software for algebraic geometry

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The IMA Volumes

in Mathematics

and its Applications

Series EditorsDouglas N. Arnold

Volume 148

Arnd Scheel


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Institute for Mathematics and

its Applications (IMA)

The Institute for Mathematics and its Applications was established by a

grant from the National Science Foundation to the University of Minnesota

in 1982. The primary mission of the IMA is to foster research of a truly inter-

disciplinary nature, establishing links between mathematics of the highest

caliber and important scientific and technological problems from other dis-

ciplines and industries. To this end, the IMA organizes a wide variety of pro-

grams, ranging from short intense workshops in areas of exceptional interest

and opportunity to extensive thematic programs lasting a year. IMA Volumes

are used to communicate results of these programs that we believe are of particular value to the broader scientific community.

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for Mathematics and its Applications:

http://www.ima.umn.edu/springer/volumes.html

Presentation materials from the IMA talks are available at

http://www.ima.umn.edu/talks/

Douglas N. Arnold, Director of the IMA

* * * * * * * * * *

IMA ANNUAL PROGRAMS

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(Continued at the back)


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Editors

Software for AlgebraicGeometry

Michael Stillman Nobuki TakayamaJan Verschelde


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© 2008 Springer Science + Business Media, LLCAll rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science + Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if theyare not identified as such, is not to be taken as an expression of opinion as to whether or not they aresubject to proprietary rights.

Camera-ready copy provided by the IMA.

9 8 7 6 5 4 3 2 1

springer.com

Series Editors

Arnd Scheel

Applications

Minneapolis, MN 55455USA

USA

Douglas N. Arnold

Institute for Mathematics and its

University of Minnesota

ISBN: 978-0-387-78132-7 e-ISBN: 978-0-387-78133-4DOI: 10.1007/978-0-387-78133-4

Michael StillmanDepartment of Mathematics

Cornell University

Nobuki TakayamaDepartment of Mathematics

Rokko, Kobe 657-8501Japan

Jan VerscheldeDepartment of MathematicsUniversity of IllinoisChicago, IL 60607-7045USA

Editors

Kobe UniversityIthaca, NY 14853-4201

Mathematics Subject Classification (2000):11R09, 11Y99, 12D05, 13P10, 14P05, 14Q05, 14Q1052A39, 52B20, 52B55, 65D18, 65F15, 65F20, 65F22, 65H10, 65H20, 65-04, 91-08

Library of Congress Control Number: 2008923357


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F O R E W O R D

This IMA Volume in Mathematics and i ts AppUcations

S O F T W A R E F O R A L G E B R A I C G E O M E T R Y

contains papers presented at a highly successful one-week workshop on the

same t i t le. The event was an integral part of the 2006-2007 IMA Thematic

Year on "Ap plications of Algeb raic Ge om etry." We are grateful to all th e

part icipants for making this workshop a very productive and st imulat ing

event . Special thanks to Michael E. Sti l lman (Department of Mathematics,

Cornel l Univers ity) , Nobuki Takayama (Dep ar tmen t of M athem at ics , K obe

Universi ty) , and Jan Verschelde (Department of Mathematics, Stat is t ics

and Computer Science, University of Illinois at Chicago) for their superb

role as workshop organizers and editors of these proceedings.

We take this oppor tuni ty to thank the Nat ional Science Foundat ion

for i ts support of the IMA.

S e r i e s Ed i t o r s

Douglas N. Arnold, Director of the IMA

Arnd Scheel , Deputy Director of the IMA


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P R E F A C E

The workshop on "Software for Algebraic Geometry" was held in the

week from 23 to 27 October 2006, as the second workshop in the thematic

year on Applicat ions of Algebraic Geometry at the IMA.

Algori thms in algebraic geom etry go han d in han d w ith software pack

ages th at im pleme nt them . Together the y have established the m od ern field

of computat ional algebraic geometry which has come to play a major role

in both theoret ical advances and applicat ions. Over the past f if teen years,

several excellent general purpose packages for computations in algebraic

geometry have been developed, such as CoCoA, Singular and Macaulay 2.

While these packages evolve continuously, incorporat ing new mathematical

advances , they bo th mo t ivate and dem and the creat ion of new m athem at ics

and smar ter algor i thms.

Surrounding the general packages, a host of specialized packages for

dedicated and focused computat ions have created a platform for the in

teract ion of algebraic geometry with numerous other areas of mathemat

ics including optimization, combinatorics, polyhedral geometry, numerical

analysis and com pute r science. Th e wo rkshop brou gh t togeth er a wide

array of theoret icians and practi t ioners interested in the development of

algori thms and software in algebraic geometry at this workshop. Such in

teractions are essential for dramatic increases in the power and appUcability

of algorithms in the field.

There were 89 registered part icipants at the workshop. At four talks

a day, 20 regular 50 m inutes talk s were scheduled. On M onday evening,

10 posters were presen ted. On Wedn esday and T hu rsda y evening we had

respectively 5 and 6 software d em on stratio ns. T h e Ust of featured software

packages includes Macaulay 2, SAGE, HomLab, Ber t ini , APAtools , PH-

Clab , PHC map le , PHC pack, K N O PP IX /M ath , D -modules for Macaulay 2 ,

Singular , Risa/Asi r , CRACK, di f falg, RIFsimp, Gambi t , FGb/RS, Co-

CoALib, 4t i2, PHoMpara, SYNAPS, DEMiCs, Magma, Kronecker , SOS-

TOOLS, Gfan, Maple 11.

The IMA systems group had instal led many of these programs on the

computers at the IMA. At the poster session, the part icipants were given

the opportunity to instal l the featured software systems on their laptop.

A demonstrat ion cluster computer of Rocketcalc was running during the

poster session and accessible to al l part icipants during the workshop.

The evening before the workshop dinner on Tuesday started with a

problem session. Pr io r to this session we m ad e a list of problem desc ription s

available on th e web si te. Th e workshop ended on Friday evening with some

addit ional problems, discussion on the posted problems, and a presentat ion

of Jiawang Nie about the applicat ion to semidefini te programming to solve

systems of algebraic equations which arise from differential equations. We


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vui PREFACE

are also happy that several new research projects were st imulated by this

problem session. Some results are going to appear elsewhere.

Instead of th e "second chances" (usual for IMA wo rkshops), the p art ic

ipants were given the opportunity to test the software systems on Wednes

day and Thursday evening. The evening session started with a one hour

plenary session, where each software system on demo in the evening was

briefly explained. Following th is plen ary session, th e pa rtic ipa nts mo ved

to the 4th i ioor of Lind Hall , to experience the software systems on the

computers in the open poster area, or in paral lel , in the classroom 409.

The IMA systems group worked hard in the weeks leading up to the

workshop to install the software systems. Their effort benefited not only the

workshop part icipants , but al l al l subsequent part icipants to the thematic

year, as they found their workstat ions equipped with the latest software

tools in algebraic geometry.

The papers in this volume describe the software packages Bert ini , PH-

Clab, Gfan, DEMiCs, SYNAPS, Tr im, Gambi t , ApaTools , and the appl i

cat ion of Risa/Asir to a conjecture on mult iple zeta values. We thank the

part icipants to the workshop, the authors and the anonymous referees. We

are grateful to the editorial staff of the IMA, Patricia V. Brick and Dzung

N .

Nguyen, for their dedication and care.

