\input texinfo @c -*-texinfo-*-
@comment $Id: singular.doc,v 1.175 2009-01-12 17:24:33 Singular Exp $
@comment this file contains the main structure of the manual
@c ------------------------
@c %**start of header
@c ------------------------
@setfilename singular.hlp
@settitle Singular Manual
@c @setchapternewpage odd
@paragraphindent 0
@c ------------------------
@c %**end of header
@c ------------------------
@c --------------------- general settings --------------------
@iftex
@afourpaper
@finalout
@pagesizes 23cm, 16.5cm
@end iftex
@set singularmanual 1
@set withplural 1
@set PSUFFIX (plural)
@set namespaces 1
@include version.texi
@include uname.texi
@c --------------------- end general setting --------------------
@c --------------------- invisible Info header ------------------------
@ifinfo
This is the texinfo file describing Singular (version @value{VERSION})
@end ifinfo
@c --------------------- end Info invisible header --------------------
@c --------------------- tex titlepage --------------------
@titlepage
@sp 2
@center @titlefont{Singular}
@sp 2
@center A Computer Algebra System for Polynomial Computations
@sp 5
@center @titlefont{Manual}
@center Version @value{VERSION}, @value{VERSION_DATE}
@sp 2
@center Singular is created and its development is directed and coordinated by
@center G.-M. Greuel, G. Pfister, and H. Sch@"onemann
@sp 2
@c @center with contributions by
@center Principal developers:
@center O. Bachmann, M. Brickenstein, W. Decker, A. Fr@"uhbis-Kr@"uger, K. Kr@"uger,
@center V. Levandovskyy, C. Lossen, W. Neumann, W. Pohl, J. Schmidt, M. Schulze,
@center T. Siebert, R. Stobbe, E. Westenberger, T. Wichmann, O. Wienand
@c @center O. Bachmann, W. Decker, A. Fr@"uhbis-Kr@"uger, C. Gorzel,
@c @center H. Grassmann, A. Heydtmann, D. Hillebrand, T. Keilen, U. Klein, K. Kr@"uger, M. Lamm, V. Levandovskyy
@c @center C. Lossen, B. Martin, M. Me@ss{}ollen, W. Neumann, T. N@"u@ss{}ler, W. Pohl, T. Siebert,
@c @center J. Schmidt, M. Schulze, R. Stobbe, M. Wenk, T. Wichmann, D. Yanovich
@sp 2
@author Fachbereich Mathematik
@author Zentrum f@"ur Computeralgebra
@author Universit@"at Kaiserslautern
@author D-67653 Kaiserslautern
@end titlepage
@c print short table of contents in tex right after titlepage
@c Detailed ToC is printed at the end of the manual
@shortcontents
@c -----------------------Top node-----------------------------------
@node Top, Preface, (dir), (dir)
@ifnothtml
@ifinfo
@center @strong{SINGULAR}
@center A Computer Algebra System for Polynomial Computations
@center User Manual for @sc{Singular} Version @value{VERSION}
@center @uref{http://www.singular.uni-kl.de}
@end ifinfo
@end ifnothtml
@ifhtml
@html
A Computer Algebra System for Polynomial Computations
HTML User Manual for Singular Version
@end html
@value{VERSION}, @value{VERSION_DATE}
@html
University of Kaiserslautern
Department of Mathematics
Centre for Computer Algebra
@end html
@end ifhtml
@menu
* Preface::
* Introduction::
* General concepts::
* Data types::
* Functions and system variables::
* Tricks and pitfalls::
@ifset withplural
* Non-commutative subsystem::
@end ifset
* Examples::
* Polynomial data::
* Mathematical background::
* SINGULAR libraries::
* Release Notes::
* Index::
@end menu
@c ----------------------------------------------------------------------------
@node Preface, Introduction, Top, Top
@chapter Preface
@cindex Preface
@include COPYING.texi
@c ----------------------------------------------------------------------------
@node Introduction, General concepts, Preface, Top
@chapter Introduction
@cindex Introduction
@include start.tex
@c ----------------------------------------------------------------------------
@node General concepts, Data types, Introduction, Top
@chapter General concepts
@cindex General concepts
@include general.tex
@c ----------------------------------------------------------------------------
@node Data types, Functions and system variables, General concepts, Top
@chapter Data types
@cindex Data types
@include types.tex
@c ----------------------------------------------------------------------------
@node Functions and system variables, Tricks and pitfalls, Data types, Top
@chapter Functions and system variables
@cindex Commands
@include reference.tex
