Sections:
Books:
Syllabus for BoANA (MA030779)
Numerical Analysis
2nd Edition, Pearson.
John P. Boyd, "Chebyshev and Fourier Spectral Methods", 2nd Ed. (free book from the web-page of the author).
How grades are assigned?
All homeworks: 250 points.Midterm exam: 100 points.
Final exam: 100 points.
Class attendance + activities: 50 points.
Total: 500 points.
Boundaries for grades (not higher than):
A = 430, B = 380, C = 330, D = 280, E = 230.
Homework regulations
HW has to be typed in what ever editor you like. Submission format: PDF (one file, reasonable size!).If you were asked to produce any output from the code - it has to be copied in the HW text.
If you were asked to prepare a script and use it, every script has to be included.
If you were asked to plot something, this plot has to be included.
Essentially, it is recommended to type all HW together with plots and cut-and-paste scripts and code outputs.
| Week # | Homework problems | Due date |
|---|---|---|
| - |
Final Exam: Friday, October 24th, 2025, usual classroom. What is covered: all topics up to and including finite difference methods for PDEs. You need to understand and be able to use definitions and algorithms. Training set of problems: Sauer, p. 19: 0.4.1; Sauer, p. 59: 1.4.5; Sauer, p. 101: 2.4.3; Sauer, p. 198: 4.1.2, 4.1.7-9; Sauer, p. 224: 4.3.1, 4.3.2 (Using both Gramm-Schmidt process and Givens' rotations); For linear algebra in general: you need to be able to estimate number of operations necessary to complete the problem, understand sources of errors (condition number), similar to problems in Midterm; Sauer, p. 156: 3.2.1-3 (also work through problems in the Midterm); Sauer, p. 164: 3.3.1, 3.3.5, 3.3.7-8; Sauer, p. 176: 3.4.7, 3.4.12; Sauer, p. 263: 5.2.1-3 (also you shall need understanding of error dependence on a grid step); Sauer, p. 278: 5.5.1-3 (work through the last HW); Sauer, p. 252: 5.1.1, 5.1.5; Sauer, p. 321: 6.4.3 (you need to be able to use all methods we studied); In all methods which we studied you need to understand errors dependence on the parameters of methods like grid step, time step etc. You are allowed to have one page two sides of A4 size paper with any formulae you like, BUT it is prohibited to have solutions of the problems there! You will need to submit this page with you Final. | - |
| 3 |
Homework 03. Help with analytical solution. |
October 26th, 2025, end of day. |
| - |
Midterm: October 13th, at the beginning of a class. What is covered: all topics up to and including interpolation. You need to understand and be able to use definitions and algorithms. Training set of problems: HW 1: 2, 3, 6-11, 14, 18 (use matrix from the problem 17); HW 2: 2, 5. You are allowed to have one page one side of A4 size paper with any formulae you like, BUT it is prohibited to have solutions of the problems there! You will need to submit this page with you Midterm. |
October 13th, 2025, class time. |
| 2 | Homework 02 (preliminary). |
October 15th, 2025, end of day. |
| 1 | Homework 01 (updated!). |
October |
Below you can find additional material.
| Week # | Lectures Notes | |||
|---|---|---|---|---|
| 4 |
Lecture 9: Taylor methods for ODEs, (TS 6.2) Finite difference methods for PDEs, Von Neumann stability analysis for linear PDEs. |
Lecture 10: Von Neimann stability analysis for nonlinear advection equation. Spectral methods, Strang splitting, spectral split-step method for NLSE. Lecture 11: More on spectral methods, Hamiltonian integration, stability. |
Final Exam. Lecture 12: Richardson's extrapolation, Romberg integration. (TS 5.2, 5.1.3, 5.3). |
|
| 3 |
Lecture 07: Midterm. Errors for trapezoid and midpoint methods. Adaptive quadrature. (TS 5.4) Gauss-Legendre quadrature. (TS 5.5) Integration of ODEs. Euler, Improved Euler, and Midpoint methods. (TS 6.1) |
Industry day. No lecture. |
Lecture 08: Runge-Kutta methods, including RK4. (TS 6.4) Existence and uniqueness of solution for ODE. (TS 6.1.2). Boundary value problems. Differential sweep method (Progonka). (some elements are in TS 7.2) Higher order ODEs. (TS 6.3) Variable stepsize methods. Embedded RK pairs. (TS 6.5) Stiff systems, Implicit methods. (TS 6.6) Multistep methods. (TS 6.7) |
|
| 2 |
Lecture 04: Interpolation. Polynomial interpolation using natural basis, Newton's and Lagrange's approaches. (TS 3.2, 3.1) Chebyshev interpolation. (TS 3.3) |
Lecture 05: Chebyshev interpolation. (TS 3.3) Cubic splines. (TS 3.4) Least squares, normal equations, QR (classical and modified Gramm-Schmidt) (TS 4.1, 4.3) |
Lecture 06: QR (Givens rotations). (TS 4.3) Nonlinear systems of equations. (TS 2.7) Numerical integration, Newton-Cotes' rules (rectangle, midpoint, trapezoid, Simpson). (TS 5.2) Errors for Newton-Cotes' rules, including composite (local and global erros). (TS 5.2) |
|
| 1 |
Lecture 01: Binary representation, floating point representation and machine epsilon. IEEE754 standard for floating point numbers, loss of significance. (TS Chapter 0) |
Lecture 02: Root finding algorithms (bisection, fixed point iterations, Newton's method). Linear and quadratic convergence. (TS Chapter 1). Systems of linear equations. (TS 2.1). |
Lecture 03: Systems of linear equations. (TS 2.2-2.4) Iterative methods for SLEs. (TS 2.5.1-2.5.2) |
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