AB adaptive #2
AB adaptive #2
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Avoid using `start` as a variable
| for j = 0:s | ||
| # Assign in correct order for multiplication below | ||
| # (a factor depending on j and s) .* (an integral of a polynomial with -(-1:s-1), except -(j-1), as roots) | ||
| p_int = polyint(poly(diagm(-[-1:j - 2; j:s-1]))) |
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the diagm is actually not needed (I think).
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So, I was working on dense output, and the problem that I am currently running into the polynomial interpolation away from the step points is not too accurate. I could change the stepsize so the value at each time in |
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? My understanding was that you get a interpolation polynomial out which should be accurate anywhere within its support to the specified order. Isn't this polynomial used for the integration on which the method is based. If it wasn't accurate inbetween, wouldn't that mean that the method doesn't work either? Maybe I need to re-read, can you point me to the relevant sections? |
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Yeah, on a second pass over, it may not be this. The reason why I started to think in this direction is because the basic interpolation is akin to an explicit method, with no Correction-Evaluation step. I was wondering if this was enough to introduce extra error, however, I'll still testing around. |
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I don't think so. And even if, at most this would be reducing the order by 1, probably not enough to notice easily. |
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Yeah, after running some graphs, I can tell it is a coding issue. I just need to rework the dense output code. |
| b_imp = am_imp_coefficients3 | ||
| b_exp = ms_coefficients4 | ||
| else | ||
| #calculating higher order coefficients for implicit Adam Multon method |
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Current coverage is 91.58% (diff: 89.44%)
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| #Based on Hairer equation 5.9. Calculates b, ϕ, ϕstar for |
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Based on Hairer equation III.5.9, p.399.
Just make the document more clear
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On tableau was in terms of 1/2 instead of 1//2, etc.
example of a user defined type
Update Simple_Differential_Equation.ipynb
We add two new solvers for ODE.jl, and relocate the old multistep solver ode_ms, namely ode_am and ode113. ode_am is a fixed step method similiar to ode_ms, but uses a PECE step. ode113 is a variable step size and variable order method using the same underlying theory as ode_am.For more documentation, refer to Hairer et al Volume 1 on Adam Methods.
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