@c ----------------------------------------------------------------------------
@ifset withplural
@node Tricks and pitfalls, Non-commutative subsystem, Functions and system variables, Top
@end ifset
@ifclear withplural
@node Tricks and pitfalls, Examples, Functions and system variables, Top
@end ifclear
@chapter Tricks and pitfalls
@cindex Tricks and pitfalls
@include tricks.tex
@c ----------------------------------------------------------------------------
@ifset withplural
@node Non-commutative subsystem, Examples, Tricks and pitfalls, Top
@chapter Non-commutative subsystem
@cindex Non-commutative subsystem
@menu
* Getting started with PLURAL::
* Letterplace::
@end menu
@include s-plural.tex
@include letterplace.tex
@end ifset
@c ----------------------------------------------------------------------------
@ifset withplural
@node Examples, Polynomial data, Non-commutative subsystem, Top
@end ifset
@ifclear withplural
@node Examples, Polynomial data, Tricks and pitfalls, Top
@end ifclear
@appendix Examples
@cindex Examples
@include examples.tex
@c ----------------------------------------------------------------------------
@node Polynomial data, Mathematical background, Examples, Top
@appendix Polynomial data
@cindex Polynomial data
@include pdata.tex
@c ----------------------------------------------------------------------------
@node Mathematical background, SINGULAR libraries, Polynomial data, Top
@appendix Mathematical background
@cindex Mathematical background
@include math.tex
@c @ifset withplural
@c @include s-plumath.tex
@c @end ifset
@c ----------------------------------------------------------------------------
@node SINGULAR libraries, Release Notes, Mathematical background, Top
@appendix SINGULAR libraries
@cindex SINGULAR libraries
@cindex LIBs
@sc{Singular} comes with a set of standard libraries. Their content is
described in the following subsections.
Use the @code{LIB} command (see @ref{LIB}) for loading of single libraries,
and the command @code{LIB "all.lib";} for loading all libraries.
@menu
* standard_lib:: extensions of Singular kernel
* General purpose::
* Linear algebra::
* Commutative algebra::
* Singularities::
* Invariant theory::
* Symbolic-numerical solving::
* Visualization::
* Coding theory::
* System and Control theory::
* Teaching::
* Tropical Geometry::
* Contributed::
@end menu
@c ----------------------------------------------------------
@node standard_lib, General purpose, SINGULAR libraries, SINGULAR libraries
@section standard_lib
The library @code{standard.lib} provides extensions to the
set of built-in commands and is automatically loaded
during the start of @sc{Singular}, unless @sc{Singular} is started up
with the @code{--no-stdlib} command line option (see
@ref{Command line options}).
@c lib standard.lib:LibInfo lib_fun lib_ex section
@c ----------------------------------------------------------------------
@c and now some routines from standard.lib:
@c ---------------------------------------------------------
@node qslimgb,par2varRing,standard_lib, standard_lib
@subsection qslimgb
@cindex qslimgb
@c lib standard.lib:qslimgb Fun subsection
@c ---------------------------------------------------------
@node par2varRing,,qslimgb, standard_lib
@subsection par2varRing
@cindex par2varRing
@c lib standard.lib:par2varRing Fun subsection
@c ----------------------------------------------------------------------------
@node General purpose, Linear algebra, standard_lib, SINGULAR libraries
@section General purpose
@cindex General purpose
@menu
* all_lib:: load all other libraries
* compregb_lib:: comprehensive Groebner base system
* general_lib:: procedures of general type
* inout_lib:: procedures for manipulating in- and output
* poly_lib:: procedures for manipulating polynomials and ideals
* random_lib:: procedures of random/sparse matrix and poly operations
* redcgs_lib:: procedures for reduced comprehesive Groebner systems
* ring_lib:: procedures for manipulating rings and maps
@end menu
@c ----------------------------------------------------------
@node all_lib, compregb_lib, General purpose, General purpose
@subsection all_lib
@cindex all_lib
@cindex all.lib
The library @code{all.lib} provides a convenient way to load all
libraries of the @sc{Singular} distribution.
