Numerical Methods Contents - SAM

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Website: http://www.math.ethz.ch/education/bachelor/lectures/hs2009/math/nummath_cse Examination Three-hour written examination involving coding problems to be done at the computer on ✗ ✔ Thu, Feb 4, 2010, 09:00 – 12:00 ✖ ✕ Dry-run for computer based examination: Mon, Jan 18, 2010, 09:00, registration via course website Pre-exam question session: Mon, Jan 18, 2010, 10:15-12:009:00, room will be announced Subject of examination: • Chapters 1 through 11 of the course, • homework problems on sheet 1 through 13. Marcus Wittberger (D-INFK) Dr. Julia Schweitzer (SAM, D-MATH) Ôº½¿ ¼º¼ Lecture documents will be available as PDF during the examination, both in four-page and onepage layout. The correspoding final version of the lecture documents will be made available in Jan, 18, 2010. The exam questions will be asked both in German and in English. Ôº½ ¼º¼ Reporting errors Please report errors in the electronic lecture notes via a wiki page ! http://elbanet.ethz.ch/wikifarm/rhiptmair/index.php?n=Main.NCSECourse (Password: CSE, please choose EDIT menu to enter information) Please supply the following information: Manfred Quack (D-INFK) Peter Kauf (SAM, D-MATH) • (sub)section where the error has been found, Classes: Mon, 8.15-10.00 (HG G5), Thu, 10.15-11.55 (HG G5) Tutorials: Mon 10.15-11.55 (HG E 21, for students of RW/CSE) Thu 8.15-10.00 (for students of computer science) • precise location (e.g, after Equation (4), Thm. 2.3.3, etc. ). Refrain from giving page numbers, • brief description of the error. Assignments. • One problem sheet will be handed out every week Ôº½ ¼º¼ Alternative (for people not savvy with wikis): E-mail an hiptmair@sam.math.ethz.ch, Subject: NUMCSE Ôº½ ¼º¼ • “Testatbedingung”: attempted solutions for a least 60% of the problems.
Extra questions for course evaluation Course number (LV-ID): 401-0663-00 Date of evaluation: 23.11.2009 D1: I try to do all programming exercises. D2: The programming exercises help understand the numerical methods. D3: The programming exercises offer too little benefit for the effort spent on them. D4: Scope and limitations of methods are properly addressed in the course. D5: Numerical examples in class provide useful insights and motivation. D6: There should be more examples presented and discussed in class. D7: Too much information is crammed onto the lecture slides D8: The course requires too much prior knowledge in linear algebra Ôº½ ¼º¼ D9: The course requires too much prior knowledge in analysis D10: My prior knowledge of MATLAB was insufficient for the course D11: More formal proofs would be desirable D12: The explanations on the blackboard promote understanding D13: The codes included in the lecture material convey useful information D14: The model solutions for exercise problems offer too little guidance. D15: The relevance of the numerical methods taught in the course is convincingly conveyed. D16: I would not mind the course being taught in English. Scoring: 6: I agree fully 5: I agree to a large extent 4: I agree partly 3: I do not quite agree 2: I disagree 1: I disagree strongly Evaluation of assistants: Assistant shortcut Julia Schweitzer Manfred Quack Peter Kauf Daniel Wright Marcelo Serrano Zanetti Marcus Wittberger Please enter the shortcut code after the LV-ID in the three separate boxes. Evaluation results: Fall semester 2009 MATLAB MATLAB ("‘Matrix Laboratory”): JUL MAN PET DAN SER MAR full fledged high level programming language (for numerical algorithms) integrated programming environment versatile collection of numerical libraries in this course used for ✄ demonstrating (implementation of) algorithms ✄ numerical experiments ✄ programming assignments This course assumes familiarity with MATLAB as acquired in the introductory linear algebra courses. Ôº½ ¼º¼ Ôº½ ¼º¼ ★ ✧ Proficiency in MATLAB through “Learning on the job” (using the very detailed online help facilities) ✥ Ôº¾¼ ¼º¼ ✦
Page 1 and 2: 2 Direct Methods for Linear Systems
Page 3: III Integration of Ordinary Differe
Page 7 and 8: 1.1.2 Matrices Matrices = two-dimen
Page 9 and 10: Remark 1.2.1 (Row-wise & column-wis
Page 11 and 12: 1.3 Complexity/computational effort
Page 13 and 14: Syntax of BLAS calls: The functions
Page 15 and 16: 4 { 5 a ssert ( this−>n==B. n &&
Page 17 and 18: 34 long r t 0 ; 35 bool bStarted ;
Page 19 and 20: Obviously, left multiplication with
Page 21 and 22: ❶: elimination step, ❷: backsub
Page 23 and 24: A direct way to LU-decomposition:
Page 25 and 26: Solution of LŨx = b: x ( ) 2ǫ = 1
Page 27 and 28: numerically equivalent ˆ= same res
Page 29 and 30: Code 2.4.8: Finding outeps in MATLA
Page 31 and 32: Terminology: Def. 2.5.5 introduces
Page 33 and 34: Example 2.5.5 (Instability of multi
Page 35 and 36: Note: sensitivity gauge depends on
Page 37 and 38: 6 for i =1:20 7 n = 2^ i ; m = n /
Page 39 and 40: 0 20 40 60 80 100 120 140 160 180 2
Page 41 and 42: Use sparse matrix format: 10 1 10 2
Page 43 and 44: Envelope-aware LU-factorization: 0
Page 45 and 46: 0 20 40 60 80 100 0 20 0 20 0 20 De
Page 47 and 48: Evident: symmetry of Ã − bbT a 1
Page 49 and 50: 9 ylabel ( ’ { \ b f c o n d i t
Page 51 and 52: Mapping a ∈ K n to a multiple of
Page 53 and 54: Then store G ij (a,b) as triple (i,
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Recall: e i ˆ= i-th unit vector Ch
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Computation of Choleskyfactorizatio
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1 ∃ (partial) cyclic row permutat
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Definition 3.1.3 (Local and global
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Example 3.1.6 (quadratic convergenc
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k |x (k) − π| L 1−L |x (k) −
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(x (k) ) k∈N0 Cauchy sequence ➤
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Termination criterion for contracti
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Given x (k) ∈ I, next iterate :=
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secant method ( MATLAB implementati
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Assuming p = 1: p > 1: ∥ C ∥e (
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This is a simple computation: DG(x)
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k x (k) ǫ k := ‖x ∗ − x (k)
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Code 3.4.14: Damped Newton method (
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MATLAB-CODE: Broyden method (3.4.11
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Algorithm 4.1.3 (Steepest descent).
