[sage-support] Re: substitution in list

[Rogerio]

Besides the pure python options pointed above, if your list is filled by 
symbolic expressions, you can also turn it into a vector and do the 
substitutions straightforwardly.

Given 

x = list(var('x', n=10))
ep = list(var('epsilon', n=10))

a random sublist

my_list = [x[randint(0,8)] for _ in range(6)]

and a substitution rule

subs_rule = dict(zip(x[4:], ep))

You can make 

my_vec_list = vector(my_list).subs(subs_rule)

and turn back to a list

my_list2 = list(my_vec_list)
Em terça-feira, 28 de dezembro de 2021 às 15:03:51 UTC-3, Nils Bruin 
escreveu:

> That's a python question. See for instance,
>
>
> https://stackoverflow.com/questions/2582138/finding-and-replacing-elements-in-a-list
>
> If you scroll down, there are some suggestions there that deal with 
> multiple replacements as well
>
> On Tuesday, 28 December 2021 at 06:09:35 UTC-8 cyrille piatecki wrote:
>
>> First I have seen that perhaps my question has an answer in AskSagemath 
>> but currently it doesn't answer.
>>
>> Suppose I have a list of variables x_1, x_2, x_3,...x_n
>>
>> I have some sub-list of variables say [x_4, x_1, x_6...]. But whenever 
>> x_4, x_5,x_6 belong to this list I want the substitution x_4 ->epsilon_0, 
>> x_5 -> epsilon_1... (this is an only an example. Is there a way to do this 
>> in the same way that we have a substitution in a fonction in Sagemath?  
>>
>

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[sage-support] Re: substitution in list

[Nils Bruin]

That's a python question. See for instance,

https://stackoverflow.com/questions/2582138/finding-and-replacing-elements-in-a-list

If you scroll down, there are some suggestions there that deal with 
multiple replacements as well

On Tuesday, 28 December 2021 at 06:09:35 UTC-8 cyrille piatecki wrote:

> First I have seen that perhaps my question has an answer in AskSagemath 
> but currently it doesn't answer.
>
> Suppose I have a list of variables x_1, x_2, x_3,...x_n
>
> I have some sub-list of variables say [x_4, x_1, x_6...]. But whenever 
> x_4, x_5,x_6 belong to this list I want the substitution x_4 ->epsilon_0, 
> x_5 -> epsilon_1... (this is an only an example. Is there a way to do this 
> in the same way that we have a substitution in a fonction in Sagemath?  
>

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[sage-support] Re: substitution of lambda variable

[Mandeep Singh]

On Thursday, May 28, 2009 at 9:31:46 PM UTC+5:30, Jason Grout wrote:
Mike Hansen wrote: 
> Hello, 
> 
> On Thu, May 28, 2009 at 8:38 AM, Paul Sargent  wrote: 
>> # Subs for lambda (have to use "lambda", but that's a keyword) 

lambda is a reserved keyword in Python, which means you can't use it 
in 
a python statement that you are writing. You'll notice below that Mike 
never writes lambda on a line (he only writes the string "lambda"). 

Yeah, I know; it's frustrating for me in linear algebra to not use 
lambda for eigenvalues. 


Can we declare lambda as a variable? I want to use it in eigenvalues 
(linear algebra).
like:
l = var("lambda");
It'll give error. So how can I declare it as a variable.

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[sage-support] Re: substitution of lambda variable

[Nils Bruin]

On Wednesday, September 16, 2015 at 1:54:40 AM UTC-7, Mandeep Singh wrote:
>
> Can we declare lambda as a variable? I want to use it in eigenvalues 
> (linear algebra).
> like:
> l = var("lambda");
> It'll give error. So how can I declare it as a variable.
>

This works. I don't know if it will keep working:

sage: l=SR.symbol('lambda')
sage: l
lambda
sage: l^2+l+1
lambda^2 + lambda + 1

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[sage-support] Re: Substitution in infinite polynomial rings

[Simon King]

Hi Bruno,

I am sorry that I (as author of the InfinitePolynomialRing stuff) did
not answer before.

On 2014-04-16, BJ brunojo...@gmail.com wrote:
 The output looks something like this:

 [-e_8 + e_4^2, -e_10 + e_6*e_4, -1382*e_12 + 2205*e_8*e_4 + 500*e_6^2 -
 1323*e_4^3, -10*e_14 + 21*e_10*e_4 + 22*e_8*e_6 - 33*e_6*e_4^2,
 45606*e_12*e_4 + 65000*e_10*e_6 + 42042*e_8^2 - 63063*e_8*e_4^2 -
 71500*e_6^2*e_4, 126126*e_14*e_4 + 212828*e_12*e_6 + 309582*e_10*e_8 -
 154791*e_10*e_4^2 - 378378*e_8*e_6*e_4 - 71500*e_6^3, 858000*e_14*e_6 +
 1337776*e_12*e_8 - 501666*e_12*e_4^2 + 760500*e_10^2 -
 1287000*e_10*e_6*e_4 - 693693*e_8^2*e_4 - 786500*e_8*e_6^2]


 These list elements should be interpreted as relations between the 
 variables e_i (for even i), so that, for instance, the first one should be 
 interpreted as e_8 = e_4^2.

 Moreover, I know that these relations should allow me to inductively 
 express each e_k in terms of e_4 and e_6.

The original purpose of InfinitePolynomialRing is to implement the algorithm of
Aschenbrenner and Hillar for the computation of symmetric Gröbner bases. This
means: You would be in a situation where any permutation of indices applied
to a relation between the e[i] results in another relation between the
e[i]. In your example, you could easily deduce that all coefficients
would be trivial if you would allow to freely permute indices:

 sage: SI = M.ideal(L)
 sage: SG = SI.groebner_basis()
 sage: SG
 [e_1]

Do you have any permutation action at all on your coefficients? Say, if
you have a relation then shifting all coefficients by 1 will result in
another relation? If I recall correctly, it is not implemented yet, but
it should be possible (without being able to guarantee termination of
the algorithm, though) to consider a subgroup of the full permutation
group.

 For instance, the second relation 
 shows that e_10 = e_4e_6. The third relation -1382*e_12 + 2205*e_8*e_4 + 
 500*e_6^2 -1323*e_4^3 = 0 allows me (using a pen and paper...) to express 
 e_12 only in terms of e_4 and e_6, by using e_8 = e_4^2.

 What I would like is to be able, for each k in the appropriate range, to 
 obtain a polynomial f_k(u,v) with rational coefficients, such that f_k(e_4, 
 e_6) = e_k.

Well, if you have any *finite* list of relations and no index
permutations on it, then you could simply use classical (non-symmetric)
Gröbner bases. In fact, by your construction, all element of your list L
are classical polynomials. Hence, it should be easy to get a
polynomial ring with the variables involved in this list, and hopefully
such that e_je_i for ji, because then the Gröbner basis will tell you
how to express e_k in terms of e_4,e_6 for k6.

Let's try:

  sage: I = ideal(L)
  sage: I
  Ideal (-e_8 + e_4^2, -e_10 + e_6*e_4, -1382*e_12 + 2205*e_8*e_4 +
  500*e_6^2 - 1323*e_4^3, -10*e_14 + 21*e_10*e_4 + 22*e_8*e_6 -
  33*e_6*e_4^2, 45606*e_12*e_4 + 65000*e_10*e_6 + 42042*e_8^2 -
  63063*e_8*e_4^2 - 71500*e_6^2*e_4, 126126*e_14*e_4 + 212828*e_12*e_6 +
  309582*e_10*e_8 - 154791*e_10*e_4^2 - 378378*e_8*e_6*e_4 -
  71500*e_6^3, 858000*e_14*e_6 + 1337776*e_12*e_8 - 501666*e_12*e_4^2 +
  760500*e_10^2 - 1287000*e_10*e_6*e_4 - 693693*e_8^2*e_4 -
  786500*e_8*e_6^2) of Multivariate Polynomial Ring in e_14, e_12, e_10,
  e_8, e_6, e_4, e_0 over Rational Field
  sage: G
  [e_14 - e_6*e_4^2, e_12 - 250/691*e_6^2 - 441/691*e_4^3, e_10 -
  e_6*e_4, e_8 - e_4^2, e_6^3 + 38367/5500*e_6*e_4^3, e_6^2*e_4 +
  1617/2000*e_4^4, e_6*e_4^4, e_4^5]
  sage: P = G.ring()
  sage: P
  Multivariate Polynomial Ring in e_14, e_12, e_10, e_8, e_6, e_4, e_0
  over Rational Field
  sage: P('e_10').reduce(G)
  e_6*e_4
  sage: P('e_12').reduce(G)
  250/691*e_6^2 + 441/691*e_4^3
  sage: P('e_14').reduce(G)
  e_6*e_4^2

Does this solve your problem?

I am very sorry for the late answer.

Best regards,
Simon



  



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[sage-support] Re: Substitution in infinite polynomial rings

[Nils Bruin]

On Wednesday, April 16, 2014 4:16:30 PM UTC-7, BJ wrote:

 I have the following code, which produces a list of polynomials in the 
 infinite number of variables e_0, e_1, ...

 M.e = InfinitePolynomialRing(QQ, implementation=sparse) 

 However, I've been having a lot of trouble figuring out how to do 
 substitutions in these polynomial rings in order to get what I want.



I think the following should be the recommended way of doing this, but 
currently it doesn't  work:

sage: f = e[1]^2+e[2]^3
sage: f.subs({e[1]: 2})   #this doesn't work
4+e_2^3

the following does work:

sage: f(e_1=2)
4+e_2^3

but it's flawed:

sage: f(e_4=2)
KeyError: 'e_4'

The problem seems to be that the standard subs routines expect the parent 
to have a finite, predetermined sequence of generators. That's of course 
not the case for your rings. I think InfinitePolynomialRing has to override 
more of the methods involved.

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[sage-support] Re: Substitution in infinite polynomial rings

[Nils Bruin]

On Thursday, April 17, 2014 9:39:09 AM UTC-7, Nils Bruin wrote:


 but it's flawed:

 sage: f(e_4=2)
 KeyError: 'e_4'


And also flawed in a different way:

 sage: f(e_2=e[4])
TypeError: unsupported operand parent(s) for '+': 'Multivariate Polynomial 
Ring in e_4, e_2, e_1 over Rational Field' and 'Multivariate Polynomial 
Ring in e_4, e_0 over Rational Field

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[sage-support] Re: substitution

[P Purkayastha]

On 01/09/2013 09:27 AM, Michael Beeson wrote:

sage: K.d,e,p,g,m,f,u,v,j,N =
FractionField(PolynomialRing(QQ,10,'depgmfuvjN'))
sage: R.s = K[]
sage: w=u
sage: u=0
sage: w
u


Why doesn't Sage answer 0 for the value of w here? More generally, if I
have some complicated expression and I assign a value to one of its
variables,
Sage knows the value I've just assigned, but it doesn't propagate into
the value of previously defined expressions using that variable. Please
comment
and please tell me how to force the substitution to take effect in
previously defined expressions.


You can not have this kind of thing. w is not a pointer to u. w in 
this case contains a copy of u.