@strong{Example:}
@smallexample
@c example
option(loadLib);
LIB "all.lib";
@c example
@end smallexample
@c ----------------------------------------------------------
@node compregb_lib, general_lib, all_lib, General purpose
@subsection compregb_lib
@c lib compregb.lib
@c ----------------------------------------------------------
@node general_lib, inout_lib, compregb_lib, General purpose
@subsection general_lib
@c lib general.lib no_ex
@c ----------------------------------------------------------
@node inout_lib, poly_lib, general_lib, General purpose
@subsection inout_lib
@c lib inout.lib
@c ----------------------------------------------------------
@node poly_lib, random_lib, inout_lib, General purpose
@subsection poly_lib
@c lib poly.lib
@c ----------------------------------------------------------
@node random_lib, redcgs_lib, poly_lib, General purpose
@subsection random_lib
@c lib random.lib
@c ----------------------------------------------------------
@node redcgs_lib, ring_lib, random_lib, General purpose
@subsection redcgs_lib
@c lib redcgs.lib
@c ----------------------------------------------------------
@node ring_lib, , redcgs_lib, General purpose
@subsection ring_lib
@c lib ring.lib
@c ----------------------------------------------------------------------------
@node Linear algebra, Commutative algebra, General purpose, SINGULAR libraries
@section Linear algebra
@cindex Linear algebra
@menu
* matrix_lib:: procedures for matrix operations
* linalg_lib:: procedures for algorithmic linear algebra
@end menu
@c ----------------------------------------------------------
@node matrix_lib, linalg_lib, Linear algebra, Linear algebra
@subsection matrix_lib
@c lib matrix.lib
@c ---------------------------------------------------------
@node linalg_lib, , matrix_lib, Linear algebra
@subsection linalg_lib
@c lib linalg.lib
@c ----------------------------------------------------------------------------
@node Commutative algebra, Singularities, Linear algebra, SINGULAR libraries
@section Commutative algebra
@cindex Commutative algebra
@menu
* absfact_lib:: absolute factorization for characteristic 0
* algebra_lib:: procedures for computing with algebras and maps
* cimonom_lib:: is the toric ideal of an affine monomial curve a complete intersection ?
* elim_lib:: procedures for elimination, saturation and blowing up
* grwalk_lib:: Groebner walk and Fraktal walk
* homolog_lib:: procedures for homological algebra
* intprog_lib:: Integer Programming
* lll_lib:: Integral LLL-Algorithm
* modstd_lib:: Groebner basis of ideals via modular computations
* mprimdec_lib:: procedures for primary decomposition of modules
* mregular_lib:: procedures for Castelnuovo-Mumford regularity
* noether_lib:: Noether normalization of an ideal
* normal_lib:: procedure for normalization
* primdec_lib:: procedures for primary decomposition
* primitiv_lib:: procedures for finding a primitive element
* realrad_lib:: procedures for finding the real radical
* reesclos_lib:: Rees Algebra and integral closure of an ideal
* resolve_lib:: Resolution of Singularities
* reszeta_lib:: Applications of Resolution of Singularities
* sagbi_lib:: Subalgebras bases Analogous to Groebner bases for ideals
* sheafcoh_lib:: Procedures for Computing Sheaf Cohomology
* toric_lib:: toric ideals
@end menu
@c ---------------------------------------------------------
@node absfact_lib, algebra_lib, Commutative algebra, Commutative algebra
@subsection absfact_lib
@c lib absfact.lib
@c ---------------------------------------------------------
@node algebra_lib, cimonom_lib, absfact_lib, Commutative algebra
@subsection algebra_lib
@c lib algebra.lib
@c ---------------------------------------------------------
@node cimonom_lib, elim_lib, algebra_lib, Commutative algebra
@subsection cimonom_lib
@c lib cimonom.lib
@c ----------------------------------------------------------
@node elim_lib, grwalk_lib, cimonom_lib, Commutative algebra
@subsection elim_lib
@c lib elim.lib
@c ----------------------------------------------------------
@node grwalk_lib, homolog_lib, elim_lib, Commutative algebra
@subsection grwalk_lib
@c lib grwalk.lib
@c ----------------------------------------------------------
@node homolog_lib, intprog_lib, grwalk_lib, Commutative algebra
@subsection homolog_lib
@c lib homolog.lib
@c ----------------------------------------------------------
@node intprog_lib, lll_lib, homolog_lib, Commutative algebra
@subsection intprog_lib
@c lib intprog.lib