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Example 4.1.8 (Convergence of gradi
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4.2.1 Krylov spaces Definition 4.2.
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Remark 4.2.3 (A posteriori terminat
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10 figure ; view ([ −45 ,28]) ; m
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Idea: Solve Ax = b approximately in
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eplaced with κ(A) ! 4.4.2 Iteratio
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For circuit of Fig. 55 at angular f
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(Linear) generalized eigenvalue pro
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10 0 10 1 matrix size n d = eig(A)
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0 50 100 150 200 250 300 350 400 45
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1 2 3 k ρ (k) EV ρ (k) EW ρ (k)
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✬ ✩ ✬ ✩ Lemma 5.3.4 (Ncut a
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In other words, roundoff errors may
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Theory: linear convergence of (5.3.
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error in eigenvalue 10 0 10 −2 10
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✬ Residuals r 0 ,...,r m−1 gene
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Algebraic view of the Arnoldi proce
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5.5 Singular Value Decomposition Re
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Illustration: columns = ONB of Im(A
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✬ Theorem 5.5.7 (best low rank ap
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Reassuring: Remark 6.0.4 (Pseudoinv
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Consider the linear least squares p
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Goal: Euclidean distance of y ∈ R
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6.5 Non-linear Least Squares If (6.
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0 2 4 6 8 10 12 14 16 value of ∥
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Definition 7.1.1 (Discrete convolut
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Expand a 0 ,...,a n−1 and b 0 , .
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(7.2.2) is a simple consequence of
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Dominant coefficients of a signal a
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11 c = f f t ( y ) ; 12 13 figure (
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Two-dimensional trigonometric basis
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8 end 9 t1 = min ( t1 , toc ) ; 10
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Step II: for k =: rq + s, 0 ≤ r <
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MATLAB-CODE Sine transform function
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△ Example 7.5.2 (Linear regressio
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[23, Ch. IX] presents the topic fro
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Code 8.1.3: Horner scheme, polynomi
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1.2 equality in (8.2.10) for y := (
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ecursive definition: p i (t) ≡ y
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a 1 = y 1 − a 0 t 1 − t 0 = y 1
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Observations: Strong oscillations o
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−1 −0.8 −0.6 −0.4 −0.2 0
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8.5.3 Chebychev interpolation: comp
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9.1 Shape preserving interpolation
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9.2.2 Piecewise polynomial interpol
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Interpolation of the function: f(x)
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2 % Plot convergence of approximati
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9.4 Splines Definition 9.4.1 (Splin
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➤ Linear system of equations with
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y i+1 t i−1 t i t i+1 y i−1 y i
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35 h= d i f f ( t ) ; 36 d e l t a
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1 0.9 Function f 1 0.9 Function f
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f 10 Numerical Quadrature Numerical
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f • n = 1: Trapezoidal rule • n
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For fixed local n-point quadrature
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Equidistant trapezoidal rule (order
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Heuristics: A quadrature formula ha
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20 Zeros of Legendre polynomials in
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|quadrature error| 10 0 Numerical q
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f f • line 9: estimate for global
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Model: autonomous Lotka-Volterra OD
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Example 11.1.6 (Transient circuit s
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y y 1 y(t) y 0 t t 0 t 1 Fig. 133 e
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for discrete evolution defined on I
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⇒ if y ∈ C 2 ([0, T]), then y(t
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The implementation of an s-stage ex
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Example 11.5.2 (Blow-up). ✸ toler
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However, it would be foolish not to
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
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4 3 2 abstol = 0.000010, reltol = 0
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✸ ✸ Motivated by the considerat
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0 1 2 3 4 5 6 u(t),v(t) 0.01 0.008
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MATLAB-CODE : Explicit integration
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Shorthand notation for Runge-Kutta
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13 Structure Preservation 13.1 Diss
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[18] G. GOLUB AND C. VAN LOAN, Matr
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linear in Gauss-Newton method, 538
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Chebychev nodes, 678 double, 634 fo
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x∗ n y ˆ= discrete periodic conv
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Gaussian elimination with pivoting
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Numerical Methods Contents - SAM

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