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

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118 A. Software A.2 Rational interpolation of vertical segments For the technique presented in Part II, in particular Chapter 6, a dimension free Matlab implementation is given. The reference documentation for the function vertsegmapprox.m is given below. VERTSEGMAPPROX Finds rational function(s) that intersect vertical segments. [N, M, Lambda, Exitflag, Output]=vertsegmapprox(X,Fl,Fh,Opts) returns the simplest rational function from R^n ro R with numerator degree set N, denominator degree set M and coefficients Lambda such that it intersects the vertical segments (X,[Fl, Fh]). - X must be a [k x n] matrix containing data [ x11,x12, ...,x1n x21,x22, ...,x2n x31,x32, ...,x3n . . ... . . . ... . . . ... . xk1,xk2,...,xkn] - Fl (resp. Fh) must be a [n x 1] column vector corresponding to the lower (resp. upper) bounds of the segments [Fl, Fh] such that Fl < Fh Opts is an optional structure that defines algorithm behavior. If Opts is empty or not given, default Options are used. Opts = vertsegmapprox(’Name’,Value,...) set values in Options structure based on the Name/Value pairs: Name Values{default} Description ’Display’ {’off’} No Display of Iteration Info ’iter’ Display each Iteration ’MaxDiag’ {100} Maximum Diagonal to explore ’StartDiag’ {0} Starting Diagonal ’Basisfun’ {’monom’} Monomial basisfunctions
A.2. Rational interpolation of vertical segments 119 function_handle function handle to custom basisfunction of the form mybasefunc = @(n1,x1) n1 must be an integer, x1 must be a real column vector (not a row vector) e.g. monom = @(n1,x1) x1.^n1; ’Polefree’ {’off’} Free poles. ’on’ Incorporates extra constraints for a guaranteed polefree approximation. Make sure that the domain of the basisfunctions is the interval[-1,1]. Opts = vertsegmapprox(Opts,’Name’,Value,...) updates the Options structure with new parameter values. N = Final Numerator Degrees Found D = Final Denominator Degrees Found Lambda = Final Coefficients Found ExitFlag = Description of algorithm success 1 normal return -1 no solution found up until MaxDiag (incr.MaxDiag) -2 no solution found There Must Not be a Unique Rational Function that intersects The Given Segments. In Case There Are More Solution, All Solutions are Returned in Output Output = Structure containing extra algorithm information Output.N Container with all Numerator Degrees Output.D Container with all Denominator Degrees Output.lambda Container with all Coefficients Output.rad Container with all radii IMPORTANT NOTE: vertsegmapprox uses the function QUADPROG to solve the quadratic programming problem. Make sure that QUADPROG can be found. See also RATEVAL.
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Faculteit Wetenschappen Departement
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iv Contents 5 Related algorithms 57
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vi Nederlandstalige samenvatting ri
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Acknowledgments “ It is, at the e
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Introduction In this first part the
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Classical rational interpolation 1
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1.3. Non-normality and block struct
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1.4. Discussion 13 Proof. For rℓ,
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16 2. Barycentric representation Th
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18 2. Barycentric representation It
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20 2. Barycentric representation as
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22 2. Barycentric representation If
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24 2. Barycentric representation Pr
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26 2. Barycentric representation an
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28 2. Barycentric representation 6
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Interpolating continued fractions 3
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3.1. Thiele continued fractions 33
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3.1. Thiele continued fractions 35
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3.2. Werner’s algorithm 37 Algori
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3.4. Non-normality 39 rj = 1, then
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3.4. Non-normality 41 For i odd, we
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3.5. Practical aspects and numerica
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3.5. Practical aspects and numerica
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48 4. Asymptotic behavior In the se
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50 4. Asymptotic behavior the denom
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52 4. Asymptotic behavior Since Bj(
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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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Page 81 and 82: Introduction In this second part, w
Page 83 and 84: Rational interpolation of vertical
Page 85 and 86: 6.2. Linearization 75 A problem sim
Page 87 and 88: 6.2. Linearization 77 Lℓ,m(Sn) =
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Page 98 and 99: 88 7. Benchmarks 80 60 40 20 0 −1
Page 100 and 101: 90 7. Benchmarks | f(x,y) − r l,m
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Page 106 and 107: 96 7. Benchmarks prone optimization
Page 108 and 109: 98 8. Case study: multidimensional
Page 122 and 123: 112 9. Conclusions and further rese
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Page 126 and 127: 116 A. Software Cfrac Continued fra
Page 130 and 131: 120 A. Software As before, together
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Page 138 and 139: 128 A. Software 2 1 0 0 1 2 0 0 0 3
Page 140 and 141: 130 B. Filter Coefficients Table B.
Page 142 and 143: 132 B. Filter Coefficients Table B.
Page 144 and 145: 134 B. Filter Coefficients (k1, k2)
Page 146 and 147: 136 B. Filter Coefficients (k1, k2)
Page 148 and 149: 138 Bibliography [BGM96] , Padé ap
Page 150 and 151: 140 Bibliography Tucker, eds.), Ann
Page 152 and 153: 142 Bibliography [SCLV05] O. Salaza
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Practical Rational Interpolation of Exact and Inexact Data Theory ...

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