@c ---------------------------------------------------------
@node lll_lib, modstd_lib, intprog_lib, Commutative algebra
@subsection lll_lib
@c lib lll.lib
@c ---------------------------------------------------------
@node modstd_lib, mprimdec_lib, lll_lib, Commutative algebra
@subsection modstd_lib
@c lib modstd.lib
@c ---------------------------------------------------------
@node mprimdec_lib, mregular_lib, modstd_lib, Commutative algebra
@subsection mprimdec_lib
@c lib mprimdec.lib
@c ----------------------------------------------------------
@node mregular_lib, noether_lib, mprimdec_lib, Commutative algebra
@subsection mregular_lib
@c lib mregular.lib
@c ----------------------------------------------------------
@node noether_lib, normal_lib, mregular_lib, Commutative algebra
@subsection noether_lib
@c lib noether.lib
@c ---------------------------------------------------------
@node normal_lib, primdec_lib, noether_lib, Commutative algebra
@subsection normal_lib
@c lib normal.lib
@c ----------------------------------------------------------
@node primdec_lib, primitiv_lib, normal_lib, Commutative algebra
@subsection primdec_lib
@c lib primdec.lib
@c ---------------------------------------------------------
@node primitiv_lib, realrad_lib, primdec_lib, Commutative algebra
@subsection primitiv_lib
@c lib primitiv.lib
@c ---------------------------------------------------------
@node realrad_lib, reesclos_lib, primitiv_lib, Commutative algebra
@subsection realrad_lib
@c lib realrad.lib
@c ---------------------------------------------------------
@node reesclos_lib, resolve_lib, realrad_lib, Commutative algebra
@subsection reesclos_lib
@c lib reesclos.lib
@c ---------------------------------------------------------
@node resolve_lib, reszeta_lib, reesclos_lib, Commutative algebra
@subsection resolve_lib
@c lib resolve.lib
@c ---------------------------------------------------------
@node reszeta_lib, sagbi_lib , resolve_lib, Commutative algebra
@subsection reszeta_lib
@c lib reszeta.lib
@c ---------------------------------------------------------
@node sagbi_lib, sheafcoh_lib , reszeta_lib, Commutative algebra
@subsection sagbi_lib
@c lib sagbi.lib
@c ---------------------------------------------------------
@node sheafcoh_lib, toric_lib, sagbi_lib, Commutative algebra
@subsection sheafcoh_lib
@c lib sheafcoh.lib
@c ---------------------------------------------------------
@node toric_lib, , sheafcoh_lib, Commutative algebra
@subsection toric_lib
@c lib toric.lib
@c ----------------------------------------------------------------------------
@node Singularities, Invariant theory, Commutative algebra, SINGULAR libraries
@section Singularities
@cindex Singularities
@menu
* alexpoly_lib:: resolution graph and Alexander polynomial
* arcpoint_lib:: truncations of arcs at a singular point
* classify_lib:: procedures for the Arnold-classifier of singularities
* curvepar_lib:: procedures for space curves
* deform_lib:: procedures for computing miniversal deformation
* equising_lib:: procedures for equisingularity strata
* gmssing_lib:: procedures for gauss-manin system of a singularity
* gmspoly_lib:: procedures for gauss-manin system of cohomologically tame polynomials
* hnoether_lib:: procedures for the Hamburger-Noether (Puiseux) development
* kskernel_lib:: procedures for kernel of the kodaira-spencer map
* mondromy_lib:: procedures to compute the monodromy of a singularity
* qhmoduli_lib:: procedures for moduli spaces of sqh-singularities
* sing_lib:: procedures for computing invariants of singularities
* spcurve_lib:: procedures for cm codimension 2 singularities
* spectrum_lib:: procedures for computing singularity spectra
@end menu
@c ---------------------------------------------------------
@node alexpoly_lib, arcpoint_lib, Singularities, Singularities
@subsection alexpoly_lib
@c lib alexpoly.lib
@c ---------------------------------------------------------
@node arcpoint_lib, classify_lib, alexpoly_lib, Singularities
@subsection arcpoint_lib
@c lib arcpoint.lib
@c ---------------------------------------------------------
@node classify_lib, curvepar_lib, arcpoint_lib, Singularities
@subsection classify_lib
@c lib classify.lib unix_only
@c ---------------------------------------------------------
@node curvepar_lib, deform_lib, classify_lib, Singularities
@subsection curvepar_lib
@c lib curvepar.lib
@c ----------------------------------------------------------
@node deform_lib, equising_lib, curvepar_lib, Singularities
