Implement ``diff`` format symbolic derivative in new symbolics

[golam-m-hossain]

Implement a diff format symbolic derivative in new symbolics as the second form of abstract derivative to be avialable in Sage. See this long thread

http://groups.google.com/group/sage-devel/browse_thread/thread/ff10f99729a74eea/73308bf626ae06b3

for rationale behind it.

Implementation:

Instructions for installing these patches (sage-4.1.1)

(1) Pynac patch
(a) Get the pynac spkg

http://sage.math.washington.edu/home/burcin/pynac/pynac-0.1.8.p2.spkg

(b) Apply the pynac patch implementing diff derivative from

http://www.math.unb.ca/~ghossain/diff-derivative-pynac.patch

(c) install the patched spkg in Sage.

OR if you are feeling lazy, you can directly install my patched copy of pynac from here

http://www.math.unb.ca/~ghossain/pynac-0.1.8.p2-with-diff.spkg

(2) Sage patch:

Apply the attached patch in Sage and build the changes ("sage -b").
If everything goes smoothly then you are ready for testing.

Testing:

Above, patches will provide two new user accessible functions
(a) set_diff_derivative_level

(b) symbolic_diff

Please see the docs for usage of these functions:

It would be good to thoroughly test the following features:

(1) Substitution of the function

(2) Derivative with or without chain rule

(3) Explicit evaluation of derivative even for some situation where chain rule has been applied and derivative is specified w.r.t. an expression

(4) Symbolic n-th derivative

(5) Typesetting

Please test diff implementation against some related bugs

#6376, #6523, #6480

as new diff implementation should avoid these.

Speed Test:

sage: f(x) = function('f',x);
sage: f(x).diff(x)
D[0](f)(x)
sage: timeit('f(x).diff(x)')
625 loops, best of 3: 124 µs per loop
sage: timeit('f(x).diff(x,100)')
125 loops, best of 3: 5.58 ms per loop

sage: set_diff_derivative_level(1)
sage: f(x).diff(x)
diff(f(x), x, 1)
sage: timeit('f(x).diff(x)')
625 loops, best of 3: 116 µs per loop
sage: timeit('f(x).diff(x,100)')
625 loops, best of 3: 1.01 ms per loop

sage: set_diff_derivative_level(2)
sage: f(x).diff(x)
diff(f(x), x, 1)
sage: timeit('f(x).diff(x)')
625 loops, best of 3: 130 µs per loop
sage: timeit('f(x).diff(x,100)')
125 loops, best of 3: 1.85 ms per loop

***** This patch removes old "dummy_diff" from
"calculus/calculus" as we now have a _real_ diff.

CC: @ncalexan @mwhansen @kcrisman

Component: symbolics

Author: Golam Mortuza Hossain

Reviewer: Nick Alexander

Issue created by migration from https://trac.sagemath.org/ticket/6756

[golam-m-hossain]

Description changed:

--- 
+++ 
@@ -1,5 +1,93 @@
-Implement diff format symbolic derivative in new symbolics as the second form of abstract derivative to be avialable in Sage. See this long thread
+Implement a diff format symbolic derivative in new symbolics as the second form of abstract derivative to be avialable in Sage. See this long thread
 
 http://groups.google.com/group/sage-devel/browse_thread/thread/ff10f99729a74eea/73308bf626ae06b3
 
 for rationale behind it.
+
+**Implementation:**
+
+Instructions for installing these patches (sage-4.1.1)
+
+**(1) Pynac patch**   
+ (a) Get the pynac spkg
+
+ http://sage.math.washington.edu/home/burcin/pynac/pynac-0.1.8.p2.spkg
+
+ (b) Apply the pynac patch implementing diff derivative from
+
+ http://www.math.unb.ca/~ghossain/diff-derivative-pynac.patch
+
+ (c) install the patched spkg in Sage.
+
+ **OR** if you are feeling lazy, you can directly install my patched copy of pynac from here
+
+ http://www.math.unb.ca/~ghossain/pynac-0.1.8.p2-with-diff.spkg
+
+
+
+**(2) Sage patch:**
+
+ Apply the attached patch in Sage and build the changes ("sage -b").
+ If everything goes smoothly then you are ready for testing.
+
+**Testing:**
+
+Above, patches will provide two new user accessible functions
+ (a) ``set_diff_derivative_level``
+
+ (b) ``symbolic_diff``
+
+Please see the docs for usage of these functions:
+
+It would be good to thoroughly test the following features:
+
+ (1) Substitution of the function
+
+ (2) Derivative with or without chain rule
+
+ (3) Explicit evaluation of derivative even for some situation where chain rule has been applied and derivative is specified w.r.t. an expression
+
+ (4) Symbolic n-th derivative
+
+ (5) Typesetting 
+
+Please test diff implementation against some related bugs
+
+#6376, #6523, #6480
+
+as new diff implementation should avoid these.
+
+
+**Speed Test:**
+
+```
+sage: f(x) = function('f',x);
+sage: f(x).diff(x)
+D[0](f)(x)
+sage: timeit('f(x).diff(x)')
+625 loops, best of 3: 124 µs per loop
+sage: timeit('f(x).diff(x,100)')
+125 loops, best of 3: 5.58 ms per loop
+
+sage: set_diff_derivative_level(1)
+sage: f(x).diff(x)
+diff(f(x), x, 1)
+sage: timeit('f(x).diff(x)')
+625 loops, best of 3: 116 µs per loop
+sage: timeit('f(x).diff(x,100)')
+625 loops, best of 3: 1.01 ms per loop
+
+sage: set_diff_derivative_level(2)
+sage: f(x).diff(x)
+diff(f(x), x, 1)
+sage: timeit('f(x).diff(x)')
+625 loops, best of 3: 130 µs per loop
+sage: timeit('f(x).diff(x,100)')
+125 loops, best of 3: 1.85 ms per loop
+```
+
+
+***** This patch removes old "dummy_diff" from 
+"calculus/calculus" as we now have a `_real_` diff.
+
+

