Implicit derivative

[miguelmarco]

Added implicit_derivative function. It allows to compute the n'th derivative of the implicit function defined by a symbolic expression.

CC: @kcrisman @sagetrac-sjg10 @burcin

Component: calculus

Keywords: implicit derivative

Author: Miguel Marco

Branch/Commit: 0a76da7

Reviewer: Kannappan Sampath, Ralf Stephan

Issue created by migration from https://trac.sagemath.org/ticket/12922

[miguelmarco]

Attachment: 16191.patch.gz

[sagetrac-sjg10]

comment:3

Reviewed, seems good. Added a check for having no y term in the expression. Was previously an uncaught DivideByZero, while code now gives a more informative error.
Was my first review, would appreciate to know whether I'm actually being any help!

[kcrisman]

comment:4

Yes, both helpful! A few comments.

• The reviewer patch should be a separate patch on top of the original patch, so that the original author is kept.
• In any case, the patch should have a meaningful commit message.
• I'm a little apprehensive about the many specific variables and functions defined here. Won't that cause some kind of conflict with (say) an already-defined yy? We shouldn't depend on people not needing those variables. Or is everything only in local scope because of the use of SR.symbol?

I'm also wondering - is it possible to have an implicit derivative method for symbolic expressions? Just curious.

[sagetrac-sjg10]

adds error handler

[miguelmarco]

comment:5

Attachment: 16191_review.patch.gz

All the variables defined in the function are local. They do not get mixed with the global variables with the same name. You can even call the function with xx and yy as parameters, and they will not be mixed with the local xx and yy.

sage: var('x,y')
(x, y)
sage: implicit_derivative(y,x,y*x/cos(x+y))
-(x*y*sin(x + y)/cos(x + y)^2 + y/cos(x + y))/(x*y*sin(x + y)/cos(x + y)^2 + x/cos(x + y))
sage: var('xx,yy')
(xx, yy)
sage: implicit_derivative(yy,xx,yy*xx/cos(xx+yy))
-(xx*yy*sin(xx + yy)/cos(xx + yy)^2 + yy/cos(xx + yy))/(xx*yy*sin(xx + yy)/cos(xx + yy)^2 + xx/cos(xx + yy))

About writing this as a method for symbolic expressions... that was my first idea; but i prefeared this aproach, due to the fact that you are actually not diferentiating the symbolic expression, but a function defined implicitely by the symbolic expression. So it seemed more natural to me this kind of syntax.

[sagetrac-sjg10]

comment:6

Calling var or SR.symbol('somename') even locally does add to the pynac_symbol_registry, yet calling SR.symbol() with no argument will create a temporary variable that will not be added to this registry. Possibly adding to the registry may cause problems if it is being referenced directly for some reason. Maybe worth just using x = SR.symbol() etc?

sage: from sage.symbolic.ring import pynac_symbol_registry
sage: pynac_symbol_registry                               
{'some_variable': some_variable, 'A': A, 'C': C, 'B': B, 'E': E, 'D': D, 'G': G, 'F': F, 'H': H, 'K': K, 'J': J, 'M': M, 'L': L, 'N': N, 'Q': Q, 'P': P, 'S': S, 'R': R, 'U': U, 'T': T, 'W': W, 'V': V, 'Y': Y, 'X': X, 'Z': Z, 'a': a, 'c': c, 'b': b, 'd': d, 'g': g, 'f': f, 'h': h, 'k': k, 'j': j, 'm': m, 'l': l, 'o': o, 'n': n, 'q': q, 'p': p, 's': s, 'r': r, 'u': u, 't': t, 'w': w, 'v': v, 'y': y, 'x': x, 'z': z}
sage: def f(): temp = SR.symbol()                         
....: 
sage: f()
sage: pynac_symbol_registry
{'some_variable': some_variable, 'A': A, 'C': C, 'B': B, 'E': E, 'D': D, 'G': G, 'F': F, 'H': H, 'K': K, 'J': J, 'M': M, 'L': L, 'N': N, 'Q': Q, 'P': P, 'S': S, 'R': R, 'U': U, 'T': T, 'W': W, 'V': V, 'Y': Y, 'X': X, 'Z': Z, 'a': a, 'c': c, 'b': b, 'd': d, 'g': g, 'f': f, 'h': h, 'k': k, 'j': j, 'm': m, 'l': l, 'o': o, 'n': n, 'q': q, 'p': p, 's': s, 'r': r, 'u': u, 't': t, 'w': w, 'v': v, 'y': y, 'x': x, 'z': z}
sage: def f(): temp = SR.symbol('temp')
....: 
sage: f()
sage: pynac_symbol_registry
{'temp': temp, 'some_variable': some_variable, 'A': A, 'C': C, 'B': B, 'E': E, 'D': D, 'G': G, 'F': F, 'H': H, 'K': K, 'J': J, 'M': M, 'L': L, 'N': N, 'Q': Q, 'P': P, 'S': S, 'R': R, 'U': U, 'T': T, 'W': W, 'V': V, 'Y': Y, 'X': X, 'Z': Z, 'a': a, 'c': c, 'b': b, 'd': d, 'g': g, 'f': f, 'h': h, 'k': k, 'j': j, 'm': m, 'l': l, 'o': o, 'n': n, 'q': q, 'p': p, 's': s, 'r': r, 'u': u, 't': t, 'w': w, 'v': v, 'y': y, 'x': x, 'z': z}

[miguelmarco]

comment:7

I have tried to use SR.symbol() instead of SR.symbol('x'), but then the result is a mess.

I can't think of a way to make this work without the said side effect.

[miguelmarco]

comment:8

I solved the pynac symbol registry pollution by saving a copy of it at the beginning of the function, and restoring it at the end. I am not sure if this is the best choice, but it is the only solution i found, and seems to work fine.

[burcin]

comment:9

Messing with the symbol registry this way is really not a good idea. That is just supposed to be internal data.

What if there is an exception in that code and the registry is not restored?

What exactly is the problem with using temporary variables? Looking briefly at the patch, it looks like we also need temporary functions.