CAAM 553 · Numerical Analysis I

Fall 2017 · Rice University


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Notes, Software, and Supplementary Material


Lecture Notes: Notes are provided here, with additional links/demos to be posted below.
For help with MATLAB used in the context of numerical analysis, Cleve Moler's book, Numerical Computing with MATLAB may be helpful (available free online).
Lecture 35 (11/27): NSODE: stiff systems of ODEs
Lecture 32 in the course lecture notes.
Stability regions for implicit methods.
Correction to lecture: the largest eigenvalue of the 2-by-2 example matrix is -1000 instead of -100.
Lecture 34 (11/20): NSODE: linear multistep methods, absolute stability
Lecture 31 in the course lecture notes.
Multistep stability regions.
Lecture 33 (11/17): NSODE: linear multistep methods, zero stability
Lecture 30,31 in the course lecture notes.
Multistep zero stability demo code.: LMM with 2 roots of char polynomial with unit magnitude).
Multistep absolute (timestep) stability demo code: instability from too large of a timestep.
Lecture 32 (11/15): NSODE: linear multistep methods: truncation error
Lecture 30,31 in the course lecture notes.
Lecture 31 (11/13): NSODE: linear multistep methods
Lecture 30 in the course lecture notes.
Multistep demo code.
Lecture 30 (11/10): NSODE: truncation and global error
Lecture 29, 30 in the course lecture notes.
Lecture 12.3 in Suli and Mayers.
Lecture 29 (11/6): Intro to numerical solution of ODEs (NSODE)
Lecture 28-29 in the course lecture notes.
Lecture 12.1-12.2 in Suli and Mayers.
Euler demo code
Lecture 28 (11/3): Eigenvalue computations: connections between QR and other eigenvalue algorithms
Lecture 37 in the course lecture notes.
Lecture 27,28 in Trefethen and Bau.
QR eigenvalue demo
Lecture 27 (11/1): Eigenvalue computations: the practical QR algorithm
Lecture 37 in the course lecture notes.
Lecture 27,28 in Trefethen and Bau.
Lecture 26 (10/30): Eigenvalue computations: the QR algorithm
Lecture 37 in the course lecture notes.
Lecture 27-28 in Trefethen and Bau.
John Francis: inventor of the QR algorithm
Retrospective on the QR algorithm: 50 years later
Lecture 25 (10/27): Eigenvalue computations: power, inverse, and Rayleigh quotient iteration
Lecture 37 in the course lecture notes.
Lecture 24 (10/25): Cholesky and LDL factorization
Lecture 36 in the course lecture notes.
Lecture 23 (10/23): LU decomposition
Lecture 35,36 in the course lecture notes.
Lecture 22 (10/20): Singular value decomposition, cont. LU factorization.
Lecture 17,18 in the course lecture notes.
Reference for SVD analysis of Congressional voting patterns
Summary of SVD analysis of voting patterns.
NYTimes article on Lawrence Sirovich.
Lecture 21 (10/18): Singular value decomposition
Lecture 17,18 in the course lecture notes.
Lecture 20 (10/16): Linear least squares with QR, conditioning and stability.
Lecture 16, 11 in the course lecture notes.
Lecture 19 (10/13): Gram-Schmidt QR, solving linear systems with QR.
Lecture 3-4 in the course lecture notes.
Lecture 4-5 in the course lecture notes.
Lecture 6-7, 16 in the course lecture notes.
Lecture 18 (10/11): Householder reflectors, QR decomposition
Lecture 3-4 in the course lecture notes.
Lecture 17 (10/6): Gaussian quadratures: error bounds. Linear algebra review
Suli and Mayers 10.2
Lecture 25 in the course lecture notes (Gauss quadrature).
Lecture 2 in the course lecture notes (lin alg).
Lecture 16 (10/4): Gaussian quadrature: construction and positivity of weights.
Lecture 24,25 in the course lecture notes.
Suli and Mayers 7.7.
Lecture 15 (10/2): Richardson extrapolation and Romberg integration; Gaussian quadrature.
Lecture 24,25 in the course lecture notes.
Suli and Mayers 7.7.
richardson_demo.m: Richardson extrapolation for finite difference approximations.
romberg_demo.m: Romberg integration demo.
Lecture 14 (9/29): Quadrature: error bounds and composite rules.
Lecture 23 in the course lecture notes.
Suli and Mayers 7.1-7.5.
Lecture 13 (9/27): Minimax theory, optimal interpolation points
Lecture 21-22 in the course lecture notes.
Lecture 11 (9/25): Orthogonal polynomials, Gram-Schmidt orthogonalization.
Lecture 19,20 in the course lecture notes.
Lecture 10 (9/20): Continuous L2 approximation, orthogonal polynomials.
Lecture 19,20 in the course lecture notes.
Suli and Mayers 9.4.
monomial_projection.m: best polynomial approximation using monomials.
legendre_projection.m: best polynomial approximation using orthogonal (Legendre) polynomials.
Lecture 9 (9/18): Spline interpolation.
Lecture 13,14 in the course lecture notes.
Lecture 8 (9/15): Piecewise polynomial interpolation.
Lecture 13,14 in the course lecture notes.
p1_interp_demo.m showing a comparison between piecewise linear and high order polynomial interpolation.
Lecture 7 (9/13): Hermite interpolation.
Lecture 13,14 in the course lecture notes.
Additional reading: Suli and Mayers 11.
hermite_interp.m showing Hermite interpolation of function and derivative values at equispaced points.
hermite_birkhoff_interp.m showing Hermite interpolation of function and higher derivative values at endpoints.
Lecture 6: Polynomial interpolation: Newton bases.
Lecture 10,11 in the course lecture notes.
Additional reading: Suli and Mayers 6.3.
Demo showing convergence of interpolation along with an error bound.
Demo showing growth of derivatives of Runge's function.
Lecture 5: Polynomial interpolation: Lagrange bases and convergence.
Lecture 9, 10 in the course lecture notes.
Additional reading: Suli and Mayers 6.1-6.3.
Demo of Vandermonde matrix and Lagrange form.
Lecture 4: Root finding: secant method, fixed point iterations.
Lecture 40 in the course lecture notes.
Additional reading: Suli and Mayers 1.1-1.3.
Code for the secant method.
Lecture 3: Root finding: convergence of Newton's method.
Lecture 39 in the course lecture notes.
Thm 1.9 in Suli/Mayers
Code to show failure of Newton's method.
Lecture 2: Root finding: bisection and Newton's method.
Piazza forum signup.
Lectures 38,39 in the course lecture notes.
Additional reading: Suli and Mayers 1.4, 1.6.
Matlab demo
Lecture 1: Introduction to Numerical Analysis:
Floating point number systems, catastrophic cancellation.
Rounding error disasters
Lectures 1,8 in the course lecture notes.