methods, interpolation, and numerical differentiation and integration are the topics studies in this class. 3 credits Prerequisites: MATH 290 Calculus III or equivalent

Goals

A. To strengthen the student's grasp of basic notions in analysis and algebra, e.g. the idea of a sequence, limit, recursion relation, definite integral, matrix techniques in algebra.

B. To help the student see the connection between an algorithm as a computational procedure and the mathematical foundations.

C. To help the student appreciate the type of algorithmic approach that enables a problem to be handled by a computer.

D. To understand the nature of a recursive formula with specific examples used to solve certain classes of problems.

E. To see how a computational procedure is developed from the mathematical theory.

F. To learn basic principles of computation as an art in so far as it pertains to matters of precision, accuracy, errors, and checking, by carrying out actual numerical calculations with specific problems.

G. To learn the basic techniques used to approximate a given function by simpler functions.

H. To understand the basic theorems concerned with convergence of sequences generated by iterative procedures.

Procedures

A. Lectures covering theory and areas of application.

B. Verification of algorithms using computing facilities.

C. Assignment of problems.

Course Content

A. Solution of Equations (10 Lessons)

1. Functional iteration

2. Convergence Theorems including Cauchy Criterion

3. Lipschitz condition

4. Aitken's delta-squared method

5. Newton-Raphson method

6. Method of false position

7. Method of chords

8. Bisection method

9. Bairstow's method for polynomial equations

10. Von Mise's method

B. Polynomial Approximation (15 Lessons)

1. Evaluation of polynomials

2. Taylor polynomial

3. Legendre polynomial

4. Least-squares approximation

5. Definitions of norms

6. Error of approximation

7. Lagrange's interpolation formula

8. Gram-Schmidt process

9. Chebyshev polynomials

10. Trigonometric approximations

11. Newton's interpolation polynomial with divided differences

12. Ordinary differences. Forward and backward difference operator

13. Trapezoidal rule

14. Simpson's rule

15. Gaussian quadrature

C. Solution of Ordinary Differential Equations (5 Lessons)

1. Numerical differentiation

2. Runge-Kutta with Runge's coefficients

3. Adams-Moulton Predictor-Corrector method

D. Matrix Algebra and Simultaneous Equations (10 Lessons)

1. Elementary operations

2. Gauss-Jordan elimination method

3. Matrix inversion

4. Gauss-Seidel iterative method

5. Eigenvalues and Eigenvectors

E. Monte Carlo (5 Lessons)

1. Random number generators

2. Solution of problems

3. Statistical analysis

Evaluation methods

1. Final examination, periodic quizzes.

2. Class Participation

3. Assignment of specific problems to be turned in.

4. Writing of programs to be tested on the computer.

Bibliography

Required Text: Conte, S.D., Elementary Numerical Analysis, 2nd Ed.. McGraw Hill Co., 1974.

Acton, Forman S., Numerical Methods That Work, Harper Collins, 1970.

Arden, Bruce W. & Astill, Kenneth N., Numerical Algorithms: Origins and Applications, Addison-Wesley, 1970.

Carnahan, Brice, Luther, H.A. & Wilkes, James O., Applied Numerical Methods, John Wiley & Sons, 1969.

Carnahan, Brice & Wilkes, J.C., Digital Computing and Numerical Methods (with FORTRAN IV, WATFOR, and WATFIV PROGRAMMING), John Wiley & Sons, 1973.

Cohen, A.M., Cutts, J.F, Feilder, R., Jones, D.E., Ribbans, J. & Stuart, E., Numerical Analysis, Halsted Publishing, 1973.

Conte, S.D., deBoor, Carl, Elementary Numerical Analysis, 2nd Ed., McGraw Hill, 1972.

Dahlquist, Germund, Ake Bjorck & Anderson, Ned, Numerical Methods, Prentice Hall, 1974.

Daniel, James W. & Moore, Ramon E., Computation and Theory in Ordinary Differential Equations, W.H. Freeman, 1970.

Davis, Phillip J. & Rabinowitz, Philip, Methods of Numerical Integration, Academic Publishing, 1975.

Dorn, W.S. & Greenberg, H.J, Mathematics and Computing with FORTRAN Programming, John Wiley & Sons, 1967.

Dorn, W.S. & McCracken, D. kD., Numerical Methods with FORTRAN IV Case Studies, John Wiley & Sons, 1972.

Forsythe, G.E, Malcolm, M.P. & Moler, C.B., Computer Methods for Mathematical Computations, Prentice Hall, 1977.

Gear, C. William, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, 1971.

Gerald, Curtis F., Applied Numerical Analysis, Addison-Wesley, 1970.

Greenspan, Donald, Introduction to Numerical Analysis and Applications, Rand Publishing, 1971.

Hamming, Richard W., Introduction to Applied Numerical Analysis, McGraw Hill, 1971.

Hamming, Richard W., Numerical Methods for Scientist and Engineers, 2nd Ed., McGraw Hill, 1973.

Hildebrand, F.G, Introduction to Numerical Analysis, 2nd Ed., McGraw Hill, 1974.