# Math 300 – Numerical Analysis.

T 12-1:50, room 412 Matheson, R 12-1:50, room 105D Korman.

Instructor: Anatolii Grinshpan
Current office hours:  TR 3:30-5 or by appointment, 270 Korman

Textbook supplements (Atkinson’s website).

Outline

Apr 3. Introduction. Taylor polynomials.

Apr 5. Basics of  MATLAB. Polynomial evaluation.

Approximation by Taylor polynomials. Example.

Problems: 2 a), 4, 14 (1.1),  1, 4, 9 a),  13 (1.2),  7 (1.3).

Apr 10. Decimal and binary number systems. Floating-point numbers.

Reading: 2.1, 2.2, Appendix E. Subnormal numbers and biased exponents.

Apr 12. Quiz: decimal and binary arithmetic (appendix E, problems on page 531).

LAB: error arithmetic. Extra credit.

Apr 17. Rootfinding.

Problems 1 a), d), g), 11 (3.1),  2 a), d), 3, 13 (3.2).

Apr 19. Quiz: bisection and Newton’s method.

LAB: rootfinding. Example from class.

Apr  24. Secant method. Fixed point iterations.

Problems: 8 (3.3), 3-7 (3.4)

Apr 26. LAB: fixed point iterations, ill-behaving rootfinding.

Examples: multiple root slow down (newton1, newton2),

root instability for polynomials (wilkin).

May 1.   Quiz: fixed point iterations.

Polynomial interpolation. Review.

May 3.   Test 1:  Chapters 1-3.  Important topics.

Focus: homework and quiz questions, in class examples.

May 8.  Lagrange and Newton interpolation.

Problems: 3, 4, 7, 12, 20, 24 (4.1).

May 10. Error in polynomial interpolation. Minimax problem.

Chebyshev polynomials.

Problems:  2, 9 (4.2).

May 15. Quiz: Lagrange and Newton interpolation.

Splines. Legendre polynomials. Least squares approximation.

Problems: Example 4.3.1; 2, 5 (4.5); Example 4.7.1; check formula (4.118) for i, j=0, 1, 2, 3.

May 17. Spline interpolation, Legendre polynomials. Legendre nodes.

Least squares approximation (continued).

Problems: 2 (4.7).

May 22. Quiz: least squares approximation.

Numerical integration, error formulas.

Problems: 6, 8 (5.3).

May 24.  Richardson extrapolation. Numerical differentiation.

Review.

Problems: 6, 8, 10 (5.4).

May 29.  Test 2:  Chapters 4 & 5.  Important topics. Sample. Sample test answers.

Focus: homework, examples (textbook, class), quizzes. Test 2 answers.

May 31.  Linear systems, Gaussian elimination.

Lab: Legendre nodes, natural cubic spline.

June 5.    Solving systems of  linear equations: LU factorization.

Problems: Gaussian elimination, LU factorization for 3x3 systems (try your own examples).

June 7.   Quiz: Gaussian elimination, LU factorization.

Solving linear systems: stability, error analysis.

Iteration methods (Jacobi and Gauss-Seidel).

Problems: 4-6 (6.5), 3, 5 (do several iteration steps by hand) (6.6).

June 12. Questions session 3:30-5 (office).

June 14.