Coefficients for Cohen-Hubbard-Oesterwinter #1733
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eggrobin
approved these changes
Feb 25, 2018
| // Definition of a Cohen-Hubbard-Oesterwinter formula. |order_| is the order of | ||
| // the approximation, that is, the error on the derivative is h^(order + 1). | ||
| // The formula requires |order_| values of the acceleration. numerators[0] | ||
| // corresponds to f"(x₀), numerators[1] to f"(x₋₁), etc. |
| using numerics::FixedVector; | ||
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| // Definition of a Cohen-Hubbard-Oesterwinter formula. |order_| is the order of | ||
| // the approximation, that is, the error on the derivative is h^(order + 1). |
| // the approximation, that is, the error on the derivative is h^(order + 1). | ||
| // The formula requires |order_| values of the acceleration. numerators[0] | ||
| // corresponds to f"(x₀), numerators[1] to f"(x₋₁), etc. | ||
| // TODO(phl): This struct shows up in many places. Factor it out. |
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We have 3 or 4 occurrences of a struct containing a FixedVector for numerators and a single value for the denominator.
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| using numerics::FixedVector; | ||
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| // Definition of a Cohen-Hubbard-Oesterwinter formula. |order_| is the order of |
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Cite the paper, document any notational differences from the paper (I think we're off-by-one compared to it?)
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| (fs[-h,\[ScriptCapitalN]+3]-fs[-2h,\[ScriptCapitalN]+3])/h+h Sum[f2s[-i h,\[ScriptCapitalN]+3]\[Beta][[i]],{i,1,Length[\[Beta]]}]//Expand | ||
| ReleaseHold[Hold[(fs[-h,\[ScriptCapitalN]+3]-fs[-2h,\[ScriptCapitalN]+3])/h+h Sum[f2s[-i h,\[ScriptCapitalN]+3]\[Beta][\[ScriptCapitalN]][[i]],{i,1,Length[\[Beta][\[ScriptCapitalN]]]}]]/.\[ScriptCapitalN]->14]//Expand |
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What is going on with this ReleaseHold and Hold?
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Mathematica magic: (1) Hold prevents the functions like fs from being evaluated at a place (2) N is being substituted in the argument of fs et al. (3) releasing the hold causes the whole thing to evaluate.
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| // Definition of a Cohen-Hubbard-Oesterwinter formula. |order_| is the order of | ||
| // the approximation, that is, the error on the derivative is h^(order + 1). | ||
| // The formula requires |order_| values of the acceleration. numerators[0] |
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Document what the formula is, and what it approximates.
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and a fix for the order of the backward difference method.