Mi c h a e l E . S t i l l ma n

Depar tment of Mathemat ics

Cornell University

ht tp : / /www.math .cornel l . edu/People /Facul ty / s t i l lman.h tml

N o b u k i T a k a y a m a

Depar tment of Mathemat ics

Kobe Universi ty

ht tp : / /www.math .sc i .kobe-u .ac . jp / t aka /

J a n Ve r s c h e l d e

Department of Mathematics, Stat is t ics aj id Computer Science

University of Illinois at Chicago

h t t p : / / www2.mat h .u i c . edu / j a n /

http://www.math.cornell.edu/People/Faculty/stillman.htmlhttp://www.math.sci.kobe-u.ac.jp/http://www.math.sci.kobe-u.ac.jp/http://www2.math.uic.edu/http://www2.math.uic.edu/http://www2.math.uic.edu/http://www.math.sci.kobe-u.ac.jp/http://www.math.cornell.edu/People/Faculty/stillman.html


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C O N T E N T S

Foreword v

Preface vii

Software for numerical algebraic geometry: A paradigm

and progress tow ards i ts implem entat ion 1

Daniel J. Bates, Jon athan D . Hauen stein,

Andrew J. Sommese, and Charles W. W ampler II

PH Clab: A M AT LAB /Octave inter face to PHC pack 15

Yun Guan and Jan Verschelde

Co mp uting Grob ner fans and tropical variet ies in Gfan 33

Anders Nedergaard Jensen

On a conjecture for the dimension of the space of

the mu lt iple zeta values 47

Masanobu Kaneko, Masayuki Noro,

and Ken'ichi Tsuruma ki

DEMiCs: A software package for computing the mixed

volum e via d5Tiamic en um eratio n of all mixed cells 59

Tomohiko Mizutani and Akiko Takeda

SYNAPS, a l ibrary for dedicated applicat ions in

symbolic num eric com puting 81

Bernard Mourrain, Jean-Pascal Pavone,

Philippe Trebuchet, Elias P. Tsigaridas, a nd Julien Wintz

Tropical impUcit ization and mixed f iber polytop es I l l

Bem d Sturmfels and Josephine Yu

Towards a black-box solver for finite games:

Co mp uting all equil ibria with Ga mb it and PH Cp ack 133

Theodore L. Turocy

ApaTools: A software toolbox for approximate

polynom ial algebra 149

Zhonggang Zeng

List of workshop pa rt icipa nts 169

ix


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SOFTWARE FOR NUMERICAL ALGEBRAIC GEOMETRY:

A PARADIGM AND PROGRESS TOWARDS ITS

IMPLEMENTATION

DANIEL

J.

BATES*, JONATHAN

D.

HAUENSTEINt,

ANDREW

J.

SOMMESE*,

AND

CHARLES

W.

WAMPLER

n§

A b st ra c t . Though numer ical methods

to

find all the isolated solutions

of

nonllnestr

systems

of

mu ltivariate polynom ials go back 30 years,

it

is only over the last decad e th at

numerical methods have been devised for the computation and manipulation

of

algebraic

sets coming from polynomJEil systems over the complex num bers. Collectively, these

algorithms and the underlying theory have come to be known as numerical algebraic

geometry. Several software packages are capable of carrying out some of the operations

of numerical algebraic geometry, although

no

one package provides

all

such capa bilities.

This paper contains

an

enum eration

of the

operations that

an

ideal software package

in this field would allow.

The

current

and

upcoming capabilities

of

Bertini,

the

most

recently released package

in

this field,

are

also described.

K e y w o rd s . Homotopy continuation, numerical algebrjiic geometry, polynomial

systems, software, Bertini.

AMS ( MO S ) su b j e c t

cla ssi fic at ion s. 65H10, 65H20, 65-04, 14Q99.

1.

Introduction.

Numerical algebraic geometry refers to the appli

cation of numerical methods to compute the solution sets of polynomial

systems, generally over C. In particular, basic numerical algebraic geome

try embodies probability one algorithms for computing all isolated solutions

of a polynomial system as well as the numerical irreducible decomposition

of an algebraic set, i.e., one or more points on each irreducible component

in each dimension. More recently, numerical algebraic geometry has grown

to include more advanced techniques which make use of the basic methods

in order to compute data of interest in both real-world applications and

pure algebraic geometry.

' I ns t i tu te

for

Ma thematics and

its

Applications, University

of

Minnesota, M inneapo

lis, MN 55122 ([email protected], www.nd .edu /~dbates l). This author was supported

by

the Duncan Chair

of

the University

of

Notre D ame,

the

University

of

Notre Dam e,

NSF

grant DMS-0410047,

the

Arthur

J.

Schmitt Foundation,

and the

Institute

for

Mathe

matics

and its

Applications

in

Minneapolis (IMA).

* Department

of

Mathematics, University

of

Notre Dame, Notre Dame,

IN

46556

([email protected], www.nd.edu/~jhauenst ). This author was supported by the D uncan

Chair of th e University of Notre Dame, NSF grant DMS-0410047, and the Institute for

Mathematics and its Applications in Minneapolis (IMA).

'•Department

of

Mathematics, University

of

Notre Dame, Notre Dame,

IN

46556

([email protected], www.nd .edu /~somm«e). This author was supported by the Du ncan

Chair of the University of Notre Dame, the U niversity of Notre Dame, NSF grant DMS-

0410047, and the Institute for M athematics and its Applications in Minneapolis (IMA).

^General Motors Research

and

Development, Mail Code 480-106-359, 30500 Mound

Road, Warren, MI 48090 ([email protected] , www.nd.edu/~cwamplel ).

This author was supported % NSF grant DMS-0410047 and the Insstitute for Math

ematics and its Applications in Minneapolis (IMA).

mailto:[email protected]:[email protected]://www.nd.edu/'%5edbateslhttp://www.nd.edu/'%5edbateslmailto:[email protected]:[email protected]://www.nd.edu/Hhauensthttp://www.nd.edu/Hhauenstmailto:[email protected]:[email protected]:[email protected]:[email protected]://www.nd.edu/~cwamplelhttp://www.nd.edu/~cwamplelhttp://www.nd.edu/~cwamplelmailto:[email protected]:[email protected]://www.nd.edu/Hhauenstmailto:[email protected]://www.nd.edu/'%5edbateslmailto:[email protected]


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2 DANIEL J. BATES ET AL.

One of the key tools used in the algori thms of numerical algebraic

geo me try is ho m oto py c on tinua tion [1, 16], a m eth od for finding all zero-

dimensional solutions of a polynomial system. Given a polynomial system

/ : C ^ —» C " to be solved by homo topy con tinuation , one f irst forms a poly

nomial system g th at is related to / in a prescribed way bu t has known, or

easi ly computable, solut ions. The systems

g

an d / are com bined to form

a homotopy, such as the l inear homotopy

H{x,t) — f • {1 — t) +'y • t • g

where 7 e C is randomly chosen. For a properly formed homotopy, there

are continuous solution paths leading from the solutions of g to those of

/ which m ay be followed using predictor-corrector me tho ds. Singular

solutions cause numerical diflttculties, so singular endgames [17, 18, 19]

are typically employed . Zero-dim ensional solving is discussed further in

Section 2.2.

Numerical algebraic geometry treats both zero-dimensional ( isolated)

solutions an d jwsit ive dimen sional solution sets . T he bu ilding blocks of the

solution set of a set of equations are the irreducible components, i .e., the

algebraic subsets of the solution set that consist of connected sets of points

with neighborhoods biholomorphic to a neighborhood of a EucUdean space.

The solution set breaks up into a union of a finite number of irreducible

com pone nts, none of which is contained in the union of the rema ining com

ponents. In numerical algebraic geometry, we associate to each component

a witness set , which is th e basic da ta stru ctu re used to num erically de scribe

and m anip ulate po si tive dimen sional solution sets . Given a mu lt ipl ici ty one

irreducible component Z of a system of polynomials

f{x)

= 0, a witn ess

set consists of a triple (/ ,

L, W),

where L is a random linear space of di

mension complementary to that of Z and where W is a set of points such

t h a t W := Z n L. There is a slightly more involved definition for the case

of components of multiplicity greater than one as described in [24].

