Opened 6 years ago
Last modified 5 years ago
#18092 needs_info enhancement
evaluating symbolic expressions (without conversion to SR, i.e., staying in ring of values)
Reported by:  dkrenn  Owned by:  

Priority:  major  Milestone:  sage6.6 
Component:  symbolics  Keywords:  sd66 
Cc:  cheuberg, mmezzarobba  Merged in:  
Authors:  Daniel Krenn  Reviewers:  
Report Upstream:  N/A  Work issues:  
Branch:  u/dkrenn/SR/eval (Commits)  Commit:  276f0f3d3583c66c7e137350c578e8588c9b236d 
Dependencies:  Stopgaps: 
Description (last modified by )
This ticket proposes a new method evaluate
which can evaluate symbolic expressions at values coming from a ring which not coerces into SR
. The result again lives in the ring of the values. This forces the calculation to be done completely in the given ring (and not in the symbolic ring, where sometimes one does not know exactly what's going on).
For example:
sage: P.<p> = ZZ[[]] sage: E = x.evaluate(x=p) sage: E, E.parent() (p, Power Series Ring in p over Integer Ring)
which is not possible with subs
sage: P.<p> = ZZ[[]] sage: x.subs(x=p) Traceback (most recent call last): ... TypeError: no canonical coercion from Power Series Ring in p over Integer Ring to Symbolic Ring sage: E = x.evaluate(x=p) sage: E, E.parent() p, Power Series Ring in p over Integer Ring)
Change History (33)
comment:1 Changed 6 years ago by
 Branch set to u/dkrenn/SR/eval
comment:2 Changed 6 years ago by
 Cc cheuberg added
 Commit set to c3696a81f65df6e64c5e2bcbdf8905d4f2d5b796
 Status changed from new to needs_review
comment:3 Changed 6 years ago by
comment:4 followup: ↓ 5 Changed 6 years ago by
+ The reason is that :meth:`subs` convert its arguments to the + symbolic ring, so we even have:: + + sage: x.subs(x=RIF(3.42)).parent() + Symbolic Ring + + The :meth:`evaluate`method prevents this conversion and
I think you misunderstand. x
is not converted, it is wrapped in an expression:
sage: x.subs(x=RIF(3.42)).pyobject().parent() Real Interval Field with 53 bits of precision
I have no idea if your idea is worth the effort, but suspect that not if it is only based on the necessity to prevent "conversion".
comment:5 in reply to: ↑ 4 Changed 6 years ago by
 Description modified (diff)
Replying to rws:
I think you misunderstand.
x
is not converted, it is wrapped in an expression:
Ok, I used the wrong word; however, this example was to point out the differences between the two commands.
sage: x.subs(x=RIF(3.42)).pyobject().parent() Real Interval Field with 53 bits of precisionI have no idea if your idea is worth the effort, but suspect that not if it is only based on the necessity to prevent "conversion".
subs
is not possible with something that does go into the symbolic ring, like power series:
sage: P.<p> = ZZ[[]] sage: x.subs(x=p) Traceback (most recent call last): ... TypeError: no canonical coercion from Power Series Ring in p over Integer Ring to Symbolic Ring sage: E = x.evaluate(x=p) sage: E, E.parent() p, Power Series Ring in p over Integer Ring)
comment:6 Changed 6 years ago by
 Keywords sd66 added
comment:7 followup: ↓ 8 Changed 6 years ago by
I thought everything coerces to SR
? Maybe this is just a coercion bug?
comment:8 in reply to: ↑ 7 Changed 6 years ago by
Replying to rws:
I thought everything coerces to
SR
? Maybe this is just a coercion bug?
IMHO, not everything coerces into SR
and this for a good reason. But this is not (or should not) under discussion here.
The following is not possible at with subs:
sage: sage: P.<p> = ZZ[[]] sage: var('a,b') (a, b) sage: (a+b).subs({a: p, b: p^2})
evaluate
can do.
comment:9 Changed 6 years ago by
 Cc mmezzarobba added
comment:10 followup: ↓ 13 Changed 6 years ago by
 Status changed from needs_review to needs_info
Hello,
Why not
sage: E = (1+x).subs(x=RIF(3.42)) sage: E.parent() sage: F = E.pyobject() sage: F 4.4200000000000000? sage: F.parent() Real Interval Field with 53 bits of precision
Vincent
comment:11 Changed 6 years ago by
 Commit changed from c3696a81f65df6e64c5e2bcbdf8905d4f2d5b796 to 7e0be7f3a76c98d5ec3e5250947aec814467048b
comment:12 Changed 6 years ago by
added a doctest and corrected a small bug
comment:13 in reply to: ↑ 10 Changed 6 years ago by
Replying to vdelecroix:
Why not
sage: E = (1+x).subs(x=RIF(3.42)) sage: E.parent() sage: F = E.pyobject() sage: F 4.4200000000000000? sage: F.parent() Real Interval Field with 53 bits of precision
Ok, I see. Maybe RIFs are not a good example since they coerce into SR. Power series are better" example; since there problems.