@subsection deform_lib
@c lib deform.lib
@c ----------------------------------------------------------
@node equising_lib, gmssing_lib, deform_lib, Singularities
@subsection equising_lib
@c lib equising.lib
@c ----------------------------------------------------------
@node gmssing_lib, gmspoly_lib, equising_lib, Singularities
@subsection gmssing_lib
@c lib gmssing.lib
@c ----------------------------------------------------------
@node gmspoly_lib, hnoether_lib, gmssing_lib, Singularities
@subsection gmspoly_lib
@c lib gmspoly.lib
@c ---------------------------------------------------------
@node hnoether_lib, kskernel_lib, gmspoly_lib, Singularities
@subsection hnoether_lib
@c lib hnoether.lib
@c ---------------------------------------------------------
@node kskernel_lib, mondromy_lib, hnoether_lib, Singularities
@subsection kskernel_lib
@c lib kskernel.lib
@c ---------------------------------------------------------
@node mondromy_lib, qhmoduli_lib, kskernel_lib, Singularities
@subsection mondromy_lib
@c lib mondromy.lib
@c ---------------------------------------------------------
@node qhmoduli_lib, sing_lib, mondromy_lib, Singularities
@subsection qhmoduli_lib
@c lib qhmoduli.lib
@c ----------------------------------------------------------
@node sing_lib, spcurve_lib, qhmoduli_lib, Singularities
@subsection sing_lib
@c lib sing.lib
@c ---------------------------------------------------------
@node spcurve_lib, spectrum_lib, sing_lib, Singularities
@subsection spcurve_lib
@c lib spcurve.lib
@c ---------------------------------------------------------
@node spectrum_lib, , spcurve_lib, Singularities
@subsection spectrum_lib
@c lib spectrum.lib
@c ----------------------------------------------------------------------------
@node Invariant theory, Symbolic-numerical solving, Singularities, SINGULAR libraries
@section Invariant theory
@cindex Invariant theory
@menu
* finvar_lib:: procedures to compute invariant rings of finite groups
* ainvar_lib:: procedures to compute invariants rings of the additive group
* rinvar_lib:: procedures to compute invariants ring of reductive groups
* stratify_lib:: algorithmic stratification by the Greuel-Pfister algorithm
@end menu
@c ----------------------------------------------------------
@node finvar_lib, ainvar_lib, Invariant theory, Invariant theory
@subsection finvar_lib
@c lib finvar.lib
@c ----------------------------------------------------------
@node ainvar_lib, rinvar_lib, finvar_lib, Invariant theory
@subsection ainvar_lib
@c lib ainvar.lib
@c ----------------------------------------------------------
@node rinvar_lib, stratify_lib, ainvar_lib, Invariant theory
@subsection rinvar_lib
@c lib rinvar.lib
@c ----------------------------------------------------------
@node stratify_lib, ,rinvar_lib, Invariant theory
@subsection stratify_lib
@c lib stratify.lib no_ex
@c ----------------------------------------------------------------------------
@node Symbolic-numerical solving, Visualization, Invariant theory, SINGULAR libraries
@section Symbolic-numerical solving
@cindex Symbolic-numerical solving
@menu
* presolve_lib:: procedures for pre-solving polynomial equations
* solve_lib:: procedures to solve polynomial systems
* triang_lib:: procedures for decomposing zero-dimensional ideals
* ntsolve_lib:: one real solution of polynomial systems (Newton iteration)
* zeroset_lib:: procedures for roots and factorization
* signcond_lib:: computing realizable sign conditions
@end menu
@c ---------------------------------------------------------
@node presolve_lib, solve_lib, Symbolic-numerical solving, Symbolic-numerical solving
@subsection presolve_lib
@c lib presolve.lib
@c ---------------------------------------------------------
@node solve_lib, triang_lib, presolve_lib, Symbolic-numerical solving
@subsection solve_lib
@c lib solve.lib
@c ---------------------------------------------------------
@node triang_lib, ntsolve_lib, solve_lib, Symbolic-numerical solving
@subsection triang_lib
@c lib triang.lib no_ex
@c ---------------------------------------------------------
@node ntsolve_lib, zeroset_lib, triang_lib, Symbolic-numerical solving
@subsection ntsolve_lib
@c lib ntsolve.lib
@c ---------------------------------------------------------
@node zeroset_lib,signcond_lib, ntsolve_lib, Symbolic-numerical solving
@subsection zeroset_lib
@c lib zeroset.lib
@c ---------------------------------------------------------
@node signcond_lib, , zeroset_lib, Symbolic-numerical solving