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Re: [sage-support] Re: substitution

[John Cremona]

Your first line defines a function field of several variables and
assigns values to the identifiers d,p,e, etc so that in particular, u
is assigned a value which is the 6th generator of that field.  When
you set w=u you have a new identifier called w whose value is a copy
of u's value, so is also equal to the 6th geberator of the field.
Then you set u=0 whcih just gives a new value to the identifier u,
namely 0;  this has no effect on w!  Before, both u and w had the same
value;  now they don't.

You can substitute values for all the variables in a different way.
After w=(some expression in d,e,...,N) you can evaluate
w([d0,e0,...,N0]), but you would have to give values to all the
variables (which could be themselves, e.g. to evaluate at u=0 you
could do w([d,e,p,g,m,f,0,v,j,N]).

This is probably differnt from what happens with a symbolic
manipulation system such as Maple, or even Sage's own symbolic ring --
but someone else would have to tell you about that, as I do not use
it.

John Cremona

On 9 January 2013 08:04, P Purkayastha ppu...@gmail.com wrote:
 On 01/09/2013 09:27 AM, Michael Beeson wrote:

 sage: K.d,e,p,g,m,f,u,v,j,N =
 FractionField(PolynomialRing(QQ,10,'depgmfuvjN'))
 sage: R.s = K[]
 sage: w=u
 sage: u=0
 sage: w
 u


 Why doesn't Sage answer 0 for the value of w here? More generally, if I
 have some complicated expression and I assign a value to one of its
 variables,
 Sage knows the value I've just assigned, but it doesn't propagate into
 the value of previously defined expressions using that variable. Please
 comment
 and please tell me how to force the substitution to take effect in
 previously defined expressions.


 You can not have this kind of thing. w is not a pointer to u. w in this
 case contains a copy of u.



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[sage-support] Re: substitution in sums and multiplications

[Simon King]

Hi Burcin,

On Jan 25, 4:41 pm, Burcin Erocal bur...@erocal.org wrote:
 ...
 In your example, (k1 + k2) is not a subexpression of f, so there is
 nothing to substitute.

In other words, (k1+k2)*2 is automatically turned into 2*k1+2*k2, but
(k1+k2)^2 is not turned into k1^2+2*k1*k2+k2^2 ?

Why?

Cheers,
Simon

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[sage-support] Re: Substitution

[Stochastix]

Thanks. Unfortunately, for my particular example, it didn't work as is

--
| Sage Version 4.2, Release Date: 2009-10-24 |
| Type notebook() for the GUI, and license() for information.|
--
sage: alpha=700
sage: mu1=0.1
sage: mu2=0.2
sage: pI=0.8
sage: l=5
sage: r=1.0
sage: b=r+l
sage: c=r*mu1+l*(1-pI)*mu2
sage: a=lambda x: alpha*x-mu1+mu2
sage: f=lambda x: (a(x)*b-c+sqrt((a(x)*b-c)^2+4.0*a(x)*b*r*mu1))/(2*a
(x)*mu1)
sage: g=lambda x: (r+l-mu1*f(x))/mu2
sage: prev=lambda x: f(x)/(f(x)+g(x))
sage: k=lambda x: diff(prev(x),x)
sage: k(x=0.03)
---
TypeError Traceback (most recent call
last)

/Applications/sage/ipython console in module()

/Applications/sage/ipython console in lambda(x)

/Applications/sage/local/lib/python2.6/site-packages/sage/calculus/
functional.pyc in derivative(f, *args, **kwds)
133 if not isinstance(f, Expression):
134 f = SR(f)
-- 135 return f.derivative(*args, **kwds)
136
137 diff = derivative

/Applications/sage/local/lib/python2.6/site-packages/sage/symbolic/
expression.so in sage.symbolic.expression.Expression.derivative (sage/
symbolic/expression.cpp:11427)()

/Applications/sage/local/lib/python2.6/site-packages/sage/misc/
derivative.so in sage.misc.derivative.multi_derivative (sage/misc/
derivative.c:2175)()

/Applications/sage/local/lib/python2.6/site-packages/sage/symbolic/
expression.so in sage.symbolic.expression.Expression._derivative (sage/
symbolic/expression.cpp:11721)()

TypeError: argument symb must be a symbol


I had to type

sage: k(x).subs(x=0.03)
0.0262047639227205


By the way, there is something still puzzling me. The equivalent Maple
code gives a value of .883.
Who should I believe ?
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[sage-support] Re: Substitution

[Alasdair]

This simple minded approach seems to work for me:

sage: a(x)=2*x
sage: b(x)=a(x)^2
sage: g(x)=diff(b(x),x)
sage: g(2)
  16

-Alasdair


On Jan 21, 2:14 am, Stochastix laurent.decreusef...@gmail.com wrote:
 I have the following problem : I have two nested functions

 a=lambda x: 2*x
 b=lambda x: a(x)^2

 Everything goes well as long as I want to compute the derivative of b
 but how can I evaluate this derivative at x=2.
 I tried

 g=lambda x: diff(b(x),x)
 g(x) returns 8*x as expected

 but g(2)=0.

 In other words, how can we substitute x=2 to the expression 8*x ?
  I guess this is somewhere in the docs but I didn't find where.
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[sage-support] Re: Substitution

[Jason Bandlow]

Hi,

I think what Minh was trying to say is that these lines:

Stochastix wrote:
 sage: a=lambda x: alpha*x-mu1+mu2
 sage: f=lambda x: (a(x)*b-c+sqrt((a(x)*b-c)^2+4.0*a(x)*b*r*mu1))/(2*a
 (x)*mu1)
 sage: g=lambda x: (r+l-mu1*f(x))/mu2
 sage: prev=lambda x: f(x)/(f(x)+g(x))
 sage: k=lambda x: diff(prev(x),x)

should instead be written as

a = alpha*x-mu1+mu2
f = (a(x)*b-c+sqrt((a(x)*b-c)^2+4.0*a(x)*b*r*mu1))/(2*a(x)*mu1)
...

Or, if you prefer,

a(x) = alpha*x-mu1+mu2
f(x) = ...

etc.  When you define functions this way, they become elements of the
'SymbolicRing' which Sage has a lot of tools to deal with.  Defining
them with 'lambda' makes them pure Python functions, which are not as
flexible.

 sage: k(x).subs(x=0.03)
 0.0262047639227205
 
 
 By the way, there is something still puzzling me. The equivalent Maple
 code gives a value of .883.
 Who should I believe ?
 

If you fix the errors as above, do you still get different answers in
Maple and Sage?


Best,
Jason

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[sage-support] Re: Substitution

[Jason Grout]

Stochastix wrote:

I have the following problem : I have two nested functions

a=lambda x: 2*x
b=lambda x: a(x)^2

Everything goes well as long as I want to compute the derivative of b
but how can I evaluate this derivative at x=2.
I tried

g=lambda x: diff(b(x),x)
g(x) returns 8*x as expected

but g(2)=0.




What is happening is that since g is a lambda function, g(something) is 
literally returning the result of diff(b(something), something).  So 
g(x) is returning the result of the command diff(b(x),x), and g(2) is 
returning the result of diff(b(2),2).


The key is not using lambda in this case.  Other posts have shown how to 
do what you want.


Thanks,

Jason


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[sage-support] Re: substitution

[Burcin Erocal]

On Wed, 7 Oct 2009 13:20:30 -0700 (PDT)
ma...@mendelu.cz ma...@mendelu.cz wrote:

 BTW: When looked at this problem, the notation D[0](alpha)(r)^2  seems
 to be unusual to me.
 Is it possible to fix sage so that it prints derivatives of functions
 in one variable
 as usual: alpha'(r) ?

There is a patch attached to #6344 to change printing to alpha'(r) in
latex mode. The syntax alpha'(r) would not be accepted on the command
line without modifying the preparser, so normal printing is still a
variant of the D notation. 

We can switch to using the diff notation when the function takes only
one argument which is a symbol. As Maple does here:

maple: diff(f(x),x);
diff(f(x),x)
maple: diff(f(x,x),x);
D[1](f)(x,x)+D[2](f)(x,x)
maple: diff(f(x^2),x);
2*D(f)(x^2)*x


 And is it possible to get sin^2(x) ans (alpha'(r))^2 in LaTeX output
 instead of sin(x)^2 and y'(r)^2?

I don't think sin^2(x) is possible, (y'(r))^2 can be done. If we switch
to that notation, I can make the necessary changes in pynac.


Thanks.

Burcin

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[sage-support] Re: substitution

[Ranjit]

Sorry I should make my question clearer. I'm not trying to substitute
for sigma^2. What I want to do is replace the various constants
multiplying the terms in sigma with a single constant. For example  I
get the following term in sigma^2:

-1/2*(1/R^2 - beta_0/(R^2*beta))*sigma(r)^2

And I want to get:

-epsilon*sigma(r)^2

And basically I want to do this for each power or derivative of sigma
in the expression I get. I've tried using subs(), but for some reason
for my case it doesn't work, but for a simple example I tried it does
and I don't see the difference.

When I try

var('epsilon')
(-1/2*(1/R^2 - beta_0/(R^2*beta))*sigma(r)^2).subs(-1/2*(1/R^2 -
beta_0/(R^2*beta))==epsilon)

I get back the expression unchanged.

But when I try:

var('a b c')
((a+b)*x^2).subs((a+b)==c)

the substitution does work. What's the difference?




On Oct 7, 4:20 pm, ma...@mendelu.cz ma...@mendelu.cz wrote:
 On 7 říj, 16:21, Ranjit rjcha...@gmail.com wrote:





  I'm trying to figure out how to do a simple substitution. I've the
  following code:

  var('r beta beta_0 R a h')
  psi=function('psi',r)
  sigma=function('sigma',r)

  H_0=function('H_0',r)

  H_grad = lambda psi : (1/2)*(beta/beta_0)*R^2*(psi.diff(r))^2
  H_0(psi) = -(1/2)*(beta/beta_0-1)*psi^2 + beta*a*psi^4 + beta*h*psi

  H_1=H_grad((1/R)*(beta_0/beta)^(1/2)*sigma)
  H=H_1+H_0((1/R)*(beta_0/beta)^(1/2)*sigma)

  H=H.simplify_radical().expand().collect(sigma^2)
  H

  Let's say I want to rewrite the coefficient on sigma^2 as simply
  epsilon, how can I do that?

 perhaps like this (I used alpha indstead of epsilon)

 alpha=function('alpha',r)
 H.subs_expr(sigma==sqrt(alpha)).subs_expr(diff(sigma)==diff(sqrt
 (alpha)))

 Not exactly what you want, but similar.

 BTW: When looked at this problem, the notation D[0](alpha)(r)^2  seems
 to be unusual to me.
 Is it possible to fix sage so that it prints derivatives of functions
 in one variable
 as usual: alpha'(r) ?

 And is it possible to get sin^2(x) ans (alpha'(r))^2 in LaTeX output
 instead of sin(x)^2 and y'(r)^2?

 Robert Marik
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[sage-support] Re: substitution

[ma...@mendelu.cz]

perhaps, no subst is necessary:

var('r beta beta_0 R a h')
psi=function('psi',r)
sigma=function('sigma',r)

H_0=function('H_0',r)

H_grad = lambda psi : (1/2)*(beta/beta_0)*R^2*(psi.diff(r))^2
H_0(psi) = -(1/2)*(beta/beta_0-1)*psi^2 + beta*a*psi^4 + beta*h*psi

H_1=H_grad((1/R)*(beta_0/beta)^(1/2)*sigma)
H=H_1+H_0((1/R)*(beta_0/beta)^(1/2)*sigma)

H=H.simplify_radical().expand().collect(sigma^2)
show(H)
coef_sigma=(H.coeff(sigma^2))
show(coef_sigma)
show(H)
H=H-(coef_sigma-epsilon)*sigma^2
show(H)
H=H.collect(sigma^2)
H

R.M.