I'll rewrite the description of the ticket and the examples.
comment:14 followup: ↓ 18 Changed 6 years ago by
The general idea is that the result of arithmetic only depends on the parents of the input data, not on the values of the input data (because the idea is that these things implement maps, which have domains and codomains). When you evaluate a SR element at a nonsymbolic value, you don't know if the result can live in the parent of the original result (e.g., (sin(x)+y).subs(y=1)
).
The appropriate solution is probably to first *convert* your symbolic expression to a parent where the parent is the desired thing, e.g.
sage: f = SR(1+x) sage: R.<t>= ZZ[[]] sage: P=R['x'] sage: P(f)(x=t^2+O(t^3))
This also has other advantages: in principle, when you do this with RIF, you might end up with an evaluation routine that takes into account that the coefficients are not exact and hence it could choose some more stable way of doing the evaluation (I think that's hypotheticallikely no such effort is made right now, but it could).
comment:15 Changed 6 years ago by
 Description modified (diff)
comment:16 Changed 6 years ago by
 Commit changed from 7e0be7f3a76c98d5ec3e5250947aec814467048b to 276f0f3d3583c66c7e137350c578e8588c9b236d
Branch pushed to git repo; I updated commit sha1. New commits:
276f0f3  rewrite documentation (examples) after discussion on trac

comment:17 Changed 6 years ago by
rewritten documentation of function
comment:18 in reply to: ↑ 14 ; followup: ↓ 19 Changed 6 years ago by
Replying to nbruin:
The general idea is that the result of arithmetic only depends on the parents of the input data, not on the values of the input data (because the idea is that these things implement maps, which have domains and codomains). When you evaluate a SR element at a nonsymbolic value, you don't know if the result can live in the parent of the original result (e.g.,
(sin(x)+y).subs(y=1)
).
True.
The appropriate solution is probably to first *convert* your symbolic expression to a parent where the parent is the desired thing, e.g.
sage: f = SR(1+x) sage: R.<t>= ZZ[[]] sage: P=R['x'] sage: P(f)(x=t^2+O(t^3))
What if
f = SR(1+2^x)
or something worse (including e.g. exp, log, sin, ... or other functions)? There are no parents (except SR) for any of these constructs.
comment:19 in reply to: ↑ 18 ; followups: ↓ 20 ↓ 31 Changed 6 years ago by
Replying to dkrenn:
What if
f = SR(1+2^x)or something worse (including e.g. exp, log, sin, ... or other functions)? There are no parents (except SR) for any of these constructs.
And indeed it's tricky to evaluate the result. What is 2^<power series>
? I guess exp(log(2)*x)
, which requires a ring that contains both log(2)
and inverses of all integers., so that doesn't work in Z[[t]]
. I think Sage is right in putting the onus on the user to first find a parent in which the expression fits and where the evaluation behaviour is the desired one.
Anyway, fast_callable
takes a best effort approach towards compiling a program that tries to perform the evaluation, so that might be your best bet.
comment:20 in reply to: ↑ 19 Changed 6 years ago by
Replying to nbruin:
Anyway,
fast_callable
takes a best effort approach towards compiling a program that tries to perform the evaluation, so that might be your best bet.
Ok, I'll make some experiments and run some tests to see if it satisfies my needs.
Thanks
comment:21 followup: ↓ 22 Changed 6 years ago by
Hello,
Would this ticket solve the following issue (from #9787)?
sage: parent(exp(1.2)) Real Field with 53 bits of precision sage: f(x) = exp(x) sage: parent(f(1.2)) Symbolic Ring
Vincent
comment:22 in reply to: ↑ 21 ; followup: ↓ 23 Changed 6 years ago by
Replying to vdelecroix:
Hello,
Would this ticket solve the following issue (from #9787)?
sage: parent(exp(1.2)) Real Field with 53 bits of precision sage: f(x) = exp(x) sage: parent(f(1.2)) Symbolic Ring
Yes.
sage: f(x).evaluate({x: 1.2}).parent() Real Field with 53 bits of precision
comment:23 in reply to: ↑ 22 ; followup: ↓ 24 Changed 6 years ago by
Replying to dkrenn:
Replying to vdelecroix:
Hello,
Would this ticket solve the following issue (from #9787)?
sage: parent(exp(1.2)) Real Field with 53 bits of precision sage: f(x) = exp(x) sage: parent(f(1.2)) Symbolic RingYes.
sage: f(x).evaluate({x: 1.2}).parent() Real Field with 53 bits of precision
Sorry. This was not my question. What would be parent(f(1.2))
? Is this modified by this ticket?
comment:24 in reply to: ↑ 23 ; followup: ↓ 25 Changed 6 years ago by
Replying to vdelecroix:
Replying to dkrenn:
Replying to vdelecroix:
Would this ticket solve the following issue (from #9787)?
sage: parent(exp(1.2)) Real Field with 53 bits of precision sage: f(x) = exp(x) sage: parent(f(1.2)) Symbolic Ring[...]
