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.

[miguelmarco]

comment:10

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]

comment:11

Replying to @miguelmarco:

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]

comment:12

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?

[miguelmarco]

comment:13

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.

[miguelmarco]

Attachment: 12922.patch.gz

Newer version, solves the pynac registry pollution without manipulating it

[miguelmarco]

comment:14

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

[KPanComputes]

New commits:

39aaf0c

Added implicit_derivative

[KPanComputes]

comment:25

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 ".
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.

[sagetrac-git]

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

3f3f949

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

[sagetrac-git]

Changed commit from 39aaf0c to 3f3f949

[rwst]

comment:27

Replying to @miguelmarco:

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.

[sagetrac-git]

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

5fe0ffe

Moved to a method in the expression class

[sagetrac-git]

Changed commit from 3f3f949 to 5fe0ffe

[miguelmarco]

comment:29

Moved it to a method in the expression class.

[rwst]

Changed branch from u/mmarco/ticket/12922 to u/rws/ticket/12922

[rwst]

comment:31

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

[rwst]

Changed reviewer from Kannappan Sampath to Kannappan Sampath, Ralf Stephan

[rwst]

Changed commit from 5fe0ffe to 9549976

[miguelmarco]

comment:32

Thanks for the review.

[mezzarobba]

comment:33

**********************************************************************
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'
**********************************************************************

[sagetrac-git]

Changed commit from 9549976 to 0a76da7

[sagetrac-git]

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

0a76da7

12922: fix doctest

[rwst]

comment:35

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

[vbraun]

Changed branch from u/rws/ticket/12922 to 0a76da7