Many of the algori thms of numerical algebraic geometry make abun

da nt use of hom otop y continuation . For exam ple, the cascade algori thm

[20],

one of the basic methods involved in computing the numerical i rre

ducible decomposit ion of an algebraic set , uses repeated applicat ions of

homotopy continuation at dif ierent dimensions in order to produce points

on each comp onent in each dimension. Mo nodromy a nd t race tes ts then

lead to the complete numerical i rreducible decomposit ion of the solution

set of / . In the end, the user obtains from the methods of numerical alge

braic geometry a wealth of information regarding the characterist ics of the

solution set of a given polyn om ial system , some of wh ich ma y be difficult to

procure by purely symb olic mea ns. For good references on zero-dimensional

solving see [13, 16] and for numerical algebraic geometry see [24].

There are several software packages available to the pubUc which carry

out some of th e ope rat ion s of num erical algebraic geometry. However,

no one package con tains all such capabil i t ies . Th ese packages include

HO M 4PS [6], PH oM [9], PO LS YS [27], PH Cp ack [28], and Ho m Lab [30].

Th e mo st recently released pack age, Bertin i [2], is und er ongo ing develop-


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SOFTWARE FOR NUMERICAL ALGEBRAIC GEOMETRY 3

me nt by the a uth ors . Altho ugh al l software packages were developed a t

different times for different reasons, they share the goal of solving polyno

mial systems by numerical means.

T he pu rpos e of th e present p ape r is two-fold. On e purpo se is to p resent

a para dig m for software in th e field of nu me rical algebraic geom etry. T he

following section contains a n elabora t ion on the aforementioned algori thm s

and an enumeration of the various operat ions required for carrying out

those algorithms, broken into four levels. Implementation-specific details,

such as da ta struc ture s, are om itted. Th e other purpose is to provide a brief

introduction to Bert ini and indicate i ts part ial fulf i l lment of the paradigm

of Section 2. T h at is th e co nte nt of Section 3, which is also broke n up

into four levels to mirro r Section 2. T he final section include s plan ne d

extensions of the Bertini software package.

2 .

A p a r a d i g m f or n u m e r i c a l a l g e b r a i c g e o m e t r y s o f t wa r e . All

good software packages share several chara cterist ics . In part icu lar , good

software should be reliable (i .e., i t provides correct output with clear signals

upon failure), as fast as possible with estimates of time remaining for large

job s, m od ular for easy modification, a nd user-friendly. In ad ditio n, goo d

numerical software must be accurate and provide error est imates for al l

solutions. ReHabili ty and accu racy general ly have an adv erse imp act on

speed, so while efficiency is important, it should not be emphasized over

finding correct answers.

Num erical accuracy may be ap proached in two ways . O ne approach is

to use a fixed level of precision and find as accurate a solution as possible,

possibly using higher levels of precision for subsequent runs to attain more

accuracy, if necessary. The other approach is to select an accuracy before

the run and adjust the precision during the run to at tain that accuracy.

Either way, i t has recently become general ly accepted that i t is important

to have available multiple levels of precision when implementing numerical

routines.

The purpose of this sect ion is to provide an enumerated paradigm

for software specifically in th e field of num erical algeb raic geom etry. Th is

detailed list is broken into four levels, beginning with very basic operations

not specific to polynomial system solving at level 0 in Section 2.1 and

moving through extensions of basic numerical algebraic geometry at level

3 in Section 2.4. The operations of each level build upon the capabilities

of th e previo us level. Ea ch of th e following four sections beg ins with a

discussion of the necessary operations of the given level and the resulting

cap abilities of th e software. Ea ch section th en conclud es w ith a brief list of

the main operat ions to be implemented at that level . All operat ions should

be implemented for various levels of precision, ideally for arbitrarily high

precision.

2 . 1 . L e v e l 0 : B a s i c o p e r a t i o n s . At the very core of a numerical

polynom ial solver, one mu st of course have access to basic ari thm etic bo th


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of complex numbers and complex matrices. I t is important to optimize the

efficiency of this arithmetic as much as possible, particularly in high preci

sion, as most operat ions in numerical algebraic geometry rely heavily upon

ar i thme t ic . In addi t ion to bas ic m atr ix ar i thme t ic , s tand ard techniques

from numerical l inear algebra are needed. Among the most important are

Gaussian elimination for linear solving, QR factorization for least squares

and orthogonal complements, and the SVD, or a related technique such

as the efficient m eth od of [14], for finding nu m eric al ra n ks . See [5, 26] for

general references on numerical linear algebra, while [14] provides a more

efficient method for determining the numerical rank of a matrix.

Random numbers play a key role in numerical algebraic geometry

as many statements hold generically, i .e., for almost all random choices,

thereb y making th e result ing algori thm s hold with probab il i ty one. An y

standard random number generator wil l suff ice, al though i t is best to have

the chosen random complex numbers of approximately unit modulus, for

stabil ity. I t is also im po rtan t to have a consistent m echanism for exte nding

the precision of a randomly chosen complex number. In part icular , upon

extending the precision of a random number

a

to make

a,

t runcat ing back

to lower precision, and then again extending the precision, one should once

again obtain St.

I t is of course necessary to somehow ob tain the polynom ials of interes t

from the user, although the specific procedure for doing so is implementa

tion-specific. To build a general solver, it is im po rta nt t o allow th e functions

to be defined as expressions built from su bex pression s. Th is is beneficial

no t only for ease of use , b ut also for efficiency a nd n um eric al stability . It

is also necessary for generality to allow for homotopies that depend on

parameters, including analytic expressions as discussed in more detai l in

th e following section. If th e user is specifying th e en tire hom otopy , it is

also necessary to have a way for the user to provide solutions to the start

system

g.

Otherwise, the automat ical ly generated s tar t sys tem should be

solved by the software.

Regardless of how the input data is provided, parsed, and stored, i t is

at t imes necessary for the software to automatical ly homogenize the poly

nom ials provided by the user . Ho mo genization is s imply the me chan ism

for moving from a product of one or more complex spaces to a product

of com plex projective spa ces. T his is a pure ly sjmnbolic op era tion wh ich

should be implemented in such a way as to be easily reversed in case the

need arises. Suppose the homogenized system involves the cross product

of

V

project ive spaces. Th en

v

random nonhomogeneous l inear equat ions ,

one for each project ive space in the product , should be appended to the

system in order to choose a patch on each complex project ive space . The se

V l inear equations are known as the patch polynomials .

Basic pa th tracking , a level 1 ope rat ion , makes heav y use of bo th func

t ion evaluation and Jacobian evaluation. Function evaluation is s traight

forward, al though i t could be optimized via special ized techniques such as


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H orne r 's m etho d. Th e Jacob ian ( i .e., th e ma trix of pa rt ial derivatives of

the polynomials) should be computed automatical ly by some form of auto

matic differentiat ion so that the user does not need to provide i t as input .

I t is general ly believed that the Jacobian should be computed explici t ly

rather than approximated numerical ly, for s tabil i ty.

Some of th e op erat ion s at this level are comm on to n um erical software

in gen eral, rega rdless of th e specific app licatio n. It should be noted th a t

exist ing l ibraries, such as LAPACK, provide robust implementat ions of

some of these operations, albeit in one level of precision.

S u mma r y o f l e v e l 0 o p e r a t i o n s :

• Com plex scalar and matr ix a r i thmet ic

• Matrix operat ions (e.g. , l inear solving, numerical ranks, and or

thogonal complements)

• Ran dom num ber generat ion

• Pars ing of input da ta

• Function homogenization

• Function evaluation

• Jaco bian evaluation

• All num erical op eratio ns should be available in multiprecision .

