Noud's profile - activity

I actually want to implement Askey-Wilson polynomials (http://en.wikipedia.org/wiki/Askey%E2%80%93Wilson_polynomials), but I want to give the polynomials some extra functions so that you can do something like: Askey_Wilson(n, z, a, b, c, d, q).three_term() and you get a triple with the three term recurrence relation. So if n is an integer I can use Expression, but if n is a variable I want to use an other object. Your answer does work, except that I don't want to call askey_wilson.I all the time, but just without the I.

Dear Sage people,

I want to create a new (mathematical) object that sometimes is an Expression and sometimes a SymbolicFunction, depending on the arguments. You can think of this for example like $f(a, b, t) = \int_0^t a^b e^{-x^2} dx$. For special values of $t$ I would like to see it as an Expression ($t=0$ or $t=\infty$), but in all other cases I want it to be a BuiltinFunction (or something alike).

In Sage I can do something like:

class MyObjectExpression(Expression):
    def __init__(self, a, b, t):
        Expression.__init__(self, integral(a**b*e**(-x**2), x, 0, t))
    # More (override) stuff below

class MyObjectFunction(BuiltinFunction):
    def __init__(self, a, b, t):
        BuiltinFunction.__init__(self, 'f(a,b,t)', nargs=1)
    # More (override) stuff below

def MyObject(a, b, t):
    if t == 0 or t == infty:
        return MyObjectExpression(a, b, t)
    else:
        return MyObjectFunction(a, b, t)

Is it possible to combine these three things into one class? So I want to create a class which is sometimes an Expression and sometimes an much more abstract class, is this possible?

Best, Noud

Edit: What I actually want to do is programming Askey-Wilson polynomials and give them extra options, like a three term recurrence relation. But this depends on $n$. I already programmed this.

class Askey_Wilson(SageObject):
    def __init__(self, SR, n, z, a, b, c, d, q):
        self.n = n
        self.z = z
        self.q = q
        self.a = a
        self.b = b
        self.c = c
        self.d = d
        self.param = [a, b, c, d]

        if self.n in ZZ:
            self.I = self.evaluate()
        else:
            self.I = var('askey_wilson')

    def __repr__(self):
        return 'p_%i(%s;%s,%s,%s,%s|%s)' % (
            self.n, self.z, self.a, self.b, self.c, self.d, self.q
        )

    def evaluate(self):
        n, q, z, a, b, c, d = [self.n, self.q, self.z] + self.param

        lc = qPochhammerSymbol(SR, [a*b, a*c, a*d], q, n) / a**n
        poly = BasicHypergeometricSeries(SR,
            [q**(-n), a*b*c*d*q**(n-1), a*z, a*z**(-1)],
            [a*b, a*c, a*d], q, q)
        return lc*poly

    def three_term_recurrence(self):
        A, B, C = 0, 0, 0
        # compute three term recurrence relation
        return A, B, C

But now every time I want to know the explicit value of the Askey-Wilson polynomials I have to call askey_wilson.I. I want to get rid of the I.

I found the answer myself. Instead of using AA you should use the SR (Symbolic Ring). An example:

sage: q = var('q')
sage: F.<a,b,c,d> = FreeAlgebra(SR, 4)
sage: (1+q)**(1/2)*b*a
(sqrt(q+1))*b*a

This SR ring is still a bit of a mystery to me, but it seems to work.

Hi,

I'm working with $q$-functions and I would like to define a free algebra over the algebraic field with an extra parameter $q$ for which I can also take roots i.e. I would like to do something like this

sage: F.<q> = FunctionAlgebra(AA)
sage: A.<a,b,c,d> = FreeAlgebra(F, 4)
sage: (1+q)**(1/2)*a*b
sqrt(1+q)*a*b

The function FunctionAlgebra is not the correct function, since it only includes (integer) powers of $q$. Is it possible to extend this FunctionAlgebra to something where sums of powers of $q$ also have roots? Or is there an other function for which I can do this?

Best, Noud

I don't think there's a method that queries the assumptions data. You can write a simple function to test this using the output of assumptions():

sage: a=var('a')
sage: assume(a, 'integer')
sage: any('a is integer' in str(x) for x in assumptions())
True
sage: any('a is real' in str(x) for x in assumptions())
False

The assumptions function returns a list of GenericDeclaration objects which are turned into strings above.

sage: type(assumptions()[0])
<class 'sage.symbolic.assumptions.GenericDeclaration'>

Beware that the only subsystem of Sage that respects (or takes account of) symbolic assumptions is Maxima. So some symbolic operations will "know" about the assumptions (the ones that use Maxima under the hood) and some won't.

I agree with you that it is indeed quite hard and maybe even impossible to do it in a nice way. Let me see if I can bury it in a new method of the symbolic expression type. This will give me some more insight in how Sage works. I'll mark your solution as the solution, since it is obvious a solution. ;) Thank you!

