Implicit derivative

[Miguel Marco]

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

[Samuel Gonshaw]

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!

[Karl-Dieter Crisman]

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.

[Samuel Gonshaw]

adds error handler

[Miguel Marco]

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.

[Samuel Gonshaw]

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}

[Miguel Marco]

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.

[Miguel Marco]

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 Erocal]

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.

[Miguel Marco]

The problem with using temporary variables without giving them a name string is this:

sage: from sage.symbolic.function_factory import SymbolicFunction
sage: var('x,y')                                                                                                                                    
(x, y)
sage: def implicit_derivative(Y,X,F,n=1):                                                                                                           
....:         x=SR.symbol()                                                                                                                         
....:         yy=SR.symbol()                                                                                                                        
....:         y=SymbolicFunction('y',1)(x)                                                                                                          
....:         f=SymbolicFunction('f',2)(x,yy)                                                                                                       
....:         Fx=f.diff(x)                                                                                                                          
....:         Fy=f.diff(yy)                                                                                                                         
....:         G=-(Fx/Fy)                                                                                                                            
....:         G=G.subs_expr({yy:y})                                                                                                                 
....:         di={y.diff(x):-F.diff(X)/F.diff(Y)}                                                                                                   
....:         R=G                                                                                                                                   
....:         S=G.diff(x,n-1)                                                                                                                       
....:         for i in range(n+1):                                                                                                                  
....:                 di[y.diff(x,i+1)(x=x)]=R                                                                                                      
....:                 S=S.subs_expr(di)                                                                                                             
....:                 R=G.diff(x,i)                                                                                                                 
....:                 for j in range(n+1-i):                                                                                                        
....:                         di[f.diff(x,i,yy,j)(x=x,yy=y)]=F.diff(X,i,Y,j)                                                                        
....:                         S=S.subs_expr(di)                                                                                                     
....:                 return S                                                                                                                      
....:                                                                                                                                               
sage: implicit_derivative(y,x,x^2+y^2-1)                                                                                                            
-D[0](f)(symbol160, y(symbol160))/D[1](f)(symbol160, y(symbol160))

I don't know why, this problem is not present if we put the strings in the definitions of the variables.

[Burcin Erocal]

Replying to mmarco:

sage: implicit_derivative(y,x,x^2+y^2-1)                                                                                  
-D[0](f)(symbol160, y(symbol160))/D[1](f)(symbol160, y(symbol160))

It looks like the substitution doesn't work at some point.

Looking at the code...

The calls y.diff(x,i+1)(x=x) and f.diff(x,i,yy,j)(x=x,yy=y) rely on the name of the variable. You should use a dictionary argument to .subs().

BTW, the subs_expr() method will go away at some point. Why not just use .subs() everywhere?

[Burcin Erocal]

Before this gets switched to positive review, we should check how it behaves if there already is a symbolic function defined with a custom method for evaluation named y or f.

Why is this not a method of symbolic expressions? Isn't it inconsistent to have the functional notation supported but not a class method?

[Miguel Marco]

Thanks for the comments. I will try to work on it.

As i said, the reason why it is not a method is that it is not clear to me what class should it be in.

Is this a method of the expression F? Well, not exactly, since F is not the function that we are trying to differentiate, but the symbollic expression that deffines it implicitly.

Of X? This is just the variable with which we differentiate on.

Of Y? Riguorilously that should be the function that we are differentiating. But doesn't sound natural at all to add a method like this for symbollic variables.

Or maybe we could define a new class ImplicitelyDefinedSymbolicFunction? for this case, bat that sounds way overkill.

[Miguel Marco]

Newer version, solves the pynac registry pollution without manipulating it

[Miguel Marco]

Ok, your suggestions seemed to work fine. I have uploaded a newer version. Please check it.

[Miguel Marco]

bump

[Kannappan Sampath]

Hi mmarco:

Thanks for writing up a patch regarding something very fundamental to calculus. Sage has since v6.0 moved to Git for its revision control. As a consequence, we work with "branches" instead of "patches" and so on. Could you kindly make the patch here into a branch?

I think this might be a bit painful depending on several factors. So, please feel free to ask for any help!

[Kannappan Sampath]

New commits:

​39aaf0c

Added implicit_derivative

[Kannappan Sampath]

For starters, I have the following comments:

1. The one-line docstring for the function is "incorrect": it computes something, yes, but it also returns it. So, please change that to "Return <whatever>".
2. The value error could return the expression F explicitly instead of saying the generic F. While you are it, please use the .format to format the string for the output instead.

Honestly, I have not looked at the math. I will get back about the math and the programming part in a bit.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​3f3f949

Reviewer's suggestions changed, plus some changes for PEP8 compliance.

[Ralf Stephan]

Replying to mmarco:

As i said, the reason why it is not a method is that it is not clear to me what class should it be in.

Is this a method of the expression F? Well, not exactly, since F is not the function that we are trying to differentiate, but the symbollic expression that deffines it implicitly.

I think Burcin wants this method part of the symbolic expression or function class. If it's not tied to a specific expression then make it a @classmethod (equivalent to a static method in C), and then create the globally accessible name for it.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​5fe0ffe

Moved to a method in the expression class

[Miguel Marco]

Moved it to a method in the expression class.

[Ralf Stephan]

Just some minor fixes. It looks good. I did no make ptestlong as it's just a new method.

---

New commits:

​7e00610	Merge branch 'develop' into t/12922/ticket/12922
​9549976	12922 reviewer's patch: optimize imports and doc cosmetics

[Miguel Marco]

Thanks for the review.

[Marc Mezzarobba]

**********************************************************************
File "src/sage/misc/sageinspect.py", line 1815, in sage.misc.sageinspect.sage_getsourcelines
Failed example:
    sage_getsourcelines(x)[0][-1]    # last line
Expected:
    '        return self / x\n'
Got:
    '        return S\n'
**********************************************************************

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​0a76da7

12922: fix doctest

[Ralf Stephan]

Interesting. That doctest depends on the last line in class Expression to remain unchanged.