ore_algebra.analytic.monodromy

Monodromy matrices

Functions

monodromy_matrices(dop, base[, eps, algorithm])

Compute generators of the monodromy group of dop with base point base.

ore_algebra.analytic.monodromy.monodromy_matrices(dop, base, eps=1e-16, algorithm='connect')

Compute generators of the monodromy group of dop with base point base.

OUTPUT:

A list of matrices, each encoding the analytic continuation of solutions along a simple positive loop based in base around a singular point of dop (with no other singular point inside the loop). Identity matrices may be omitted. The precise choice of elements of the fundamental group corresponding to each matrix (position with respect to the other singular points, order) are unspecified.

EXAMPLES:

sage: from ore_algebra import *
sage: from ore_algebra.analytic.monodromy import monodromy_matrices
sage: Dops, x, Dx = DifferentialOperators()

sage: monodromy_matrices(Dx*x*Dx, 1)
[
[  1.0000...  [6.2831853071795...]*I]
[          0               1.0000...]
]

sage: monodromy_matrices(Dx*x*Dx, 1, algorithm="loop")
[
[[1.0000...] + [+/- ...]*I  [+/- ...] + [6.283185307179...]*I]
[  [+/- ...] + [+/- ...]*I           [1.0000...] + [+/- ...]*I]
]

sage: monodromy_matrices(Dx*x*Dx, 1/2)
[
[ [1.0000...] + [+/- ...]*I  [+/- ...] + [3.1415926535897...]*I]
[   [+/- ...] + [+/- ...]*I           [1.0000...] + [+/- ...]*I]
]