Numerical Methods Contents - SAM

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spline cubic, 722 spline cubic, locality, 731 spline shape preserving, 735 inverse interpolation, 292 inverse iteration, 439 preconditioned, 444 inverse matrix, 70 invertible matrix, 70, 71 iteration Halley’s, 287 Euler’s, 287 quadratical inverse interpolation, 287 iteration function, 240, 258 iterative method convergence, 240 IVP, 829 Jacobi preconditioner, 376 Jacobian, 265, 300 kinetics of chemical reaction, 920 Kirchhoff law, 67 line search, 337 linear complexity, 43 linear correlation, 492 linear electric circuit, 67 linear filter, 541 linear operator diagonalization, 609 linear ordinary differential equation, 394 linear regression, 46 linear system of equations, 66 multiple right hand sides, 82 local a posteriori erorr estimation for adaptive quadrature, 813 local convergence Newton method, 318 local linearization, 300 local mesh refinement for adaptive quadrature, 813 Lotka-Volterra ODE, 824 low pass filter, 574 lower triangular matrix, 86 LU-decomposition Konvergenz Algebraische, Quadratur, 810 Kronecker symbol, 24 Krylov space, 353 for Ritz projection, 460 L-stable, 931 Lagrange multiplier, 526 Landau-O, 41 Lapack, 78 Least squares with linear constraint, 525 least squares, 503 total, 524 least squares problem conditioning, 505 Lebesgue constant, 678 Lebesgue constant, 643 Levinson algorithm, 622 limit cycle, 922 limiter, 716 blocked, 94 computational costs, 92 envelope aware, 169 existence, 88 in place, 92 LU-factorization envelope aware, 169 of sparse matrices, 152 machine number, 108 exponent, 108 machine numbers, 109 distribution, 108 machine precision, 112 mantissa, 108 Markov chain, 411, 619 stationary distribution, 413 Matlab, 19 Matrix adjoint, 26 Hermitian, 400 Hermitian transposed, 26 Ôº ½¿º Ôº ½¿º normal, 399 obere Dreiecks, 227 obere Hessenberg, 221 skew-Hermitian, 400 transposed, 26 unitary, 400 matrix banded, 164 condition number, 132 dense, 139 diagonal, 86 envelope, 166 Fourier, 563 Hermitian, 179 Hessian, 306 lower triangular, 86 normalized, 86 orthogonal, 194 positive definite, 179 positive semi-definite, 179 rank, 70 sine, 605 Quasi-Newton, 323 midpoint rule, 767, 862 Milne rule, 770 min-max theorem, 435 minimal residual methods, 384 model function, 277 Modellfunktionsverfahren, 277 modification technique QR-factorization, 220 monomial representation of a polynomial, 631 monomials, 631 multi-point methods, 278, 288 multiplicity geometric, 397 NaN, 110 Ncut, 430 nested subspaces, 350 Newton basis, 656 sparse, 139 structurally symmetric, 172 symmetric, 179 tridiagonal, 164 unitary, 194 upper triangular, 86 matrix algebra, 39 matrix block, 26 matrix faktorization, 84 matrix norm, 116 column sums, 117 row sums, 117 matrix product, 31 matrix storage envelope oriented, 172 Matrixmultiplikation Blockweise, 39 Matrixnorm, 116 Submultiplikativität, 116 mesh in time, 848 Method damping, 319 damping factor, 319 monotonicity test, 320 simplified method, 307 Newton correction, 300 simplified, 317 Newton iteration, 300 numerical Differentiation, 307 termination criterion, 315 Newton method 1D, 278 damped, 318 local convergence, 318 local quadratic convergence, 312 region of convergence, 318 nodal analysis, 67, 235 nodal potentials, 67 node in electric circuit, 67 quadrature, 764 nodes, 633 Chebychev, 676 Ôº ½¿º Ôº¼ ½¿º
