Numerical Methods Contents - SAM

More documents

Recommendations

Info

Examples and Remarks LU-decomposition of sparse matrices, 152 h-adaptive numerical quadrature, 817 [Bad behavior of global polynomial interpolants, 692 [Conditioning of row transformations, 191 [From higher order ODEs to first order systems, L-stable implicit Runge-Kutta methods, 932 fft Efficiency, 592 2-norm from eigenvalues, 348 3-Term recursin for Legendre polynomials, 803 3-term recursion for Chebychev polynomials, 673 834 A posteriori error bound for linearly convergent [Stable solution of LSE by means of QR-decomposition, iteration, 255 210 A posteriori termination criterion for linearly convergent iterations, 254 ode45 for stiff problem, 901 “Butcher barriers” for explicit RK-SSM, 867 A posteriori termination criterion for plain CG, “Failure” of adaptive timesteppin, 890 361 “Low” and “high” frequencies, 571 B = B H Accessing rows and columns of sparse matrices, s.p.d. mit Cholesky-Zerlegung, 401 144 Adapted Newton method, 283 Adaptive integrators for stiff problems in LAB, 935 Adaptive quadrature in MATLAB, 820 MAT- Adaptive timestepping for mechanical problem, 896 Adding EPS to 1, 113 Affine invariance of Newton method, 304 Analytic solution of homogeneous linear ordinary differential equations, 394 approximation Bernstein, 751 uses of, 629 Approximation by polynomials, 661 Arnoldi process Ritz projection, 474 Banach’s fixed point theorem, 264 bandwidth, 164 Bernstein approximation, 751 Block Gaussian elimination, 80 block LU-decomposition, 94 Blockweise Matrixmultiplikation:, 39 Blow-up, 873 Blow-up of explicit Euler method, 903 Broyden method for a large non-linear system, 330 Broydens Quasi-Newton method: convergence, 326 Butcher scheme for some explicit RK-SSM, 865 Cancellation in decimal floating point arithmetic, 311 CG convergence and spectrum, 370 Changing entries/rows/columns of a matrix, 216 Characteristic parameters of IEEE floating point numbers, 110 Chebychev interpolation error, 678 Chebychev polynomials on arbitrary interval, 675 Chebychev representation of built-in functions, 687 Chebychev vs uniform nodes, 676 Chemical reaction kinetics, 920 Choice of quadrature weights, 790 Ôº¿ ½¿º Ôº ½¿º Choice of unitary/orthogonal transformation, 204 Complexity of Householder QR-factorization, 205 Computational effort for eigenvalue computations, 405 Computing Gauss nodes and weights, 805 condition extended system, 512 Conditioning and relative error, 133 Conditioning of normal equations, 509 Conditioning of the least squares problem, 505 Construction of simple Runge-Kutta methods, 862 Convergence of CG as iterative solver, 364 Convergence of equidistant trapezoidal rule, 784 Convergence of gradient method, 345 Convergence of Hermite interpolation, 711 Convergence of Hermite interpolation with exact slopes, 708 Convergence of Krylov subspace methods for non-symmetric system matrix, 389 Convergence of Newton’s method, 312 Convergence of PINVIT, 447 Efficient associative matrix multiplication, 43 Efficient Initialization of sparse matrices in MAT- LAB, 146 Eigenvalue computation with Arnoldi process, 478 Eigenvectors of circulant matrices, 555 Envelope of a matrix, 166 Envelope oriented matrix storage, 172 Error estimates for polynomial quadrature, 771 Error of Gauss quadrature, 806 Euler methods for stiff logistic IVP, 926 Explicit Euler method as difference scheme, 842 Explicit Euler method for damped oscillations, 918 Explicit integrator in MATLAB, 868 Explicit trapzoidal rule for decay equation, 907 Exploring convergence, 666 Extremal properties of natural cubic spline interpolant, 723 Failure of damped Newton method, 322 Failure of Krylov iterative solvers, 388 Fast matrix multiplication, 42 Fast Toeplitz