2 . 2 . Le v e l 1 : Ba s i c t r a c k i n g a n d z e r o - d i me n s i o n a l s o l v i n g . I t

is now possible to glue togeth er th e cap abilities of level 0 to build algo rithm s

leading up to a full zero-dimensional solver, i .e., a method for computing

aU isolated solutions of / . T he con cep t of "solving" a system could be

inter pre ted tw o wa ys. Given a desired acc urac y e 6 R"^, let S := {s £

C ^ : | / ( s ) | < e}. Th is set may break up into

k

disconnected pieces, say

5 = 5 i U . . . U 5fc. The n on e inte rpr eta tion of solving is to find a. z € Si

for each i. Th e second way to inte rpre t "solving" / is to find, for each

s £ C^ such th a t / ( s ) = 0 and s has mult ipl ici ty m as a root of / , a set

{zi,...,Zm}

C C ^ s uch t ha t

\zi — s\ < e

for all

i.

Th e solutions in the

former sense of solving depend upon the scaling of the poljmomials while

those of the lat ter depend upon the scal ing of the variable. The lat ter is

more widely accepted as correct and is thus the defini t ion used throughout

this paper.

As discussed in Section 1, hom otopy co ntinuation c asts th e system /

in a family of polynomial systems, another of which is

g.

Such parame

terized families sometimes arise naturally and are of great interest in some

applications. Other families are artificially constructed for the specific pur

pose of solving / by con tinua tion, in a so-called ab initio homotopy. Often

a natural ly parameterized family can be used to define a homotopy that

has fewer paths to follow than an

ab initio

homotopy. As a result, software

in numerical algebraic geometry should address both ab initio and na tu

ral parameter homotopies . The natural parameter spaces may be complex

analytic and thus the need for evaluating complex analjrtic expressions of

parameters and complex numbers as mentioned in the previous sect ion.


15/180


At i ts core, homotopy continuation is s imply a sequence of s teps ad

vancing the path variable and updating the solution vector x to match.

Each step forward consists of a predictor step foUowed by a sequence of

corrector s teps. The predict ion is commonly carr ied out by Euler 's method,

which steps along the tang en t to the pa th . Th is is general ly regarde d a s

an acceptable approach, al though secant predict ion can also be employed,

as can higher order meth od s such as Ru ng e-K utta . Once a predict ion is

made to a new value of t, it is necessary to refine the point back towards

the solution curve. This may be accomplished by f ixing

t

and running a

Newton-Hke corrector unti l the norm of the Newton residual has dropped

below a prespecified tolerance.

Naturally, there are times when steps will fail , where failure is declared

wh en the corre ctor does not conv erge sufficiently fast. Such step failures

need not t r igger path fai lure. Rather, adaptive steplength should be ut i

l ized. Upon step fai lure when using adaptive steplength, the steplength is

decreased by a prespecified factor in the hope that convergence will occur

for a smaller step . On ly if progress along the p at h b ecom es excessively slow

is the path declared a failure. Conversely, if the steps are progressing well,

i t is worthwhile to try to increase the step length . M ore detai ls rega rding

predict ion, correct ion, and ada ptive step length ma y be found in [1, 16, 24].

Path fai lure may occur for a number of reasons, but the presence

of a singularity, pa rticu larly a t t = 0, is a com mo n cau se. Th ere ar e

several sophist icated algori thms known as endgames that help to speed

convergence at t = 0 for both no nsingular an d singular end po ints . The se

endgames are typical ly employed for every path, so i t is important to have

at least one implemented in any software package for numerical algebraic

geom etry. D etails regard ing end gam es may be found in [17, 18, 19].

For zero-dimensional solving, polynomial systems given by the user

could be nonsquare with

n > N.

Fortunately, Bert ini 's theorem [24] guar

antees th a t a new system co nsisting of

N

generic linear combinations of the

original n poljoiomials will have among its solutions all the isolated solu

t ions of th e original system, tho ug h possibly with increased m ult iphcity. I t

may also have nonsingular isolated extraneous solutions. The extraneous

solutions are easily detected as they do not satisfy the original system.

Unless the user chooses to specify a parameter homotopy, the software

must be able to automat ical ly produce an appropr iate s tar t sys tem g, solve

it , and at ta ch i t to th e homogen eous, square system / in order to cre ate

the homotopy

H.

Th ere are several m etho ds for produ cing start system s,

al though the general rule is that the computat ional cost increases in order

to produce s tar t sys tems wi th fewer paths to t rack. Among the common

choices, total degree start systems, consisting of polynomials of the form

x^' —

1, where

di

is the degree of the i*' ' polynomial in / , are trivial to

build and solve but have the largest number of paths to be tracked ( that

being the p rodu ct of th e degrees of the functions) . At th e othe r end of the

spectrum, the construction and solution of sophist icated polyhedral homo-


16/180

SOFTWAR E FOR NUMERICAL ALGEBRAIC GEOMETR Y 7

topies involve far more computat ion t ime but may result in far fewer paths

(the number of which is the mixed volume). I t is not clear a priori which

type of s tar t system is best-suited for an arbitrary polynomial system, so

it is important to have mult iple types of s tar t systems available.

Once al l (or most) of the aforementioned operat ions have been imple

mented, i t is possible to compute the zero-dimensional solutions of a given

polynomial system. Two other useful tools belong at this level. First, defla

tion [11] is a m ean s of co nstru ctin g a new polyn om ial system / from / such

th a t / has a nonsingu lar solution in place of a part icu lar s ingular solution

of / . Th is ma kes i t possible to com pute singular solutions mo re ac curate ly

without relying on higher precision. The major drawback of implementing

deflat ion is that decisions must be made about the rank of the Jacobian

matrix at the solution point before the solution point is known accurately.

The use of endgames can improve the accuracy of the solut ion es t imate

before deflation, helping to ensure that the correct deflation sequence is

performed but adding the cost of endgame computat ions. Explorat ion of

the numerical stabihty and efiiciency of deflation and endgames is a topic

of ongoing research. T he big adv antag e of deflat ion comes wh en dealing

with posi t ive dimensional components of mult ipl ici ty greater than one.

The other useful tool at this level is a post-processor to manipulate

and display the solutions computed by the solver as well as any stat is t ics

ga the red d uring tracking. As the functionali ty of such a tool is app licat ion-

specific, no more details will be discussed here.


• Differential equ atio n solving, e.g., Eu ler 's m eth od

• Newton's method

• Basic pa th trackin g with adap tive stepleng th control

• Ad aptive precision pa th trackin g

• Squa ring of systems

• Star t sys tem and homotopy generat ion

• Start system solving

• Pull zero-dim ensional solving

• Endgames

• Deflation

• Post-processing of zero-dimensional da ta.

2 . 3 . Le v e l 2 : Po s i t i v e - d i me n s i o n a l s o l v i n g . The solut ion set

Z

of a polynom ial system / m ay have several com pon ents and these m ay

no t al l have th e sam e dimen sion. Lett ing D := dixnZ, the irreducible

decomposi t ion may be wri t ten as Z =

U ^ Q Z J

=

U ^ Q

Ujeii ^i,jj where

each

Zij

is an irreducible com pone nt of dimension

i,

and accordingly each

Zi

is the pure i-dimensional component of

Z.

(Sym bol Ij in th e above is

just an index set for the components of dimension.)

One of the key objectives in numerical algebraic geometry is to find

a numerical i rreducible decomposit ion of

Z,

which consists of sets

Wij —


17/180


Zij n L^-i, w here Ljv- j is a gene ric linear subsp ace of dimen sion N

—

i.

Wij,

together with

L^-i,

is know n as a w itness set for

Zij,

and by abuse of

notat ion,

Wi = Uj£l^WiJ

is called a w itne ss set for Zj . We briefly d esc ribe

the algori thms for com puting the irreducible decom posit ion below. Full

details may be found in the references cited below or in [24].