Chebychev nodes, 678 double, 634 for interpolation, 635 non-linear data fitting, 529 non-normalized numbers, 110 norm, 115 L 1 , 642 L 2 , 642 ∞-, 115 1-, 115 energy-, 333 Euclidean, 115 of matrix, 116 Sobolev semi-, 669 supremum, 641 normal equations, 508 extended, 526 normalized lower triangular matrix, 85 normalized triangular matrix, 86 not a number, 110 Nullstellenbestimmung Modellfunktionsverfahren, 277 orthogonal polynomials, 798 orthonormal basis, 207, 400 overflow, 110 page rank, 411 stochastic simulation, 412 partial pivoting, 98, 100 Partition of unity, 755 PCA, 481 PCG, 375 Peano Theorem of, 837 periodic sequence, 546 permutation, 102 permutation matrix, 102 Permutationsmatrix, 228 perturbation lemma, 131 Petrov-Galerkin condition, 387 phase space of an ODE, 830 Picard-Lindelöf numerical algorithm, 119 <strong>Numerical</strong> differentiation roundoff, 309 numerical Differentiation Newton iteration, 307 numerical quadrature, 761 numerical rank, 519 Obere Hessenbergmatrix, 221 ODE, 829 scalar, 831 Ohmic resistor, 68 one-point methods, 278 order of quadrature formula, 775 order of convergence, 247 fractional, 290 ordinary differential equation linear, 394 ordinary differential equation (ODE), 829 oregonator, 870 orthogonal matrix, 194 Theorem of, 837 PINVIT, 444 Pivot -wahl, 100 choice of, 99 pivot, 73, 75, 76 pivot row, 73, 76 pivoting, 95 point spread function, 586 polynomial Bernstein, 751 characteristic, 397 Lagrange, 635 polynomial space, 631 polynomiales fit, 502 potentials nodal, 67 power spectrum of a signal, 576 preconditioned CG method, 375 preconditioned inverse iteration, 444 preconditioner, 373 Ôº½ ½¿º Ôº¾ ½¿º preconditioning, 372 predator-prey model, 824 principal axis transformation, 399 principal component, 493 principal component analysis, 481 problem ill conditioned, 136 saddle point, 526 sensitivity, 136 well conditioned, 136 product rule, 306 Punkt stationär, 826 pwer method direct, 422 QR algorithm, 402 QR-algorithm with shift, 403 QR-decomposition computational costs, 209 QR-factorization, QR-decomposition, 202 quadratic convergence, 272 numerical, 519 of a matrix, 70 row rank, 71 rank-1 modification, 79 rank-1-modifications, 216 rate of convergence, 243 Rayleigh quotient, 423, 435 Rayleigh quotient iteration, 442 Rechenaufwand Gausselimination, 78 recursion 3-term, 673 regular matrix, 70 relative tolerance, 878 rem:Fspec, 564 residual quantity, 445 Riccati differential equation, 830, 842 richt hand side of an ODE, 830 right hand side vector, 66, 119 Ritz projection, 456, 459 quadratic eigenvalue problem, 392 quadratic functional, 334 quadratic inverse interpolation, 294 quadratical inverse interpolation, 287 quadrature adaptive, 812 polynomial formulas, 767 quadrature formula, 764 order, 775 quadrature node, 764 quadrature numerical, 761 quadrature weight, 764 Quasi-Newton Method, 324 Quasi-Newton method, 323 Radau RK-method order 3, 932 order 5, 932 radiative heat transfer, 548 rank column rank, 71 computation, 488 Ritz value, 461 Ritz vector, 461 root of unity, 560 rounding up, 111 roundoff for numerical differentiation, 309 row major matrix format, 27 row permutation, 233 row sum norm, 117 row transformation, 72, 83 Rundung, 111 Runge-Kutta increments, 864, 928 Runge-Kutta method, 864, 928 L-stable, 931 Runge-Kutta methods stability function, 909, 930 saddle point problem, 526 scalar ODE, 831 scaling of a matrix, 37 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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35 h= d i f f ( t ) ; 36 d e l t a
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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
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Page 225 and 226: 4 3 2 abstol = 0.000010, reltol = 0
Page 227 and 228: ✸ ✸ Motivated by the considerat
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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: linear in Gauss-Newton method, 538
Page 243 and 244: x∗ n y ˆ= discrete periodic conv
Page 245 and 246: Gaussian elimination with pivoting
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