Sorry. This was not my question. What would be
parent(f(1.2))
? Is this modified by this ticket?
No, not modified by this ticket.
comment:25 in reply to: ↑ 24 ; followup: ↓ 26 Changed 6 years ago by
Replying to dkrenn:
Replying to vdelecroix:
Replying to dkrenn:
Replying to vdelecroix:
Would this ticket solve the following issue (from #9787)?
...
Sorry. This was not my question. What would be
parent(f(1.2))
? Is this modified by this ticket?No, not modified by this ticket.
I saw too late your answer on #9878 ;)
By the way, let me repeat another question from #9878. I found the behavior of evaluate
in your comment:22 very weird. I thought it was a modification of .subs
in order to take care of the parent. But
sage: f(x) = 2*x sage: f.subs(x=3) x > 6
ie, f
remains a function. It is hopefully not changed into a number.
comment:26 in reply to: ↑ 25 ; followup: ↓ 27 Changed 6 years ago by
Replying to vdelecroix:
By the way, let me repeat another question from #9878. I found the behavior of
evaluate
in your comment:22 very weird. I thought it was a modification of.subs
in order to take care of the parent.
In the following it does the same as subs:
sage: sage: f(x) = 2*x sage: sage: f(x).subs(x=3) 6 sage: sage: f(x).evaluate(x=3) 6
But
sage: f(x) = 2*x sage: f.subs(x=3) x > 6ie,
f
remains a function. It is hopefully not changed into a number.
Indeed, this changes (I wasn't aware of this up to now):
sage: sage: f.evaluate(x=3) 6
This is because evaluate
uses
sage: f.operator() <builtin function mul> sage: f.operands() [x, 2]
From this, f
is equal to 2*x
. Sage sees these two as equal as well:
sage: bool(f == 2*x) True
comment:27 in reply to: ↑ 26 ; followup: ↓ 29 Changed 6 years ago by
Replying to dkrenn:
Replying to vdelecroix:
By the way, let me repeat another question from #9878. I found the behavior of
evaluate
in your comment:22 very weird. I thought it was a modification of.subs
in order to take care of the parent.From this,
f
is equal to2*x
. Sage sees these two as equal as well:sage: bool(f == 2*x) True
Argh. Definitely a bug to me. Another bug is that the variable defining a function should be transparent. And currently
sage: f(x) = 2*x sage: g(y) = 2*y sage: bool(f == g) False
Vincent
comment:28 Changed 6 years ago by
Please Cc: me with any ticket you open regarding Expression.nonzero()
or pynac.
comment:29 in reply to: ↑ 27 Changed 6 years ago by
Replying to vdelecroix:
sage: f(x) = 2*x
From this,
f
is equal to2*x
. Sage sees these two as equal as well:sage: bool(f == 2*x) TrueArgh. Definitely a bug to me. Another bug is that the variable defining a function should be transparent. And currently
sage: f(x) = 2*x sage: g(y) = 2*y sage: bool(f == g) False
This is now #18259.
comment:30 followup: ↓ 32 Changed 5 years ago by
As to the original subs
error, nbruin has explained why there is no general solution, a workaround for polynomials would be
sage: P.<p> = ZZ[[]] sage: x.power_series(ZZ) x + O(x^2) sage: P(_) p + O(p^2)
I believe a more general way to have all possibilities of both SR
and the series rings is to fix conversions between them, and use a series ring over SR
. This depends on #17659, please review.
comment:31 in reply to: ↑ 19 Changed 5 years ago by
Replying to nbruin:
Anyway,
fast_callable
takes a best effort approach towards compiling a program that tries to perform the evaluation, so that might be your best bet.
I had another instance where I needed a version of .subs
like in this ticket, because there is no coercion from a number field to the symbolic ring.
sage: K.<omega> = NumberField(x^4 + 1) sage: var('u') sage: z = u/(u + 1)^2 sage: z.subs(u=omega) Traceback (most recent call last): ... TypeError: no canonical coercion from Number Field in omega with defining polynomial x^16 + 1 to Symbolic Ring
Using fast_callable
works in this case:
sage: fast_callable(z, vars=[u])(omega) 1/2*omega^3  1/2*omega + 1
It works, but the solution is hard to find and the notation a bit cumbersome.
I see several solutions:
 adding a link to
fast_callable
and some examples from this ticket to the documentation ofsubs
.  indeed create a method as proposed here which acts as a wrapper for
fast_callable
.
Opinions?
New commits:
create method eval
rename eval to evaluate
docstring: output clearified
rename ring to convert_to
implement automatic detection when convert_to=None
examples rewritten
add seealsoblock
fix typo
docstring for helper function _evaluate_
fix links in doc