The main steps in computing a numerical i rreducible decomposi

t ion are:

• find witness supersets Wi D Wi for each dimension i,

• prune out "junk points" from

Wi

to extract the witness sets

Wi,

and

• break

Wi

into dist inct sets

Wij,

th e witne ss sets for th e irreducible

component s Zij.

The wi tness supersets

Wi

are generated by the applicat ion of zero-dimen

sional solving to find the isolated points in the slice

Z

n

Lj^-i.

All of

the Wi, for 0 < J < D, can be obtained using the cascade algori thm

[20], s tar t ing at i = D a nd cascading sequential ly down to i = 0. Th e

junk points in

Wi

must l ie on some

Zj

wi th

j > i.

Thus , the junk may

be removed by test ing each point

p € W i

for membership in a higher-

dimensional component. This can be done using continuation on sl ices to

see if any of the witness sets Wj, j > i connect to p as the slicing linear

space LN-I is moved continuou sly until it con tains p. The f inal break up

of

Wi

into irreducibles is accomplished by first using monodromy, which

comes down to discovering con nections between w itness poin ts as th e l inear

slicing space is moved a roun d a closed loop in the assoc iated G rassm ann ian

[21]. This is followed by checking if the connected groups so discovered are

complete, by means of the trace test [22]. The trace method may also be

used to complete a par t ial decomposi t ion. Both monodromy and the t race

method involve specialized continuation of slices and careful bookkeeping.

Squaring is a concern for positive-dimensional solving just as it is for

zero-dime nsional solving. A lthou gh m uch carries over, one difference is th a t

the size to which the system should be squared depends on the dimension

of the com pon ent, e.g., for com pon ents of dimension k, the defining system

should be randomized to a system oi N

—

k equat ions .

Given witness data for an algebraic set Z, there are three operat ions

of part icu lar interest for users. Fir st , a user migh t wa nt to find m any

poin ts on a specif ic co m pon ent. Th is is known as samp ling and is very

closely related to monodromy, as both use the continuation of sHces to

move witness points around on the component. Second, a user might want

to know if a given point lies on an irreducible component of Z. Th is is

the same component membership test used in the junk removal s tage of

com puting an irreducible decom posit ion. Finally, a user migh t wa nt th e

accuracy of some end point sha rpen ed. Th is is jus t a m atte r of running

New ton's m etho d app ropriately, perh ap s after deflat ing / . Of course, as in


18/180


the previous sect ion, a post-processor would be appropriate, al though the

exact functionality is again application-specific.

Deflat ion is part icu larly valuable as a m etho d for mult iple co mp one nts,

e.g. , to track intersections of a mult ipl ici ty greater than one component

with a one-parameter family of l inear spaces of complementary dimension,

as is done to samp le a mu lt iple com pon ent. Rou ghly speaking, if

Z

is

a fc-dimensional component of the solution set of a system f — 0, then

the computat ion of the deflat ion oi Z f) L ioi f restr icted to a generic

fe-codimensional Unear space gives rise to a "deflation" of the whole com

ponent. At the expense of increasing the number of variables, this al lows

us to numerical ly treat al l components as mult ipl ici ty one components.

S u m m a r y o f l e v e l 2 o p e r a t i o n s :

• Co ntin ua tion of slices

• Monodromy

• Traces

• Sq uaring of syste m s for positive-dim ension al solving

• C ascade algori thm

• Pull num erical i rreducible decom posit ion

• Sampling

• Component membership

• Endpoint sharpening

• Post-processing of witness da ta

• Deflation for components.

2 .4 . L e v e l 3 : Ex t e n s i o n s a n d a p p l i c a t i o n s o f t h e b a s i c s . This

highest level consists of operat ions that make use of the basic numerical

algebraic geometry maneuvers described in the previous three levels . For

example, for two algebraic sets X and Y that are the solution sets of

polynomial systems / and g, respectively, suppose one ha s witness sets Wx

and

WY

for

X

and

Y

but would Uke a witness set W for X n

Y.

Th ere is a

now a method [23] for computing the numerical i rreducible decomposit ion

of such an intersection, the inclusion of which, while not essential for basic

software in numerical algebraic geometry, will be important as the field

continues to develop.

There are several other advanced algori thms that should be included

in a complete state-of-the-art implementat ion. Another such technique is

that of [25], which pro vides a way of finding exc eption al se ts, i .e., the sets of

points in the parameter space above which the f iber has dimension higher

th an th e generic fiber dimension. Fibe r pr od ucts play a key role in this

algorithm and therefore need to be available in the software before the

me thod for f inding exce ptional sets m ay be imp lemen ted. Also, in real-

world applicat ions, real solutions are often of more interest than complex

solutions, so the ex tract io n of real solutions from th e nu me rical i rreducible

decomposition, for example, by an extension of the method of [15], would

be very useful.


19/180


This is not intended to be a complete l is t , and i t is anticipated that

m any more ope rat ions could be added in th e near future. One cap abil

i ty, though, that is important now and wil l only become more essential

over t ime , is paral lel izat ion. Althou gh n ot every algori thm of nume rical

algebraic geometry is fully parallelizable, basic path tracking is easily par

allelized so great savings can be made throughout levels 1, 2, and 3 by

doing so [10, 12, 27, 29].


• Intersection of components

• Fiber products

• Finding exceptional sets

• Ex tract in g real solution sets from com plex comp one nts

• Paral lel izat ion.

3 . B e r t i n i .

Be rtini [2] is a new software packag e un der on going devel

opment by the authors for computat ion in the f ield of numerical algebraic

geometry. Bertini itself has evolved from a program called Polysolve cre

ated by Ba tes, C. Monico (Texas Tech Universi ty) , Sommese and W am pler,

al though nothing substantial remains in Bert ini from Polysolve.

Bert ini is wri t ten in the C programming language and makes use of

several specialized Mbraries, particularly lex and yacc for parsing and GMP

and MPFR for mult iple precision support . The beta version of Bert ini was

released in October 2006 to coincide with the Software for Algebraic Ge

ometry workshop of the Inst i tute for Mathematics and i ts Applicat ions. I t

is currently anticipated that Bert ini 1.0 wil l be made available to the pub-

He som etime in 2007. Be rt ini is currently only available as an exec utable

file for 32- or 64-bit Linux and for Windows, via Cygwin. Specific instruc

t ions for using Be rt ini are included with th e distr ibution and on the Be rt ini

website.

Among other things, Bert ini is capable of producing al l complex iso

lated solutions of a given polynomial and witness sets for each positive-

dimensional i rreducible com pon ent. T he goal of the Be rt ini developm ent

team is to eventually include in Bert ini al l operat ions described above

in Sections 2.1 thr ou gh 2.4. T h e pmrpose of th is section is to ind icate

briefly which of those operations are already available in the beta version

of Bert ini . The specif ic algori thms implemented are also described, when

appropriate. Detai ls regarding the development plans for Bert ini 1.0 and

beyond may be found in Sections 4.1 and 4.2, respectively.

3 . 1 .

L ev el 0. B y default , Bert ini uses IE E E double precision, al

though it also allows for any fixed level of precision available in MPFR.

In particular, precision is available starting from 64 bits, increasing in 32

bit incremen ts. Fu rtherm ore, the be ta version of Bert ini al lows th e user

to select adaptive precision for zero-dimensional solving. In adaptive pre

cision mode, Bertini begins in double precision and increases precision as

necessary, determined by the algorithm described in [3].


20/180


All matrix operat ions described in Section 2.1 have already been im

plemented in Bert ini with the exception of the QR algori thm (i .e. , the QR

decom posit ion of a ma trix ) , which will be implem ented b y the n ext release.