@subsection signcond_lib
@c lib signcond.lib
@c ---------------------------------------------------------
@node Visualization, Coding theory, Symbolic-numerical solving, SINGULAR libraries
@section Visualization
@cindex Visualization
@menu
* graphics_lib:: procedures to draw with Mathematica
* latex_lib:: procedures for typesetting in TeX
* resgraph_lib:: visualization of resolution data
* surf_lib:: interface to the surf programm
* surfex_lib:: Procedures for Visualizing Surfaces
@end menu
@c ---------------------------------------------------------
@node graphics_lib, latex_lib, Visualization, Visualization
@subsection graphics_lib
@c lib graphics.lib no_ex
@c ---------------------------------------------------------
@node latex_lib, resgraph_lib, graphics_lib, Visualization
@subsection latex_lib
@c lib latex.lib
@c ---------------------------------------------------------
@node resgraph_lib, surf_lib, latex_lib, Visualization
@subsection resgraph_lib
@c lib resgraph.lib
@c ---------------------------------------------------------
@node surf_lib,surfex_lib, resgraph_lib, Visualization
@subsection surf_lib
This library requires the external programs @code{surf}
and/or @code{surfer} (should be in your search path).
They can be found at:
@*@code{surf}: @uref{http://surf.sf.net}
@*@code{surfer}: @uref{http://www.imaginary2008.de/surf.php}.
@c lib surf.lib no_ex
@c ---------------------------------------------------------
@node surfex_lib,, surf_lib, Visualization
@subsection surfex_lib
@c lib surfex.lib no_ex
@c ----------------------------------------------------------------------------
@node Coding theory, System and Control theory, Visualization, SINGULAR libraries
@section Coding theory
@cindex Coding theory
@menu
* brnoeth_lib:: Brill-Noether algorithm, Weierstrass semigroups and AG codes
* decodegb_lib:: Generating and solving systems of polynomial equations for decoding and finding the minimum distance of linear codes
@end menu
@c ---------------------------------------------------------
@node brnoeth_lib, decodegb_lib, , Coding theory
@subsection brnoeth_lib
@c lib brnoeth.lib
@c ---------------------------------------------------------
@node decodegb_lib, ,brnoeth_lib, Coding theory
@subsection decodegb_lib
@c lib decodegb.lib
@c ----------------------------------------------------------------------------
@node System and Control theory, Teaching, Coding theory, SINGULAR libraries
@section System and Control theory
@cindex System and Control theory
@cindex Control theory
@menu
* Control theory background:: An overview of the theory
* control_lib:: Algebraic analysis tools for System and Control Theory
* jacobson_lib:: Algorithms for Smith and Jacobson Normal Form
@end menu
@c ---------------------------------------------------------
@node Control theory background, control_lib, ,System and Control theory
@subsection Control theory background
Control systems are usually described by differential (or
difference) equations, but their properties of interest are
most naturally expressed in terms of the system trajectories
(the set of all solutions to the equations). This is formalized
by the notion of the system @emph{behavior}. On the other hand,
the manipulation of linear system equations can be formalized
using algebra, more precisely module theory. The relationship
between modules and behaviors is very rich and leads to deep
results on system structure.
The key to the module-behavior correspondence
is a property of some signal spaces that are modules
over the ring of differential (or difference) operators,
namely, @emph{the injective cogenerator property}.
This property makes it possible to translate any statement on the
solution spaces that can be expressed in terms of images and kernels,
to an equivalent statement on the modules. Thus analytic properties
can be identified with algebraic properties, and conversely, the
results of manipulating the modules using computer algebra can
be re-translated and interpreted using the language of systems theory.
This duality (@emph{algebraic analysis}) is widely used in behavioral
systems and control theory today.
For instance, a system is @strong{controllable} (a fundamental property
for any control system) if and only if the associated module
is torsion-free. This concept can be refined by the so-called
controllability degrees. The strongest form of controllability
(@emph{flatness}) corresponds to a projective (or even free) module.
