# Way to symbolically integrate function fit with table data?

**URL:** <https://discourse.julialang.org/t/way-to-symbolically-integrate-function-fit-with-table-data/4752>\
**Category:** General Usage\
**Created:** [July 9, 2017, 10:43pm UTC](https://discourse.julialang.org/t/way-to-symbolically-integrate-function-fit-with-table-data/4752 "2017-07-09T22:43:26Z")\
**Posts on this page:** 6\
**Page:** 1

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**Author:** ![djsegal](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/djsegal/32/13752_2.png) [@djsegal](https://discourse.julialang.org/u/djsegal)\
**Post date:** [July 9, 2017, 10:43pm UTC](https://discourse.julialang.org/t/way-to-symbolically-integrate-function-fit-with-table-data/4752/1 "2017-07-09T22:43:26Z")

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Is there a way to symbolically integrate a function fit with table data?

…and not have it take more than `O(seconds)`

// note: already using [SymPy.jl](https://github.com/JuliaPy/SymPy.jl), so can’t change that

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**Author:** ![djsegal](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/djsegal/32/13752_2.png) [@djsegal](https://discourse.julialang.org/u/djsegal)\
**Post date:** [July 10, 2017, 3:43am UTC](https://discourse.julialang.org/t/way-to-symbolically-integrate-function-fit-with-table-data/4752/2 "2017-07-10T03:43:56Z")

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breaking example from [github](https://github.com/JuliaPy/SymPy.jl/issues/181),

1. lookup table

```julia
table_T = [
     0.00
     1.00
     2.50
     3.00
     5.00
     6.50
     7.00
     7.50
     9.00
    10.00
    15.00
    20.00
    25.00
]

table_sigma_v = [
  0.0000e-00
  5.4835e-21
  7.6096e-19
  1.7128e-18
  1.2891e-17
  3.1494e-17
  3.9797e-17
  4.9111e-17
  8.2598e-17
  1.0886e-16
  2.6542e-16
  4.2434e-16
  5.5940e-16
]

```

* * *

1. functions that should interpolate and integrate table data

```julia
using SymPy

@syms T_0

function foo(rho)
  SymPy.interpolate(table_T, table_sigma_v, T_0 / ( 1 - rho ) ^ 2 ) * rho ^ 3 + rho
end

@syms rho_0

function bar()
  b = 3
  c = 1.3

  SymPy.integrate(foo(rho_0), (rho_0, 0.0, 1.0)) * 1e2 * b / c
end

```

* * *

1. calling `bar()` returns this error:

```julia
ERROR: PyError (ccall(@pysym(:PyObject_Call), PyPtr, (PyPtr, PyPtr, PyPtr), o, arg, C_NULL)) <class 'sympy.polys.polyerrors.PolynomialDivisionFailed'>
PolynomialDivisionFailed()
  File "/Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/Conda/deps/usr/lib/python2.7/site-packages/sympy/integrals/integrals.py", line 1280, in integrate
    risch=risch, manual=manual)

...

  File "/Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/Conda/deps/usr/lib/python2.7/site-packages/sympy/polys/densearith.py", line 1096, in dup_prem
    raise PolynomialDivisionFailed(f, g, K)

Stacktrace:
 [1] pyerr_check at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/PyCall/src/exception.jl:56 [inlined]
 [2] pyerr_check at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/PyCall/src/exception.jl:61 [inlined]
 [3] macro expansion at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/PyCall/src/exception.jl:81 [inlined]
 [4] #_pycall#66(::Array{Any,1}, ::Function, ::PyCall.PyObject, ::SymPy.Sym, ::Vararg{Any,N} where N) at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/PyCall/src/PyCall.jl:620
 [5] _pycall(::PyCall.PyObject, ::SymPy.Sym, ::Vararg{Any,N} where N) at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/PyCall/src/PyCall.jl:608
 [6] #pycall#70(::Array{Any,1}, ::Function, ::PyCall.PyObject, ::Type{PyCall.PyAny}, ::SymPy.Sym, ::Vararg{Any,N} where N) at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/PyCall/src/PyCall.jl:642
 [7] pycall(::PyCall.PyObject, ::Type{PyCall.PyAny}, ::SymPy.Sym, ::Vararg{Any,N} where N) at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/PyCall/src/PyCall.jl:642
 [8] #call_sympy_fun#538(::Array{Any,1}, ::Function, ::PyCall.PyObject, ::SymPy.Sym, ::Vararg{Any,N} where N) at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/SymPy/src/SymPy.jl:252
 [9] call_sympy_fun(::PyCall.PyObject, ::SymPy.Sym, ::Vararg{Any,N} where N) at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/SymPy/src/SymPy.jl:252
 [10] #sympy_meth#539(::Array{Any,1}, ::Function, ::String, ::SymPy.Sym, ::Vararg{Any,N} where N) at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/SymPy/src/SymPy.jl:257
 [11] sympy_meth(::String, ::SymPy.Sym, ::Vararg{Any,N} where N) at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/SymPy/src/SymPy.jl:256
 [12] #integrate#326(::Array{Any,1}, ::Function, ::SymPy.Sym, ::Tuple{SymPy.Sym,Float64,Float64}, ::Vararg{Tuple{SymPy.Sym,Float64,Float64},N} where N) at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/SymPy/src/SymPy.jl:193
 [13] integrate(::SymPy.Sym, ::Tuple{SymPy.Sym,Float64,Float64}) at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/lib/v0.6/SymPy/src/SymPy.jl:193
 [14] bar() at ./REPL[7]:5

```

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**Post date:** [July 10, 2017, 5:59am UTC](https://discourse.julialang.org/t/way-to-symbolically-integrate-function-fit-with-table-data/4752/3 "2017-07-10T05:59:36Z")

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It is unclear what you are trying to do. If the function is interpolated, integration will be numerical.

There are various methods, depending on the kind of interpolation you want to use. For univariate functions, they should be extremely fast. Eg see [the example for ApproxFun.jl](http://juliaapproximation.github.io/ApproxFun.jl/latest/faq.html#How-do-I-interpolate-a-function-at-a-specified-grid?-1).

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**Author:** ![djsegal](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/djsegal/32/13752_2.png) [@djsegal](https://discourse.julialang.org/u/djsegal)\
**Post date:** [July 10, 2017, 6:25am UTC](https://discourse.julialang.org/t/way-to-symbolically-integrate-function-fit-with-table-data/4752/4 "2017-07-10T06:25:23Z")

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As mentioned above, I’m trying to symbolically integrate a function defined by table data

The emphasis is on symbolic (using SymPy.jl). This needs to get piped into a numerical solver at the end that finds T\_0

// Some example code highlighting the problem is in the first reply

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**Post date:** [July 10, 2017, 4:53pm UTC](https://discourse.julialang.org/t/way-to-symbolically-integrate-function-fit-with-table-data/4752/5 "2017-07-10T16:53:38Z")

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I don’t know anything about SymPy, but you should definitely rescale your data before doing _anything_ with it.  
And the error you’re getting seems to indicate something going wrong inside of SymPy (`PolynomialDivisionFailed`), so you could maybe try a very simple function first?

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**Post date:** [September 24, 2017, 2:57am UTC](https://discourse.julialang.org/t/way-to-symbolically-integrate-function-fit-with-table-data/4752/6 "2017-09-24T02:57:17Z")

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You are starting with numerical data and finishing with a numerical root finder, so there doesn’t seem to be a good reason to use a true symbolic package in the middle.

As has already been stated, just using a suitable polynomial interpolant should be enough?