Numerical Methods Contents - SAM

More documents

Recommendations

Info

Heuristics: the timestep h is small ➥ “higher order terms” O(h p+2 ) can be ignored. ✎ Ψ h ky(t k ) − Φ h . ky(t k ) = ch p+1 k + O(h p+2 k ) , ˜Ψ hk y(t k ) − Φ h . ky(t k ) = O(h p+2 k ) . . notation: = equality up to higher order terms in h k ⇒ EST k . = ch p+1 k . (11.5.6) • line 9: compute presumably better local stepsize according to (11.5.9), • line 10: decide whether to repeat the step or advance, • line 11: extend output arrays if current step has not been rejected. . EST k = ch p+1 k ⇒ c = . EST k h p+1 . (11.5.7) k △ Available in algorithm, see (11.5.3) Remark 11.5.11 (Stepsize control in MATLAB). For the sake of accuracy (stipulates “EST k < TOL”) & efficiency (favors “>”) we aim for EST k ! = TOL := max{ATOL, ‖y k ‖RTOL} . (11.5.8) Ψ ˆ= RK-method of order 4 ˜Ψ ˆ= RK-method of order 5 ode45 What timestep h ∗ can actually achieve (11.5.8), if we “believe” in (11.5.6) (and, therefore, in (11.5.7))? "‘Optimal timestep”: (stepsize prediction) (11.5.7) & (11.5.8) ⇒ TOL = EST k h p+1 k h p+1 ∗ . h ∗ = h p+1 √ TOL EST k . (11.5.9) (A): In caseEST k > TOL ➣ repeat step with stepsize h ∗ . adjusted stepsize (A) suggested stepsize (B) Ôº¿ ½½º Specifying tolerances for MATLAB’s integrators: options = odeset(’abstol’,atol,’reltol’,rtol,’stats’,’on’); [t,y] = ode45(@(t,x) f(t,x),tspan,y0,options); (f = function handle, tspan ˆ= [t 0 , T], y0 ˆ= y 0 , t ˆ= t k , y ˆ= y k ) Example 11.5.12 (Adaptive timestepping for mechanical problem). △ Ôº ½½º (B): IfEST k ≤ TOL ➣ use h ∗ as stepsize for next step. Movement of a point mass in a conservative force field: t ↦→ y(t) ∈ R 2 ˆ= trajectory Code 11.5.10: refined local stepsize control for single step methods 1 function [ t , y ] = o d e i n t s s c t r l ( Psilow , p , Psihigh , T , y0 , h0 , r e l t o l , abstol , hmin ) 2 t = 0 ; y = y0 ; h = h0 ; % 3 while ( ( t ( end ) < T ) ( h > hmin ) ) % 4 yh = Psihigh ( h , y0 ) ; % 5 yH = Psilow ( h , y0 ) ; % 6 est = norm( yH−yh ) ; % 7 8 t o l = max( r e l t o l ∗norm( y ( : , end ) ) , a b s t o l ) ; % 9 h = h∗max( 0 . 5 , min ( 2 , ( t o l / est ) ^ ( 1 / ( p+1) ) ) ) ; % 10 i f ( est < t o l ) % 11 y0 = yh ; y = [ y , y0 ] ; t = [ t , t ( end ) + min ( T−t ( end ) , h ) ] ; % 12 end 13 end Newton’s law: acceleration ÿ = F(y) := − 2y ‖y‖ 2 . (11.5.10) 2 Equivalent 1st-order ODE, see Rem. 11.1.7: with velocity v := ẏ (ẏ ) ( ) v = ˙v − 2y . (11.5.11) ‖y‖ 2 2 Initial values used in the experiment: y(0) := ( ) −1 0 force ( ) 0.1 , v(0) := −0.1 Comments on Code 11.5.9 (see comments on Code 11.5.2 for more explanations): • Input arguments as for Code 11.5.2, except for p ˆ= order of lower order discrete evolution. Ôº ½½º Adaptive integrator: ode45(@(t,x) f,[0 4],[-1;0;0.1;-0.1,],options): ➊ options = odeset(’reltol’,0.001,’abstol’,1e-5); ➋ options = odeset(’reltol’,0.01,’abstol’,1e-3); Ôº ½½º
4 3 2 abstol = 0.000010, reltol = 0.001000 y 1 (t) (exakt) y 2 (t) (exakt) v 1 (t) (exakt) v 2 (t) (exakt) 5 abstol = 0.000010, reltol = 0.001000 0.2 y 1 (t k ) (Naeherung) y 2 (t k ) (Naeherung) v 1 (t k ) (Naeherung) v 2 (t k ) (Naeherung) 0.8 0.6 0.4 0.2 0.8 0.6 0.4 0.2 1 0 −1 −2 −3 −4 0 0.5 1 1.5 2 2.5 3 3.5 4 t y i (t) 0 −5 0 0.5 1 1.5 2 2.5 3 3.5 4 t 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 Zeitschrittweite y 2 0 −0.2 −0.4 −0.6 −0.8 Exakte Bahn Naeherung −1 −1 −0.5 0 0.5 1 y 1 y 2 0 −0.2 −0.4 −0.6 −0.8 Exakte Bahn Naeherung −1 −1 −0.5 0 0.5 1 y 1 reltol=0.001, abstol=1e-5 reltol=0.01, abstol=1e-3 Observations: Ôº ½½º ☞ Fast changes in solution components captured by adaptive approach through very small timesteps. ☞ Completely wrong solution, if tolerance reduced slightly. Ôº ½½º 4 3 2 1 0 −1 −2 −3 abstol = 0.001000, reltol = 0.010000 y 1 (t) (exakt) y 2 (t) (exakt) v 1 (t) (exakt) v 2 (t) (exakt) −4 0 0.5 1 1.5 2 2.5 3 3.5 4 t y i (t) 5 0 abstol = 0.001000, reltol = 0.010000 0.2 y (t ) (Naeherung) 1 k y (t ) (Naeherung) 2 k v (t ) (Naeherung) 1 k v (t ) (Naeherung) 2 k −5 0 0.5 1 1.5 2 2.5 3 3.5 4 t 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 Zeitschrittweite An inevitable consequence of time-local error estimation: ✗ ✖ Absolute/relative tolerances do not allow to predict accuracy of solution! ✔ ✕ ✸ Ôº ½½º Ôº¼¼ ½½º
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: 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
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 and 244: x∗ n y ˆ= discrete periodic conv
Page 245 and 246: Gaussian elimination with pivoting
show all

Numerical Methods Contents - SAM

Create successful ePaper yourself

Delete template?

Save as template?