thomas.pdf, 68 KB, 21 Nov 2017.
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The composite of several coarser integrations is extrapolated to obtain solutions that are equivalent to a single integration with many more steps, then the solutions are extrapolated again to obtain the asymptotic solution.
In the calculations a is the acceleration, j is da/dt, and k is dj/dt = d2a/dt2. Vectors are usually represented as the product of a magnitude and a unit vector, with a = a â = a au, etc.
extrap.a is the top level program. It calls the steps.a program to perform each of the coarser integrations, then extrapolates the composite.
extrap.a program listing With links to library functions
extrap.a program listing Plain text
steps.a program listing With links to library functions
steps.a program listing Plain text
This is the sample calculation in section II of the paper in more detail. The subscripts used in the paper are not present in the computer code. The subscripts are displayed as 0, but that is not their actual value.
The example in section II of the paper
The solutions for an accelerated clock. These solutions are all the same, except that they are for different powers of velocity and different numbers of integration steps.
v^10 a^1 solution for n and n+1 steps, with n=1
v^10 a^1 solution for n and n+1 steps, with n=10
v^25 a^1 solution for n and n+1 steps, with n=1
The solutions for a jerked clock. The v^5 solution (not shown) is not invariant.
v^3 a^2 j^1, n=1, fourth order
v^4 a^2 j^1, n=1, fifth order
v^4 a^2 j^1, n=10, fifth order (same solution as for n=1)
The yanked clock. These three solutions are identical, even though all of the calculations except the last are very different. The solution in the format displayed at the end of the file is best suited for importing into other computer programs.
v^4 a^3 j^2 k^1, for n=1, ninth order
v^4 a^3 j^2 k^1, for n=2, ninth order
v^4 a^3 j^2 k^1, for n=3, ninth order
Last update 21 Nov 2017 Revision History