Wave Propagation in Linear Media | re-examined

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6Towards a quadrature routine the best results that are in uenced chie y by the implementation details and the associated numerical e ects. This region is characterised by extrapolation errors that are constant for ne + ns = const and thus depend on the total number of sequence terms regardless of how many are actually used for extrapolation. The contour plot ( g. 6.6) also shows that it is di cult to decide where to begin the extrapolation if not too many terms are to be used. For 40 terms, for example, the optimum lies somewhere at ne = 15, ns = 25. The results we obtain when choosing the -algorithm with WynnDegree->2 are shown in g. 6.7. The inconvenient peaks have disappeared (which is the same for all even values of WynnDegree, but not for the odd ones), yet it is hard to nd optimum values for ne and ns. The setting WynnDegree->Infinity yields the best extrapolation results ( g. 6.8). It is still true that there are pairs of sequences that give the same estimates for the limit S e(2m+1) n and eS (2m+2) n+1 , respectively, which can be seen from the bands running in parallel to the ns-axis. The respective contour plot (6.9) shows that the best way to improve the extrapolation result is to increase ne, perhaps with a few terms summed up explicitly. We have thus con rmed the suspicion that the limiting process 1 from the last section is indeed superior. For comparison we compute the extrapolation error if we do not use approximate partition, but take the zeros of the integrand as partition points. Just the determination of the partial integrals is di erent. The rest of the evaluation remains the same as with the asymptotic partition before. In[25]:= a = 20 Pi; firstval = N[Integrate[Sin[Sqrt[x^2 - a^2]]/(x^2 - a^2) x, fx,a,Sqrt[Pi^2+a^2]g],40]; seq = Table[NIntegrate[Sin[Sqrt[x^2 - a^2]]/(x^2 - a^2) x, fx,Sqrt[(i Pi)^2+a^2],Sqrt[((i+1) Pi)^2+a^2]g], fi,1,300g]; We nd that the results are far better and practically independent of the value we choose for WynnDegree. We thus conclude that when implementing an automatic quadrature routine with Mathematica, we ought to consider the following points: The setting WynnDegree->Infinity gives the best results for asymptotic partition and reasonably good ones for standard (zero) partition, so it is a suitable choice for a default value of this option. To increase the accuracy of the extrapolation result, the number of terms included in the extrapolation (NSumExtraTerms) should be increased, whereas NSumTerms may be left unchanged. Whenever feasible, the zeros of the integrand should be chosen as the partition points. This yields more accurate results with fewer terms involved. 152
6.5 Considerations for a Mathematica implementation 0 -5 -10 -15 0 20 ne 40 Figure 6.5: Negative number of correct digits, log e S (k) n , S , for varying n (ns) and k (ne), computed with the 2 algorithm (WynnDegree->1). 60 50 40 30 20 10 0 0 20 10 20 30 40 50 60 Figure 6.6: Contour plot of g. 6.5, abscissa: ne, ordinate: ns. 153 40 ns 60
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DISSERTATION Wave Propagation in Li
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Kurzfassung Seit der Entdeckung des
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Nullum est iam dictum, quod non sit
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Preface Our popular writers and rep
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the quest for superluminality and t
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Contents Part I Wave propagation ph
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7.3.4 PartitionPoints . . . . . . .
