Numerical Methods Contents - SAM

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[37] M. STEWART, A superfast toeplitz solver with improved numerical stability, SIAM J. Matrix Anal- Bandbreite BLAS, 48 ysis Appl., 25 (2003), pp. 669–693. Zeilen-, 166 axpy, 49 [38] J. STOER, Einführung in die Numerische Mathematik, Heidelberger Taschenbücher, Springer, 4 ed., 1983. [39] V. STRASSEN, Gaussian elimination is not optimal, Numer. Math., 13 (1969), pp. 354–356. [40] M. STRUWE, Analysis für informatiker. Lecture notes, ETH Zürich, 2009. https://moodleapp1.net.ethz.ch/lms/mod/resource/index.php?id=145. [41] F. TISSEUR AND K. MEERBERGEN, The quadratic eigenvalue problem, SIAM Review, 43 (2001), pp. 235–286. [42] L. TREFETHEN AND D. BAU, <strong>Numerical</strong> Linear Algebra, SIAM, Philadelphia, PA, 1997. [43] P. VERTESI, On the optimal lebesgue constants for polynomial interpolation, Acta Math. Hungaria, 47 (1986), pp. 165–178. [44] , Optimal lebesgue constant for lagrange interpolation, SIAM J. Numer. Aanal., 27 (1990), pp. 1322–1331. Ôº ½¿º bandwidth, 164 lower, 164 minimizing, 173 upper, 164 barycentric interpolation formula, 647 basic linear algebra subroutines, 48 basis cosine, 614 orthonormal, 207, 400 sine, 605 trigonometric, 562 Bauelementgleichungen, 67 Belousov-Zhabotinsky reaction, 870 Bernstein polynomial, 751 Besetzungsmuster, 176 Bestapproximation durch Niedrigrangmatrizen, 496 bicg, 387 BiCGStab, 387 block LU-decomposition, 94 blow-up, 873 blurring operator, 589 Broyden Quasi-Newton Method, 324 Broyden-Verfahren ceonvergence monitor, 327 Butcher scheme, 865, 929 cancellation, 309, 311 for Householder transformation, 199 capacitance, 68 capacitor, 68 cardinal spline, 731 causal filter, 541 CG convergence, 370 preconditioned, 375 Ôº½ ½¿º bisection, 275 termination criterion, 361 CG algorithm, 358 of SVD, 488 chain rule, 306 composite quadrature, 772 characteristic polynomial, 397 compressed row storage, 141 Index Chebychev nodes, 676, 678 Chebychev polynomials, 366 3-term recursion, 673 computational costs LU-decomposition, 92 QR-decomposition, 209 Chebychev-interpolation, 670 computational effort, 296 LU-decomposition algebraic dependence, 46 chemical reaction kinetics, 920 eigenvalue computation, 405 existence, 103 algorithm Cholesky decomposition, 186 condition number 3-term recursion Clenshaw, 685 costs, 187 of a matrix, 132 for Chebychev polynomials, 673 arrow matrix, 154, 156 Cholesky factorization, 223 consistency 5-points-star-operator, 609 asymptotic complexity, 41 circuit simulation of iterative methods, 240 a posteriori error bound, 255 A-inner product, 333 A-orthogonal, 354 absolute tolerance, 878 adaptive multigrid quadrature, 813 adaptive quadrature, 812 AGM, 249 Aitken-Neville scheme, 648 algebra, 39 asymptotic error behavior, 662 asymptotic rate of linear convergence, 269 Axiom of roundoff analysis, 112 AXPY operation, 359 axpy operation, 49 back substitution, 74 backward error analysis, 121 backward substitution, 93 Banach’s fixed point theorem, 264 Ôº¼ ½¿º transient, 833 Classical Runge-Kutta method Butcher scheme, 866, 867 Clenshaw algorithm, 685 coil, 68 column major matrix format, 27 column sum norm, 117 complexity, 41 asymptotic, 41 linear, 43 fixed point iteration, 259 constant Lebesgue, 678 convergence algebraic, 665 asymptotic, 291 exponential, 663, 665, 680 global, 241 iterative method, 240 linear, 243 Ôº¾ ½¿º
linear in Gauss-Newton method, 538 local, 241 local quadratic in damped Newton method, 537 quadratic, 249 rate, 243 convergence monitor of Broyden method, 327 convolution discrete, 541, 545 discrete periodic, 548 of sequences, 545 cosine basis, 614 transform, 614 cosine matrix, 614 cosine transform, 614 costs Cholesky decomposition, 187 Crout’s algorithm, 90 CRS, 141 CRS format diagonal, 143 divided differences, 658 dot product, 31 double precision, 109 economical singular value decomposition, 487 efficiency, 296 eigenspace, 397 eigenvalue, 397 eigenvector, 397 electric circuit, 67, 235 resonant frequencies, 392 elementary arithmetic operations, 106, 112 energy norm, 333 envelope matrix, 166 Equation non-linear, 238 ergodicity, 420 error behavior asymptotic, 662 error estimator a posteriori, 255 cubic Hermite interpolation, 641 damped Newton method, 318 damping factor, 319 deblurring, 585 definite, 115 dense matrix, 139 DFT, 555, 565 two-dimensional, 584 Diagonal dominance, 182 diagonal matrix, 86 diagonalization of a matrix, 400 diagonalization of local translation invariant linear operators, 609 difference quotient, 308 difference scheme, 842 differential in non-linear least squares, 306 direct power method, 422 discrete convolution, 541, 545 discrete Fourier transform, 555, 565 discrete periodic convolution, 548 Euler method explicit, 841 implicit, 844 semi implicit, 933 Euler polygon, 842 Euler’s iteration, 287 expansion asymptotic, 652 explicit Euler method, 841 Butcher scheme, 866 explicit midpoint rule Butcher scheme, 866 for ODEs, 862 explicit Runge-Kutta method, 864 explicit trapzoidal rule Butcher scheme, 866 exponential convergence, 680 extended state space of an ODE, 830 extrapolation, 652 fast Fourier transform, 594 Ôº¿ ½¿º Ôº ½¿º FFT, 594 fill-in, 153 filter high pass, 574 low pass, 574 finite filter, 541 fit polynomial, 502 fixed point, 259 fixed point form, 259 fixed point iteration consistency, 259 floating point numbers, 107 forward substitution, 93 Fourier matrix, 563 Fourier transform discrete, 555, 565 