Wave Propagation in Linear Media | re-examined

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5 Numerical quadrature and extrapolation Apart from linear transformations (like the Euler transformation [139]), nonlinear transformations are widely used for convergence acceleration. One of them is Aitken's 2 process with the transformation rule [113, 137] Tn = SnSn+2 , S 2 n+1 Sn+2 , 2 Sn+1 + Sn (Sn+1 , Sn) 2 = Sn , ; (5.7) Sn+2 , 2 Sn+1 + Sn the second expression being preferable for implementation due to its better numerical stability. This transformation is known to accelerate sequences with linear convergence, that is [138, 139], Sn+1 , S lim n!1 Sn , S and is exact (i. e. Tn = S 8n) for sequences of the form = ; 62 f1; 0g (5.8) Sn = S + a n ; a 6= 0; j j
5.2 Convergence acceleration (1) ,1 =0 (2) ,1 =0 (3) ,1 =0 (4) ,1 =0 (5) ,1 =0 . (0) 0 (1) 0 (2) 0 (3) 0 (4) 0 = S0 = S1 = S2 = S3 = S4 . (0) 1 (1) 1 (2) 1 (3) 1 . (0) 2 = S e(2) 0 (1) 2 = S e(2) 1 (2) 2 = S e(2) 2 . (0) 3 (1) 3 . (0) 4 = S e(4) 0 Figure 5.1: The -array. Only the values (n) 2k in the even columns are valid estimates for the sequence limit S. then the -algorithm gives the exact result (0) 2p of the original sequence are required. . . .. = S [113], which means that 2p + 1 elements In any case, a sequence transformation may be seen as a functional relation between a member of the transformed sequence and a certain subset of the original sequence. Since our goal is to obtain a better estimate for the limit of the sequence, we normally desire only one result from the function and not a transformed sequence. Thus we can write Tn = e S (k) n = F (Sn;Sn+1 :::Sn+k) ; (5.13) with k +1 being the number of sequence members involved in the transformation and n the starting index. If the original sequence comprised N members, then we would typically compute S e(k) N,k or S e(N) 0 as the (hopefully) best approximation to the exact limit S. 131
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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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Page 97 and 98: 4.3 Examples of scattering processe
Page 109 and 110: 4.4 The square barrier they have va
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Page 113 and 114: 4.5 Tunnelling time de nitions for
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Page 117 and 118: 4.6 Examples of tunnelling events P
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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: 5.2 Convergence acceleration one, t
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
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
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7.5 Test of the quadrature routine
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8.1 Preparation of the wave integra
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8.2 Outline of the program structur
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8.2 Outline of the program structur
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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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