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| Lecture Notes: |
Notes for each lecture are provided below (with additional links/demos),
or you can download all the notes here,
or lecture-by-lecture:
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40
I am grateful for students and web readers who have offered
corrections over the years.
Please send any comments/corrections
to embree@rice.edu.
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| MATLAB Help: |
For help with MATLAB used in the context of numerical analysis,
I highly recommend Cleve Moler's book,
Numerical Computing
with MATLAB
(available free online, or for puchase from SIAM).
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| Root finding: |
(Useful for Problem Set 7, but not on exam)
- lecture38.pdf
- lecture39.pdf
- lecture40.pdf
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| Lecture 37: |
Eigenvalue computations
- lecture37.pdf
- G. Golub and F. Uhlig, "The QR algorithm: 50 years later", IMA J. Num. Anal. 29 (2009) 467-485.
- Biography of Evariste Galois
- B. N. Parlett, "The QR Algorithm", Comp. in Sci. & Eng., 2000. (Top 10 Algorithms of the 20th Century)
- D. S. Watkins,
"Understanding the QR Algorithm", SIAM Revew 24 (1982) 427-440.
- Who was Hessenberg and why do the matrices bear his name? See this page by J.-P. Zemke.
- qrmovie_a.m: a graphical illustration of the basic QR algorithm
- qrmovie_b.m: a graphical illustration of the QR algorithm with Hessenberg reduction
- qrmovie_c.m: a graphical illustration of the QR algorithm with Hessenberg reduction and shifts
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| Lecture 36: |
Cholesky factorization for symmetric positive definite matrices
- lecture36.pdf
- "Who is Cholesky?" from the NA Digest
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| Lecture 35: |
Introduction to LU factorization/Gaussian elimination
- lecture35.pdf
- lugui.m: Cleve Moler's LU factorization GUI from
Numerical Computing with MATLAB SIAM, Philadelphia 2004.
- Nick Gould, "On Growth in Gaussian Elimination with Complete Pivoting", SIAM J. Matrix Anal. Appl. 12 (1991) 3
54-361.
- Leslie V. Foster, "Gaussian Elimination with Parital Pivoting Can Fail in Practice",
SIAM J. Matrix Anal. Appl. 15 (1994) 1354-1362.
- Tobin A. Driscoll and Kara L. Maki, "Searching for Rare Growth Factors Using Multicanonical Monte Carlo Methods",
SIAM Review 49 (2007) 673-692.
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| Lecture 34: |
A glimpse at second order equations: geomeric integration; boundary value problems
- lecture34.pdf
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| Lecture 33: |
Stiff differential equations
- lecture33.pdf
- C. F. Curtiss and J. O. Hirschfelder, "Integration of Stiff Equations", Proc. Nat. Acad. Sci. 38 (1952) 235-243. |
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| Lecture 32: |
Linear multistep methods: absolute stability theory
- lecture32.pdf
- See also: Süli and Mayers, Chapter 12
- stabregions.m: code for computing stability regions of ten integrators (see notes) |
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| Lecture 31: |
Linear multistep methods: zero stability, Dahlquist equivalence theorem
- lecture31.pdf
- See also: Süli and Mayers, Chapter 12
- Obituary for Germund Dahlquist (RIP 1 May 2005)
- stab_demo1.m
stab_demo2.m: examples of stable and unstable methods
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| Lecture 30: |
Linear multistep methods: truncation error
- lecture30.pdf
- See also: Süli and Mayers, Chapter 12
- Biography of John Couch Adams
- Biography of Forest Ray Moulton, and notes from the US National Park Service
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| Lecture 29: |
One step methods: global error, adaptive step
- lecture29.pdf
- See also: Süli and Mayers, Section 12.2
- L. F. Shampine and M. W. Reichelt, "The MATLAB ODE Suite", SIAM J. Sci. Comp 18 (1997) 1-22.