Controllability means that one can switch from one system trajectory
to another without violating the system law (concatenation of
trajectories). For one-dimensional systems (ODE) that evolve in time,
this is usually interpreted as switching from a given past trajectory
to a desired future trajectory. Thus the system can be forced to
behave in an arbitrarily prescribed way.
The extreme case opposed to controllability is @strong{autonomy}: autonomous systems evolve independently according to their law, without being influenceable
from the outside. Again, the property can be refined in terms
of autonomy degrees.
@c ---------------------------------------------------------
@node control_lib, jacobson_lib, Control theory background, System and Control theory
@subsection control_lib
@c lib control.lib
@c ---------------------------------------------------------
@node jacobson_lib, , control_lib, System and Control theory
@subsection jacobson_lib
@c lib jacobson.lib
@c ----------------------------------------------------------------------------
@node Teaching, Tropical Geometry, System and Control theory, SINGULAR libraries
@section Teaching
@cindex Teaching
The libraries in this section are intended to be used for teaching purposes
but not for serious computations.
@menu
* aksaka_lib:: Algorithms for primality testing in polynomial time
* atkins_lib:: Procedures for teaching cryptography
* crypto_lib:: Procedures for teaching cryptography
* hyperel_lib:: Divisors in the jacobian of hyperelliptic curves
* teachstd_lib:: Procedures for teaching standard bases
* weierstr_lib:: Procedures for the Weierstrass Theorems
* rootsmr_lib:: counting the number of real roots
* rootsur_lib:: counting number of real roots of univariate polynomial
@end menu
@c ---------------------------------------------------------
@node aksaka_lib, atkins_lib, Teaching, Teaching
@subsection aksaka_lib
@c lib aksaka.lib
@c ---------------------------------------------------------
@node atkins_lib, crypto_lib, aksaka_lib, Teaching
@subsection atkins_lib
@c lib atkins.lib
@c ---------------------------------------------------------
@node crypto_lib, hyperel_lib, atkins_lib, Teaching
@subsection crypto_lib
@c lib crypto.lib
@c ---------------------------------------------------------
@node hyperel_lib, teachstd_lib, crypto_lib, Teaching
@subsection hyperel_lib
@c lib hyperel.lib
@c ---------------------------------------------------------
@node teachstd_lib, weierstr_lib, hyperel_lib, Teaching
@subsection teachstd_lib
@c lib teachstd.lib
@c ---------------------------------------------------------
@node weierstr_lib,rootsmr_lib, teachstd_lib, Teaching
@subsection weierstr_lib
@c lib weierstr.lib
@c ---------------------------------------------------------
@node rootsmr_lib,rootsur_lib, weierstr_lib, Teaching
@subsection rootsmr_lib
@c lib rootsmr.lib
@c ---------------------------------------------------------
@node rootsur_lib,,rootsmr_lib, Teaching
@subsection rootsur_lib
@c lib rootsur.lib
@c ----------------------------------------------------------------------------
@node Tropical Geometry, Contributed, Teaching, SINGULAR libraries
@section Tropical Geometry
@cindex Tropical Geometry
@menu
* polymake_lib:: interface to polymake and TOPCOM
* tropical_lib:: interface to gfan
@end menu
@c ----------------------------------------------------------
@node polymake_lib,tropical_lib,Tropical Geometry, Tropical Geometry
@subsection polymake_lib
@cindex polymake_lib
@c lib polymake.lib no_ex
@c ----------------------------------------------------------
@node tropical_lib,, polymake_lib, Tropical Geometry
@subsection tropical_lib
@cindex tropical_lib
@c lib tropical.lib no_ex
@c ----------------------------------------------------------------------------
@node Contributed,,Tropical Geometry, SINGULAR libraries
@section Contributed
@cindex Contributed libraries
@menu
* phindex_lib::
@end menu
@c ----------------------------------------------------------
@node phindex_lib,,, Contributed
@subsection phindex_lib
@cindex phindex_lib
@c lib phindex.lib
@c ----------------------------------------------------------
@node Release Notes, Index, SINGULAR libraries, Top
@chapter Release Notes
@cindex Release Notes
@include platform.tex
@c ----------------------------------------------------------
@node Index, , Release Notes, Top
@chapter Index
@cindex Index
@printindex cp
@c ---------------------------------------------------------
@c generate table of content and short table of content
@contents
@bye