Unit Tests: Theoretical Results for (integrated) Autocorrelations

[flipdazed]

Aim Need to iron out any (remaining) bugs and validate the theory as the equations are correct

Method

• Verify theory in Mathematica between each case of GHMC
• Use verified Mathematica script to produce test cases for each case

• Split the Laplace Transform into polynomial coefficient arrays so the function is defined by:

  n = [0, 1, 2] # 0 + x + 2*x**2
  d = [5, 1, 4] # 5 + x + 4*x**2
  # f = (0 + x + 2*x**2) / (5 + x + 4*x**2)
  f = lambda x: (n*x**np.arange(n.size)).sum()/(d*x**np.arange(d.size)).sum()

• Validate numerators and denominators separately

Remaining Issues

• Some typos in the python functions but I can just use the C++ version so no fix urgent
• M2_Exp $\theta=\pi/2$, $P_{\text{acc}} \ne 1$

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Exponential Issues Funit != F

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Verified all with each other through the Package form.

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Exponential Autocorrelations - all results are inexpValidations.nb`

• None of the derived form agrees with the form reported in the paper
• All the equations in the paper agree with each other
• The derived form agrees with the HMC case of itself at least
• Verified the derived form has no typos by copying out a couple of times

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Fixed Length Trajectories - all results agree with each other

• Haven't checked against derived form
• Autocorrelations are unavailable as currently it is not clear that there is a numerically plottable solution

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Unit Test Data

• Generated by fixUnitTest.nb and expUnitTest.nbto .csv data files
• Need to load into python and check against python models

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C++ functions

• c++ code was created by using a rule based system based on my flakey c++ knowledge
• The approach is good for validation as there can be no typographical errors due to the direct transfer link I created from Mathematica
• Use cython or the like to import and wrap into a python-friendly form and compare alongside pure python functions for the unit test data

[flipdazed]

Mathematica-Python Link

• A Mathematica-python link was created and now I can run any Mathematica code directly from python
• Now the functions are directly exported to c++ its not really that useful but still good for a direct sympy alternative

[flipdazed]

unit tests written - python fails for certain cases

waiting on solution for parsing full GHMC quations to c++ directly from Mathematica

• need method of solving the Root objects and reinserting

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Had previously forgotten that Mathematica returns a quintic root which cannot be solved without a full numerical reduction by input parameters.

The best solution is therefore given by python else I would have to code a root finder in boost and ceebs

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Current status

• Possible* functions implemented for Fixed trajectory case (*i.e. not Inverse Laplace Transforms)
  • Mathematica - all implemented
  • Python - all functions implemented and an approximate form for $\mathcal{L}^{-1}F$
  • C++ all implemented
• Functions implemented for Exponential trajectories
  • Mathematica - all implemented
  • Python - all functions implemented - some bugs remain ^^
  • C++ all except ghmc **

_Unable to implement directly from Mathematica into C++ as the ghmc() function requires a Root[] object to be evaluated EACH_ time the function is called as no direct analytic solution exists for quintic roots (see above). Hence, manually implementing a root solver into C++ will be non-trivial and time consuming. The python version of ghmc works perfectly well with numpy's root solver.

^^ The bugs that remain are not present in the C++ functions so leaving so now and just using C++ when theory is required

[flipdazed]

shouldn't have closed