implicitly defining a sequence of variables

4

parzan
977 ●6 ●15 ●36

To define a general polynomial in Maple one writes

p := sum(a[i]*x^i,i=0..n);

and gets $p = \sum _{i=0}^{n}a_{{i}}{x}^{i}$ .

So the "a[i]" are implicitly understood as variables, and their number (n) is also a variable. Or perhaps "a" is implicitly understood as a sequence of variables? I don't know what happens behind the scenes here, but it is very usefull.

Trying to accomplish this in sage I reached

sage: var('x,i,n')
(x, i, n)
sage: a = function('a')
sage: p = sum(a(i)*x^i,i,0,n);p
sum(x^i*a(i), i, 0, n)

Is this the right way? It doesn't behave as nice as in maple. Trying series, taylor, and diff only taylor works correctly:

sage: p.series(x==0,3)
---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
....
RuntimeError: power::eval(): division by zero
sage: p.taylor(x,0,3) 
x^3*a(3) + x^2*a(2) + x*a(1) + a(0)
sage: p.diff(x)              
i*x^(i - 1)*a(i)*D[0](sum)(x^i*a(i), i, 0, n)

In Maple they all give good results.

Am I going at this the right way? Is there a way to implicitly define variables as in Maple?

Comments

How does Maple deal with formal polynomials? I think we'd need some other sort of formal polynomial class to deal with them, as what you're doing above in Sage is only constructing an explicit symbolic polynomial.

Jason Grout ( 13 years ago )

I really don't know how Maple does this. I am certain it is not by a class of formal polynomials, but simply a symbolic sum of symbolic expressions (involving indexing). One could also take products instead of sums, or combinations of such. I assume Maplesoft would not explain how this is implemented, but Maxima also knows how to do this (see my answer below), and it is open source, so hopefully one day someone would implement this in sage, or at least wrap the maxima implementation.

parzan ( 13 years ago )

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4

answered 13 years ago

niles
4539 ●24 ●72 ●133 http://nilesjohnson.net/

I think the best way to do this is to use python's list comprehension/generator syntax to define the sequence of variables. Something like the following provides a formal sum which works as expected for diff, series, and taylor:

sage: a = var(','.join('a%s'%i for i in range(4))); a
(a0, a1, a2, a3)
sage: p = sum(a[i]*x^i for i in range(4)); p
a3*x^3 + a2*x^2 + a1*x + a0

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Comments

a = list(var("a%d" % i) for i in range(4))?

DSM ( 13 years ago )

By the way, in Python 3 this is going to get even more confusing, with the {0} notation we'll need looking vaguely like subscript notation...

kcrisman ( 13 years ago )

then maybe someone will improve `var` to automatically generate a sequence of variables ;)

niles ( 13 years ago )

Thanks for the ideas! This is nice, but it's not exactly what I want, since it bounds the polynomial degree. What I am looking for is a way to represent general polynomials, i.e., the "4" in the example should somehow be replaced by a variable 'n'.

parzan ( 13 years ago )

Maybe this would be possible with a generator...

kcrisman ( 13 years ago )

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1

answered 4 years ago

Sylvain
68 ●2 ●7

I use the following:

sage: x = SR.var("x", 10)
sage: sage: p = sum(a[i]*x^i for i in range(10)); p                                                  
a9*x^9 + a8*x^8 + a7*x^7 + a6*x^6 + a5*x^5 + a4*x^4 + a3*x^3 + a2*x^2 + a1*x + a0

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answered 13 years ago

Jason Grout
3505 ●9 ●36 ●81

updated 13 years ago

How about something like this?

class VariableGenerator(object):
    def __init__(self, prefix):
        self.__prefix = prefix

    @cached_method    
    def __getitem__(self, key):
        return SR.var("%s%s"%(self.__prefix,key))
a=VariableGenerator('a')
a[0], a[1], a[2] # all variables

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Comments

Note that this doesn't solve the "number of variables is a variable" problem, only the "how do I easily generate a number of variables" problem.

Jason Grout ( 13 years ago )

This is nice. Perhaps you could make "VariableGenerator" inherit "sage.symbolic.expression.Expression" so that it would be a variable with indexing? and even better, if the key could be a symbolic expression itself - maybe that way we could achieve maple's functionality?

parzan ( 13 years ago )

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0

answered 4 years ago

Emmanuel Charpentier
7804 ●6 ●51 ●155

You may define a symbolic function implicitly accepting an integer argument ; bonus : you may also define print and latex_print functins to get a "pleasant" output.

See function?.

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answered 13 years ago

parzan
977 ●6 ●15 ●36

updated 13 years ago

I found out that maxima does offer this capability, so one can use it in sage:

sage: p = maxima('sum(a[i]*x^i,i,0,n)')
sage: p
'sum(a[i]*x^i,i,0,n)
sage: p.taylor(x,0,3)                  
a[0]+a[1]*x+a[2]*x^2+a[3]*x^3
sage: p.diff(x)
'sum(i*a[i]*x^(i-1),i,0,n)

sadly:

sage: p.sage()
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
...
TypeError: unable to make sense of Maxima expression 'sum(a[i]*x^i,i,0,n)' in Sage

Also, it is not as strong as maple:

sage: p = maxima('product(1+a[i]*x+b[i]*x^2,i,0,n)')
sage: p.taylor(x,0,3)                               
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
taylor: unable to expand at a point specified in:
'product(b[i]*x^2+a[i]*x+1,i,0,n)
 -- an error. To debug this try: debugmode(true);

whereas in maple this works -

series(product(1+a[i]*x+b[i]*x^2,i=1..k),x=0,3);

gives the desired answer $1+\sum _{i=1}^{k}a_{{i}}x+ \left( \sum _{i=1}^{k}b_{{i}}-1/2\cdot{a_{{i} }}^{2}+1/2\cdot \left( \sum _{i=1}^{k}a_{{i}} \right) ^{2} \right) {x}^{2 }+O \left( {x}^{3} \right)$ .

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