Numerical Methods Contents - SAM

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interpolants, approx. complete cubic, 732 natural cubic, locality, 731 shape preserving quadratic interpolation, 742 Stabilty of Arnoldi process, 476 Stepsize control detects instability, 911 Stepsize control in MATLAB, 895 Storing the Q-factor, 212 Strongly attractive limit cycle, 922 Subspace power methods, 457 SVD and additive rank-1 decomposition, 485 Tangent field and solution curves, 830 Taylor approximation, 628 Termination criterion for contrative fixed point iteration, 272 Termination criterion for direct power iteration, 426 Termination criterion in pcg, 382 Termination of PCG, 381 Testing for near singularity of a matrix, 214 Transformation of quadrature rules, 764 Transient circuit simulation, 833 Transient simulation of RLC-circuit, 912 Tridiagonal preconditioner, 378 Understanding the structure of product matrices, 33 Uniqueness of SVD, 486 Unitary similarity transformation to tridiagonal form, 403 Visualization of explicit Euler method, 841 Why using K = C?, 560 Wilkinson’s counterexample, 124, 134 Zerlegung Teil-LU, 94 Ôº½ 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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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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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 and 238: [18] G. GOLUB AND C. VAN LOAN, Matr
Page 239 and 240: linear in Gauss-Newton method, 538
Page 241 and 242: Chebychev nodes, 678 double, 634 fo
Page 243 and 244: x∗ n y ˆ= discrete periodic conv
Page 245: Gaussian elimination with pivoting
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