fractional order of convergence, 290 frequency filtering, 568 Funktion shandles, 302 Halley’s iteration, 280, 287 harmonic mean, 716 heartbeat model, 826 Hermite interpolation cubic, 641 Hessian matrix, 306 high pass filter, 574 homogeneous, 115 Horner scheme, 632 Householder reflection, 197 IEEE standard 754, 109 ill conditioned, 136 image segmentation, 427 implicit Euler method, 844 impulse response of a filter, 541 in place, 78, 90, 92 in situ, 92 in-situ, 78 increment equations linearized, 934 fzero, 277 Gauss Quadrature, 792 Gauss-Newton method, 533 Gauss-Seidel preconditioner, 376 Gaussian elimination, 72 block version, 80 for non-square matrices, 81 instability, 134 Gerschgorin circle theorem, 398 Givens rotation, 200, 220 Givens-Rotation, 211, 230, 233 global solution of an IVP, 837 GMRES, 386 restarted, 386 Golub-Welsch algorithm, 805 gradient, 306, 338 Gram-Schmidt Orthonormalisierung, 471 Gram-Schmidt orthogonalization, 356, 471 grid function, 610 increments Runge-Kutta, 864, 928 inductance, 68 inductor, 68 inf, 110 infinity, 110 initial guess, 240, 258 initial value problem stiff, 925 initial value problem (IVP), 829 inner product A-, 333 intermediate value theorem, 275 interpolation barycentric formula, 647 Chebychev, 670 complete cubic spline, 728 general polynomial, 634 Hermite, 634 Lagrange, 633 natural cubic spline, 728 periodic cubic spline, 729 Ôº ½¿º Ôº ½¿º
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2 Direct Methods for Linear Systems
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III Integration of Ordinary Differe
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Extra questions for course evaluati
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1.1.2 Matrices Matrices = two-dimen
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Remark 1.2.1 (Row-wise & column-wis
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1.3 Complexity/computational effort
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Syntax of BLAS calls: The functions
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4 { 5 a ssert ( this−>n==B. n &&
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34 long r t 0 ; 35 bool bStarted ;
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Obviously, left multiplication with
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❶: elimination step, ❷: backsub
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A direct way to LU-decomposition:
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Solution of LŨx = b: x ( ) 2ǫ = 1
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numerically equivalent ˆ= same res
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Code 2.4.8: Finding outeps in MATLA
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Terminology: Def. 2.5.5 introduces
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Example 2.5.5 (Instability of multi
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Note: sensitivity gauge depends on
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6 for i =1:20 7 n = 2^ i ; m = n /
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0 20 40 60 80 100 120 140 160 180 2
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Use sparse matrix format: 10 1 10 2
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Envelope-aware LU-factorization: 0
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0 20 40 60 80 100 0 20 0 20 0 20 De
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Evident: symmetry of Ã − bbT a 1
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9 ylabel ( ’ { \ b f c o n d i t
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Mapping a ∈ K n to a multiple of
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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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Page 189 and 190: 1 0.9 Function f 1 0.9 Function f
Page 191 and 192: f 10 Numerical Quadrature Numerical
Page 193 and 194: f • n = 1: Trapezoidal rule • n
Page 195 and 196: For fixed local n-point quadrature
Page 197 and 198: Equidistant trapezoidal rule (order
Page 199 and 200: Heuristics: A quadrature formula ha
Page 201 and 202: 20 Zeros of Legendre polynomials in
Page 203 and 204: |quadrature error| 10 0 Numerical q
Page 205 and 206: f f • line 9: estimate for global
Page 207 and 208: Model: autonomous Lotka-Volterra OD
Page 209 and 210: Example 11.1.6 (Transient circuit s
Page 211 and 212: y y 1 y(t) y 0 t t 0 t 1 Fig. 133 e
Page 213 and 214: for discrete evolution defined on I
Page 215 and 216: ⇒ if y ∈ C 2 ([0, T]), then y(t
Page 217 and 218: The implementation of an s-stage ex
Page 219 and 220: Example 11.5.2 (Blow-up). ✸ toler
Page 221 and 222: However, it would be foolish not to
Page 223 and 224: 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Page 225 and 226: 4 3 2 abstol = 0.000010, reltol = 0
Page 227 and 228: ✸ ✸ Motivated by the considerat
Page 229 and 230: 0 1 2 3 4 5 6 u(t),v(t) 0.01 0.008
Page 231 and 232: MATLAB-CODE : Explicit integration
Page 233 and 234: Shorthand notation for Runge-Kutta
Page 235 and 236: 13 Structure Preservation 13.1 Diss
Page 237: [18] G. GOLUB AND C. VAN LOAN, Matr
Page 241 and 242: Chebychev nodes, 678 double, 634 fo
Page 243 and 244: x∗ n y ˆ= discrete periodic conv
Page 245 and 246: Gaussian elimination with pivoting
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Numerical Methods Contents - SAM

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