Singular

D.15.19 VecField_lib

Library:

VecField.lib

Purpose:

vector fields, with algorithms for jordan and diagonal forms

Authors:

Adrian Rettich, rettich@mathematik.uni-kl.de
Raul Epure, epure@mathematik.uni-kl.de

References:

[1] Kyoji Saito, Quasihomogene isolierte
Singularitaeten von Hyperflaechen, 1971

Overview:

Implements a class VecField, represented by a vector. For example, 'VecField V = [x3,xy]' declares the vector field v = x3 d_x+xy d_y. Instead of a vector, an nx1 matrix is also accepted. The vector can be recovered as V.vec.
Supports coordinate transformations (via maps), which are represented by tracking a map 'V.coordinates' which maps the standard coordinates to those in which V is currently represented. V.dimension stores the vector field's dimension, which is just nvars(basering), and V.lin yields the linear part of V. You may set an additional parameter V.precision, which dictates the degree to which operations on the vector field should be exact.
The default precision is 1. Precision is preserved across transformations, additions, and all other manipulations of vector fields.

Procedures:

D.15.19.1 applyVecField		apply V to a poly p / an ideal I as an operator; you can also use 'V*p'/'V*I'. If an integer n is passed, consider V up to degree n.
D.15.19.2 changeCoordinates		transform V by psi; you can also use 'V*phi'
D.15.19.3 jordanVecField		transform V s.t. the linear part is in Jordan normal form
D.15.19.4 diagonalizeVecFieldLin		l a list of VecFields. Change coordinates s.t. all linear parts are diagonal simultaneously.
D.15.19.5 SaitoBase		algorithm to find a basis where the semisimple and nilpotent parts are easily read off
D.15.19.6 diagonalizeVecField		diagonalize all VecFields in l simultaneously
D.15.19.7 vecFieldToMatrix		matrix representation of V in the basis W
D.15.19.8 decomposeVecField		split a vectorfield V / all entries of a list of vectorfields l into semisimple and nilpotent components
D.15.19.9 diagonalizeMatrixSimul		find transformation which simultaneously diagonalizes all matrices in l
D.15.19.10 invertAlgebraMorphism		return inverse of p exact up to degree n