- MATLAB code: os_gerr.m demonstrates global error of one-step methods
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| Lecture 28: |
One step methods: truncation error
- lecture28.pdf
- See also: Süli and Mayers, Section 12.1
- Biography of Carle Runge
- Biography of Martin Kutta
- Adaptive
step Runge-Kutta method by Cash and Karp (1990)
- MATLAB codes: heun.m,
rk4.m
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| Lecture 27: |
Introduction to numerical ODE solvers; Euler's method
- lecture27.pdf
- MATLAB code: euler.m
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| Lecture 26: | Gaussian quadrature
- lecture26.pdf (draft)
- G. H. Golub
and J. H. Welsh, "Calculation of Gauss Quadrature Rules", Math. Comp. 23 (1969) 221--230.
- MATLAB demo of Gauss-Legendre quadrature: gl_demo.m,
gausslegendre.m |
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| Lecture 25: |
Gaussian quadrature (orthogonal polynomials)
- lecture25.pdf
- See also: Süli and Mayers, Chapter 10
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| Lecture 24: |
Richardson extrapolation
- lecture24.pdf
- Biography of Lewis Fry Richardson
- MATLAB demo of Richardson extrapolation: deriverr.m
- MATLAB demo of Romberg integration: romberg_demo.m,
romberg.m
- To learn more about the poems shown in class, see Piet Hein's Wikipedia page
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| Lecture 23: |
Quadrature: Peano kernel analysis [553 only - lecture date to be announced]
- lecture23b.pdf
- Biography of Giuseppe Peano
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| Lecture 23: |
Interpolatory quadrature
- lecture23.pdf
- See also: Süli and Mayers, Chapter 7
- Biography of Thomas Simpson
- W. Gander and W. Gautschi, "Adaptive quadrature - revisisted," BIT 40 (2000) 84-101.
- MATLAB code for composite trapezoid rule: trapezoid.m
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| Lecture 22: |
Minimax aproximation: Chebyshev polynoials and optimal interpolation points
- lecture22.pdf
- Biography of Pafnuty Lvovich Chebyshev
- COCA:
a MATLAB package for minimax approximation in the complex plane
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| Lecture 21: |
Minimax aproximation
- lecture21.pdf
- See also: Süli and Mayers, Sections 8.3
- Biography of de la Vallée Poussin
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| Lecture 20: |
Continuous least squares approximation, orthogonal polynomials
- lecture20.pdf
- See also: Süli and Mayers, Sections 9.4 - 9.5
- MATLAB demo: cls_mono.m
- MATLAB demo: cls_monopert.m
- MATLAB demo: cls_legendre.m
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| Lecture 19: |
Continuous least squares approximation
- lecture19.pdf
- See also: Süli and Mayers, Sections 9.1 - 9.3
- M.-D. Choi, "Tricks or treats with the Hilbert matrix," American Math. Monthly 90 (1983) 301-312. |
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| Lecture 18: |
Singular value decomposition: fundamental subspaces, low-rank approximation
- lecture18.pdf
- See also: Trefethen and Bau, Lecture 5
- Compress an image of the founders of numerical linear algebra at the 1964 Gatlinburg conference:
svd_gatlin_demo.m
- Fritz Bauer reminisces
about Alston Householder and the Gatlinburg conferences.
