M471 Introduction to Numerical Methods
Winter 2020
Instructor:
Shixu Meng
Office:
East Hall 3060
When:
Section 001: MoWeFr 8:00AM - 9:00AM
Section 002: MoWeFr 9:00AM - 10:00AM
Where:
1084 EH
Office Hours:
MF: 10:00 AM-11:30 AM
Course updates: We have switched to remote teaching starting from Mar 16. Check your Canvas for additional details and updates.
Course Description: This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. Some convergence theorems and error bounds are proved. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming.
Prerequisites: Math 216, 286, or 316; Math 214, 217, 417, or 419; and a working knowledge of one high-level computer language.
Syllabus: M471-syllabus
Grader (GSI): Yuxuan Bao
Tips about Matlab programming: To use Matlab, go to the University’s software download page to get a copy to install on your computer. Here is a Matlab user guide: Interactive Matlab Course, Introduction to Matlab can also be useful. This is a Cheatsheet.
Week 1
01/08(W)
1.1 Review of Calculus
HW 1 (due on 01/15): Section 1.1: 2(c), 4(b), 6(c)
01/10(F)
1.1 Review of Calculus
HW1 (due on 01/15): Section 1.1: 19, 25(b)
Week 2
01/13(M)
1.2 Round-off Errors and Computer Arithmetic
HW2 (due on 01/22): Section 1.2: 19(a,b), 20[for numbers given in 19(a,b)]
01/15(W)
1.2 Round-off Errors and Computer Arithmetic
HW2 (due on 01/22): Section 1.2: 3(b), 4(a), 6(c), 14(a, b, d[find relative error obtained from 14(b)])
01/17(F)
1.3 Algorithms and Convergence
HW2 (due on 01/22): Section 1.2: 16(b), 21, 25(a), 26
Matlab code on Canvas
Week 3
01/20(M)
No class meeting
01/22(W)
1.3 Algorithms and Convergence
HW3(due on 01/29): Section 1.3: 6(b), 7(b), 11
01/24(F)
2.1 The Bisection Method
HW3(due on 01/29): Section 2.1: 1, 11(a,b), 14[find approximation correct to within 0.01]
Matlab code on Canvas
Week 4
01/27(M)
2.2 Fixed-Point Iteration
HW4(due on 02/05): Section 2.2: 1(d), 4(a)[only compute p1, p2, p3, p4], 10[Replace 10^-4 by 10^-3 in the problem statement. Hence, for the fixed pt algorithm, stopping criterion is |p_n-p_n-1|<10^-3.]
Matlab code on Canvas
01/29(W)
2.3 Newton’s Methods and its Extensions
HW4(due on 02/05): Section 2.3: 1, 6(a)
Matlab code on Canvas
01/31(F)
2.3 Newton’s Methods and its Extensions
HW4 (due on 02/05): Section 2.3: 4(a), 30
Week 5
02/03(M)
Midterm I review
02/05(W)
Midterm I
02/07(F)
2.4 Error Analysis for Iterative Methods
HW5 (due on 02/19): Section 2.4: 10[hint: write f(x)=(x-p)^m q(x)], 14
Matlab code on Canvas
Week 6
02/10(M)
2.5 Accelerating Convergence
HW5 (due on 02/19): Section 2.5: 4,6
Matlab code on Canvas
02/12(W)
3.1 Interpolation and Lagrange Polynomial
HW5 (due on 02/19): Section 3.1: 2(d),9, 23
02/14(F)
3.1 Interpolation and Lagrange Polynomial
F: no homework
Second Computer Project (due on Wed 03/18)
Matlab code on Canvas
Week 7
02/17(M)
3.3 Divided Differences
HW6 (due on 02/26): Section 3.3: 7(a), 14, 16
02/19(W)
3.4 Hermite Interpolation
HW6 (due on 02/26): Section 3.4: 2(a & b)[Use Theorem 3.9], 4(a & b)
02/21(F)
3.4 Hermite Interpolation
HW6 (due on 02/26): Section 3.4: 2(a & b) [Use divided difference], 7[do H3 only], 10
Week 8
02/24(M)
3.5: Cubic Spline Interpolation(Thomas Algorithm for Tridiagonal Linear System)
HW7 (due on 03/11): Section 3.5: 11,13
02/26(W)
HW7 (due on 03/11): Section 3.5: 4(c),6(c)
Matlab code on Canvas
02/28(F)
HW7 (due on 03/11): Section 3.5: 21(a),(c)[Use Theorem 3.13 to estimate max|f(x)-s(x)| only]
Week 9
03/02 –03/06: Spring Break
Week 10
03/09(M)
4.1 Numerical Differentiation
HW8 (due on 03/20): Section 4.1: 2(b),4(b), 6(c), 8(c), 10(a)
03/11(W)
4.3 Elements of Numerical Integration
HW8 (due on 03/20): Section 4.3: 1(a,c), 3[Find error bound and actual error for 1(a,c); hint: for 1(c), antiderivative is x^3 (ln(x)/3-1/9)], 5[repeat 1(a,c) using Simpson’s rule]
03/13(F)
Class canceled
Week 11
03/16(M)
4.3 Elements of Numerical Integration
HW8 (due on 03/20): Section 4.3: 19, 22, 23
03/18(W)
Midterm II open book
03/20(F)
4.4 Composite Numerical Integration
HW9 (due on 04/01): Section 4.4: 1(a,c), 3[for 1(a,c)], 13(a,b).
Week 12
03/23(M)
4.7 Gaussian Quadrature
HW9 (due on 04/01): Section 4.7: 1(a,c)
03/25(W)
5.1 The Elementary Theory of Initial Value Problems
HW9 (due on 04/01): Section 5.1: 1(a), 2(a), 3(a)
03/27(F)
5.2 Euler’s Method
HW9 (due on 04/01): Section 5.2: 1(a), 3(a)
Week 13
03/30(M)
5.3 High-order Taylor Methods
HW10 (due on 04/08): Section 5.3: 1(a), 10(a), Discussion problem 1
04/01(W)
5.4 Runge-Kutta Methods
HW10 (due on 04/08): Section 5.4: 2(d), 7[repeat for 2(d)], 15[repeat for 2(d)]
04/03(F)
5.6 Multistep Methods
HW10(due on 04/08): Section 5.6: 1(a)[use AB2 and AB3 methods only].
Week 14
04/06(M)
5.10: Zero-Stability
HW11(due on 04/15): Section 5.10: 5(a) [Hint: for analyzing consistency by local truncation error, do 3rd order Taylor expansion for y_{i-2}, y_{i-1} and y_{i+1} about y_i respectively. In the difference equation, replace the approximate solution by exact values and plug these Taylor expansions into the equation. See what you have after some cancellation.]
04/08(W)
5.11: Stiff Differential Equations
04/10(F)
5.11: Stiff Differential Equations
HW11(due on 04/15): Section 5.11: 10.
Week 15
04/13(M)
6.1: Linear System of Equations
6.2: Pivoting Strategies
No Hw
04/15(W)
7.3: Jacobi iterative method
No Hw
04/17(F)
7.3: Gauss-Siedel iterative method
No Hw