Partial derivative of an interpolated function (with Mathematica)

[evgenx]

Hi,

I faced a problem (in Mathematica) when trying to plot a partial
derivative of a functiona (of two variables) obatined by "Interpolation".
More precisely, here is my input:

surf=Interpolation[{
{{160.0, 160.0}, 2.852688},
{{160.0, 170.0}, 2.827547},
{{160.0, 180.0}, 2.818931},
{{160.0, 190.0}, 2.826640},
{{160.0, 200.0}, 2.851483},

{{170.0, 160.0}, 2.861634},
{{170.0, 170.0}, 2.832750},
{{170.0, 180.0}, 2.822240},
{{170.0, 190.0}, 2.830275},
{{170.0, 200.0}, 2.858395},

{{180.0, 160.0}, 2.862344},
{{180.0, 170.0}, 2.831671},
{{180.0, 180.0}, 2.820786},
{{180.0, 190.0}, 2.831775},
{{180.0, 200.0}, 2.862605},

{{190.0, 160.0}, 2.857940},
{{190.0, 170.0}, 2.830020},
{{190.0, 180.0}, 2.822137},
{{190.0, 190.0}, 2.832657},
{{190.0, 200.0}, 2.861574},

{{200.0, 160.0}, 2.850865},
{{200.0, 170.0}, 2.826201},
{{200.0, 180.0}, 2.818547},
{{200.0, 190.0}, 2.827127},
{{200.0, 200.0}, 2.852228}
}]

s1=Function[{a,b},surf[a,b]]

da=Function[{a,b},D[s1[a,b],a]]

Plot3D[da[a, b], {a, 175.0, 185.0}, {b, 175.0, 185.0}]

When running this input Mathematica gives the error:
General::ivar: 175.00071499999999` is not a valid variable

Any help will be greatly appreciated !
Many thanks!


Evgeniy

 

[Dale]

The problem is here:

da=Function[{a,b},D[s1[a,b],a]]

When you evaluate da[170,175] it translates this to D[s1[170,175],170]. Obviously, you cannot differentiate wrt 170, so it throws the error. What you want is for it to evaluate the partial derivative in advance and construct a new interpolating function object and simply evaluate that function as needed. You can do that as follows:

da=Function[{a,b},Evaluate[D[s1[a,b],a]]]

 

[evgenx]

Hi,

Many thanks for your reply. It works now fine.

Well, there is an additional related question.
Namely, I'd like to find a point on the surf[a,b]
(i.e. values of a and b) where both dsurf/da=0
and d²surf/da²=0. Just simply solving:

s1=Function[{a,b},surf[a,b]]

da=Function[{a,b},Evaluate[D[s1[a,b],a]]]
dada=Function[{a,b},Evaluate[D[s1[a,b],{a,2}]]]

sol=FindRoot[{da==0,dada==0},{a, 180.0},{b, 165.0}]


gives the error:

The function value {Function[{a, b},
InterpolatingFunction[{{160., 200.}, {160., 200.}}, <>][a, b]],
Function[{a, b}, InterpolatingFunction[{{160., 200.}, {160., 200.}}, <>][
a, b]]} is not a list of numbers with dimensions {2} at {a, b} =
{180., 165.}.


What can I do wrong?
Thanks!


Evgeniy

 

[Dale]

sol=FindRoot[{da==0,dada==0},{a, 180.0},{b, 165.0}]

should be

sol=FindRoot[{da[a,b]==0,dada[a,b]==0},{a, 180.0},{b, 165.0}]

Since da and dada are pure functions you need to supply the arguments.

 

[evgenx]

Great, everything works! Thanks a lot!

A-a, just the very last question ... When I tried to plot
da and dada (with Plot3D) they looked like the function (surf) has
discontinuities at a=180.0. I guess it has to do with the quality of
interpolation. After I added Method->"Spline" (as an option for Interpolation)
da became better but dada still has a crease at a=180.0. Don't you happen
to know how to improve it.

Best wishes,
Evgeniy

 

[Dale]

You may want to increase the order of the interpolation using the InterpolationOrder option. However, you should realize that what you are doing, taking the second derivative of a function with only 5 support points, is inherently unlikely to give very good results.

 

[evgenx]

Ok, I see.
Thanks again for your help!