- Photos
from the 1977 Gatlinburg conference |
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| Lecture 17: |
Singular value decomposition: derivation
- lecture17.pdf
- See also: Trefethen and Bau, Lecture 4
- G. W. Stewart,
"On the early history of the singular value decomposition," SIAM Review 35 (1993) 551-566 |
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| Lecture 16: |
Discrete least squares
- lecture16.pdf
- See also: Trefethen and Bau, Lecture 11
- Biography
of Carl Friedrich Gauss
- Biography
of Adrien-Marie Legendre
- MATLAB demo comparing the polynomial interpolation to discrete least squares approximation
for Runge's function:
rungle_ls_demo.m
- MATLAB demo comparing the normal equations and QR for a least squares problem:
ne_vs_qr.m |
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| Lecture 15: |
Trigonometric interpolation
- lecture15.pdf
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| Lecture 14: |
Matrix formulation of spline interpolants
- lecture14.pdf
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| Lecture 13: |
Basis splines
- lecture13.pdf
- James Epperson's brief History of Splines
- Paul Davis on "B-Splines and Geometric Design"
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| Lecture 12: |
Hermite and piecewise linear interpolation
- lecture12.pdf
- Süli and Mayers, Section 6.4, Chapter 11
- MATLAB demo: pwfit_demo.m
- Biography of Charles Hermite
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| Lecture 11: |
Interpolation error bound; Hermite interpolation
- lecture11.pdf
- runge_deriv.m: Shows derivatives of Runge's function (requires the Symbolic Toolbox)
- Runge.nb: Mathematica notebook examine derivatives of Runge's function
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| Lecture 10: |
Polynomial Interpolation in the Newton and Lagrange basis
- lecture10.pdf
- mono_basis.m
- newton_basis.m
- lagrange_basis.m
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| Lecture 9: |
Polynomial Interpolation in the monomial basis
- lecture9.pdf
- monomial_fit.m
- Biography
of Alexandre-Thoeophile Vandermonde
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| Lecture 8b: |
Floating Point Error Analysis
- Cleve Moler's floatgui.m
- Biography of James Hardy Wilkinson
- N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002.
- Essex, Davison, Schulzky, Numerical Monsters, ACM SIGSAM Bulletin 34 (2000) 16-32.
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| Lecture 8: |
IEEE Floating Point Arithmetic
- lecture8.pdf
- See this householder_diary
- Doug Arnold's catalog of numerical analysis disasters
- M. L. Overton,
Numerical Computing with IEEE Floating Point Arithmetic, SIAM, 2001.
- "An Interview
with the Old Man of Floating-Point" (William Kahan) by Charles Severance
- Kahan's 1981 paper, "Why do we need a floating-point arithmetic standard?"
- Floating point conversion tool (thanks, DCS!)
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| Lecture 7: |
Solving linear systems; conditioning
- lecture7.pdf
- Read Trefethen and Bau, Lecture 12
- RationalLinearSystem.nb: Mathematica demonstration of an
exact solution of a linear system
- Von Neumann and Goldstine, Numerical inverting
of matrices of high order, Bull. Amer. Math. Soc. 53 (1947) 1021-1099. (Historical interest)
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| Lecture 6: |
Gram-Schmidt QR, Solving Systems via QR
- lecture6.pdf
- cgs_qr.m: classical Gram-Schmidt algorithm
- mgs_qr.m: modified Gram-Schmidt algorithm
- qr_test.m
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| Lecture 5: |
Householder QR factorization, QR via Gram-Schmidt
- lecture5.pdf (updated 12 Sept)
- householder_qr.m
- qrdemo.m
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| Lecture 4: |
Householder reflectors
- lecture4.pdf
- Biography
of Alston S. Householder
- SIAM News article about the Top 10 Algorithms of the 20th Century
- Householder's
landmark paper on the QR factorization (J. ACM 5 (1958) 339-342.)
- MATLAB code: slow_householder_qr.m
- See Trefethen and Bau, Lectures 10
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| Lecture 3: |
Matrix Norms, Projectors, Reflectors
- lecture3.pdf
- house3d.m: demo of Householder reflector in 3d
- See Trefethen and Bau, Lectures 3, 6.
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| Lecture 2: |
Matrix Analysis and Norms
- lecture2.pdf (updated 27 August)
- norm_demo.m (See Figure 3.1 in Trefethen and Bau)
- Gilbert Strang,
"The Fundamental Theorem of Linear Algebra", American Math. Monthly, 100 (1993) 848-855.
- See Trefethen and Bau, Lectures 1, 2, 3.
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| Lecture 1: |
Introduction to Numerical Analysis by way of history and applications
- lecture1.pdf
- orbit.m
- Read Trefethen's Definition of Numerical Analysis,
(Trefethen & Bau, pp. 321-327)
- Consider subscribing to the
NA Digest
free weekly electronic newsletter
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