Struggling to use nsimplify and N properly

[Andrew Spielberg]

Hi all,

Recent adopter, first time caller, bear with me if I have some newbish misunderstandings about sympy.

I am working on a project where we are trying to use sympy to do a lot of symbolic geometric computation.  The project predates sympy, so I cannot really change too much code to accomodate it.

Our calculations create matrices, and sometimes, the expressions can seem large, like the one seen below.  I know that this is actually supposed to be a linear expression, and I need to get it into a linear form.

Matrix([

[                                                                      1.0*leg1.beamwidth - (2.05432527401305e-33*leg1.beamwidth**3 + 0.166666666666667*leg1.beamwidth**2*total_length + 0.166666666666667*leg1.beamwidth**2*(1.23259516440783e-32*leg1.beamwidth + total_length/2))/(6.16297582203915e-33*leg1.beamwidth**2 + 0.25*leg1.beamwidth*total_length + 0.5*leg1.beamwidth*(1.23259516440783e-32*leg1.beamwidth + total_length/2)),                                                                                       1.0*leg1.beamwidth - (2.05432527401305e-33*leg1.beamwidth**3 + 0.166666666666667*leg1.beamwidth**2*total_length + 0.166666666666667*leg1.beamwidth**2*(1.23259516440783e-32*leg1.beamwidth + total_length/2))/(6.16297582203915e-33*leg1.beamwidth**2 + 0.25*leg1.beamwidth*total_length + 0.5*leg1.beamwidth*(1.23259516440783e-32*leg1.beamwidth + total_length/2)),                                                 1.0*leg1.beamwidth - (2.05432527401305e-33*leg1.beamwidth**3 + 0.166666666666667*leg1.beamwidth**2*total_length + 0.166666666666667*leg1.beamwidth**2*(1.23259516440783e-32*leg1.beamwidth + total_length/2))/(6.16297582203915e-33*leg1.beamwidth**2 + 0.25*leg1.beamwidth*total_length + 0.5*leg1.beamwidth*(1.23259516440783e-32*leg1.beamwidth + total_length/2)),                                 1.0*leg1.beamwidth - (2.05432527401305e-33*leg1.beamwidth**3 + 0.166666666666667*leg1.beamwidth**2*total_length + 0.166666666666667*leg1.beamwidth**2*(1.23259516440783e-32*leg1.beamwidth + total_length/2))/(6.16297582203915e-33*leg1.beamwidth**2 + 0.25*leg1.beamwidth*total_length + 0.5*leg1.beamwidth*(1.23259516440783e-32*leg1.beamwidth + total_length/2))],

[-6.12323399573677e-17*leg1.beamwidth - (-2.53215139886928e-65*leg1.beamwidth**3 + 0.0416666666666667*leg1.beamwidth*total_length**2 + 0.166666666666667*leg1.beamwidth*(1.23259516440783e-32*leg1.beamwidth + total_length/2)*(1.23259516440783e-32*leg1.beamwidth + total_length))/(6.16297582203915e-33*leg1.beamwidth**2 + 0.25*leg1.beamwidth*total_length + 0.5*leg1.beamwidth*(1.23259516440783e-32*leg1.beamwidth + total_length/2)), 6.12323399573677e-17*leg1.beamwidth + total_length/2 - (-2.53215139886928e-65*leg1.beamwidth**3 + 0.0416666666666667*leg1.beamwidth*total_length**2 + 0.166666666666667*leg1.beamwidth*(1.23259516440783e-32*leg1.beamwidth + total_length/2)*(1.23259516440783e-32*leg1.beamwidth + total_length))/(6.16297582203915e-33*leg1.beamwidth**2 + 0.25*leg1.beamwidth*total_length + 0.5*leg1.beamwidth*(1.23259516440783e-32*leg1.beamwidth + total_length/2)), total_length/2 - (-2.53215139886928e-65*leg1.beamwidth**3 + 0.0416666666666667*leg1.beamwidth*total_length**2 + 0.166666666666667*leg1.beamwidth*(1.23259516440783e-32*leg1.beamwidth + total_length/2)*(1.23259516440783e-32*leg1.beamwidth + total_length))/(6.16297582203915e-33*leg1.beamwidth**2 + 0.25*leg1.beamwidth*total_length + 0.5*leg1.beamwidth*(1.23259516440783e-32*leg1.beamwidth + total_length/2)), -(-2.53215139886928e-65*leg1.beamwidth**3 + 0.0416666666666667*leg1.beamwidth*total_length**2 + 0.166666666666667*leg1.beamwidth*(1.23259516440783e-32*leg1.beamwidth + total_length/2)*(1.23259516440783e-32*leg1.beamwidth + total_length))/(6.16297582203915e-33*leg1.beamwidth**2 + 0.25*leg1.beamwidth*total_length + 0.5*leg1.beamwidth*(1.23259516440783e-32*leg1.beamwidth + total_length/2))],