On 8 říj, 15:27, Ranjit rjcha...@gmail.com wrote:
 Sorry I should make my question clearer. I'm not trying to substitute
 for sigma^2. What I want to do is replace the various constants
 multiplying the terms in sigma with a single constant. For example  I
 get the following term in sigma^2:

 -1/2*(1/R^2 - beta_0/(R^2*beta))*sigma(r)^2

 And I want to get:

 -epsilon*sigma(r)^2


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[sage-support] Re: substitution

[Francois Maltey]

Ranjit a écrit :
 I'm trying to figure out how to do a simple substitution. I've the
 following code:

 var('r beta beta_0 R a h')
 psi=function('psi',r)
 sigma=function('sigma',r)

 H_0=function('H_0',r)

 H_grad = lambda psi : (1/2)*(beta/beta_0)*R^2*(psi.diff(r))^2
 H_0(psi) = -(1/2)*(beta/beta_0-1)*psi^2 + beta*a*psi^4 + beta*h*psi

 H_1=H_grad((1/R)*(beta_0/beta)^(1/2)*sigma)
 H=H_1+H_0((1/R)*(beta_0/beta)^(1/2)*sigma)

 H=H.simplify_radical().expand().collect(sigma^2)
 H
   
The formula.subs_expr (equation) doesn't recognize all sub-product or 
all sub-sum, but recognize ONE term in a sum, even a complex one, so I 
test with succes the code :

coef = H.coefficient(sigma(r), 2)
H.subs_expr(coef*sigma(r)^2==epsilon*sigma(r)^2)

epsilon*sigma(r)^2 + sqrt(beta)*sqrt(beta_0)*h*sigma(r)/R + 
a*beta_0^2*sigma(r)^4/(R^4*beta) + 1/2*D[0](sigma)(r)^2

Do I answer to your question ?

F.

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[sage-support] Re: substitution

[Ranjit]

This works, though it would have been nicer if epsilon actually
equalled what it was replacing. I just tried

(-1/2*(1/R^2 - beta_0/(R^2*beta))*sigma(r)^2).subs((1/R^2 - beta_0/
(R^2*beta))==epsilon)

which works, where

(-1/2*(1/R^2 - beta_0/(R^2*beta))*sigma(r)^2).subs((1/2)*(1/R^2 -
beta_0/(R^2*beta))==epsilon)

does not. So I guess there's an upper limit to the number of
operations that the expression being replaced can have to work in subs
().


On Oct 8, 12:52 pm, Francois Maltey fmal...@nerim.fr wrote:
 Ranjit a écrit :



  I'm trying to figure out how to do a simple substitution. I've the
  following code:

  var('r beta beta_0 R a h')
  psi=function('psi',r)
  sigma=function('sigma',r)

  H_0=function('H_0',r)

  H_grad = lambda psi : (1/2)*(beta/beta_0)*R^2*(psi.diff(r))^2
  H_0(psi) = -(1/2)*(beta/beta_0-1)*psi^2 + beta*a*psi^4 + beta*h*psi

  H_1=H_grad((1/R)*(beta_0/beta)^(1/2)*sigma)
  H=H_1+H_0((1/R)*(beta_0/beta)^(1/2)*sigma)

  H=H.simplify_radical().expand().collect(sigma^2)
  H

 The formula.subs_expr (equation) doesn't recognize all sub-product or
 all sub-sum, but recognize ONE term in a sum, even a complex one, so I
 test with succes the code :

 coef = H.coefficient(sigma(r), 2)
 H.subs_expr(coef*sigma(r)^2==epsilon*sigma(r)^2)

 epsilon*sigma(r)^2 + sqrt(beta)*sqrt(beta_0)*h*sigma(r)/R +
 a*beta_0^2*sigma(r)^4/(R^4*beta) + 1/2*D[0](sigma)(r)^2

 Do I answer to your question ?

 F.
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[sage-support] Re: substitution

[ma...@mendelu.cz]

On 8 říj, 19:41, Ranjit rjcha...@gmail.com wrote:

 does not. So I guess there's an upper limit to the number of
 operations that the expression being replaced can have to work in subs
 ().

Perhaps true, but IMHO very risky guess which is not true, if pattern
matching in Maxima is behind the scene when doing substitution.

Robert Marik

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[sage-support] Re: substitution

[ma...@mendelu.cz]

On 7 říj, 16:21, Ranjit rjcha...@gmail.com wrote:
 I'm trying to figure out how to do a simple substitution. I've the
 following code:

 var('r beta beta_0 R a h')
 psi=function('psi',r)
 sigma=function('sigma',r)

 H_0=function('H_0',r)

 H_grad = lambda psi : (1/2)*(beta/beta_0)*R^2*(psi.diff(r))^2
 H_0(psi) = -(1/2)*(beta/beta_0-1)*psi^2 + beta*a*psi^4 + beta*h*psi

 H_1=H_grad((1/R)*(beta_0/beta)^(1/2)*sigma)
 H=H_1+H_0((1/R)*(beta_0/beta)^(1/2)*sigma)

 H=H.simplify_radical().expand().collect(sigma^2)
 H

 Let's say I want to rewrite the coefficient on sigma^2 as simply
 epsilon, how can I do that?


perhaps like this (I used alpha indstead of epsilon)

alpha=function('alpha',r)
H.subs_expr(sigma==sqrt(alpha)).subs_expr(diff(sigma)==diff(sqrt
(alpha)))

Not exactly what you want, but similar.

BTW: When looked at this problem, the notation D[0](alpha)(r)^2  seems
to be unusual to me.
Is it possible to fix sage so that it prints derivatives of functions
in one variable
as usual: alpha'(r) ?

And is it possible to get sin^2(x) ans (alpha'(r))^2 in LaTeX output
instead of sin(x)^2 and y'(r)^2?

Robert Marik

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[sage-support] Re: substitution of lambda variable

[Jason Grout]

Mike Hansen wrote:
 Hello,
 
 On Thu, May 28, 2009 at 8:38 AM, Paul Sargent psa...@gmail.com wrote:
 # Subs for lambda (have to use lambda, but that's a keyword)

 sage: e1.subs(lambda = 3)
 
   File ipython console, line 1
 e1.subs(lambda = Integer(3))
^
 SyntaxError: invalid syntax

 For some reason I don't understand, whilst .solve() will take the
 symbolic variable as a parameter to solve for (e.g. t), .subs()
 requires the variable name (e.g. theta). Often these are the same,
 but not always, and in the case where I have a variable lambda it
 causes a syntax error.


lambda is a reserved keyword in Python, which means you can't use it in 
a python statement that you are writing.  You'll notice below that Mike 
never writes lambda on a line (he only writes the string lambda).

Yeah, I know; it's frustrating for me in linear algebra to not use 
lambda for eigenvalues.

Jason





 Any way to get around this? Is this the intended syntax for subs(), as
 it seems a little confused?
 
 The subs() method can take various types of input -- you can look a
 the docstring to see examples of these.  Here's one way to do what you
 want:
 
 sage: l = var(lambda)
 sage: t = var(theta)
 sage: e1 = t == l^2
 sage: e2 = e1.solve(l)
 sage: (e2[0].subs({t:5}), e2[1].subs({t:5}))
 (lambda == -sqrt(5), lambda == sqrt(5))
 sage: sage: e1.subs({l:3})
 theta == 9
 
 In Sage 4.0, you'll be able to do
 
 sage: sage: e1.subs(l==3)
 theta == 9
 
 --Mike
 
  
 


-- 
Jason Grout


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[sage-support] Re: Substitution with symbolic equations

[Stan Schymanski]

Not sure if this is what you are after, but the following would give you 
the solution:
sage: solve([e1,e2],c,a)
[[c == (d - b*e)/e, a == d/e]]

You can give n equations to solve and solve for n variables. Solve will 
insert one into another automatically.

An equation has a different syntax to a substitution rule, so if you 
want to substitute what is in an equation, you could write
sage: e2(a=eq1.rhs())

Not sure if there is a better way of transforming equations into 
substitution rules. Perhaps the dict() function could help? You can 
substitute a dictionary of variables, e.g.
sage: vars = dict(a = b + c)
sage: e2.subs(vars)

However, again the dict() arguments have a single equal sign, not a 
double one as an equation. I'm sure someone can explain this much more 
elegantly.

Regards,
Stan


Paul Sargent wrote:
 Hi,

 I keep running into a road block which I think means either I'm
 missing something simple, or I'm thinking about things the wrong way.
 I'm fairly new to sage, and CAS in general, so either is possible.

 Here's a simple example of what I'm doing.

 Lets give ourselves two symbolic equations:

 sage: var(a b c d e)
 sage: e1 = a == b + c
 sage: e2 = d == e * a

 Now, lets say I want to know what c is in terms of b, d  e. By hand
 I'd substitute e1 in e2, and then solve for c.

 sage: e3=e2.subs(a=b+c)
 sage: solve(e3, c)
 [c == (d - b*e)/e]

 All fine, but note that I had to enter e1 into the substitution
 explicitly. I've yet to find a way of substituting one symbolic
 equation into another.

 Is there a way?
 Where am I going wrong?

 Thanks

 Paul

 
   


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[sage-support] Re: Substitution with symbolic equations

[Mike Hansen]

Hello,

On Fri, May 15, 2009 at 5:06 AM, Paul Sargent psa...@gmail.com wrote:
 Lets give ourselves two symbolic equations:

 sage: var(a b c d e)
 sage: e1 = a == b + c
 sage: e2 = d == e * a

 Now, lets say I want to know what c is in terms of b, d  e. By hand
 I'd substitute e1 in e2, and then solve for c.

 sage: e3=e2.subs(a=b+c)
 sage: solve(e3, c)
 [c == (d - b*e)/e]

 All fine, but note that I had to enter e1 into the substitution
 explicitly. I've yet to find a way of substituting one symbolic
 equation into another.

 Is there a way?
 Where am I going wrong?

In Sage 4.0 which will be released within the week, you'll be able to
do the following:

sage: var(a b c d e)
(a, b, c, d, e)
sage: e1 = a == b + c
sage: e2 = d == e * a
sage: e3 = e2.subs(e1); e3
d == (b + c)*e

--Mike

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[sage-support] Re: Substitution with symbolic equations

[Paul Sargent]

On 15 May 2009, at 16:33, Mike Hansen wrote:

 In Sage 4.0 which will be released within the week, you'll be able to
 do the following:

 sage: var(a b c d e)
 (a, b, c, d, e)
 sage: e1 = a == b + c
 sage: e2 = d == e * a
 sage: e3 = e2.subs(e1); e3
 d == (b + c)*e

Well, that's what I call service!

;-)

Seriously, that'll make some of the manipulations I'm doing flow a lot  
better.

