Revision history [back]

Series Reversion

I have a rational polynomial y(x) = P(x)/Q(x) which I want to revert (I'm trying to find the inverse of y).

y(x) is a symbolic expression in SAGE. In order to use g = y.reversion() I have to convert the symbolic expression to a power series.

When I do y(x).power_series(QQ) SAGE returns

[sage]: TypeError: unable to convert 45*(256*sqrt(2))*e to a rational

When I try y(x).power_series(RR) it tells me

[sage]: TypeError: denominator must be a unit

Am I missing some algebra?

Series Reversion

I have a rational polynomial y(x) = P(x)/Q(x) which I want to revert (I'm trying to find the inverse of y).

y(x) is a symbolic expression in ~~SAGE.~~ Sage. In order to use g = y.reversion() I have to convert the symbolic expression to a power series.

When I do y(x).power_series(QQ) ~~SAGE~~ Sage returns

[sage]: TypeError: unable to convert 45*(256*sqrt(2))*e to a rationalrational

When I try y(x).power_series(RR) it tells me

[sage]: TypeError: denominator must be a unitunit

Am I missing some algebra?

Series Reversion

I have a rational polynomial y(x) = P(x)/Q(x) which I want to revert (I'm trying to find the inverse of y).

y(x) is a symbolic expression in ~~Sage.~~ SAGE. In order to use g = y.reversion() I have to convert the symbolic expression to a power series.

y(x) looks like this

x^2*f1((3/x-1)^n,(1/x-1)^n) / (const + f2((3/x-1)^n,(1/x-1)^n)))

The numerator has higher degree than the denominator.

~~When I do y(x).power_series(QQ) ~~Sage~~ SAGE returns~~

~~[sage]: TypeError: unable to convert 45*(256*sqrt(2))*e to a rational rational~~

~~When I try y(x).power_series(RR) it tells me~~

~~[sage]: TypeError: denominator must be a unit unit~~

Am I missing some algebra?

Series Reversion

I have a rational polynomial y(x) = P(x)/Q(x) which I want to revert (I'm trying to find the inverse of y).

y(x) is a symbolic expression in SAGE. In order to use g = y.reversion() I have to convert the symbolic expression to a power series.

y(x) looks like this

x^2*f1((3/x-1)^n,(1/x-1)^n) x^2*f1((3/x-1)^m,(1/x-1)^m) / (const + f2((3/x-1)^n,(1/x-1)^n)))

The numerator has higher degree than the denominator.

When I do y(x).power_series(QQ) SAGE returns

[sage]: TypeError: unable to convert 45*(256*sqrt(2))*e to a rational

When I try y(x).power_series(RR) it tells me

[sage]: TypeError: denominator must be a unit

Am I missing some algebra?

Series Reversion

I have a rational polynomial ~~y(x) = P(x)/Q(x)~~ f(z) = P(z)/Q(z) which I want to revert (I'm trying to find the inverse of ~~y).~~

y(x) is a symbolic expression f).

I'll pick a simple example, the function f(z) = (1/z-1)^4

[sage:] z = var('z');
[sage:] z = FractionField(PolynomialRing(QQ, 'z')).objgen();
[sage:] f(z) = (1/z-1)^4;
[sage:] g = f(z).taylor(x,2,4); #I expand f in SAGE. In order to use g = y.reversion() I have to convert the symbolic expression to a power series.
 y(x) looks like this 
 x^2*f1((3/x-1)^m,(1/x-1)^m) / (const + f2((3/x-1)^n,(1/x-1)^n)))
 The numerator a Taylor series just to make sure it's a series
[sage:] f.expand().reversion();
           AttributeError: 'sage.symbolic.expression.Expression' object has higher degree than the denominator.
  When I do y(x).power_series(QQ) SAGE returns
 
 [sage]: TypeError: unable to convert 45*(256*sqrt(2))*e to a rational
  When I try y(x).power_series(RR) it tells me
 
 [sage]: no
           attribute 'reversion'
[sage:] g1 = g.power_series(QQ);
           TypeError: denominator must be a unitunit
[sage:] g.reversion();
           AttributeError: 'sage.symbolic.expression.Expression' object has no
           attribute 'reversion'
 Am

 A power series P(z) is not a rational function P(z)/Q(z). That's why the TypeError: denominator must be a unit appears. 
 Is it possible that SAGE cannot invert rational polynomials? In the SAGE help text the computation is passed to pari first, before using Lagrangian inversion.
 I missing some algebra?thought SAGE uses FLINT. Why is it not used for computing the inverse of rational polynomials?