[                                                                                                                                                                                                                                                                                                                                                                                                                        -1.0*leg1.beamwidth,                                                                                                                                                                                                                                                                                                                                                                                                                                         -1.0*leg1.beamwidth,                                                                                                                                                                                                                                                                                                                                                                                                                     0,                                                                                                                                                                                                                                                                                                                                                                                                     0]])

If I perform nsimplify on this with a tolerance = 1e-5, I get something reasonable:

Matrix([

[                                                                 leg1.beamwidth/2,                                                                 leg1.beamwidth/2, leg1.beamwidth/2, leg1.beamwidth/2],

[-612323399573677*leg1.beamwidth/10000000000000000000000000000000 - total_length/4, 612323399573677*leg1.beamwidth/10000000000000000000000000000000 + total_length/4,   total_length/4,  -total_length/4],

[                                                                  -leg1.beamwidth,                                                                  -leg1.beamwidth,                0,                0]])

But I can't simplify it further for some reason.  Even though -612323399573677*leg1.beamwidth/10000000000000000000000000000000  = -6e-17, I can't seem to make that term disappear.

This was a simple case, but in more complex cases, sympy fails to cancel out terms and the solution becomes highly nonlinear.  As an example, this won't simplify with nsimplify for some reason, even though certain terms clearly dominate others:

(leg1.beamwidth*total_length*(24492935982947*leg2.beamwidth/200000000000000000000000000000 + total_length/2)**2/2 - 306161699786839*leg2.beamwidth*total_length**2*(24492935982947*leg2.beamwidth/200000000000000000000000000000 + total_length/2)/10000000000000000000000000000000 + 153080849893419*total_length**3*(24492935982947*leg2.beamwidth/200000000000000000000000000000 + total_length/2)/10000000000000000000000000000000)/(total_length*(24492935982947*leg2.beamwidth/200000000000000000000000000000 + total_length/2)**2)

Lastly, things sometimes get worse.  If I try to play around with simplify, N, and nsimplify to try to "force" it to simplify things, I sometimes get the error attached.

Is there a proper way to get do the simplifications I need?  And, why would I get that value error I displayed attached.

Thanks for any and all help,

Andy S.

[Aaron Meurer]

I think you want the chop option, like expr.eval(chop=True) or N(expr, chop=True).

Aaron Meurer

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[Andrew Spielberg]

Thanks a ton - yes, using N and then specifying a tolerance with N worked.

Out of curiosity then, what is the point of the nsimplify tolerance?  What does that do?  I appear to be misunderstanding it.

-Andy S.

[Aaron Meurer]

nsimplify tries to convert floating point numbers into exact numbers, like sqrt(2) or pi + 1. The tolerance flag specifies how close the exact number needs to be to the floating point number. For example:

In [8]: nsimplify(pi.evalf(), tolerance=0.1)

Out[8]: 22/7

In [9]: nsimplify(pi.evalf(), tolerance=0.01)

Out[9]: 22/7

In [10]: nsimplify(pi.evalf(), tolerance=0.001)

Out[10]:

355

───

113

In [11]: nsimplify(pi.evalf(), tolerance=0.0001)

Out[11]:

355

───

113

Aaron Meurer

To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/d8685eaf-c936-4c6e-8a3d-ea8c101e8245%40googlegroups.com.

[Andrew Spielberg]

Aha, thanks so much!

-Andy S.

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[Chris Smith]

Note also that your matrix will simplify without evaluation:

>>> m.applyfunc(simplify)

Matrix([

[ 0.5*leg1_beamwidth,  0.5*leg1_beamwidth, 0.5*leg1_beamwidth, 0.5*leg1_beamwidth],

[ -0.25*total_length,   0.25*total_length,  0.25*total_length, -0.25*total_length],

[-1.0*leg1_beamwidth, -1.0*leg1_beamwidth,                  0, 0]])

(I replaced the .beam with _beam, but otherwise you see that it simplifies just fine.)