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

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24 2. Barycentric representation Proof. From 1 − ˛ n ˛ ˛i=0 1 n ui (1 + θ x − xi i n+2 )(1 + δui ) i=0 = n ui + x − xi i=0 1 n ui (θ x − xi i n+2 + δui + θi n+2δui ) i=0 = n ui x − xi 1 + i=0 1 ≤ n ui x − xi i=0 1 − 1 n ui (θ x − xi i n+2 + δui + θi n+2δui ) n ui x − xi i=0 ˛ n ˛ ˛i=0 i=0 1 ui (θ x − xi i n+2 + δui + θi ˛ ˛ ˛ ˛ n+2δui ) ˛ ˛ ˛ ˛ ˛ ˛ n ˛ ˛ ui ˛ ˛ ˛ ˛ ˛ x − xi ˛ ˛i=0 1 ui (θ x − xi i n+2 + δui + θi ˛ ˛ ˛ ˛ n+2δui ) ˛ ˛ ˛ ˛ ˛ ˛ n ˛ ˛ ui ˛ ˛ ˛ ˛ ˛ x − xi ˛ i=0 ˛ ⎛ n ui ∞ ⎜ (θ ⎜ x − xi = 1 + ⎜ i=0 ⎜ j=1 ⎝ i n+2 + δui + θi ⎞j n+2δui ) ⎟ n ⎟ ⎟ , ui ⎠ x − xi using the upper-bound θ i n+2 ≤ γn+2 = (n + 2)ǫmach 1 − (n + 2)ǫmach = i=0 ˛ ∞ (n + 2) k ǫ k mach k=1
2.4. Numerical stability 25 and with slight abuse of notation |δui | = O(κU ǫmach) = ǫmach O(κU), we also have that n ui (θ x − xi i=0 i n+2 + δui + θi n ui n+2δui ) x − xi i=0 n ≤ ǫmach (n + 2 + O(κU)) n ui ui x − xi x − xi i=0 i=0 + O(ǫ 2 mach ). Collecting appropriate terms in ǫmach and O(ǫ2 mach ) finishes the proof. The main result of this Section is the following. Proposition 2.4.3. The relative forward error for the computed value rn(x) of (2.1) satisfies n ui fi |rn(x) − rn(x)| x − xi i=0 ≤ ǫmach (n + 4 + O(κU)) |rn(x)| n ui fi x − xi i=0 n ui x − xi i=0 + ǫmach (n + 2 + O(κU)) n + O(ǫ ui x − xi 2 mach ) (2.10) Proof. We have rn(x) rn(x) ≤ n ui fi(1 + θ x − xi i=0 i n ui n+4 )(1 + δui ) x − xi i=0 n ui (1 + θ x − xi i n . ui n+2 )(1 + δui ) fi x − xi i=0 Since n ui fi(1 + θ x − xi i=0 i n+4 )(1 + δui ) ≤ n ui fi x − xi i=0 n ui + ǫmach (n + 4 + O(κU)) x − xi i=0 i=0 i=0 fi + O(ǫ2mach )
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 and 14: Introduction In this first part the
Page 15 and 16: Classical rational interpolation 1
Page 17 and 18: 1.3. Non-normality and block struct
Page 23: 1.4. Discussion 13 Proof. For rℓ,
Page 26 and 27: 16 2. Barycentric representation Th
Page 28 and 29: 18 2. Barycentric representation It
Page 30 and 31: 20 2. Barycentric representation as
Page 32 and 33: 22 2. Barycentric representation If
Page 36 and 37: 26 2. Barycentric representation an
Page 38 and 39: 28 2. Barycentric representation 6
Page 41 and 42: Interpolating continued fractions 3
Page 43 and 44: 3.1. Thiele continued fractions 33
Page 45 and 46: 3.1. Thiele continued fractions 35
Page 47 and 48: 3.2. Werner’s algorithm 37 Algori
Page 49 and 50: 3.4. Non-normality 39 rj = 1, then
Page 51 and 52: 3.4. Non-normality 41 For i odd, we
Page 53 and 54: 3.5. Practical aspects and numerica
Page 55: 3.5. Practical aspects and numerica
Page 58 and 59: 48 4. Asymptotic behavior In the se
Page 60 and 61: 50 4. Asymptotic behavior the denom
Page 62 and 63: 52 4. Asymptotic behavior Since Bj(
Page 64 and 65: 54 4. Asymptotic behavior of Propos
Page 66 and 67: 56 4. Asymptotic behavior Note that
Page 68 and 69: 58 5. Related algorithms Algorithm
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Page 76 and 77: 66 5. Related algorithms In the gen
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Page 81 and 82: Introduction In this second part, w
Page 83 and 84: 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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