# Math 301 – Numerical Analysis 2, Winter 2009

TR 3:30-4:50, Matheson 411.

Instructor: Anatolii Grinshpan
Office hours:  M 4-6, W 4-5 (Korman 247) or by appointment.

Elementary Numerical Analysis (K. Atkinson and W. Han).

Numerical Computing with MATLAB (C. Moler).

Outline

Jan 6. Introduction to the course. Order and rate. Splines.

An example: spline, graph.

Jan 8. Natural cubic spline: construction, minimization property.

A & H notes: 4.3.

Jan 13. Complete cubic spline. Variants for boundary conditions.

A & H notes: 5.1. Homework 1, due January 20.

Jan 15. Numerical integration. Riemann sums.

A & H notes: 5.2.

Jan 20. Matlab  and homework discussion.

Homework 1 codes: linear spline, quadratic spline, natural cubic, complete cubic, joint plot.

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Jan 22. Quiz (numerical integration). Periodic integrands. Euler-Maclaurin formula.

A & H notes: 5.2.

A & H notes: 5.3.

Jan 29. Matlab examples:  asymptotic error constant, complete spline error, oscillatory error pattern.

Feb 3.  Midterm 1 (lectures of Jan. 6 - 29). Answers. Plots: 1, 3.

Feb 5.  Midterm discussion. Numerical derivatives via interpolation. Error bound.

Feb 10. Numerical derivatives via undetermined coefficients. Sensitivity to function values.

A & H notes: 5.4. Homework 4, due February 17/19.

Feb 12. Quiz (numerical differentiation). Systems of linear equations. Gaussian elimination.

Feb 17. Gaussian elimination, partial pivoting, backward substitution.

A & H notes: 6.3. Homework 5, due February 24.

Feb 19. Quiz (Gaussian elimination). LU decomposition.

Homework answers. A & H notes: 6.4.

Feb 24. Cholesky decomposition Error analysis.

Homework answers. A & H notes: 6.5. Homework 6, due March 3.

Feb 26. Vector and matrix norms. Conditioning of a system.

Matlab examples: numerical derivative, LU and Cholesky, matrix inversion, sparsity plot.

Mar 3.  Midterm 2 (lectures of  Feb. 5 - 26).

Mar 5. Midterm discussion. Iterative methods for solving linear systems (Jacobi, Gauss-Seidel, SOR).

Richardson extrapolation. A & H notes: 6.6. Extra: Conjugate Gradient method.

Mar 10. Examples of  the Jacobi, Gauss-Seidel, and SOR methods. Multivariable Newton’s method.

Euler’s method for solving ODE. A & H notes: 7.3, 8.2. Exercises, last set.

Mar 12. Euler’s method: forward, backward, and integral forms. Error estimate.

Applications to PDE: discretization of the heat equation.

A & H notes: 9.1, 9.2.

Mar 19. Office hours: 12-1+. Answers to exercises. Review.

Mar 20. Final Exam, 3:30-5:30 P.M., Matheson 405. Answers.