Sorry to say this to you, but I think this is an extremely ugly solution. Is there an other solution to give a variable properties like being positive, or having real part, etc. So I might should have asked, is there a way to give variables extra properties, like being real, having real part, being imaginary, etc?

Dear Sage,

I define the following variable

sage: a = var('a')
sage: assume(a, 'integer')

How do I check that is variable is an integer? The following doese not seem to work in sage 5.2.

sage: a in ZZ
False
sage: 2 in ZZ
True
sage: assumptions()
[a is integer]

How should I check the assumption that a is an integer?

Best, Noud

Nice! Thank you for this fix!

Some debug output after the lines pasted in the question:

sage: %debug                                     
> /home/burcin/sage/ask_qpoch.py(28)_eval_()
     27         if len(args) != self.m + 2:
---> 28             raise RuntimeError, args
     29 

ipdb> print self
qPochhammer(3)
ipdb> self.name()
'qPochhammer(3)'

Note that the function in use is not qPochhammer(3). ATM, we expect each BuiltinFunction to be instantiated once. In other words, each class corresponds to exactly one symbolic function. This is evident in the check on line 827 of sage/symbolic/function.pyx. Changing this to also check for the name of the function fixes your problem.

Patch waiting for review on trac ticket #13586.

Dear Sage,

I'm trying to program qPochhammer functions which can interact with sage in a nice way. However I encounter some weird behavior of these new BuiltinFunctions. This is what I made: from sage.symbolic.function import BuiltinFunction

class qPochhammer(BuiltinFunction):

    def __init__(self, m=1, expand_exp=False):
        self.m = m
        self.expand_exp = expand_exp
        self.has_arguments = False

        BuiltinFunction.__init__(self, 'qPochhammer(%i)' % self.m, nargs=(m+2))

    def _latex_(self):
        if self.has_arguments:
            return '(%s; %s)_{%s}' % (self.a, self.q, self.n)
        return 'qPochhammer'

    def _qPochhammer1(self, a, q, n):
        if n == 0:
            return 1
        return (1 - a*q**(n-1))*self._qPochhammer1(a, q, n-1)

    def _expand(self):
        return prod([self._qPochhammer1(elm, self.q, self.n) \
                     for elm in self.a])

    def _eval_(self, *args):
        if len(args) != self.m + 2:
            raise RuntimeError, args

        if self.expand_exp:
            return self.evaluate(args[:self.m], args[-2], args[-1])

        return None

    def _evalf_(self, *args, **kwds):
        if type(self.a) == tuple:
            return self._expand()
        return self._qPochhammer1(self.a, self.q, self.n)

    def evaluate(self, a, q, n):
        self.a = a
        self.q = q
        self.n = n

        self.has_arguments = True

        if type(self.a) == tuple:
            return self._expand()
        return self._qPochhammer1(self.a, self.q, self.n)

Now I try the following the the sage console:

sage: a, b, c, q = var('a b c q')
sage: qshift3 = qPochhammer(m=3, expand_exp=True)
sage: qshift3(a, b, c, q, 2)
(c - 1)*(b - 1)*(a - 1)*(c*q - 1)*(b*q - 1)*(a*q - 1)
sage: qshift2 = qPochhammer(m=2, expand_exp=True)
sage: qshift2(a, b, q, 2)
---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)

/home/noud/work/sage/local/lib/python2.7/site-packages/sage/all_cmdline.pyc in  <module>()
----> 1 
      2 
      3 
      4 
      5 

/home/noud/work/sage/local/lib/python2.7/site-packages/sage/symbolic/function.so in sage.symbolic.function.Function.__call__ (sage/symbolic/function.cpp:4941)()
    430 
    431 
--> 432 
    433 
    434 

/home/noud/work/sage/local/lib/python2.7/site-packages/sage/all_cmdline.pyc in _eval_(self, *args)
     27 
     28 
---> 29 
     30 
     31 

RuntimeError: (a, b, q, 2)

Can someone explain to my why I get a RuntimeError and how to solve this?

Best regards, Noud

Thank you Simon! I'm looking forward to an answer on the sage-combinat-devel mailinglist.

I tried to do this, but I do not know how to make the generators of infinite order. I define `FreeAlgebra(QQ, 2n, 'x')` and define with `FreeAlgebraQuotient` relations $x_i x_{2n-i} = 1 = x_{2n-i} x_i$. But how do you define that $x_i^k = x_i^k$ (this looks a bit ambiguous)? In the examples in this reference everything has finite order.

Hello Sage,

I would like to make a ring over $\mathbb{Q}$ with $n$ variables which are non-commutative and also include their inverses. So I want to generate the free algebra over $\mathbb{Q}$ with generates $x_1, x_2, ..., x_n$ and $x_1^{-1}, x_2^{-1}, ..., x_n^{-1}$. How can I do this?

Best regards, Noud