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Part I Wave propagation phenomena S
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1.1 Phase and group velocity 1.1 Ph
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1.1 Phase and group velocity ! !c v
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1.2 A few notes on dispersion He co
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1.2 A few notes on dispersion v/c 6
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1.3 Signal velocity dipoles with a
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1.3 Signal velocity The arbitrarine
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1.4 Energy velocity For electromagn
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1.5 Other velocity de nitions For n
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1.5 Other velocity de nitions evide
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2.1 Superluminal wave propagation 2
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2.1 Superluminal wave propagation a
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2.1 Superluminal wave propagation t
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2.2 Quantum mechanical tunnelling e
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2.2 Quantum mechanical tunnelling R
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Chapter 3 Wave propagation in elect
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3.1 Model of a transmission line th
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3.2 Excursion: a delay line section
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3.3 Re ection due to termination mi
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3.4 A simple thought experiment I0
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3.5 A dispersive system: the lossle
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3.6 Inhomogeneous transmission line
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3.7 Turn-on e ects in a lossless pl
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3.8 Turn-on e ects in a wave guide
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3.8 Turn-on e ects in a wave guide
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3.9 A Gaussian pulse in plasma Like
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3.9 A Gaussian pulse in plasma 250
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3.9 A Gaussian pulse in plasma 250
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3.9 A Gaussian pulse in plasma Note
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4.1 The potential step 4.1 The pote
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4.1 The potential step Inside the b
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4.2 Initial wave forms -60 -50 -40
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4.2 Initial wave forms -60 -50 -40
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4.3 Examples of scattering processe
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4.4 The square barrier they have va
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4.4 The square barrier 1 0.8 0.6 0.
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4.5 Tunnelling time de nitions for
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4.5 Tunnelling time de nitions for
Page 117 and 118: 4.6 Examples of tunnelling events P
Page 119 and 120: 4.6 Examples of tunnelling events 2
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Page 123 and 124: 4.6 Examples of tunnelling events t
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Page 135 and 136: Interlude Wave functions in graphic
Page 137 and 138: Wave functions in graphical represe
Page 139 and 140: Part II Numerical aspects of wave e
Page 141 and 142: 5.1 Univariate numerical quadrature
Page 143 and 144: 5.1 Univariate numerical quadrature
Page 145 and 146: 5.2 Convergence acceleration one, t
Page 147 and 148: 5.2 Convergence acceleration (1) ,1
Page 149 and 150: 6.1 Partitioning the integration in
Page 155 and 156: 6.2 Choosing the rst partition poin
Page 157 and 158: 6.3 How to compute the rst integral
Page 159 and 160: 6.3 How to compute the rst integral
Page 161 and 162: 6.4 Asymptotic partition consuming
Page 163 and 164: 6.4 Asymptotic partition of the int
Page 165 and 166: 6.5 Considerations for a Mathematic
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Page 171 and 172: 6.6 Controlling the accuracy of the
Page 177 and 178: Chapter 7 Mathematica implementatio
Page 179 and 180: 7.1 User interface of the function
Page 181 and 182: 7.3 Implementation of OscInt and re
Page 189 and 190: 7.4 Auxiliary functions While[itera
Page 191 and 192: 7.4 Auxiliary functions Linear appr
Page 193 and 194: 7.4 Auxiliary functions For the sak
Page 195 and 196: 7.4 Auxiliary functions Out[8]= 3 f
Page 197 and 198: 7.5 Test of the quadrature routine
Page 207 and 208: 8.1 Preparation of the wave integra
Page 213 and 214: 8.2 Outline of the program structur
Page 215 and 216: 8.2 Outline of the program structur
Page 217 and 218: 8.3 Implementation of the quadratur
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8.3 Implementation of the quadratur
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8.4 Test of the package 8.4 Test of
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8.4 Test of the package -200 -150 -
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8.4 Test of the package 0.4 0.2 -0.
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8.4 Test of the package 8.4.2 Forma
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8.4 Test of the package Out[24]= 0
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A.1 Numerical quadrature Asymptotic
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A.1 Numerical quadrature WynnDegree
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A.1 Numerical quadrature Which[ft =
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A.2 Solutions for the step potentia
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A.3 Solutions for the square barrie
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A.3 Solutions for the square barrie
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A.4 Electromagnetic waves function
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A.5 Utilities for displaying points
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Bibliography Bibliography [1] James
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Bibliography [29] Kurt Edmund Oughs
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Bibliography [59] Ch. Spielmann, R.
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Bibliography [91] C. R. Leavens and
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Bibliography [123] T. O. Espelid an
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Index Index absorption, 7, 9 accura
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Index saddle point integration, 11,
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Curriculum vitae Dipl.-Ing. Thilo S
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