Practical Rational Interpolation of Exact and Inexact Data Theory ...

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4 Introduction ues, but excluding the point at infinity, which is added in Chapter 4. Such a treatment is crucial for understanding the different aspects and possible degeneracies that are connected with the classical rational interpolation problem. Except for the straightforward method, no algorithm is given. In Chapter 2 a specific representation for interpolating rational functions is considered: the barycentric form. The barycentric form has a remarkable property which makes it attractive for numerical implementation, namely exact interpolation even in the presence of rounding errors. We review how classical rational interpolation fits into this representation and also present some other common uses of the barycentric formula. Then the numerical stability of the barycentric formula is considered. It turns out that, due to catastrophic cancellation, the evaluation of the barycentric form may give very inaccurate results when evaluated outside the interpolation interval. For the incorporation of asymptotic behavior (interpolation at infinity), this is unacceptable. For that reason we abandon the barycentric form of the rational interpolant and turn to another representation in Chapter 3 and consider interpolating continued fractions. First, the construction of a Thiele continued fraction is given. Such a continued fraction is the basis for a more advanced algorithm due to Werner. Werner’s algorithm constructs a more general form of a Thiele continued fraction and also deals with degenerate situations. It is one of the few algorithms for rational interpolation for which a stability analysis exists. If a special pivoting strategy is incorporated, then it has even been proven to be backward stable [GM80]. Therefore it is a suitable algorithm for numerical implementation. With the pivoting strategy, it also allows for poles (vertical asymptotes) to be prescribed, but in its standard form, interpolation at infinity (including horizontal and oblique asymptotes) is not supported. In Chapter 4 the interpolation condition at infinity is added. First a condition to detect degeneracy for the point at infinity is given if the points are ordered such that all finite data come first and the point at infinity is last. For this ordering of the data, we show how the algorithm of Werner can be modified to interpolate also at infinity. An illustration of the usefulness of interpolation at infinity is given. In the last Chapter, we review other algorithms for rational interpolation which are more or less related to Werner’s algorithm. The more practical algorithms are most useful in a symbolic environment rather than a numerical one. The results in this part have been presented partially in [SCLV05, SCV08].
Classical rational interpolation 1 In this first Chapter we start by introducing some well-known theoretical results on rational interpolation, excluding the point at infinity, which is added in Chapter 4, but allowing infinite function values. Such a treatment is crucial for understanding the different aspects that are connected with the rational interpolation problem. We do not treat the problem in the same generality as in [VBB92, Gut93a, Gut93b], where also derivatives at interpolation points are taken into account. 1.1 Problem statement Let Xn = {x0,... ,xn} be a set of n + 1 distinct complex numbers and f(x) be a given function. We allow f(x) to be unbounded in some points, such that poles can be prescribed. Let p(x) and q(x) be polynomials of degree at most ℓ and m respectively, with n + 1 = ℓ + m + 1 and such that the rational function r(x) = p(x)/q(x) is irreducible. The rational interpolation problem consists in finding r(x) for which the interpolation conditions r(xi) = p(xi) q(xi) = f(xi) = fi i = 0,... ,n (1.1) are satisfied. Define the following set of polynomial couples R(ℓ,m) = {(p(x),q(x)) |∂p ≤ ℓ and ∂q ≤ m}. A non-trivial polynomial couple (p(x),q(x)) ∈ R(ℓ,m) satisfying the linearized interpolation conditions fiq(xi) − p(xi) = 0 if f(xi) is finite q(xi) = 0 if f(xi) is ∞ i = 0,... ,n. (1.2) 5
Page 1: Faculteit Wetenschappen Departement
Page 4 and 5: iv Contents 5 Related algorithms 57
Page 6 and 7: vi Nederlandstalige samenvatting ri
Page 9: Acknowledgments “ It is, at the e
Page 13: Introduction In this first part the
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Page 41 and 42: Interpolating continued fractions 3
Page 43 and 44: 3.1. Thiele continued fractions 33
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54 4. Asymptotic behavior of Propos
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56 4. Asymptotic behavior Note that
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58 5. Related algorithms Algorithm
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60 5. Related algorithms In compari
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62 5. Related algorithms Algorithm
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64 5. Related algorithms or in full
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66 5. Related algorithms In the gen
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68 5. Related algorithms Example 5.
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Introduction In this second part, w
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Rational interpolation of vertical
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6.2. Linearization 75 A problem sim
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6.2. Linearization 77 Lℓ,m(Sn) =
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6.3. Solution of the existence and
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6.3. Solution of the existence and
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6.4. Algorithmic aspects 83 The fol
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6.5. Conclusion 85 Uλ ≥ 0 is sma
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88 7. Benchmarks 80 60 40 20 0 −1
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90 7. Benchmarks | f(x,y) − r l,m
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92 7. Benchmarks • In the previou
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94 7. Benchmarks are respectively g
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96 7. Benchmarks prone optimization
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98 8. Case study: multidimensional
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112 9. Conclusions and further rese
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114 9. Conclusions and further rese
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116 A. Software Cfrac Continued fra
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118 A. Software A.2 Rational interp
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120 A. Software As before, together
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122 A. Software 1.5 1 0.5 0 −0.5
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124 A. Software preference labels e
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126 A. Software 000 3 2 1 001 010 1
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128 A. Software 2 1 0 0 1 2 0 0 0 3
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130 B. Filter Coefficients Table B.
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132 B. Filter Coefficients Table B.
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134 B. Filter Coefficients (k1, k2)
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136 B. Filter Coefficients (k1, k2)
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138 Bibliography [BGM96] , Padé ap
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140 Bibliography Tucker, eds.), Ann
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142 Bibliography [SCLV05] O. Salaza
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