26 lines
1.8 KiB
Text
26 lines
1.8 KiB
Text
{{Wikipedia|Thiele's interpolation formula}}
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'''[[wp:Thiele's_interpolation_formula|Thiele's interpolation formula]]''' is an interpolation formula for a function ''f''(•) of a single variable. It is expressed as a [[continued fraction]]:
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:: <big><big><math> f(x) = f(x_1) + \cfrac{x-x_1}{\rho_1(x_1,x_2) + \cfrac{x-x_2}{\rho_2(x_1,x_2,x_3) - f(x_1) + \cfrac{x-x_3}{\rho_3(x_1,x_2,x_3,x_4) - \rho_1(x_1,x_2) + \cdots}}} </math></big></big>
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<big><big><math>\rho</math></big></big> represents the [[wp:reciprocal difference|reciprocal difference]], demonstrated here for reference:
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:: <big><big><math>\rho_1(x_0, x_1) = \frac{x_0 - x_1}{f(x_0) - f(x_1)}</math></big></big>
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:: <big><big><math>\rho_2(x_0, x_1, x_2) = \frac{x_0 - x_2}{\rho_1(x_0, x_1) - \rho_1(x_1, x_2)} + f(x_1)</math></big></big>
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:: <big><big><math>\rho_n(x_0,x_1,\ldots,x_n)=\frac{x_0-x_n}{\rho_{n-1}(x_0,x_1,\ldots,x_{n-1})-\rho_{n-1}(x_1,x_2,\ldots,x_n)}+\rho_{n-2}(x_1,\ldots,x_{n-1})</math></big></big>
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Demonstrate Thiele's interpolation function by:
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# Building a '''32''' row ''trig table'' of values for <big><big><math> x </math></big></big> from '''0''' by '''0.05''' to '''1.55''' of the trig functions:
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#* '''sin'''
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#* '''cos'''
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#* '''tan'''
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# Using columns from this table define an inverse - using Thiele's interpolation - for each trig function;
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# Finally: demonstrate the following well known trigonometric identities:
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#* <big><big> 6 × sin<sup>-1</sup> ½ = <math>\pi</math></big></big>
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#* <big><big> 3 × cos<sup>-1</sup> ½ = <math>\pi</math></big></big>
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#* <big><big> 4 × tan<sup>-1</sup> 1 = <math>\pi</math></big></big>
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<br><br>
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