Math 284 Numerical Analysis (Prof. Leise)

Fall 2016 Schedule (Syllabus)

  MWThF 10am Merrill 131 -- Textbook: Numerical Analysis by J. Leader

Date Topic
Handouts and other links
Assigned Exercises
Due Date
9/6-9/9




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Introduction: root-finding and computing tools Tutorial for MATLAB (and its free clones) For numerical answers, please give ~12 decimal places  
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1.1-1.2: Newton's method Square root intro 1.1 #6
1.2 #12abc
Fri 9/16
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1.3 Fixed-Point Theorem

Secant method example
Newton's method example

1.3 #3 Fri 9/16
9/12-9/16
     
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1.4 Convergence Iterative method example 1.4 #4 (use function in example 1.4.2), 5ab Wed 9/21
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1.5 Variants Newton's method as function
Convergence order example
1.5 #1 Wed 9/21
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1.6 Brent's method Muller's method example 1.6 #1 (Muller's method), 15 Fri 9/23
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1.7 Finite precision Quadratic formula 1.7 #8ab (for Freemat use realmax=1.797693134862314*10^308) Fri 9/23
9/19-9/23
       
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1.8 Newton's method for systems Newton's method for systems function 1.8 #7, 10 Wed 9/28
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1.9 Broyden's method Broyden's method example 1.9 #3ab (use inverse of A1 for B1) Wed 9/28
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2.1 Partial pivoting   2.1 #1a, 4abd (not c), 7abc Fri 9/30
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2.2 LU decomposition   2.2 #2ac, 3abc Fri 9/30
9/26-9/30
       
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2.3 LU + pivoting

Hilbert matrix script
Inverse Hilbert matrix script

2.3 #2ab, 3, 9 Wed 10/5
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2.4 Cholesky decomposition   2.4 #6ab (by hand), 7a, 9a Wed 10/5
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2.5 Condition numbers   2.5 #3, 6, 8 Fri 10/7
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2.6 QR decomposition   2.6 #1a, 4abcd Fri 10/7
10/3-10/7
       
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2.7 Householder triangularization   2.7 #6 (by hand) Fri 10/14
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2.8 Gram-schmidt   2.8 #2ac Fri 10/14
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2.9 Singular value decomposition   2.9 #8 (use computer, not by hand) Fri 10/14
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3.1 Jacobi and Gauss-Siedel iteration Example script for iteration methods 3.1 #10 (give spectral radius for optimal w in SOR) Fri 10/14
10/12-10/14
       
W 3.2 Sparsity
(Mon class schedule)
3.2 no problems assigned  
Th
3.3 Iterative refinement Iterative refinement example
LU scripts
3.3 #7 Thurs 10/20
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3.4 Preconditioning Preconditioning example 3.4 #7a Thurs 10/20
10/17-10/21
       
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3.5 Krylov space methods GMRES example 3.5 #2 (4 iterations), 4a (add 5th row to A: 0 2 2 -1 1, do 5 iterations) Fri 10/28
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3.6 Numerical eigenproblems Power method and Hotelling's deflation example 3.6 #1ab, 10abcd Fri 10/28
Th
Review (take-home exam due Wed) Solutions to some proof exercises    
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4.1 Lagrange interpolating polynomials 4.1 #1, 2abcde Fri 10/28
10/24-10/28
       
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4.1, continued

Polynomial examples    
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4.2 Piecewise linear interpolation   4.2 #1abc, 5 (for plots, can email as png) Wed 11/2
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4.3 Cubic splines Cubic spline example 4.3 #2, 5, 7 (typo in book's answer to #5) Wed 11/2
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5.1 Closed Newton-Cotes formulas   5.1 #7a (choose n by trial and error to get error ~10e-6), #12ab (use n=100 and report error; compare f'' for a vs b) Fri 11/4
10/31-11/4
       
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5.1 cont'd Project info    
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5.2 Open Newton-Cotes formulas Adaptive Trapezoidal Rule 5.2 #7, 15a Wed 11/9
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5.3 Gaussian quadrature
5.6 Adaptivity
Mathematica examples 5.3 #1, 4, 8ad for only m=1, f(x)=sqrt(abs(x)), 4 point rule Fri 11/11
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6.1 Numerical differentiation   6.1 #1 (plot for x=1:0.05:2, rather than writing out values), 9 (want O(h^2) error) Fri 11/11
11/7-11/11
       
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6.2 Euler's method
6.3 Improved Euler's method
Euler's method example Plot solutions on [0,1] for y'=2xy, y(0)=1 using h=0.1 and h=0.01 for Euler's method and improved Euler's method (along with the true solution). State error at x=1 for each case. Fri 11/18
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6.4 Analysis of one-step methods Euler's method function and example 6.4 no problems assigned  
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6.5 Runge-Kutta method

Runge-Kutta function

Plot solutions on [0,1] for y'=2xy, y(0)=1 using h=0.1 and h=0.01 for RK4 (along with the true solution). State error at x=1 for each case.

Fri 11/18

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Review (take-home exam due Wed)      
11/14-11/18
       
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7.1 Nonlinear optimization Golden section search method example Use the golden search method to calculate the locations of all four local extrema of f(x)=x.^2.*sin(x)-2*x on the interval [-3,3] to within 1e-8 Fri 12/2
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7.2 Steepest descent Golden section search fn for use in steepest descent
Steepest descent example
Use steepest descent with golden search to calculate the locations of the two minima of f(x)=8*y.^4+x.*y+x.^2-3*y.^2-y.^3 with residual error < 1e-8 Fri 12/2
Th
7.3 Newton methods Newton's method fn Use the Newton method with Hessian to calculate the locations of the two minima of f(x)=8*y.^4+x.*y+x.^2-3*y.^2-y.^3 with residual error < 1e-8 Fri 12/2
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7.4 Multiple random start Project topic due Apply the MRS method to the Rastrigin function for d=2 on the region [-5.12,5.12]x[-5.12,5.12]. Run the method 20 times, each time using 100 random starts. How often does it find the true global min? (to within 1e-8) Fri 12/2
11/19-11/27 Thanksgiving break    
11/28-12/2
       
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7.5 Direct search methods 7.5 no problems assigned  
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No class      
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No class      
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7.6 Nelder-Mead method

sortrows
Project outline due
Nelder-Mead example

7.6 #8 (do 100 iterations, compare to actual min found using calculus) Wed 12/7
12/5-12/9        
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7.7 Conjugate direction methods   7.7 #7 Wed 12/14
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Finish chapter 7      
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Presentations      
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Presentations      
12/12-14
       
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Presentations      
W Presentations      
 

Final written reports due 12/19
Take-home exam due 12/21