solvers, 624 Convergence of simple Runge-Kutta methods, 862 Convergence of subspace variant of direct power method, 456 Convergence rates for CG method, 369 Convergence theory for PCG, 376 Conversion into autonomous ODE, 832 Convolution of sequences, 545 Cosine transforms for compression, 616 CRS format, 141 cubic Hermite interpolation, 641 Damped Newton method, 321 Deblurring by DFT, 585 Decimal floating point numbers, 107 Details of Householder reflections, 198 Differentiation repetition, 305 Direct power method, 423 Divided differences and derivatives, 660 Domain of definition of solutions of IVPs, 837 Efficiency of iterative methods, 298 Efficiency of fft, 592 Feasibility of implicit Euler timestepping, 844 FFT algorithm by matrix factorization, 597 FFT based on general factrorization, 600 FFT for prime vector length, 602 Finite linear time-invariant causal channel, 541 fit linear, 502 Fit of hyperplanes, 520 Fixed points in 1D, 265 Fractional order of convergence of secant method, 290 Frequency filtering by DFT, 575 Frequency identification with DFT, 569 Function representation, 629 Gain through adaptivity, 885 Gaining efficiency through usage of BLAS, 51 Gaussian elimination, 73 Gaussian elimination and LU-factorization, 84 Gaussian elimination for non-square matrices, 81 Gaussian elimination via rank-1 modifications, 79 Ôº ½¿º Ôº ½¿º
Gaussian elimination with pivoting for 3×3-matrix, 98 Generalized eigenvalue problems and Cholesky factorization, 401 Generalized normalization, 452 Gradient method in 2D, 342 Group property of autonomous evolutions, 839 Growth with limited resources, 822 Halley’s iteration, 280 Heartbeat model, 826 Heating generation in electrical circuits, 762 Hermite interpolation, 704 theorem, 668 Horner scheme, 632 IEEE standard special cases, 109 IEEE standard 754 for machine numbers, 109 Image compression, 498 Image segmentation, 427 Impact of choice of norm, 243 Impact of data access patterns on runtime, 28 Krylov subspace methods for generalized EVP, 480 Lanczos process for eigenvalue computation, 467 Lebesgue constant, 643 Linear filtering of periodic signals, 546 linear regression, 501 Linear regression for stationary Markov chains, 619 Lineare zeitinvariante Systeme, 617 Linearization of increment equations, 933 Linearly convergent iteration, 245 Local convergence of Newton’s method, 318 local convergence of secant method, 292 Machine precision for MATLAB, 112 Many sequential solutions of LSE, 93 Matrix algebra, 39 Matrix storage formats, 27 Midpoint rule, 767 Multidimensional fixed point iteration, 269 Multiplication of polynomials, 544 Multiplication of sparse matrices, 149 Impact of roundoff errors on CG, 362 Impact of roundoff on Lanczos process, 469 Implicit Euler timestepping for decay equation, 925 Initial guess for power iteration, 426 Input errors and rounding errors, 110 Instability of multiplication with inverse, 129 Instability of normal equations, 509 interpolation auxiliary construction for shape preserving quadratic spline, 739 cubic Hermite, 704 cubic spline- locality, 731 piecewise, 699 piecewise cubic monotonicity preserving, 717 shape preserving quadratic spline, 742 interpolation error, 662 Intersection of lines in 2D, 137 Keeping track of unitary transformations, 207 Krylov methods for complex s.p.d. system matrices, 333 Necessary condition for L-stability, 931 Necessity of iterative approximation, 238 Newton simplified method, 307 Newton method and minimization of quadratic functional, 532 Newton method in 1D, 279 Newton method in 2D, 302 Newton method, modifizied , 287 Newton-Cotes formulas, 768 Nodal analysis of (linear) electric circuit, 67 Non-linear data fitting, 529 Non-linear data fitting (II), 536 Non-linear electric circuit, 235 Non-linear