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[sage-support] Re: substitution problem

[V]

On Mar 30, 6:07 pm, Jason Grout jason-s...@creativetrax.com wrote:
 V wrote:
  Hi,

  I'm fairly new to sage with some background in maxima.

  My workbook is shared at
 http://www.sagenb.org/home/pub/410/

  the last three lines show the error I get.

  Basically, I derive an equilibrium condition that I would like to use
  in the previous stage of my game (solving by backwards induction), but
  maxima fails (line -2) with simplifying after the substitution is
  made, even though the substitution is successful (line -3).

  I expect to receive the line in the last paragraph as defined by
  D1p.

  Is this a bug or did I made a mistake?

 In this line,

 solve([q1==q1.substitute(q2=sol2[0][1].right()).simplify_full()], q1)

 you are trying to solve for q1, but q1 is not a variable, it's the
 expression:

 f2^alpha*p1^(alpha - 1)*p2^(beta - alpha*beta)*q2^alpha*A^(1 - alpha)/f1

 Did you mean to solve for a variable in the solve statement above?


The expression in the solve line would be a demand function where q
stands for quantity. But as it is derived from a later stage of the
game, q is present on the right hand side as well, and I would like
to express the demand function explicitly, that is I have q=f(q,x)
where x are all the other parameters and q is defined implicitly, and
I would like to have q=g(x).

I thought solve would do this for me, even though I already understand
why it doesn't, I still don't know how to do it. Could you give me a
hint, please?

Thanks, V
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[sage-support] Re: substitution problem

[Jason Grout]

V wrote:
 On Mar 30, 6:07 pm, Jason Grout jason-s...@creativetrax.com wrote:
 V wrote:
 Hi,
 I'm fairly new to sage with some background in maxima.
 My workbook is shared at
 http://www.sagenb.org/home/pub/410/
 the last three lines show the error I get.
 Basically, I derive an equilibrium condition that I would like to use
 in the previous stage of my game (solving by backwards induction), but
 maxima fails (line -2) with simplifying after the substitution is
 made, even though the substitution is successful (line -3).
 I expect to receive the line in the last paragraph as defined by
 D1p.
 Is this a bug or did I made a mistake?
 In this line,

 solve([q1==q1.substitute(q2=sol2[0][1].right()).simplify_full()], q1)

 you are trying to solve for q1, but q1 is not a variable, it's the
 expression:

 f2^alpha*p1^(alpha - 1)*p2^(beta - alpha*beta)*q2^alpha*A^(1 - alpha)/f1

 Did you mean to solve for a variable in the solve statement above?

 
 The expression in the solve line would be a demand function where q
 stands for quantity. But as it is derived from a later stage of the
 game, q is present on the right hand side as well, and I would like
 to express the demand function explicitly, that is I have q=f(q,x)
 where x are all the other parameters and q is defined implicitly, and
 I would like to have q=g(x).
 
 I thought solve would do this for me, even though I already understand
 why it doesn't, I still don't know how to do it. Could you give me a
 hint, please?


I see.

Disclaimer: I don't know the mathematics that you are doing, so forgive 
me if I mess things up below...

When you do q1==q1.subs(...), what the computer sees is:

f2^alpha*p1^(alpha - 1)*p2^(beta - alpha*beta)*q2^alpha*A^(1 - alpha)/f1 
== f1^(alpha^2 - 1)*p1^((alpha - alpha^2)*beta + alpha - 1)*p2^((1 -
alpha)*beta + alpha^2 - alpha)*q1^alpha^2*A^(1 - alpha^2)

This doesn't seem like what you want; you want the left side to be a 
single variable, and the right side to have q1 be that same single 
variable.  So maybe you can do something like:

sage: var('q')
sage: q==q1.substitute(q2=sol2[0][1].right()).simplify_full().subs(q1=q)
q == f1^(alpha^2 - 1)*p1^((alpha - alpha^2)*beta + alpha - 1)*p2^((1 -
alpha)*beta + alpha^2 - alpha)*q^alpha^2*A^(1 - alpha^2)
sage: 
solve(q==q1.substitute(q2=sol2[0][1].right()).simplify_full().subs(q1=q), q)
[q == f1^(alpha^2 - 1)*p1^((alpha - alpha^2)*beta + alpha - 1)*p2^((1 -
alpha)*beta + alpha^2 - alpha)*q^alpha^2*A^(1 - alpha^2)]

So it doesn't look like maxima is able to handle this.

Jason


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[sage-support] Re: substitution problem

[V]

This is exactly what I wanted. Thanks for your help! :)

I'll look up the maxima manuals to see if I can force it to think
harder. Thanks anyway!

Have a nice day!
V

On Mar 31, 3:35 pm, Jason Grout jason-s...@creativetrax.com wrote:
 V wrote:
  On Mar 30, 6:07 pm, Jason Grout jason-s...@creativetrax.com wrote:
  V wrote:
  Hi,
  I'm fairly new to sage with some background in maxima.
  My workbook is shared at
 http://www.sagenb.org/home/pub/410/
  the last three lines show the error I get.
  Basically, I derive an equilibrium condition that I would like to use
  in the previous stage of my game (solving by backwards induction), but
  maxima fails (line -2) with simplifying after the substitution is
  made, even though the substitution is successful (line -3).
  I expect to receive the line in the last paragraph as defined by
  D1p.
  Is this a bug or did I made a mistake?
  In this line,

  solve([q1==q1.substitute(q2=sol2[0][1].right()).simplify_full()], q1)

  you are trying to solve for q1, but q1 is not a variable, it's the
  expression:

  f2^alpha*p1^(alpha - 1)*p2^(beta - alpha*beta)*q2^alpha*A^(1 - alpha)/f1

  Did you mean to solve for a variable in the solve statement above?

  The expression in the solve line would be a demand function where q
  stands for quantity. But as it is derived from a later stage of the
  game, q is present on the right hand side as well, and I would like
  to express the demand function explicitly, that is I have q=f(q,x)
  where x are all the other parameters and q is defined implicitly, and
  I would like to have q=g(x).

  I thought solve would do this for me, even though I already understand
  why it doesn't, I still don't know how to do it. Could you give me a
  hint, please?

 I see.

 Disclaimer: I don't know the mathematics that you are doing, so forgive
 me if I mess things up below...

 When you do q1==q1.subs(...), what the computer sees is:

 f2^alpha*p1^(alpha - 1)*p2^(beta - alpha*beta)*q2^alpha*A^(1 - alpha)/f1
 == f1^(alpha^2 - 1)*p1^((alpha - alpha^2)*beta + alpha - 1)*p2^((1 -
 alpha)*beta + alpha^2 - alpha)*q1^alpha^2*A^(1 - alpha^2)

 This doesn't seem like what you want; you want the left side to be a
 single variable, and the right side to have q1 be that same single
 variable.  So maybe you can do something like:

 sage: var('q')
 sage: q==q1.substitute(q2=sol2[0][1].right()).simplify_full().subs(q1=q)
 q == f1^(alpha^2 - 1)*p1^((alpha - alpha^2)*beta + alpha - 1)*p2^((1 -
 alpha)*beta + alpha^2 - alpha)*q^alpha^2*A^(1 - alpha^2)
 sage:
 solve(q==q1.substitute(q2=sol2[0][1].right()).simplify_full().subs(q1=q), q)
 [q == f1^(alpha^2 - 1)*p1^((alpha - alpha^2)*beta + alpha - 1)*p2^((1 -
 alpha)*beta + alpha^2 - alpha)*q^alpha^2*A^(1 - alpha^2)]

 So it doesn't look like maxima is able to handle this.

 Jason
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[sage-support] Re: substitution problem

[Jason Grout]

V wrote:
 Hi,
 
 I'm fairly new to sage with some background in maxima.
 
 My workbook is shared at
 http://www.sagenb.org/home/pub/410/
 
 the last three lines show the error I get.
 
 Basically, I derive an equilibrium condition that I would like to use
 in the previous stage of my game (solving by backwards induction), but
 maxima fails (line -2) with simplifying after the substitution is
 made, even though the substitution is successful (line -3).
 
 I expect to receive the line in the last paragraph as defined by
 D1p.
 
 Is this a bug or did I made a mistake?
 


In this line,

solve([q1==q1.substitute(q2=sol2[0][1].right()).simplify_full()], q1)

you are trying to solve for q1, but q1 is not a variable, it's the 
expression:

f2^alpha*p1^(alpha - 1)*p2^(beta - alpha*beta)*q2^alpha*A^(1 - alpha)/f1

Did you mean to solve for a variable in the solve statement above?

Jason



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[sage-support] Re: Substitution

[Alex Ghitza]

Hi,

On Fri, Mar 13, 2009 at 7:09 PM, hpon peter.norli...@gmail.com wrote:


 Hi,

 What is the easiest way to make a mathematical substitution in Sage?

 For example:  We have

 eqn1 = F == a + b


I'm not sure I understand what you are trying to do in this line.  If you
just want to define F as the sum of a and b, use

var('a b')  # to tell Sage that a and b are symbolic
variables
F = a + b



 and would like to use a = 0 and b = 3 to calculate F.


Now F is a symbolic expression, and you can perform substitutions easily as
follows:

F.substitute(a=0, b=3)  # and Sage will answer: 3

If you do F.substitute?  you will get the documentation for the substitution
method, with more examples of usage.

Best,
Alex


-- 
Alex Ghitza -- Lecturer in Mathematics -- The University of Melbourne --
Australia -- http://www.ms.unimelb.edu.au/~aghitza/

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[sage-support] Re: Substitution

[hpon]

Excellent.  Thank you Alex!

/hpon

On 13 Mar, 09:16, Alex Ghitza aghi...@gmail.com wrote:
 Hi,

 On Fri, Mar 13, 2009 at 7:09 PM, hpon peter.norli...@gmail.com wrote:

  Hi,

  What is the easiest way to make a mathematical substitution in Sage?

  For example:  We have

  eqn1 = F == a + b

 I'm not sure I understand what you are trying to do in this line.  If you
 just want to define F as the sum of a and b, use

 var('a b')                      # to tell Sage that a and b are symbolic
 variables
 F = a + b



  and would like to use a = 0 and b = 3 to calculate F.

 Now F is a symbolic expression, and you can perform substitutions easily as
 follows:

 F.substitute(a=0, b=3)      # and Sage will answer: 3

 If you do F.substitute?  you will get the documentation for the substitution
 method, with more examples of usage.

 Best,
 Alex

 --
 Alex Ghitza -- Lecturer in Mathematics -- The University of Melbourne --
 Australia --http://www.ms.unimelb.edu.au/~aghitza/
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[sage-support] Re: substitution in and integration of piecewise functions

[Ondrej Certik]

 One of the issues right now is that sage's piecewise is a completely
 separate class then the rest of the calculus library. I think it
 should descend from symbolic expression and be on the same level as,
 e.g. sin and addition (and probably even have a pynac counterpart).
 Letting the operands be the conditions (as symbolic equations, giving
 much more flexibility then we now have) and the corresponding
 expressions (as symbolic equations) would allow it to fit into the
 symbolic ring nicely. This would have the advantage that one could
 leverage all the generic SR functionality like subs, pow,
 composition, etc. without having to re-implement it for every method
 that's missing.