Gaussian elimination with scaled partial pivoting is used for linear solving.

The SVD algori thm is based on the descript ion provided in [26].

Bert ini uses the basic random number generator found in the C stan

dard l ibrary. Each rando m n um ber i s scaled to be of the ap prop r iate m od

ulus and stored as an exact rat ional number with the denominator a power

of ten. Thus, to increase precision, all new digits are set to zero, although

this is not ideal . I t would be better to increase precision using randomly

chosen digits such that subsequent truncations and extensions result in the

same additional digits. This will be changed in a future version of Bertini.

The user provides poljTiomials (ei ther the entire homotopy or just the

target system) to Bertini in a file which is parsed using lex and yacc. The

polynomials are stored in straight-line format for easy automatic differenti

ation and homogenization as well as efficient evaluation. Any optimization,

such as using Horner 's method, is the responsibil i ty of the user . Homoge

nization is carried out as the polynomials are being parsed, and differenti

at ion is carr ied out af ter the entire system has been parsed, currently via

forward automatic differentiation as described in [8].

Bert ini al lows the user to include constants and subfunctions when

defining the target system and also the path variable and parameters when

defining a homotopy. There are various restr ict ions placed on the homoge

neous structures al lowed depending upon the scenario, but mult iple vari

able groups are generally supported. All coefficients, constants, and other

numbers are stored as exact rat ional numbers, as described for random

numbers above, and are assumed to be exact . As a result , function and Ja-

cobian evaluation are available at any level of precision. Analytic functions

of constants and parameters are not yet suppor ted.

3 . 2 .

Leve l 1 .

All level 1 operations of Section 2.2 have been imple

m ented in Bert ini with the exception of deflation. Eu ler 's m etho d is used as

the predictor , and New ton's m etho d as th e corrector . T he fractional pow er

series endgame [19] is available in basic precision, fixed higher precision,

and adaptive precision.

Al l polynomial sys tems are automat ical ly squared to the appropr iate

number of polynomials and variables, al though the detai ls of the squaring

me chan ism are excluded from th e present pa pe r for brevity. If th e user

specif ies only the target system and variable groupings, Bert ini mult iho-

mogenizes the system according to the groupings and at taches a compatible

m-homogeneous start system via a l inear homotopy. Each polynomial in

the start system consists of a product of random linear functions matching

the homogeneous structure of the corresponding target polynomial . If the

user specifies only one variable group, the result is a total degree start sys

tem. Bert ini also has a post-processor which collects the endpoint data for


21/180


the run and creates a number of f i les , some of which are human-readable

and some of which are better suited for machine reading. The files provide

lis ts of s ingular endpoints , nonsingular endpoints , real endpoints , and so

on. Important path characterist ics , such as an est imate of the error in the

endpoint, are also supplied in these files. Bertini also indicates whether any

paths fai led (and why they did) and whether any path crossing occurred

prior to the endgame.

3 . 3 .

L e v e l 2 . All of th e op era tion s of level 2 desc ribed in Section 2.3

have been implemented in double precision in Bert ini , with the exceptions

of endpoint sharpening and deflat ion. Users have the option of computing

the numerical i rreducible decomposit ion of the polynomial system input

f i le, sampling a component of a system for which witness data has pre

viously been computed, or performing component membership on such a

system. Th e algori thms are the n carr ied out as described in Section 2.3

and [24].

In the case of computing the numerical i rreducible decomposit ion,

Be rt ini s tores all d at a in mem ory unti l the end of th e run. At the en d

of the run, a postprocessor sorts the data and creates a number of f i les , as

in the case of a zero-dimensional run. The structure of the algebraic set is

also provided in a table on the screen, namely the number of i rreducible

com pon ents of each degree in each dimension. Ad dit ional detai ls may be

found in the Bertini documentation available in [2].

3.4 . L ev el 3 . None of th e level 3 ope rat ions described in Section 2.4

are available in the beta version of Bertini. Much of the further develop

ment of Bertini will involve level 3 operations.

4 . T h e fu tu re for B e r t in i . I t i s of course impossible to predict wh at

wil l happen with ei ther Bert ini or numerical algebraic geometry in the

distan t future. However, the re is a sho rt- term d evelopmen t plan for Bert ini .

The following sections describe the development plan for Bertini 1.0 and

further anticipated developments beyond Bertini 1.0.

4 . 1 . B e r t i n i 1.0. Th ere are several key developments current ly in the

process of being implem ented in order to m ove Be rt ini from the be ta version

to version 1.0. The four main improvements are the inclusion of the cascade

algo rithm in fixed m ultiple precision as well as ada ptiv e precision, deflation

in both the zero- and posit ive-dimensional cases, endpoint sharpening (as

described in Section 2.2) , and the QR decomposit ion. Other improvements

are minor, such as several small changes to the ou tpu t fi les and an improve d

post-processor. T he go al is to have mo st of the o pera t ions of levels 0 , 1 , and

2 available in Bertini 1.0 in fixed user-specified multiprecision and adaptive

precision.

4 . 2 .

B e y o n d B e r t i n i 1.0. Whi le there will cer tainly be a num ber of

chan ges to the core functions of Bertini after v ersion 1.0 is released (such a s

improving the way that random numbers are handled) , many of the major


22/180

S O F T W A R E FOR N U M E R I C A L A L G E B R A I C G E O M E T R Y

13

changes will involve

the

implementat ion of

the

operat ions at level 3

and

new algori thms currently under development.

Other planned extensions include al lowing

the

user

to

specify an aly tic

functions of parameter s

and

constants ,

the

inclusion of polyhedral me th

ods,

e.g.,

using M ixedVol [7],

and

various forms of paral lel izat ion. Be rt ini

current ly o perates

by

set t ing

up

files with input data, calling Bertini,

and

reading the results from files. It is anticipated th at Bert ini will eventu

ally include a script ing language or make

use

of software th at provides

an interact ive environm ent. Be rt ini wil l also und ergo a move from

the

C

programming language to C4-I- sometime after

the

release of version 1.0,

bo t h for increased m odu lar i ty and for easier exten sion. Finally, B ertin i will

eventually make

use

of more efficient multiprecision numerical libraries as

the y becom e available

and

will includ e interfaces

to

other software packages

comm only used

in the

m athem at ics communi ty.

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SOMMESE

AND J.

VERSCHELDE,

Nun,erical homotopies

to

compute generic

points

on

positive dimen sional algebra ic sets, J.

Complex i ty , 1 6(3) :572-6 02 ,

2000.

[21] A. J. S O M M E S E , J. V E R S C H E L D E , AND C.W. W A M P L E R ,

Using monodromy to de

compose solution seta of polynomial systems into irreducible com ponents,

in

proceed ings

of

Appl ica t ions

of

a lgebra ic g eomet ry

to

codin g theory, physic s

and computa t ion (E i la t , 2001) , NATO Sci. Ser.

II

M a t h . P h y s . C h e m . ,

36,

2001.

[22]

,

Symm etric functions applied to decompo sing solution sets of polynomial

systems, SIAM

J. Num.

An al . , 40{6):20 26-204 6, 2002.

[23]

,

Homotopies for intersecting solution compon ents of polynomial systems,

SIAM

J.

Num. AnaJ . , 42(4 ) :1552 -1571 , 2004 .

[24]

A.J.

SOMMESE AND

C.W.

WAMPLER,

Numerical solution

of

systems of polynomials

arising in engineering and science, World Scient i f ic , Singapore , 2005.

[25]

,

Exceptional sets

and

fiber produ cts,

to

a p p e a r F o u n d a t i o n s

of

C o m p u t a

t i o n a l M a t h e m a t i c s .

