Math 300 – Numerical Analysis, Spring 2012


Instructor: Anatolii Grinshpan
Office hours:  TWR 4-6, Korman 247,  or by appointment.

Course information   Elementary Numerical Analysis   Academic Calendar    Math resource center  

Apr 4. Introduction to the course. Number systems: decimal and binary arithmetic.

           A & H: Overview, binary numbers.

 

Apr 6. Octal and hexadecimal systems. Babylonian and Mayan numerals.

           Approximation of irrational numbers. Error, absolute and relative error. Significant digits.

           Homework 1, due April 13.

 

Apr 11. Propagation of error in arithmetic. Examples. Error in function evaluation.

             A & H: Propagation of error.

 

Apr 13. Quiz 1 (number systems and conversion). Mean-value theorem.

             A & H: Polynomial evaluation. Nested multiplication and operation count.

             Homework 2, due April 20.

 

Apr 18. Nested multiplication (continued).  Modes of rounding.  Example on rounding.  Example on summation.

             Fixed-point and floating-point representations.

 

Apr 20. Quiz 2 (significant digits, error arithmetic). IEEE single-precision format: subnormals and exponent bias. 1-byte floating-point model.

             A & H: Error accumulation in sums. Notes on IEEE: Overton, Goldberg. Arithmetic simulator: addition/subtraction, multiplication/division.

             Homework 3, due April 25.

 

Apr 25. Computer arithmetic. Examples. The relative error of floating-point representation.

 

Apr 27. Midterm I (lectures of April 4 – April 25).

 

May 2. Taylor polynomials. Lagrange remainder formula.

 

May 4. Taylor polynomials (continued). Irrationality of e.

            A & H: Calculation of functions.

            Homework 4, due May 11.

 

May 9. Computing derivatives and integrals via Taylor polynomials.

            Error sequences: the order and rate of convergence. Linear convergence. Bisection.

            A & H: Rootfinding.

 

May 11. Quiz 3 (Taylor approximation). Newton’s method. Examples and error analysis.

              Division via Newton’s iteration.  A & H: Rootfinding (continued).

              Homework 5, due May 18.

 

May 16. Newton’s method (continued): error analysis, cycles, basins of attraction. Example from class.

              Secant method: error analysis. When to stop an iteration.

              A & H: The secant method. A quartic with a thousand roots.

 

May 18. Quiz 4 (Bisection and Newton’s method). Multiple root. Stability of roots.

              Fixed-point iteration. A & H: Multiple roots. Perturbation analysis.

              Homework 6, due May 23.

             

May 23. Contractive mapping. Examples. Classification of fixed points. Logistic map. An applet.

              A & H: Fixed-point iteration. Chaotic search for i.

 

May 25. Midterm II (lectures of May 2 – May 23).

 

May 30. Bit shifting and the logistic map. Lagrange interpolation formula.

              A & H: Interpolation.

 

Jun 1.    Divided differences and polynomial interpolation. Interpolation error. The minimax problem.

              A & H: An error formula.  Homework 7, due June 8.

 

June 6.  Quiz 5 (Lagrange interpolation, divided differences).  Newton interpolation formula.  Chebyshev’s theorem.

             A & H: Best approximation, Chebyshev polynomials.

                                     

June 8.  Applications to numerical differentiation and integration.

 

Jun 13.  Final Exam (all lectures): 10:30-12:30  Drexel Plaza  GL 20.