### D.15.32 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.15.32.1 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.15.32.2 changeCoordinates |  | transform V by psi; you can also use 'V*phi'
> D.15.32.3 jordanVecField |  | transform V s.t. the linear part is in Jordan normal form
> D.15.32.4 diagonalizeVecFieldLin |  | l a list of VecFields. Change coordinates s.t. all linear parts are diagonal simultaneously.
> D.15.32.5 SaitoBase |  | algorithm to find a basis where the semisimple and nilpotent parts are easily read off
> D.15.32.6 diagonalizeVecField |  | diagonalize all VecFields in l simultaneously
> D.15.32.7 vecFieldToMatrix |  | matrix representation of V in the basis W
> D.15.32.8 decomposeVecField |  | split a vectorfield V / all entries of a list of vectorfields l into semisimple and nilpotent components
> D.15.32.9 diagonalizeMatrixSimul |  | find transformation which simultaneously diagonalizes all matrices in l
> D.15.32.10 invertAlgebraMorphism |  | return inverse of p exact up to degree n

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