[26] G.W. STEWART, Matrix algorithms. Vol. I. Basic decompo sitions, S I A M , P h i l a d e l

p h i a , 1998.

[27] H.- J. Su, J. M C C A R T H Y , M . S O S O N K I N A , AND L . T. W A T S O N ,

Algorithm 857: POL-

SYS-GLP - A parallel general linear product homotopy code for solving poly

nomial systems of equations, ACM Tran s . Math . Sof tware , 32(4) :56 1-579 ,

2006. Software avai lable

at

w w w . v r a c . i a s t a t e . e d u / ~ h a i j u n s u.

[28]

J.

VERSCHELDE, Algorithm 795: PHCpack: a general-purpose solver for poly

nomial systems by homotopy continuation, ACM Tra ns . M ath . Sof tware ,

25(2) :251-276, 1999. Software avai lable at w w w . m a t h . u i c . e d u / ~ j a n .

[29]

J.

VERSCHELDE

AND

Y . ZHUANG, Parallel implementation of the polyhedral homo

topy

method,

in

p roceed ings

of

I C P P W

'06

( In te rna t iona l Confe rence W ork-

shojjs on Para l l e l P rocess ing ) , IEE E Co mp ute r Soc ie ty, W ash ing ton , DC, 4 8 1 -

488, 2006.

[30] C.W. WAMPLER, Hom Lab: Homotopy Continuation Lab, avai lable at w w w . n d . e d u /

'^^cwamplel .

http://www.vrac.iastate.edu/~haijunsuhttp://www.vrac.iastate.edu/~haijunsuhttp://www.math.uic.edu/~janhttp://www.math.uic.edu/~janhttp://www.nd.edu/http://www.nd.edu/http://www.math.uic.edu/~janhttp://www.vrac.iastate.edu/~haijunsu


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P H C L A B : A M A T L A B / O C T A V E I N T E R F A C E T O P H C P A C K *

Y U N G U A N t A N D J A N V E R S C H E L D E *

A b s t r a c t . P H C p a c k i s a s o ft w a re p a ck a g e fo r P o l y n o m i a l H o m o t o p y C o n t i n u a t i o n ,

to numer ica l ly so lve sys tem s of po lynom ia l equa t ion s . Th e execu tab le p ro gram "phc"

pro duc ed by PH Cp ack h as several op t ion s ( the mos t po pu lar one "- b" offers a blackbox

solver) and is menu dr iven. PHClab is a col lect ion of scr ipts which cal l phc f rom within

a MA TL AB or Octav e sess ion. I t provides an interface to th e blackbox solver for f inding

i so la ted so lu t ions . We execu ted the PHClab func t ions on our c lus te r compute r us ing

the M PI ToolBox (M PIT B) fo r Oc tave to so lve a l is t o f po lynom ia l sys tems . PH Cl ab

also interfaces to the numerical i r reducible decomposi t ion, g iving access to the tools to

represent , factor , and intersect posi t ive dimensional solut ion sets .

1 .

I n t r o d u c t i o n . Po lyno m ial system s arise in variou s fields of science

and engineering, e.g.: the design of a robot arm [13] so its hands passes

through a prescribed sequence of points in space requires the solution of

a polynom ial system. Ho mo topy continuation m etho ds are efficient nu

merical algori thms to approximate al l isolated solutions of a polynomial

system [10]. Recently homotopies have been developed to describe positive

dimensional solution sets [20].

This paper documents an interface PHClab to use the functionali ty

provided by PHCpack [21] from within a MATLAB or Octave session.

The main executable program provided by PHCpack is phc, available for

downloading on a wide variety of com puters and op erat ing system s. T he

program phc requires no compilat ion. I ts most popular mode of operat ion

is via the blackbox opt ion, i .e .: as phc -b in p u t o u tp u t . Recent ly the

program has been updated wi th tools to compute a numer ical i r reducible

decomposion [18].

The main motivation for PHClab is to make i t easier to use phc by

automatic conversions of the formats for poljaiomial systems (on input)

and solutions (on ou tp ut ) . M anu al or adho c conversions can be tedious

and lead to errors. As PHCpack has no script ing language on i ts own, the

second advantage of PHClab is help the user to systematical ly use the ful l

capabil i t ies of PH Cp ack . As M AT LA B (an d i ts freely available co un terpa rt

Oc tave) is a very po pu lar scientific software sy stem , P H C lab will be a useful

addi t ion to PHCpack.

*This ma te r ia l i s based upo n work sup por te d by the Na t iona l Sc ience Foun da t ion

unde r Gr an t No . 0105739 and Gra n t No . 0134611 .

t De par tm ent o f M athem at ics , S ta t i s t i c s , and C om pute r Sc ience, Un ivers i ty of

I l l inois a t Chicag o, 851 So uth M org an ( M /C 249) , Chicago , IL 60607-7045, US A

( g u a n a m a t h . u i c . e d u ) .

* De par tm ent o f M athem at ics , S ta t i s t i c s , and C om pute r Sc ience, Univer

s i ty of I l linois a t Chicago , 851 Sou th Morg an ( M /C 249) , C hica go, IL

60607-7045, USA ( j a n 9 m a t h . u i c . e d u or j a n . v e r s c h e l d e a n a - n e t . o m l . g o v ; h t t p : / /

www . m a th . u i c . e d u / ~ j an).

15

http://jan9math.uic.edu/http://jan9math.uic.edu/http://jan.verscheldeana-net.oml.gov/http://jan.verscheldeana-net.oml.gov/http://jan.verscheldeana-net.oml.gov/http://jan9math.uic.edu/


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16 YUN GUAN AND JAN VERSCH ELDE

Another feature of PHClab is the possibiUty of developing parallel code

at a high level, when using the MPI ToolBox (MPITB [2]) for Octave.

2 .

T h e

d e s i g n

o f P H C l a b .

The first interface from a C program to

phc was wr it ten by Nobuki Takay am a and is s t il l available via th e PH Cp ack

download web si te. Via this interface, phc became part of OpenXM [11]

(see also [12]). We used the same idea to build PHCmaple [7, 8], defining a

Maple interface to PHCpack.

All that is needed to make the interface work is the executable form of

p h c .

PHClab is a collect ion of scripts wri t ten in the language of MATLAB

and Octave. These scr ipts cal l phc wi th the appropr iate opt ions and menu

choices.

3 .

D o w n l o a d i n g a n d i n s t a ll i n g . PHClab was tes ted on Mat lab 6.5

and Octave 2.1.64 on computers running Windows and Linux. On an Apple

laptop running Mac OS X version 10.3.7, we executed PHClab in Octave

2.1.57.

The most recent vers ion of PHCpack and PHClab can be ret r ieved

from

h t t p : / /www.

m a t h . u i c . e d u / ^ j a n / d o w n l o a d . h t m l

which we fi-om now on call th e dow nload w eb site. To install and use

PHClab, execute the fol lowing steps:

1. Prom the download web si te, ei ther download the source code for

phc (a makefile is provided with the code), or select an executable

version of phc. Currently, phc is available in executable form on

Windows, workstat ions from IBM (running AIX 5.3) and SUN

(running SunOS 5.8) , and PCs running Linux and Mac OS X 10.3.

Except for Windows (which comes just as a plain phc.exe), one

has to run gim zi p followed by t a r xpf on the dow nloaded f ile.

2.

Th e P H Cla b d is t ribut ion is available as PHClab. t a r . gz fi rom the

download web s i te . To ins tal l PHClab in the di rectory / tmp, save

PHC lab. t a r . gz f irs t in / tm p, a nd th en execute th e following se

quence of commands:

cd / tmp ; mk di r PHClab; mv / tm p/PH Cla b. t a r . gz PHClab;

c d / t m p /P H C l a b ; g u n z i p P H C l a b . t a r . g z ; t a r x pf P H C l a b . t a r .