Exactly. In fact, that is how Piecewise works in sympy. So I think
pynac has to know how to properly dispatch stuff like .subs(),
.series() and other things into the Piecewise class (after it is
integrated into the calculus). And I know it is possible to do that
with Cython. Btw, how about pattern matching? Will you extend the C++
code in ginac to handle Piecewise functions?

The other option is to reimplement the .subs(), .series() and pattern
matching in Python. Burcin, please don't take my comments as sarcastic
--- I am really interested how to handle this and how to cooperate as
much as possible. Also, how will you approach to fix the bug in pynac
series I reported here:

http://groups.google.com/group/sage-devel/msg/a59f3c5d0586766b

E.g. will you fix ginac, or rather reimplement it in Cython/Python?
You can actually still use Maxima for that, so that's not a problem in
fact. And later just reimplement it in Python/Cython.

Ondrej

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[sage-support] Re: substitution in and integration of piecewise functions

[Robert Bradshaw]

On Nov 24, 2008, at 5:04 AM, Ondrej Certik wrote:

 On Mon, Nov 24, 2008 at 1:25 PM, David Joyner [EMAIL PROTECTED]  
 wrote:
 On Mon, Nov 24, 2008 at 6:19 AM, Ondrej Certik [EMAIL PROTECTED]  
 wrote:
 Hi,

 when I use regular expressions, I can use .subs():

 sage: e = x+y
 sage: e.subs(x=y)
 2*y

 but not with Piecewise:

 sage: var(h H x y)
 (h, H, x, y)
 sage: u = Piecewise([((0, h), x/h), ((h, H), 1)])


 I don't think variable endpoints are implemented in Sage's Piecewise.
 Maybe one could add a method=sympy option to use sympy's
 Piecewise in place of the Maxima interface?

 Well, even fixed endpoints doesn't seem to work properly:

 sage: u = Piecewise([((0, 1), x/h), ((1, 2), 1)])
 sage: u**2
 -- 
 -
 TypeError Traceback (most recent  
 call last)

 /home/ondra/ext/sage-3.1.4-debian32-IntelXeon-x86-i686-Linux/ipython
 console in module()

 /home/ondra/ext/sage/local/lib/python2.5/site-packages/sage/rings/ 
 integer.so
 in sage.rings.integer.Integer.__pow__ (sage/rings/integer.c:9379)()

 TypeError: unsupported operand type(s) for ** or pow(): 'instance'  
 and 'int'


 I don't see how pynac is relevant here. Pynac will only provide a
 different internal format for representing symbolic expressions, the
 piecewise class will still be the same.

 Well, if you want to do stuff like u**2, where u is piecewise, or use
 the .subs() that is implemented in the symbolic engine, you can either
 implement everything by hand (e.g. implement .subs(), implement
 __pow__, ...), but that's imho not the way, or, you can integrate it
 inside the symbolic engine, so that you can manipulate with it like
 with any other function -- that's the approach we took with sympy.

+1



 You can open a trac ticket if you need this functionality. It is safe
 to change piecewise.py, and add a .subs() method. Since the user
 interface for the pynac expressions will be almost the same as the
 current ones, it should still work when the underlying representation
 changes.

 I think it should be discussed more first. My vision is that the
 symbolic engine should treat everything as first class citizens ---
 e.g. if you make series expansion, it does the right thing.

One of the issues right now is that sage's piecewise is a completely  
separate class then the rest of the calculus library. I think it  
should descend from symbolic expression and be on the same level as,  
e.g. sin and addition (and probably even have a pynac counterpart).  
Letting the operands be the conditions (as symbolic equations, giving  
much more flexibility then we now have) and the corresponding  
expressions (as symbolic equations) would allow it to fit into the  
symbolic ring nicely. This would have the advantage that one could  
leverage all the generic SR functionality like subs, pow,  
composition, etc. without having to re-implement it for every method  
that's missing.

- Robert


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[sage-support] Re: substitution in and integration of piecewise functions

[William Stein]

On Tue, Nov 25, 2008 at 3:15 AM, Ondrej Certik [EMAIL PROTECTED] wrote:

 One of the issues right now is that sage's piecewise is a completely
 separate class then the rest of the calculus library. I think it

Just for the record, this is because Piecewise was implemented long
long ago by David Joyner, probably a year before any of the rest of
the calculus library was implemented.It definitely needs to be
somehow integrated better with the
calculus code as explained here.

 should descend from symbolic expression and be on the same level as,
 e.g. sin and addition (and probably even have a pynac counterpart).
 Letting the operands be the conditions (as symbolic equations, giving
 much more flexibility then we now have) and the corresponding
 expressions (as symbolic equations) would allow it to fit into the
 symbolic ring nicely. This would have the advantage that one could
 leverage all the generic SR functionality like subs, pow,
 composition, etc. without having to re-implement it for every method
 that's missing.

 Exactly. In fact, that is how Piecewise works in sympy. So I think
 pynac has to know how to properly dispatch stuff like .subs(),
 .series() and other things into the Piecewise class (after it is
 integrated into the calculus). And I know it is possible to do that
 with Cython. Btw, how about pattern matching? Will you extend the C++
 code in ginac to handle Piecewise functions?

 The other option is to reimplement the .subs(), .series() and pattern
 matching in Python. Burcin, please don't take my comments as sarcastic
 --- I am really interested how to handle this and how to cooperate as
 much as possible. Also, how will you approach to fix the bug in pynac
 series I reported here:

 http://groups.google.com/group/sage-devel/msg/a59f3c5d0586766b

 E.g. will you fix ginac, or rather reimplement it in Cython/Python?
 You can actually still use Maxima for that, so that's not a problem in
 fact. And later just reimplement it in Python/Cython.

 Ondrej

 




-- 
William Stein
Associate Professor of Mathematics
University of Washington
http://wstein.org

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[sage-support] Re: substitution in and integration of piecewise functions

[David Joyner]

On Tue, Nov 25, 2008 at 3:13 PM, William Stein [EMAIL PROTECTED] wrote:

 On Tue, Nov 25, 2008 at 3:15 AM, Ondrej Certik [EMAIL PROTECTED] wrote:

 One of the issues right now is that sage's piecewise is a completely
 separate class then the rest of the calculus library. I think it

 Just for the record, this is because Piecewise was implemented long
 long ago by David Joyner, probably a year before any of the rest of
 the calculus library was implemented.It definitely needs to be
 somehow integrated better with the
 calculus code as explained here.


I'll try working on this some over the winter break, ie, in 2-3 weeks.



 should descend from symbolic expression and be on the same level as,
 e.g. sin and addition (and probably even have a pynac counterpart).
 Letting the operands be the conditions (as symbolic equations, giving
 much more flexibility then we now have) and the corresponding
 expressions (as symbolic equations) would allow it to fit into the
 symbolic ring nicely. This would have the advantage that one could
 leverage all the generic SR functionality like subs, pow,
 composition, etc. without having to re-implement it for every method
 that's missing.

 Exactly. In fact, that is how Piecewise works in sympy. So I think
 pynac has to know how to properly dispatch stuff like .subs(),
 .series() and other things into the Piecewise class (after it is
 integrated into the calculus). And I know it is possible to do that
 with Cython. Btw, how about pattern matching? Will you extend the C++
 code in ginac to handle Piecewise functions?

 The other option is to reimplement the .subs(), .series() and pattern
 matching in Python. Burcin, please don't take my comments as sarcastic
 --- I am really interested how to handle this and how to cooperate as
 much as possible. Also, how will you approach to fix the bug in pynac
 series I reported here:

 http://groups.google.com/group/sage-devel/msg/a59f3c5d0586766b

 E.g. will you fix ginac, or rather reimplement it in Cython/Python?
 You can actually still use Maxima for that, so that's not a problem in
 fact. And later just reimplement it in Python/Cython.

 Ondrej

 




 --
 William Stein
 Associate Professor of Mathematics
 University of Washington
 http://wstein.org

 


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[sage-support] Re: substitution in and integration of piecewise functions

[Burcin Erocal]

On Tue, 25 Nov 2008 12:15:41 +0100
Ondrej Certik [EMAIL PROTECTED] wrote:

 
  One of the issues right now is that sage's piecewise is a completely
  separate class then the rest of the calculus library. I think it
  should descend from symbolic expression and be on the same level as,
  e.g. sin and addition (and probably even have a pynac counterpart).
  Letting the operands be the conditions (as symbolic equations,
  giving much more flexibility then we now have) and the corresponding
  expressions (as symbolic equations) would allow it to fit into the
  symbolic ring nicely. This would have the advantage that one could
  leverage all the generic SR functionality like subs, pow,
  composition, etc. without having to re-implement it for every method
  that's missing.
 
 Exactly. In fact, that is how Piecewise works in sympy. So I think
 pynac has to know how to properly dispatch stuff like .subs(),
 .series() and other things into the Piecewise class (after it is
 integrated into the calculus). And I know it is possible to do that
 with Cython. Btw, how about pattern matching? Will you extend the C++
 code in ginac to handle Piecewise functions?

Ok. Now I see what you mean, first let me explain why I failed to
understand you before.

Sage treats piecewise functions as functions, and in Sage, once you
have a function, it remains a function even when you do arithmetic with
it, unless you apply it to values and get an expression. E.g.,

sage: var('x,y,z,n')
(x, y, z, n)
sage: f(x) = 2*x^n
sage: f
x |-- 2*x^n
sage: g = f + y; g
x |-- y + 2*x^n
sage: g.subs(x=z)
2*z^n + y
sage: e = g(x,z); e
y + 2*x^n
sage: type(e)
class 'sage.calculus.calculus.SymbolicArithmetic'

So, when we were talking about piecewise functions, I thought that
would also stay as a piecewise function object all along. No matter
what arithmetic you do with it.

When you start having piecewise (or conditional?) expressions, things
do get complicated as you have been trying to point out all this
time. :)


Note that we already call python functions from c++ in pynac. Just look
at the sage.symbolic.function.SFunction class. You can specify python
functions for evaluation, derivation, conjugation, etc. just as ginac
lets you do this with c++ functions.

sage: from sage.symbolic.function import function as nfunction
sage: var('x,y',ns=1)
(x, y)
sage: def npow(x, power_param=None): return x*power_param 

sage: foo = nfunction(foo, 1, power_func=npow) 
sage: foo(y)^(x+y)
(x + y)*y

sage: def nseries(*args, **kwds): return sum(args[0]^i for i in
range(kwds['order']))

sage: foo = nfunction(foo, 1, series_func=nseries) 
sage: foo(y).series(y, 5) 
y^4 + y^3 + y^2 + y + 1

Once a couple more glitches are fixed, this should allow people to
implement new special functions using only python.

It should be straightforward to add a new class to ginac, which also
dispatches substitution and pattern matching to custom functions as
well. I'll have to think more about the best way to do this. 

As you might have noticed from the lack of progress on the pynac front,
I don't have much time to spend on pynac these days. I will have at
least a week to work on this in december, than more january. I hope to
have most of the lower level stuff sorted out by then, so we can start
moving things over to pynac.

 The other option is to reimplement the .subs(), .series() and pattern
 matching in Python. Burcin, please don't take my comments as sarcastic
 --- I am really interested how to handle this and how to cooperate as
 much as possible. 

