Thiele's interpolation formula: Difference between revisions
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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]]: |
'''[[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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:<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><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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:<math>\rho_1(x_0, x_1) = \frac{x_0 - x_1}{f(x_0) - f(x_1)}</math> |
:: <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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:<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><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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:<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><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: |
Demonstrate Thiele's interpolation function by: |
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# Building a 32 row ''trig table'' of values |
# 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; |
# 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: |
# Finally: demonstrate the following well known trigonometric identities: |
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#* 6 × sin<sup>-1</sup> ½ = |
#* <big><big> 6 × sin<sup>-1</sup> ½ = <math> \pi </math> </big></big> |
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#* 3 × cos<sup>-1</sup> ½ = |
#* <big><big> 3 × cos<sup>-1</sup> ½ = <math> \pi </math> </big></big> |
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#* 4 × tan<sup>-1</sup> 1 = |
#* <big><big> 4 × tan<sup>-1</sup> 1 = <math> \pi </math> </big></big> |
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