1 | initial version |

This is more a comment, addressing rather the optimal setting to get many iterations in time. (It became an answer since there is no fit in space.)

The **short** answert to the question is to take `j( x, 0 )`

, which is still has the parent `Multivariate Power Series Ring in x, y over Rational Field`

. See the detailed answer of tmonteil .

The **long answer** following is hopefully not so long.
The limit is of course known, we solve in the fix point equation $y=f(x,y)$ for $y$. I will take for $f$ the version with the plus sign, $f(x,y)=x^3+xy^2$. The solution to this equation, rewritten as $xy^2 -y +x^3=0$ is *formally*
$$ y =frac 1{2x}( 1\pm \sqrt{ 1-4x^4} )\ ,$$
which can also be easily implemented. The nonsingular solution is obtained for the choice of the minus sign above. The following is the code for the iterations
$$y_0=y\ , \qquad y_{n+1}=f(x,y_n)\ ,$$
showing from them the terms that are established from the expected formal limit, plus a four in the valuation. (Next, unestablished term is also shown.) It is important in code to have a truncation when defining `f`

, else the 8.th iteration is still hard to compute in a glance.

The following code, written in the spirit of the posted question shows the linear convergence:

Code:

```
PREC = 50
STEPS = 7
R.<x, y> = PowerSeriesRing( QQ, default_prec=PREC )
f = x^3 + x*y^2 + R.O( PREC )
LIM = ( 1 - exp( log( 1-4*x^4 ) / 2 ) ) / 2 / x + R.O( PREC )
print "Expected limit:\nLIM = %s" % LIM
fn = y # f0 - the start of the iteration process
for n in [ 1..STEPS ]:
fn = f( x, fn )
yn = fn( x, 0 )
val = ( LIM - yn ).valuation()
print "f%s = %s" % ( n, yn + R.O( val+4 ) )
```

Results:

```
Expected limit:
LIM = x^3 + x^7 + 2*x^11 + 5*x^15 + 14*x^19 + 42*x^23 + 132*x^27 + 429*x^31 + 1430*x^35 + 4862*x^39 + 16796*x^43 + 58786*x^47 + O(x, y)^49
f1 = x^3 + O(x, y)^11
f2 = x^3 + x^7 + O(x, y)^15
f3 = x^3 + x^7 + 2*x^11 + x^15 + O(x, y)^19
f4 = x^3 + x^7 + 2*x^11 + 5*x^15 + 6*x^19 + O(x, y)^23
f5 = x^3 + x^7 + 2*x^11 + 5*x^15 + 14*x^19 + 26*x^23 + O(x, y)^27
f6 = x^3 + x^7 + 2*x^11 + 5*x^15 + 14*x^19 + 42*x^23 + 100*x^27 + O(x, y)^31
f7 = x^3 + x^7 + 2*x^11 + 5*x^15 + 14*x^19 + 42*x^23 + 132*x^27 + 365*x^31 + O(x, y)^35
```

Only seven iterations that fit in the line were done and shown here.

Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.