implicitly defining a sequence of variables

Question

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?

Answer 1

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2

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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

link

posted Jun 23 '11

niles
3354 ● 5 ● 38 ● 92
http://nilesjohnson.net/

a = list(var("a%d" % i) for i in range(4))? DSM (Jun 23 '11)

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 (Jun 23 '11)

then maybe someone will improve `var` to automatically generate a sequence of variables ;) niles (Jun 23 '11)

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 (Jun 23 '11)

Maybe this would be possible with a generator... kcrisman (Jun 23 '11)

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Answer 2

1

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

link

posted Jul 04 '11

Jason Grout
3185 ● 7 ● 27 ● 71

updated Jul 04 '11

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 (Jul 04 '11)

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 (Jul 13 '11)

Answer 3

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) $.

implicitly defining a sequence of variables

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