[golam-m-hossain]

Author: Golam Mortuza Hossain

[ncalexan]

comment:2

Attachment: trac_6756-diff-derivative-sage.patch.gz

I am interested in reviewing this, but I'm not sure that I can at this time. I have a heavily modified Sage Version 4.1.rc1, Release Date: 2009-07-07 tree that I can't really upgrade right now. I have applied this patch and installed your modified spkg. I am getting the following doctest failures

-*- mode: sage-test; default-directory: "~/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/" -*-
sage-test started at Fri Aug 21 10:18:46

/Users/ncalexan/bin/sage -b >/dev/null && /Users/ncalexan/bin/sage -tp 4 /Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py

real	0m2.361s
user	0m1.297s
sys	0m0.591s
Global iterations: 1
File iterations: 1
Using cached timings to run longest doctests first.
Doctesting 1 files doing 4 jobs in parallel
sage -t  derivative.py
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 51:
    sage: f(x).diff(x)
Expected:
    diff(f(x), x, 1)
Got:
    D[0](f)(x)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 53:
    sage: diff(f(x), x, 1)
Expected:
    diff(f(x), x, 1)
Got:
    D[0](f)(x)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 68:
    sage: h.diff(x)
Expected:
    diff(f(x), x, n + 1)
Got:
    D[0](diff)(f(x), x, n)*D[0](f)(x) + D[1](diff)(f(x), x, n)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 112:
    sage: f(x^2).diff(x)
Expected:
    diff(f(x^2), x, 1)
Got:
    2*x*D[0](f)(x^2)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 118:
    sage: f(x^2).diff(x)
Expected:
    2*x*diff(f(x^2), x^2, 1)
Got:
    2*x*D[0](f)(x^2)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 121:
    sage: f(sin(x^2)).diff(x)
Expected:
    2*x*cos(x^2)*diff(f(sin(x^2)), sin(x^2), 1)
Got:
    2*x*D[0](f)(sin(x^2))*cos(x^2)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 124:
    sage: g(x,f(x)).diff(x)
Expected:
    diff(f(x), x, 1)*diff(g(x, f(x)), f(x), 1) + diff(g(x, f(x)), x, 1)
Got:
    D[0](f)(x)*D[1](g)(x, f(x)) + D[0](g)(x, f(x))
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 129:
    sage: symbolic_diff(f(x^2), x, 1)
Expected:
    diff(f(x^2), x, 1)
Got:
    2*x*D[0](f)(x^2)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 138:
    sage: g(x,x).diff(x)
Expected:
    2*diff(g(x, x), x, 1)
Got:
    D[0](g)(x, x) + D[1](g)(x, x)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 147:
    sage: h = f(x).diff(x); h
Expected:
    diff(f(x), x, 1)
Got:
    D[0](f)(x)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 149:
    sage: h.subs(f(x)==x^4)
Expected:
    4*x^3
Got:
    D[0](f)(x)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 155:
    sage: h = f(x^2).diff(x); h
Expected:
    2*x*diff(f(x^2), x^2, 1)
Got:
    2*x*D[0](f)(x^2)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 157:
    sage: h.subs(f(x^2)==x^4)
Expected:
    4*x^3
Got:
    2*x*D[0](f)(x^2)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 159:
    sage: symbolic_diff(f(x)^2, f(x), 1)
Expected:
    2*f(x)
Got:
    diff(f(x)^2, f(x), 1)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 186:
    sage: h = f(x).diff(x) + f(x^2).diff(x); h
Expected:
    2*x*diff(f(x^2), x^2, 1) +  diff(f(x), x, 1)
Got:
    2*x*D[0](f)(x^2) + D[0](f)(x)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 217:
    sage: f(x,y).diff(x,y)
Expected:
    diff(f(x, y), x, 1, y, 1)
Got:
    D[0, 1](f)(x, y)
**********************************************************************
File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 219:
    sage: f(x,y).diff(x,2)
Expected:
    diff(f(x, y), x, 2)
Got:
    D[0, 0](f)(x, y)
**********************************************************************
2 items had failures:
  15 of  40 in __main__.example_1
   2 of  10 in __main__.example_2
***Test Failed*** 17 failures.
For whitespace errors, see the file /Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/tmp/.doctest_derivative.py
	 [3.8 s]
 
