Singular

D.8.8 rootisolation_lib

Library:

rootisolation.lib

Purpose:

implements an algorithm for real root isolation using interval arithmetic

Authors:

Dominik Bendle ([email protected])
Janko Boehm ([email protected]), supervisor Fachpraktikum
Clara Petroll ([email protected])

Overview:

In this library the interval arithmetic from interval.so is used. The new type ivmat, a matrix consisting of intervals, is implemented as newstruct. There are various functions for computations with interval matrices implemented, such as Gaussian elimination for interval matrices.

Interval arithmetic, the interval Newton Step and exclusion methods are used to implement the procedure rootIsolation, an algorithm which finds boxes containing elements of the vanishing locus of an ideal. This algorithm is specialised for zero-dimensional radical ideals. The theory about the interval Newton Step is detailed in [2].

Note that interval arithmetic and the aforementioned procedures are intended for rational or real polynomial rings.

References:

[1] Cloud, Kearfott, Moore: Introduction to Interval Analysis, Society for Industrial and Applied Mathematics, 2009
[2] Eisenbud, Grayson, Herzog, Stillman, Vasconcelos: Computational Methods in Commutative Algebra and Algebraic Geometry, Springer Verlag Berlin-Heidelberg, 3. edition 2004
[3] Andrew J. Sommese and Charles W. Wampler: The Numerical Solution of Systems of Polynomials - Arising in Engineering and Science, World Scientific Publishing Co. Pte. Ltd., 2005

Overloads:

[ ivmatGet indexing
print ivmatPrint printing
nrows ivmatNrows number of rows
ncols ivmatNcols number of columns
* ivmatMultiplyGeneral matrix multiplication

Procedures:

D.8.8.1 bounds		creates a new interval with given bounds
D.8.8.2 length		returns Euclidean length of interval
D.8.8.3 boxSet		returns box B with B[i]==I
D.8.8.4 ivmatInit		returns m x n [0,0]-matrix
D.8.8.5 ivmatSet		returns matrix A where A[i][j]=I
D.8.8.6 unitMatrix		returns m x m unit matrix where 1 = [1,1]
D.8.8.7 ivmatGaussian		computes M^(-1) using Gaussian elimination for intervals
D.8.8.8 evalPolyAtBox		returns evaluation of polynomial at a box
D.8.8.9 evalJacobianAtBox		jacobian matrix of A where polynomials are evaluated at B
D.8.8.10 rootIsolationNoPreprocessing		computes boxes containing unique roots of I lying in L
D.8.8.11 rootIsolation		slims down input box B and calls rootIsolationNoPreprocessing
D.8.8.12 rootIsolationPrimdec		runs a primary decomposition primdecGTZ before root isoation