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[jaydfox]

Are you talking about recentering the Abel function or about my method of shift-conjugation of the original function? Computing the Carleman matrix of exp at some other place than 0 is unexpectedly more time consuming, as I already mentioned. 100 terms take perhaps 10-20 min at my place. While 400 at 0 takes perhaps 5min (without your acceleration method of course)

How are you computing the Carleman matrix? I use iterative matrix multiplication, so it takes only slightly more time to move the development point. I've been doing 250 term matrices in only a couple minutes, give or take. Here is some SAGE code I've been using.

Code:

def Carleman(field, expr, x, size):
    carl = matrix(field, size+1, size+1);
    mulm = matrix(field, size+1, size+1);
    cof = taylor(expr, x, 0, size).coeffs();
    for rrow in xrange(size+1):
        for ccol in xrange(len(cof)):
            col2 = cof[ccol][1]+rrow;
            if col2 < size+1:
                mulm[rrow, col2] = field(cof[ccol][0]);
    vec = vector(field, size+1);
    vec[0] = field(1);
    carl[0] = vec;
    for ii in xrange(size):
        vec = vec * mulm;
        carl[ii+1] = vec;
    return carl;

# Small test run
Flt = ComplexField(27);
size = 5;
shft = Flt(1, 0);
x = var('x');
Cf = Carleman(Flt, e^(x+shft)-shft, x, size)
print Cf;
# Calculate the Abel function
Flt = ComplexField(512);
size = 250;
shft = Flt(1, 0);
time Cf = Carleman(Flt, e^(x+shft)-shft, x, size);
# time CI = Carleman(Flt, x, x, size);
CI = MatrixSpace(RR, size+1, size+1).identity_matrix();
M = Cf.submatrix(1,0,size,size) - CI.submatrix(1,0,size,size);
time A = vector(Flt, M.solve_left(vector(Flt, [1] + [0 for kk in xrange(size-1)]))[0]);
print A.change_ring(CC)[0:20];

Output:

Code:

[  1.000000          0          0          0          0          0]
[  1.718282   2.718282   1.359141  0.4530470  0.1132617 0.02265235]
[  2.952492   9.341549   12.05983   8.945981   4.699514   1.925110]
[  5.073214   24.07712   50.12800   62.18783   53.35036   34.82992]
[  8.717212   55.16170   158.4776   278.1409   340.3287   314.9432]
[  14.97863   118.4792   434.1022   987.6331   1582.217   1922.546]
Time: CPU 52.15 s, Wall: 52.51 s
Time: CPU 20.40 s, Wall: 20.41 s
(0.915948856840961, -0.208605479751221, -0.0545257718234947,
0.0713217792410870, -0.0199817585842940, -0.0110084938578771,
0.0119893515581789, -0.00267961632960444, -0.00264867908203897,
0.00236244492078691, -0.000354279433811006, -0.000647870949133354,
0.000488726691175665, -0.0000401511878585745, -0.000157929045310358,
0.000108539033152715, -2.24888735429443e-6, -0.0000505453252195363,
0.0000483511387610123, -0.0000269817432526313)

Note that I used a ComplexField. A RealField would have sufficed and would have been faster, but this shows that we can now easily step away from the real line if we wanted to.