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

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18 2. Barycentric representation It is shown in [AA86] that the rank of such a matrix, for n ′ large enough, is related to the degree of the minimal degree interpolating rational function. Explicit conditions are stated for which this minimal degree interpolant is unique (up to a multiplicative constant). Otherwise there follows a parameterization of all minimal and nonminimal interpolating rational functions in the form (2.4). Although the approach in [AA86] is mainly of theoretical importance, it shows an alternative approach to the rational interpolation problem by relaxing the degree requirements ℓ + m = n. We come back to this way of formulation the rational interpolation problem in Section 5.2. 2.2 Unattainability and non-normality An advantage of the barycentric representation (2.1) is that it exposes common factors of the form (x − xi), xi ∈ Xn directly. Multiplication of the numerator and denominator by ℓ(x) reveals that a common zero of the form (x − xi), xi ∈ Xn can only occur if and only if ui = 0. In the case where the weights are determined by the condition ℓ + m = n, even more can be said. Since the weights ui are defined following (2.3), i.e. from the Lagrangian representation of the numerator and denominator polynomials, and because wi = 0, one can immediately derive from (2.3) that ui = 0 if and only if qi = 0. Hence ui = 0 is a necessary condition for an unattainable point. However, it is not a sufficient condition, i.e. the point xi need not be unattainable unless the Lagrangian denominator (2.2) is truly the minimum degree solution. Since all of the aforementioned approaches [SW86, BM97, ZZ02] basically solve a homogeneous system of dimension ν × (ν + 1) where ν depends on n, ℓ and m, they suffer from the same difficulties as explained in Example 1.3.1. That is, they can only guarantee to deliver the minimal degree solution if and only if the corresponding coefficient matrix has full rank, hence if and only if the kernel of the homogeneous system is one-dimensional. In the degenerate case of a multidimensional kernel, the denominator is not guaranteed to be minimal anymore and the obtained rational function may contain redundant common factors of the form (x − xi), without indicating unattainability. In order to find a minimal, unique solution again, Berrut et al. [BM97] propose to modify the original problem by simultaneously increasing the numerator degree and decreasing the denominator degree until the coefficient matrix has full rank again. This last approach clearly solves a problem which is different from the original one.
2.3. Pole free rational interpolation 19 2.3 Pole free rational interpolation Because the interpolation property is fulfilled for arbitrary nonzero weights, other conditions [BBN99, BM00, AA86] can be imposed on an interpolant in barycentric form. As alternative for the condition ℓ + m = n, we already mentioned the approach of [AA86]. Another condition can be a pole free approximation. Real poles are both the blessing and the curse of rational interpolation. It may happen that unwanted real poles creep into the interpolation interval and ruin the approximation. The barycentric formula allows for a simple characterization of the poles. Throughout this Section, it is assumed that the elements of the set Xn are indexed such that a = x0 < x1 < ... < xn = b. (2.6) Proposition 2.3.1 ([SW86, BM97]). The barycentric weights ui of an irreducible rational function r(x) = p(x)/q(x) which interpolates in x0,...,xn and which has no poles in the interval [a,b] oscillate in sign. Proof. From the Lagrangian representation of p(x) and q(x) follows that ui = q(xi)wi (i = 0,... ,n). Since the wi oscillate in sign for the supposed ordering of the xi (see for example [Sti63, p.218]) and q(x) does not change sign in [a,b], also the ui oscillate in sign. An attractive pole free rational interpolation scheme is proposed by Floater and Horman [FH07]. They construct an interpolant by blending polynomials in the following way. For d (0 ≤ d ≤ n) fixed, let pi(x) (i = 0,... ,n − d) be the polynomial of degree ∂pi ≤ d that interpolates in xi,... ,xi+d. The interpolant considered in [FH07] is where λi(x) = n−d λi(x)pi(x) i=0 n−d λi(x) i=0 (−1) i (x − xi) · · · (x − xi+d) . , (2.7) The interpolant (2.7) has some very attractive properties. It is shown in [FH07] that for every d (0 ≤ d ≤ n), the interpolant (2.7) has no real poles and for d ≥ 1 it has approximation order O(h d+1 ) as h → 0, where h = max 0≤i≤n−1 (xi+1 − xi)
Page 1: Faculteit Wetenschappen Departement
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Page 13 and 14: Introduction In this first part the
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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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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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