Thanks for pointing these out. I don't use symbolics much, and was
stuck with the idea of piecewise functions for some reason. I
guess I hadn't looked at your initial example carefully. So piecewise
functions are on my todo list now:

http://wiki.sagemath.org/symbolics/pynac_todo

Please suggest anything else that you think should be in there. I don't
use symbolics enough to know about different use cases and common
problems. Your experience here is definitely needed.


 Also, how will you approach to fix the bug in pynac
 series I reported here:
 
 http://groups.google.com/group/sage-devel/msg/a59f3c5d0586766b
 
 E.g. will you fix ginac, or rather reimplement it in Cython/Python?

This was already on my todo list, see item 1/gamma(-1) = 0 in the
above link. I will fix ginac, as this annoys me too. This is also on
the ginac todo list (see the 4th item):

http://www.ginac.de/ToDo.html

 You can actually still use Maxima for that, so that's not a problem in
 fact. And later just reimplement it in Python/Cython.

I would like to use pynac as much as possible and avoid calling maxima.
I don't think I need to reimplement anything in cython to fix this
specific problem though. I just need to go around and modify a few eval
functions to take infinity into account. Covering all the possible
cases might take a couple of days, but it's definitely within reach. In
fact, I already did something similar to get around ginac's shortcuts in

[sage-support] Re: substitution in and integration of piecewise functions

[Ondrej Certik]

On Tue, Nov 25, 2008 at 11:00 PM, Burcin Erocal [EMAIL PROTECTED] wrote:

 On Tue, 25 Nov 2008 12:15:41 +0100
 Ondrej Certik [EMAIL PROTECTED] wrote:


  One of the issues right now is that sage's piecewise is a completely
  separate class then the rest of the calculus library. I think it
  should descend from symbolic expression and be on the same level as,
  e.g. sin and addition (and probably even have a pynac counterpart).
  Letting the operands be the conditions (as symbolic equations,
  giving much more flexibility then we now have) and the corresponding
  expressions (as symbolic equations) would allow it to fit into the
  symbolic ring nicely. This would have the advantage that one could
  leverage all the generic SR functionality like subs, pow,
  composition, etc. without having to re-implement it for every method
  that's missing.

 Exactly. In fact, that is how Piecewise works in sympy. So I think
 pynac has to know how to properly dispatch stuff like .subs(),
 .series() and other things into the Piecewise class (after it is
 integrated into the calculus). And I know it is possible to do that
 with Cython. Btw, how about pattern matching? Will you extend the C++
 code in ginac to handle Piecewise functions?

 Ok. Now I see what you mean, first let me explain why I failed to
 understand you before.

 Sage treats piecewise functions as functions, and in Sage, once you
 have a function, it remains a function even when you do arithmetic with
 it, unless you apply it to values and get an expression. E.g.,

 sage: var('x,y,z,n')
 (x, y, z, n)
 sage: f(x) = 2*x^n
 sage: f
 x |-- 2*x^n
 sage: g = f + y; g
 x |-- y + 2*x^n
 sage: g.subs(x=z)
 2*z^n + y
 sage: e = g(x,z); e
 y + 2*x^n
 sage: type(e)
 class 'sage.calculus.calculus.SymbolicArithmetic'

 So, when we were talking about piecewise functions, I thought that
 would also stay as a piecewise function object all along. No matter
 what arithmetic you do with it.

 When you start having piecewise (or conditional?) expressions, things
 do get complicated as you have been trying to point out all this
 time. :)


 Note that we already call python functions from c++ in pynac. Just look
 at the sage.symbolic.function.SFunction class. You can specify python
 functions for evaluation, derivation, conjugation, etc. just as ginac
 lets you do this with c++ functions.

 sage: from sage.symbolic.function import function as nfunction
 sage: var('x,y',ns=1)
 (x, y)
 sage: def npow(x, power_param=None): return x*power_param

 sage: foo = nfunction(foo, 1, power_func=npow)
 sage: foo(y)^(x+y)
 (x + y)*y

 sage: def nseries(*args, **kwds): return sum(args[0]^i for i in
 range(kwds['order']))

 sage: foo = nfunction(foo, 1, series_func=nseries)
 sage: foo(y).series(y, 5)
 y^4 + y^3 + y^2 + y + 1

 Once a couple more glitches are fixed, this should allow people to
 implement new special functions using only python.

 It should be straightforward to add a new class to ginac, which also
 dispatches substitution and pattern matching to custom functions as
 well. I'll have to think more about the best way to do this.

 As you might have noticed from the lack of progress on the pynac front,
 I don't have much time to spend on pynac these days. I will have at
 least a week to work on this in december, than more january. I hope to
 have most of the lower level stuff sorted out by then, so we can start
 moving things over to pynac.

 The other option is to reimplement the .subs(), .series() and pattern
 matching in Python. Burcin, please don't take my comments as sarcastic
 --- I am really interested how to handle this and how to cooperate as
 much as possible.

 Thanks for pointing these out. I don't use symbolics much, and was
 stuck with the idea of piecewise functions for some reason. I
 guess I hadn't looked at your initial example carefully. So piecewise
 functions are on my todo list now:

 http://wiki.sagemath.org/symbolics/pynac_todo

 Please suggest anything else that you think should be in there. I don't
 use symbolics enough to know about different use cases and common
 problems. Your experience here is definitely needed.

I only wanted to do stuff in this issue:

http://code.google.com/p/sympy/issues/detail?id=1214

e.g. substitue in the Piecewise function and then integrate it. Since
the integration failed in sympy, I was curious how far Sage is on this
front, and it is not far either. So the aim is that one should be able
to do everything in the issue.



 Also, how will you approach to fix the bug in pynac
 series I reported here:

 http://groups.google.com/group/sage-devel/msg/a59f3c5d0586766b

 E.g. will you fix ginac, or rather reimplement it in Cython/Python?

 This was already on my todo list, see item 1/gamma(-1) = 0 in the
 above link. I will fix ginac, as this annoys me too. This is also on
 the ginac todo list (see the 4th item):

 http://www.ginac.de/ToDo.html

 You can actually still use Maxima for that, so that's not a problem in
 fact. And later 

[sage-support] Re: substitution in and integration of piecewise functions

[Burcin Erocal]

On Mon, 24 Nov 2008 12:19:55 +0100
Ondrej Certik [EMAIL PROTECTED] wrote:

 Hi,
 
 when I use regular expressions, I can use .subs():
 
 sage: e = x+y
 sage: e.subs(x=y)
 2*y
 
 but not with Piecewise:
 
 sage: var(h H x y)
 (h, H, x, y)
 sage: u = Piecewise([((0, h), x/h), ((h, H), 1)])
 sage: u.subs(x=y)
 ---
 AttributeErrorTraceback (most recent call
 last)
 
 /home/ondra/ext/sage-3.1.2-debian32-intel-i686-Linux/ipython console
 in module()
 
 AttributeError: PiecewisePolynomial instance has no attribute 'subs'

This means that substitution is simply not implemented for piecewise
functions. I suppose no one needed this until now.

snip
 Am I doing something wrong, or is it just not implemented yet? What is
 the plan of implementing this in pynac?
 
I don't see how pynac is relevant here. Pynac will only provide a
different internal format for representing symbolic expressions, the
piecewise class will still be the same. 

You can open a trac ticket if you need this functionality. It is safe
to change piecewise.py, and add a .subs() method. Since the user
interface for the pynac expressions will be almost the same as the
current ones, it should still work when the underlying representation
changes.


Cheers,

Burcin

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[sage-support] Re: substitution in and integration of piecewise functions

[David Joyner]

On Mon, Nov 24, 2008 at 6:19 AM, Ondrej Certik [EMAIL PROTECTED] wrote:
 Hi,

 when I use regular expressions, I can use .subs():

 sage: e = x+y
 sage: e.subs(x=y)
 2*y

 but not with Piecewise:

 sage: var(h H x y)
 (h, H, x, y)
 sage: u = Piecewise([((0, h), x/h), ((h, H), 1)])


I don't think variable endpoints are implemented in Sage's Piecewise.
Maybe one could add a method=sympy option to use sympy's
Piecewise in place of the Maxima interface?



 sage: u.subs(x=y)
 ---
 AttributeErrorTraceback (most recent call last)

 /home/ondra/ext/sage-3.1.2-debian32-intel-i686-Linux/ipython console
 in module()

 AttributeError: PiecewisePolynomial instance has no attribute 'subs'


 Ok. Now I'd like to integrate that:

 sage: u.integral()
 TypeError: Computation failed since Maxima requested additional
 constraints (use assume):
 Is  h  positive, negative, or zero?

 sage: assume(h0)
 sage: u.integral()
 TypeError: Computation failed since Maxima requested additional
 constraints (use assume):
 Is  ?h+h  positive, negative, or zero?


 What does ?h+h mean?

 sage: u.integral(x)
 ---
 AttributeErrorTraceback (most recent call last)

 /home/ondra/ext/sage-3.1.2-debian32-intel-i686-Linux/ipython console
 in module()

 /home/ondra/ext/sage-3.1.2-debian32-intel-i686-Linux/local/lib/python2.5/site-packages/sage/functions/piecewise.py
 in integral(self, x)
647 invs = self.intervals()
648 n = len(funcs)
 -- 649 return sum([funcs[i].integral(x,invs[i][0],invs[i][1])
 for i in range(n)])
650
651 def convolution(self,other):

 AttributeError: 'sage.rings.integer.Integer' object has no attribute 
 'integral'



 Am I doing something wrong, or is it just not implemented yet? What is
 the plan of implementing this in pynac?

 For comparison, in sympy you can do this:

 In [1]: var(h)
 Out[1]: h

 In [2]: u = Piecewise( (0, x=0), (1, x=h), (x/h, True))

 In [3]: integrate(u, x)
 Out[3]:
 ⎧ 0   for x ≤ 0
 ⎪
 ⎪ x   for h ≤ x
 ⎪
 ⎨  2
 ⎪ x
 ⎪───  otherwise
 ⎪2⋅h
 ⎩


 Substitution works like this:

 In [4]: e = (u-u.subs(x, y))**2/(x-y)**2

 In [5]: e
 Out[5]:
 2
 ⎛- ⎧0  for y ≤ 0 + ⎧0  for x ≤ 0⎞
 ⎜  ⎪   ⎪⎟
 ⎜  ⎪1  for h ≤ y   ⎪1  for h ≤ x⎟
 ⎜  ⎨   ⎨⎟
 ⎜  ⎪y  ⎪x   ⎟
 ⎜  ⎪─  otherwise   ⎪─  otherwise⎟
 ⎝  ⎩h  ⎩h   ⎠
 ──
2
 (x - y)


 Now I'd like to integrate the resulting expression and that doesn't
 yet work in sympy:

 http://code.google.com/p/sympy/issues/detail?id=1214

 so I was curious what is Sage's strategy to handle Piecewise functions
 in a regular symbolic expression.


 Ondrej

 


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[sage-support] Re: substitution in and integration of piecewise functions

[David Joyner]

On Mon, Nov 24, 2008 at 7:25 AM, David Joyner [EMAIL PROTECTED] wrote:
 On Mon, Nov 24, 2008 at 6:19 AM, Ondrej Certik [EMAIL PROTECTED] wrote:
 Hi,

 when I use regular expressions, I can use .subs():

 sage: e = x+y
 sage: e.subs(x=y)
 2*y

 but not with Piecewise:

 sage: var(h H x y)
 (h, H, x, y)
 sage: u = Piecewise([((0, h), x/h), ((h, H), 1)])


 I don't think variable endpoints are implemented in Sage's Piecewise.
 Maybe one could add a method=sympy option to use sympy's
 Piecewise in place of the Maxima interface?