          6    No.6 Revision 
      
    
        
        updated 2014-07-23 05:47:16 +0100  
    slelievre 
 17704  ●22 ●160 ●350  
        http://carva.org/samue...
         
 
 
 
  
 
 
 
  Series Reversion
 I have a rational polynomial f(z) = P(z)/Q(z) which I want to revert (I'm trying to find the inverse of f).
 I'll pick a simple example, the function f(z) = (1/z-1)^4
 [sage:] sage: z = var('z');
[sage:] var('z')
sage: z = FractionField(PolynomialRing(QQ, 'z')).objgen();
[sage:] 'z')).objgen()
sage: f(z) = (1/z-1)^4;
[sage:] (1/z-1)^4
sage: g = f(z).taylor(x,2,4); #I expand f in a Taylor series just to make sure it's a series
[sage:] f.expand().reversion();
           sage: f.expand().reversion()
AttributeError: 'sage.symbolic.expression.Expression' object has no
           no attribute 'reversion'
[sage:] sage: g1 = g.power_series(QQ);
           g.power_series(QQ)
TypeError: denominator must be a unit
[sage:] g.reversion();
           sage: g.reversion()
AttributeError: 'sage.symbolic.expression.Expression' object has no
           no attribute 'reversion'
 A power series P(z) is not a rational function P(z)/Q(z). That's why the TypeError: denominator must be a unit unit appears. 
 Is it possible that SAGE Sage cannot invert rational polynomials? In the SAGE Sage help text the computation is passed to pari first, before using Lagrangian inversion.
 I thought SAGE Sage uses FLINT. Why is it not used for computing the inverse of rational polynomials?
 
 
 
          7    No.7 Revision 
      
    
        
        updated 2014-07-23 05:52:04 +0100  
    slelievre 
 17704  ●22 ●160 ●350  
        http://carva.org/samue...
         
 
 
 
  
 
 
 
  Series Reversion
 I have a rational polynomial f(z) = P(z)/Q(z) which I want to revert (I'm trying to find the inverse of f).
 I'll pick a simple example, the function f(z) = (1/z-1)^4
 sage: z = var('z')
sage: z = FractionField(PolynomialRing(QQ, 'z')).objgen()
sage: f(z) = (1/z-1)^4
sage: g = f(z).taylor(x,2,4); #I expand f in a Taylor series just to make sure it's a series
sage: f.expand().reversion()
AttributeError: 'sage.symbolic.expression.Expression' object has no attribute 'reversion'
sage: g1 = g.power_series(QQ)
TypeError: denominator must be a unit
sage: g.reversion()
AttributeError: 'sage.symbolic.expression.Expression' object has no attribute 'reversion'
 A power series P(z) is not a rational function P(z)/Q(z). That's why the TypeError: denominator must be a unit appears. 
 Is it possible that Sage cannot invert rational polynomials? In the Sage help text the computation is passed to pari first, before using Lagrangian inversion.
 I thought Sage uses FLINT. Why is it not used for computing the inverse of rational polynomials?
  
 EDIT (after answer by slelievre):
 My problem now is how to avoid using the Symbolic Ring. I'm trying to do the following:
 I want to invert an approximation formula for the Gamma-function. I am using Spouge's approximation. It has terms in the numerator and the denominator. Ithink I can compute them separetely per se, but what if I wanted to do a Pade approximation of the function
 sage: f = (z+13)^(z+1/2)*exp(z)

sage: g = f.pade(5,5)
 (it's the term ahead of the sum term in Spouge's approximation)
 How do I get separate polynomials for the numerator and the denominator that are defined in the right algebraic structure?