interpolation, 719 Normal equations vs. orthogonal transformations method, 520 Notation for single step methods, 849 numeric differentiation through extrapolation, 652 <strong>Numerical</strong> integration of logistic ODE in LAB, 870 MAT- Ôº ½¿º Ôº ½¿º Options for fixed point iterations, 260 Orders of simple polynomials quadrature formulas, 776 Oregonator reaction, 870 oscillating interpolation polynomial, 644 Overdetermined linear systems, 502 Page rank algorithm, 411 piecewise polynomial interpolation, 699 piecewise cubic Hermite interpolation, 704 piecewise polynomial interpolation from nodes, 698 Pivoting and numerical stability, 96 Pivoting destroys sparsity, 161 polynomial interpolation, 699 Polynomial evaluation: timing, 649 Polynomials in Matlab, 632 Predator-prey model, 824 Principal component analysis, 481 Principal component analysis for data analysis, 491 Region of convergence of Newton method, 318 regression linear (fit), 502 Relevance of asymptotic complexity, 47 Removing a singularity by transformation, 783 Resistance to currents map, 214 Resonances of linear electrical circuits, 392 Restarted GMRES, 386 Ritz projections onto Krylov space, 461 Roundoff errors and difference quotients, 307 Row and column transformations, 38 Row swapping commutes with forward elimination, 104 Row-wise & column-wise view of matrix product, 33 Runge’s example, 664 Runtimes of eig, 405 s.p.d. Hessian, 179 S.p.d. Hessians, 179 S.p.d. matrices from nodal analysis, 180 Scalings, 37 Pseudoinverse, 505 Pseudoinverse and SVD, 519 QR Orthogonalisierung, 206 QR-Algorithm, 402 QR-based solution of tridiagonal LSE, 211 Quadratic convergence], 249 Quadratic functional in 2D, 334 quadratic inverse interpolation, 295 Quadratur Gauss-Legendre Ordnung 4, 793 Quadrature errors for composite quadrature rules, 777 Radiative heat transfer, 548 Rationale for adaptive quadrature, 812 Rayleigh quotient iteration, 442 Reading off complexity, 46 Recursive LU-factorization, 92 Reducing fill-in by reordering, 175 reduction to periodic convolution, 552 Refined local stepsize control, 891 secant method, 289 Seeing linear convergence, 244 Sensitivity of linear mappings, 190 Shape preservation, 730 Shifted inverse iteration, 440 Simple adaptive stepsize control, 881 Simple adaptive timestepping for fast decay, 906 Simple composite polynomial quadrature rules, 773 Simple preconditioners, 376 Simplified Newton method, 307 Small residuals by Gaussian elimination, 126 Solving LSE in the case of rank-1-modification, 218 Sound filtering by DFT, 576 Sparse LU-factors, 154 Ôº ½¿º Spectrum of Fourier matrix, 564 Speed of convergence of explicit Euler method, 850 Ôº¼ ½¿º spline auxiliary construction for shape preserving quadratic interpolation, 739
Page 1 and 2:
2 Direct Methods for Linear Systems
Page 3 and 4:
III Integration of Ordinary Differe
Page 5 and 6:
Extra questions for course evaluati
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,
Page 55 and 56:
Recall: e i ˆ= i-th unit vector Ch
Page 57 and 58:
Computation of Choleskyfactorizatio
Page 59 and 60:
1 ∃ (partial) cyclic row permutat
Page 61 and 62:
Definition 3.1.3 (Local and global
Page 63 and 64:
Example 3.1.6 (quadratic convergenc
Page 65 and 66:
k |x (k) − π| L 1−L |x (k) −
Page 67 and 68:
(x (k) ) k∈N0 Cauchy sequence ➤
Page 69 and 70:
Termination criterion for contracti
Page 71 and 72:
Given x (k) ∈ I, next iterate :=
Page 73 and 74:
secant method ( MATLAB implementati
Page 75 and 76:
Assuming p = 1: p > 1: ∥ C ∥e (
Page 77 and 78:
This is a simple computation: DG(x)
Page 79 and 80:
k x (k) ǫ k := ‖x ∗ − x (k)
Page 81 and 82:
Code 3.4.14: Damped Newton method (
Page 83 and 84:
MATLAB-CODE: Broyden method (3.4.11
Page 85 and 86:
Algorithm 4.1.3 (Steepest descent).