Replying to my own post:
I am happy to copy my code over to sympy's piecewise class
(or have someone else do it). In other words, even though it is
GPL'd I'm happy to relicense it under BSD for sympy, if there is
any interest.  I might have time for that over the winter break.




 sage: u.subs(x=y)
 ---
 AttributeErrorTraceback (most recent call last)

 /home/ondra/ext/sage-3.1.2-debian32-intel-i686-Linux/ipython console
 in module()

 AttributeError: PiecewisePolynomial instance has no attribute 'subs'


 Ok. Now I'd like to integrate that:

 sage: u.integral()
 TypeError: Computation failed since Maxima requested additional
 constraints (use assume):
 Is  h  positive, negative, or zero?

 sage: assume(h0)
 sage: u.integral()
 TypeError: Computation failed since Maxima requested additional
 constraints (use assume):
 Is  ?h+h  positive, negative, or zero?


 What does ?h+h mean?

 sage: u.integral(x)
 ---
 AttributeErrorTraceback (most recent call last)

 /home/ondra/ext/sage-3.1.2-debian32-intel-i686-Linux/ipython console
 in module()

 /home/ondra/ext/sage-3.1.2-debian32-intel-i686-Linux/local/lib/python2.5/site-packages/sage/functions/piecewise.py
 in integral(self, x)
647 invs = self.intervals()
648 n = len(funcs)
 -- 649 return sum([funcs[i].integral(x,invs[i][0],invs[i][1])
 for i in range(n)])
650
651 def convolution(self,other):

 AttributeError: 'sage.rings.integer.Integer' object has no attribute 
 'integral'



 Am I doing something wrong, or is it just not implemented yet? What is
 the plan of implementing this in pynac?

 For comparison, in sympy you can do this:

 In [1]: var(h)
 Out[1]: h

 In [2]: u = Piecewise( (0, x=0), (1, x=h), (x/h, True))

 In [3]: integrate(u, x)
 Out[3]:
 ⎧ 0   for x ≤ 0
 ⎪
 ⎪ x   for h ≤ x
 ⎪
 ⎨  2
 ⎪ x
 ⎪───  otherwise
 ⎪2⋅h
 ⎩


 Substitution works like this:

 In [4]: e = (u-u.subs(x, y))**2/(x-y)**2

 In [5]: e
 Out[5]:
 2
 ⎛- ⎧0  for y ≤ 0 + ⎧0  for x ≤ 0⎞
 ⎜  ⎪   ⎪⎟
 ⎜  ⎪1  for h ≤ y   ⎪1  for h ≤ x⎟
 ⎜  ⎨   ⎨⎟
 ⎜  ⎪y  ⎪x   ⎟
 ⎜  ⎪─  otherwise   ⎪─  otherwise⎟
 ⎝  ⎩h  ⎩h   ⎠
 ──
2
 (x - y)


 Now I'd like to integrate the resulting expression and that doesn't
 yet work in sympy:

 http://code.google.com/p/sympy/issues/detail?id=1214

 so I was curious what is Sage's strategy to handle Piecewise functions
 in a regular symbolic expression.


 Ondrej

 



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[sage-support] Re: substitution in and integration of piecewise functions

[Ondrej Certik]

On Mon, Nov 24, 2008 at 1:25 PM, David Joyner [EMAIL PROTECTED] wrote:
 On Mon, Nov 24, 2008 at 6:19 AM, Ondrej Certik [EMAIL PROTECTED] wrote:
 Hi,

 when I use regular expressions, I can use .subs():

 sage: e = x+y
 sage: e.subs(x=y)
 2*y

 but not with Piecewise:

 sage: var(h H x y)
 (h, H, x, y)
 sage: u = Piecewise([((0, h), x/h), ((h, H), 1)])


 I don't think variable endpoints are implemented in Sage's Piecewise.
 Maybe one could add a method=sympy option to use sympy's
 Piecewise in place of the Maxima interface?

Well, even fixed endpoints doesn't seem to work properly:

sage: u = Piecewise([((0, 1), x/h), ((1, 2), 1)])
sage: u**2
---
TypeError Traceback (most recent call last)

/home/ondra/ext/sage-3.1.4-debian32-IntelXeon-x86-i686-Linux/ipython
console in module()

/home/ondra/ext/sage/local/lib/python2.5/site-packages/sage/rings/integer.so
in sage.rings.integer.Integer.__pow__ (sage/rings/integer.c:9379)()

TypeError: unsupported operand type(s) for ** or pow(): 'instance' and 'int'


 I don't see how pynac is relevant here. Pynac will only provide a
 different internal format for representing symbolic expressions, the
 piecewise class will still be the same.

Well, if you want to do stuff like u**2, where u is piecewise, or use
the .subs() that is implemented in the symbolic engine, you can either
implement everything by hand (e.g. implement .subs(), implement
__pow__, ...), but that's imho not the way, or, you can integrate it
inside the symbolic engine, so that you can manipulate with it like
with any other function -- that's the approach we took with sympy.


 You can open a trac ticket if you need this functionality. It is safe
 to change piecewise.py, and add a .subs() method. Since the user
 interface for the pynac expressions will be almost the same as the
 current ones, it should still work when the underlying representation
 changes.

I think it should be discussed more first. My vision is that the
symbolic engine should treat everything as first class citizens ---
e.g. if you make series expansion, it does the right thing.

Well, I just found series expansion of piecewise functions doesn't
work in sympy. :)

http://code.google.com/p/sympy/issues/detail?id=1216

But you see the pattern --- we just override the
Piecewise._eval_nseries() and it will start working.

btw. it doesn't work in Sage either:

sage: taylor(u, x, 0.5)
---
TypeError Traceback (most recent call last)

/home/ondra/ext/sage-3.1.4-debian32-IntelXeon-x86-i686-Linux/ipython
console in module()

TypeError: taylor() takes exactly 4 arguments (3 given)
sage: taylor(u, x, 0.5, 4)
---
TypeError Traceback (most recent call last)

/home/ondra/ext/sage-3.1.4-debian32-IntelXeon-x86-i686-Linux/ipython
console in module()

/home/ondra/ext/sage/local/lib/python2.5/site-packages/sage/calculus/functional.pyc
in taylor(f, v, a, n)
332 
333 if not isinstance(f, SymbolicExpression):
-- 334 f = SR(f)
335 return f.taylor(v=v,a=a,n=n)
336

/home/ondra/ext/sage/local/lib/python2.5/site-packages/sage/calculus/calculus.pyc
in __call__(self, x)
452 msg, s, pos = err.args
453 raise TypeError, %s: %s !!! %s % (msg,
s[:pos], s[pos:])
-- 454 return self._coerce_impl(x)
455
456 def _coerce_impl(self, x):

/home/ondra/ext/sage/local/lib/python2.5/site-packages/sage/calculus/calculus.pyc
in _coerce_impl(self, x)
508 return self(x._sage_())
509 else:
-- 510 raise TypeError, cannot coerce type '%s' into a
SymbolicExpression.%type(x)
511
512 def _repr_(self):

TypeError: cannot coerce type 'type 'instance'' into a SymbolicExpression.



So imho it is related to pynac, because if you use ginac's series
expansion, it should be able to handle Piecewise. If you choose to
implement series expansion yourself (that's what we did in sympy), you
need to integrate Piecewise in somehow.


Ondrej

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[sage-support] Re: substitution in and integration of piecewise functions

[Ondrej Certik]

On Mon, Nov 24, 2008 at 2:15 PM, David Joyner [EMAIL PROTECTED] wrote:
 On Mon, Nov 24, 2008 at 7:25 AM, David Joyner [EMAIL PROTECTED] wrote:
 On Mon, Nov 24, 2008 at 6:19 AM, Ondrej Certik [EMAIL PROTECTED] wrote:
 Hi,

 when I use regular expressions, I can use .subs():

 sage: e = x+y
 sage: e.subs(x=y)
 2*y

 but not with Piecewise:

 sage: var(h H x y)
 (h, H, x, y)
 sage: u = Piecewise([((0, h), x/h), ((h, H), 1)])


 I don't think variable endpoints are implemented in Sage's Piecewise.
 Maybe one could add a method=sympy option to use sympy's
 Piecewise in place of the Maxima interface?


 Replying to my own post:
 I am happy to copy my code over to sympy's piecewise class
 (or have someone else do it). In other words, even though it is
 GPL'd I'm happy to relicense it under BSD for sympy, if there is
 any interest.  I might have time for that over the winter break.

Thanks! Btw, feel free to take any code from sympy as well and
incorporate it in Sage. But it's a design issue --- I want to work on
these things with Sage together, because clearly currently we have
two, incompatible Piecewise functions implementations, and neither in
Sage nor in sympy it works 100%.

Ondrej

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[sage-support] Re: substitution in and integration of piecewise functions

[Burcin Erocal]

On Mon, 24 Nov 2008 14:04:53 +0100
Ondrej Certik [EMAIL PROTECTED] wrote:

 
 On Mon, Nov 24, 2008 at 1:25 PM, David Joyner [EMAIL PROTECTED]
 wrote:
  On Mon, Nov 24, 2008 at 6:19 AM, Ondrej Certik [EMAIL PROTECTED]
  wrote:
  Hi,
 
  when I use regular expressions, I can use .subs():
 
  sage: e = x+y
  sage: e.subs(x=y)
  2*y
 
  but not with Piecewise:
 
  sage: var(h H x y)
  (h, H, x, y)
  sage: u = Piecewise([((0, h), x/h), ((h, H), 1)])
 
 
  I don't think variable endpoints are implemented in Sage's
  Piecewise. Maybe one could add a method=sympy option to use
  sympy's Piecewise in place of the Maxima interface?
 
 Well, even fixed endpoints doesn't seem to work properly:
 
 sage: u = Piecewise([((0, 1), x/h), ((1, 2), 1)])
 sage: u**2
 ---
 TypeError Traceback (most recent call
 last)
 
 /home/ondra/ext/sage-3.1.4-debian32-IntelXeon-x86-i686-Linux/ipython
 console in module()
 
 /home/ondra/ext/sage/local/lib/python2.5/site-packages/sage/rings/integer.so
 in sage.rings.integer.Integer.__pow__ (sage/rings/integer.c:9379)()
 
 TypeError: unsupported operand type(s) for ** or pow(): 'instance'
 and 'int'

Again, this is not implemented, patches are welcome.

  I don't see how pynac is relevant here. Pynac will only provide a
  different internal format for representing symbolic expressions, the
  piecewise class will still be the same.
 
 Well, if you want to do stuff like u**2, where u is piecewise, or use
 the .subs() that is implemented in the symbolic engine, you can either
 implement everything by hand (e.g. implement .subs(), implement
 __pow__, ...), but that's imho not the way, or, you can integrate it
 inside the symbolic engine, so that you can manipulate with it like
 with any other function -- that's the approach we took with sympy.

Are you saying that your class which represents a piecewise function
doesn't implement it's .subs() method, which in turn calls the .subs()
methods of each of the pieces (and the endpoints in your case)?
Similarly for the __pow__ method.