----------------------------------------------------------------------

The following tests failed:

	sage -t  devel/sage-main/sage/symbolic/derivative.py # 17 doctests failed
----------------------------------------------------------------------
Total time for all tests: 4.1 seconds

sage-test finished with failing tests at Fri Aug 21 10:18:53

When I work from the command line, I get things like

sage: default_level=set_diff_derivative_level()
sage: set_diff_derivative_level(1)
sage: f(x) = function('f', x)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)

/Users/ncalexan/.sage/temp/mero.local/93112/_Users_ncalexan__sage_init_sage_1.py in <module>()

/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/local/lib/python2.6/site-packages/sage/calculus/var.so in sage.calculus.var.function (sage/calculus/var.c:709)()

/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/local/lib/python2.6/site-packages/sage/symbolic/function.so in sage.symbolic.function.SFunction.__call__ (sage/symbolic/function.cpp:4655)()

TypeError: Symbolic function f takes exactly 2 arguments (1 given)

So there must be some calculus or symbolic patches that I'm missing. Any suggestions?

[golam-m-hossain]

comment:3

Replying to @ncalexan:

I am interested in reviewing this, but I'm not sure that I can at this time. I have a heavily modified Sage Version 4.1.rc1, Release Date: 2009-07-07 tree that I can't really upgrade right now. I have applied this patch and installed your modified spkg. I am getting the following doctest failures

 
 **********************************************************************
 File "/Users/ncalexan/sage-4.1-OSX-10.5-Intel-64bit-i386-Darwin/devel/sage-main/sage/symbolic/derivative.py", line 51:
     sage: f(x).diff(x)
 Expected:
     diff(f(x), x, 1)
 Got:
     D[0](f)(x)

Thanks Nick for trying it out. From this error, it seems "diff_derivative_level" flag wasn't set to
value greater than "0". It could be that "set_diff_derivative_level()" failed to set it properly.

When I work from the command line, I get things like

sage: default_level=set_diff_derivative_level()
sage: set_diff_derivative_level(1)
sage: f(x) = function('f', x)
---------------------------------------------------------------------------
TypeError: Symbolic function f takes exactly 2 arguments (1 given)

This looks really weired to me. Does it work before calling "set_diff_derivative_level()"?

I can suggest you to check three things

(1) Ensure in "symbolic/pynac.pyx" you have a line:

      "cdef public int diff_derivative_level = 0"

May be you can set it to "1", to enable "diff" by default.

(2) In "symbolic/all.py" you have a line:

     "from derivative import symbolic_diff"

(3) In "calculus/calculus.py" the following line is commented out or removed

      "syms['diff'] = dummy_diff"

If these three lines are fine then it should work. Some doctests may it still
fail if "set_diff_derivative_level()" isn't working properly but you can
always change value in (1) manually without using it.

Please let me know if that works.

Best,

[ncalexan]

comment:4

Sorry, same problems with a new 4.1.1 binary. It's possible I'm building in the wrong order (patch, failed tests, then new spkg, touch patch, rebuild) but that's two systems not working.

[ncalexan]

Attachment: trac_6756-referee-patch-ncalexan-1.patch.gz

[ncalexan]

comment:5

The second patch makes this at least work for me, and fixes (maybe an ordering issue?) a test that doesn't pass on my box. I'm not done reviewing yet, but at least it works as advertised. Question: how do I pattern match on this?

[ncalexan]

Reviewer: Nick Alexander

[ncalexan]

comment:6

Aha, very interesting, you pattern match via the symbolic_diff object. This is pretty slick, but it irritates me that there's no way to match any derivative. Ie, handle diff(f(x, y), x, 2) and diff(f(x, y), x, 1, y, 1) uniformly for pattern matching.

[ncalexan]

comment:7

I would like to evaluate various derivatives numerically, but there's no obvious way to do so.

sage: def f(v, n): print v; return v**n
....: 
sage: g(x, y).derivative(x, 1).subs( symbolic_diff(g(x, y), w0, w1) == f(w0, w1) )
$0
x

I would like to have my f examine w0 and w1 to determine the correct variable, derivative, etc. But if you look, f gets evaluated with the wild card objects (it must -- that's how python works.)

So I need to match the wild cards, then calculate the function, then substitute. Just longer, that's all.

[golam-m-hossain]

comment:8

Hi Nick,

As you suggested, I moved the "set_diff_derivative_level" into pynac library.
Now I just wrap the function from cython code. I will update the patches,spkg after
you finish your first round of reviewing.