Page 87 and 88:
Example 4.1.8 (Convergence of gradi
Page 89 and 90:
4.2.1 Krylov spaces Definition 4.2.
Page 91 and 92:
Remark 4.2.3 (A posteriori terminat
Page 93 and 94:
10 figure ; view ([ −45 ,28]) ; m
Page 95 and 96:
Idea: Solve Ax = b approximately in
Page 97 and 98:
eplaced with κ(A) ! 4.4.2 Iteratio
Page 99 and 100:
For circuit of Fig. 55 at angular f
Page 101 and 102:
(Linear) generalized eigenvalue pro
Page 103 and 104:
10 0 10 1 matrix size n d = eig(A)
Page 105 and 106:
0 50 100 150 200 250 300 350 400 45
Page 107 and 108:
1 2 3 k ρ (k) EV ρ (k) EW ρ (k)
Page 109 and 110:
✬ ✩ ✬ ✩ Lemma 5.3.4 (Ncut a
Page 111 and 112:
In other words, roundoff errors may
Page 113 and 114:
Theory: linear convergence of (5.3.
Page 115 and 116:
error in eigenvalue 10 0 10 −2 10
Page 117 and 118:
✬ Residuals r 0 ,...,r m−1 gene
Page 119 and 120:
Algebraic view of the Arnoldi proce
Page 121 and 122:
5.5 Singular Value Decomposition Re
Page 123 and 124:
Illustration: columns = ONB of Im(A
Page 125 and 126:
✬ Theorem 5.5.7 (best low rank ap
Page 127 and 128:
Reassuring: Remark 6.0.4 (Pseudoinv
Page 129 and 130:
Consider the linear least squares p
Page 131 and 132:
Goal: Euclidean distance of y ∈ R
Page 133 and 134:
6.5 Non-linear Least Squares If (6.
Page 135 and 136:
0 2 4 6 8 10 12 14 16 value of ∥
Page 137 and 138:
Definition 7.1.1 (Discrete convolut
Page 139 and 140:
Expand a 0 ,...,a n−1 and b 0 , .
Page 141 and 142:
(7.2.2) is a simple consequence of
Page 143 and 144:
Dominant coefficients of a signal a
Page 145 and 146:
11 c = f f t ( y ) ; 12 13 figure (
Page 147 and 148:
Two-dimensional trigonometric basis
Page 149 and 150:
8 end 9 t1 = min ( t1 , toc ) ; 10
Page 151 and 152:
Step II: for k =: rq + s, 0 ≤ r <
Page 153 and 154:
MATLAB-CODE Sine transform function
Page 155 and 156:
△ Example 7.5.2 (Linear regressio
Page 157 and 158:
[23, Ch. IX] presents the topic fro
Page 159 and 160:
Code 8.1.3: Horner scheme, polynomi
Page 161 and 162:
1.2 equality in (8.2.10) for y := (
Page 163 and 164:
ecursive definition: p i (t) ≡ y
Page 165 and 166:
a 1 = y 1 − a 0 t 1 − t 0 = y 1
Page 167 and 168:
Observations: Strong oscillations o
Page 169 and 170:
−1 −0.8 −0.6 −0.4 −0.2 0
Page 171 and 172:
8.5.3 Chebychev interpolation: comp
Page 173 and 174:
9.1 Shape preserving interpolation
Page 175 and 176:
9.2.2 Piecewise polynomial interpol
Page 177 and 178:
Interpolation of the function: f(x)
Page 179 and 180:
2 % Plot convergence of approximati
Page 181 and 182:
9.4 Splines Definition 9.4.1 (Splin
Page 183 and 184:
➤ Linear system of equations with
Page 185 and 186:
y i+1 t i−1 t i t i+1 y i−1 y i
Page 187 and 188:
35 h= d i f f ( t ) ; 36 d e l t a
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 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: x∗ n y ˆ= discrete periodic conv
show all

Numerical Methods Contents - SAM

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?