Can you explain your design a little?

 
  You can open a trac ticket if you need this functionality. It is
  safe to change piecewise.py, and add a .subs() method. Since the
  user interface for the pynac expressions will be almost the same as
  the current ones, it should still work when the underlying
  representation changes.
 
 I think it should be discussed more first. My vision is that the
 symbolic engine should treat everything as first class citizens ---
 e.g. if you make series expansion, it does the right thing.

Note that in the design of GiNaC, which AFAIK, you also follow in SymPy,
these methods, e.g., arithmetic, substitution, etc. are implemented in
each class of the hierarchy.

I don't see why building on this hierarchy using python after a certain
point is a problem.

 Well, I just found series expansion of piecewise functions doesn't
 work in sympy. :)
 
 http://code.google.com/p/sympy/issues/detail?id=1216
 
 But you see the pattern --- we just override the
 Piecewise._eval_nseries() and it will start working.

Same in Sage. I don't see your point.

 btw. it doesn't work in Sage either:
 
 sage: taylor(u, x, 0.5)
 ---
 TypeError Traceback (most recent call
 last)
...
 TypeError: cannot coerce type 'type 'instance'' into a
 SymbolicExpression.

Can you open a ticket and list these things you found that don't work
with piecewise functions. I.e.,

- substitutions
- __pow__
- series expansion

 So imho it is related to pynac, because if you use ginac's series
 expansion, it should be able to handle Piecewise. If you choose to
 implement series expansion yourself (that's what we did in sympy), you
 need to integrate Piecewise in somehow.

Pynac will only provide the basic operations one needs from
symbolic expressions as a library. This is quite similar to your
efforts of implementing a sympy core in Cython. 

http://groups.google.com/group/sympy/browse_thread/thread/aa3f4263bc3f7e23

The road plan for SymPy says this will be included in version 0.7:

http://wiki.sympy.org/wiki/Plan_for_SymPy_1.0

So I really don't see what you mean above.


From another message:
On Mon, 24 Nov 2008 14:27:04 +0100
Ondrej Certik [EMAIL PROTECTED] wrote:
 
 Thanks! Btw, feel free to take any code from sympy as well and
 incorporate it in Sage. But it's a design issue --- I want to work on
 these things with Sage together, because clearly currently we have
 two, incompatible Piecewise functions implementations, and neither in
 Sage nor in sympy it works 100%.
 

Piecewise functions are a great example of a way SymPy and Sage can work
together. Since the interfaces to the underlying symbolic expressions
are the same, the same python implementation can be used for both the
projects. If SymPy had a stable and fast core at 

[sage-support] Re: substitution in and integration of piecewise functions

[Ondrej Certik]

On Mon, Nov 24, 2008 at 3:08 PM, Burcin Erocal [EMAIL PROTECTED] wrote:

 On Mon, 24 Nov 2008 14:04:53 +0100
 Ondrej Certik [EMAIL PROTECTED] wrote:


 On Mon, Nov 24, 2008 at 1:25 PM, David Joyner [EMAIL PROTECTED]
 wrote:
  On Mon, Nov 24, 2008 at 6:19 AM, Ondrej Certik [EMAIL PROTECTED]
  wrote:
  Hi,
 
  when I use regular expressions, I can use .subs():
 
  sage: e = x+y
  sage: e.subs(x=y)
  2*y
 
  but not with Piecewise:
 
  sage: var(h H x y)
  (h, H, x, y)
  sage: u = Piecewise([((0, h), x/h), ((h, H), 1)])
 
 
  I don't think variable endpoints are implemented in Sage's
  Piecewise. Maybe one could add a method=sympy option to use
  sympy's Piecewise in place of the Maxima interface?

 Well, even fixed endpoints doesn't seem to work properly:

 sage: u = Piecewise([((0, 1), x/h), ((1, 2), 1)])
 sage: u**2
 ---
 TypeError Traceback (most recent call
 last)

 /home/ondra/ext/sage-3.1.4-debian32-IntelXeon-x86-i686-Linux/ipython
 console in module()

 /home/ondra/ext/sage/local/lib/python2.5/site-packages/sage/rings/integer.so
 in sage.rings.integer.Integer.__pow__ (sage/rings/integer.c:9379)()

 TypeError: unsupported operand type(s) for ** or pow(): 'instance'
 and 'int'

 Again, this is not implemented, patches are welcome.

  I don't see how pynac is relevant here. Pynac will only provide a
  different internal format for representing symbolic expressions, the
  piecewise class will still be the same.

 Well, if you want to do stuff like u**2, where u is piecewise, or use
 the .subs() that is implemented in the symbolic engine, you can either
 implement everything by hand (e.g. implement .subs(), implement
 __pow__, ...), but that's imho not the way, or, you can integrate it
 inside the symbolic engine, so that you can manipulate with it like
 with any other function -- that's the approach we took with sympy.

 Are you saying that your class which represents a piecewise function
 doesn't implement it's .subs() method, which in turn calls the .subs()
 methods of each of the pieces (and the endpoints in your case)?

No, it does implement exactly that.

Actually, because of docstrings (and possibly changing the interface
of things to the end user), we have one .subs() method implemented in
Basic, with a nice docstring and the subclasses (including Piecewise)
just implement _eval_subs(). The user is supposed to just call
.subs(). But that's minor.

But in pynac, so do you call ginac's subs method, or do you implement
it on your own? I know you can pass Sage objects inside ginac, thanks
to cython. I just want to clarify this is what you do.

 Similarly for the __pow__ method.

We only implement __pow__ in Basic by default. The subclasses can
override it for efficiency reasons, but don't have to.


 Can you explain your design a little?

Yes -- please ask if more clarification is needed besides the above.


 
  You can open a trac ticket if you need this functionality. It is
  safe to change piecewise.py, and add a .subs() method. Since the
  user interface for the pynac expressions will be almost the same as
  the current ones, it should still work when the underlying
  representation changes.

 I think it should be discussed more first. My vision is that the
 symbolic engine should treat everything as first class citizens ---
 e.g. if you make series expansion, it does the right thing.

 Note that in the design of GiNaC, which AFAIK, you also follow in SymPy,
 these methods, e.g., arithmetic, substitution, etc. are implemented in
 each class of the hierarchy.

Some of, yes.


 I don't see why building on this hierarchy using python after a certain
 point is a problem.

It is not a problem, I am only curious if this is your plan.


 Well, I just found series expansion of piecewise functions doesn't
 work in sympy. :)

 http://code.google.com/p/sympy/issues/detail?id=1216

 But you see the pattern --- we just override the
 Piecewise._eval_nseries() and it will start working.

 Same in Sage. I don't see your point.

 btw. it doesn't work in Sage either:

 sage: taylor(u, x, 0.5)
 ---
 TypeError Traceback (most recent call
 last)
 ...
 TypeError: cannot coerce type 'type 'instance'' into a
 SymbolicExpression.

 Can you open a ticket and list these things you found that don't work
 with piecewise functions. I.e.,

 - substitutions
 - __pow__
 - series expansion

 So imho it is related to pynac, because if you use ginac's series
 expansion, it should be able to handle Piecewise. If you choose to
 implement series expansion yourself (that's what we did in sympy), you
 need to integrate Piecewise in somehow.

 Pynac will only provide the basic operations one needs from
 symbolic expressions as a library. This is quite similar to your
 efforts of implementing a sympy core in Cython.

 

[sage-support] Re: substitution in a matrix (newby)

[Roger Mason]

William Stein [EMAIL PROTECTED] writes:

 On 7/26/07, Roger Mason [EMAIL PROTECTED] wrote:
  You should do m.substitute(a=1) just like you're doing -- unfortunately,
  nobody implemented that yet, and it's doing some dumb generic behavior.
  [...]
  Fortunately, I just implemented this for SAGE-2.7.1, which I'll release 
  very
  very soon.

 Thank you both very much.  It seems I need a careful read of the user
 manual, I had not picked up on lambda expressions.  I am compiling
 2.7.1 so will soon be able to test blah.substitute(a=1).

 I don't know if lambda expressions are discussed in the user manual.
 The SAGE language is Python, and Python has a very nice lambda
 expression.  Thus this nicely written book also applies to SAGE:
 http://docs.python.org/tut/
 The section on lambda expressions is here:
 http://docs.python.org/tut/node6.html#SECTION00675

The compilation of 2.7.1 just finished and I had started looking for
lambda in the docs when I saw your e-mail!

Many thanks,

Roger


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URLs: http://sage.math.washington.edu/sage/ and http://sage.scipy.org/sage/
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[sage-support] Re: substitution in a matrix (newby)

[Roger Mason]

David  William,

David Joyner [EMAIL PROTECTED] writes:

 I would use the lambda notation:

 sage: m = lambda x: matrix([[x[0],x[1]],[x[2],x[3]]])
 sage: m([1,2,3,4])

 [1 2]
 [3 4]
 sage: m([1,2,3,a])

 [1 2]
 [3 a]
 sage: m([a,b,3,4])

 [a b]
 [3 4]


 There might be better ways though.

 

William Stein [EMAIL PROTECTED] writes:

 You should do m.substitute(a=1) just like you're doing -- unfortunately,
 nobody implemented that yet, and it's doing some dumb generic behavior.
 [...]
 Fortunately, I just implemented this for SAGE-2.7.1, which I'll release very
 very soon.

Thank you both very much.  It seems I need a careful read of the user
manual, I had not picked up on lambda expressions.  I am compiling
2.7.1 so will soon be able to test blah.substitute(a=1).

Roger


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[sage-support] Re: substitution in a matrix (newby)

[William Stein]

On 7/26/07, Roger Mason [EMAIL PROTECTED] wrote:
  You should do m.substitute(a=1) just like you're doing -- unfortunately,
  nobody implemented that yet, and it's doing some dumb generic behavior.
  [...]
  Fortunately, I just implemented this for SAGE-2.7.1, which I'll release very
  very soon.

 Thank you both very much.  It seems I need a careful read of the user
 manual, I had not picked up on lambda expressions.  I am compiling
 2.7.1 so will soon be able to test blah.substitute(a=1).

I don't know if lambda expressions are discussed in the user manual.
The SAGE language is Python, and Python has a very nice lambda
expression.  Thus this nicely written book also applies to SAGE:
http://docs.python.org/tut/
The section on lambda expressions is here:
http://docs.python.org/tut/node6.html#SECTION00675

William

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[sage-support] Re: substitution in a matrix (newby)

[David Joyner]

I would use the lambda notation:

sage: m = lambda x: matrix([[x[0],x[1]],[x[2],x[3]]])
sage: m([1,2,3,4])

[1 2]
[3 4]
sage: m([1,2,3,a])

[1 2]
[3 a]
sage: m([a,b,3,4])

[a b]
[3 4]


There might be better ways though.



On 7/24/07, Roger Mason [EMAIL PROTECTED] wrote:

 Hello,

 Suppose I have a symbolic matrix:

 sage: m = matrix([[a,b],[d,e]])

 and I wish to substitute some or all elements of m with numbers.

 The obvious

 sage: m.substitute(a=1)

 returns

 [a b]
 [d e]

 so the substitution (which works fine on a list) does not work on a
 matrix.

 How do I do such substitutions?

 Thanks,

 Roger


 


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