3.

Ei ther launch MATLAB or Octave in the di rectory PHClab, or

add the name of the di rectory which contains PHClab to the path

of MATLAB or Octave.

The f irst command of PHClab to be executed is s e t _ p h c p a t h . This

command takes one argument: the ful l path name of the f i le name which

contains the executable program

phc.

For exam ple, if ph c was saved in

/ t m p , th e n a se ssio n w i th P H C l a b s t a rt s w i th s e t _ p h c p a t h ( ' / t m p / p h c ' ) .

4 . S o l v i n g p o l y n o m i a l s y s te m s . In this section we define the bas ic

commands to solve polynomial systems using PHClab. We first define the

inp ut /ou tput formats , in troducing the function m ak e_ sy s t em to conver t a


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PHCLA B: A MATLAB/OCTAVE INTERFACE TO PHCPACK 17

m atrix format for a polynom ial system into a s jmibolic inpu t format to p hc .

The blackbox solver of PHCpack is cal led by the command so lve_sys te in .

Besides the solution vectors, the solver returns extra diagnostical informa

t ion about the quahty of each solution.

Path tracking typical ly starts from a generic system (without any sin

gular solutions) to a mo re specific system . We use th e system w e first

solved by the blackbox solver as start system to solve a system with spe

cific coefficients, using the function track. Be cau se of ou r specific choice

of the coefficients, we generated a polynomial system with a double solu

tion, i .e.: of multiphcity two. Via the function ref ine_sols and def la t ion ,

we respectively refine a solution and deflate its multiplicity into a regular

problem.

The last function we introduce in this section is m i x e d _ v o l u m e , to

compute the mixed volume for a polynomial system and (optionally) create

and solve a random coefficient s tar t system. The mixed volume equals the

number of roots without zero components of a polynomial system with

suf

ficiently generic coefficients. The function m i x e d . v o l u m e cal ls the trans

late d form of th e co de M ixed Vol [3].

4 . 1 . I / O f o r ma t s a n d t h e b l a c k b o x s o l v e r . The input to the

solver is a system of m ultiva riate equ atio ns with com plex floating-point

coefficients. For example, consider the system g{x) = 0:

9ixi,x, = ^ 2 . 1 x 1 - 1 . 9 x 1 = 0 ' ^ i t l ^* = v ^ - (4-1)

This system is encoded as a matrix, with in i ts rows the terms of each

polynom ial . A zero row in the m atrix m arks the end of a p olynom ial

in the system. A nonzero row in the m atrix represen ts a term as the

coefficient followed by the exponents for each variable. For example 4.7x2

is represented by the row 4 .7 0 2. If n is the num ber of variables and

m

the total number of terms, then the matr ix encoding the sys tem has m + n

rows and n + 1 columns.

To solve the system g{x.) = 0 us ing PHClab, we may execute the

following sequence of instructions:

7 , t a b l e a u input for a sys tem :

t = [ 1 .3 2 0 ; 4 .7 0 2 ; - 3 . 1 + 2 .3 * i 0 0 ; 0 0 0 ;

2.1 0 2; -1.9 1 0 ; 0 0 0];

make_system(t) % shows symbolic format of the system

s =

solve_system(t);

% call the blackbox solver

ns = size(s,2) % check the number of solutions

s3 = s(3) % look at the 3rd solution

Then we see the following output on screen:

ans =

' + 1.3*xl**2 + 4.7*x2**2 + (-3.1+2.3*1)'


27/180

18 YUN GUAN AND JAN VERSCHELDE

' + 2 .1 * x 2 * *2 - 1 . 9 * x l '

ns =

4

s3 =

t i m e : 1

m u l t i p l i c i t y : 1

e r r : 4 . 0 3 4 0 e - 1 6

r c o : 0 . 1 2 4 3

r e s :

2 . 7 7 6 0 e - 1 6

x l :

- 3 . 9 1 8 0 + 0 . 3 8 7 6 1

x 2 : 0 .0930 + 1.88511

We see the coordinates of the solution are in the last fields (displayed

by default in short format, we may see more in format long) and extra

diagnostics in the first five fields, briefly explained below.

t i m e is the end value of th e continu ation pa ram eter . If this value is no t

equal to one, then i t means tha t the p ath t racker did not m anage to

reach the end of the path. This may happens with paths diverging

to infinity or with highly singular solutions.

m u l t i p l i c i t y

is th e multiplicity of th e solution. A solution is reg ular w hen

th e mult ipl ici ty is one. W hen th e approx ima tion for a solution

is not yet accurate enough, then the mult ipl ici ty might s t i l l be

reported as one, al though the value for rco might be close to the

threshold.

er r i s the magni tude of the las t update Newton's method made to the

solution. At singular solutions, the polynomial functions exhibit a

typical "f lat" behavior. Although the residual may then be already

very small, the value for this err can be still large.

rc o

is an est im ate for th e inverse of a condit ion num ber of the Jac ob ian ma

tr ix evaluated at the approximate solution. A solution is deemed

singular when th is num ber drop s below the thresh old v alue of 10~^.

Mult iple solutions are singular . The condit ion number

C

of the Ja

cobian matrix measures the forward error, i .e.: if the coefficients

are given with D digi ts precision, then the error on the approxi

mate solut ion can be as large as C x 10"^.

res is the residual , or the magnitude of the polynomial system evaluated

at the approximate solution. This residual measures the backward

error: how much one should change the coefiicients of the given

system to have the computed approximation as the exact solution.

The values of the coordinates of the solutions are by default displayed in

MA TLA B's (or Octave 's ) form at sh o r t . By form at lo n g e we can see

the full length in scientific fo rm at. For the solution a bov e, th e values of

er r , rco, and res indicate an excel lent qual i ty of the computed solut ion.


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PHCLAB : A MATLAB/OCTAVE INTERFACE TO PHCPACK 19

4 . 2 . P a t h t r a c k i n g f r o m a g e n e r i c t o a s p e c i f i c s y s t e m . T h e

four solutions of the system we solved are all very well conditioned, so we

may use them as start solutions to solve a system with the same coefficient

structure, but with more specific coefficients:

n-̂ -̂ )

= {4U+T-tzl wt z

= v ^. (4.2)

Geometrical ly, the polynomials in the system /(x) = 0 respectively de

fine an el l ipse and a parabola, posi t ioned in such a way that their real

intersection point is a double solution.

In the sequence of instruction s below we use the function tr a c k , using

the new sys tem double ( the sys tem / (x) = 0) as target sys tem and the

system t we solved as sta rt system (we called it g( x) = 0) . N ote t h a t

before cal l ing track, we must set the value of t ime in every solution to

zero, so s contains proper star t solut ions.

do ub le = [1 .0 2 0 ; 4 .0 0 2 ; -4 . 0 0 0 ; 0 0 0 ;

- 2 .0 0 2 ; +1 .0 1 0 ; - 2 .0 0 0 ; 0 0 0 ] ;

mak e_s ys t em( do ub l e ) % s hows s ys t em

s ( l ) . t im e = 0 ; s ( 2 ) . t i m e = 0 ; '/. i n i t i a l i z e ti m e f o r e v e r y

s ( 3 ) . t i m e = 0 ; s ( 4 ) . t i m e = 0 ; % s t a r t s o l u t i o n t o z e r o

s o l s = t r a c k ( d o u b l e , t , s ) ; % c a l l t h e p a t h t r a c k e r s

n s = s i z e ( s o l s , 2 ) % c h ec k n um ber of s o l u t i o n s

s 2 = s o l s ( 2 ) % l o o k a t t h e 2nd s o l u t i o n

Th e choice of th e second solution was done on p urpose because th is solution

needs ext ra processing. In general however, we have no control over the

order in which the solutions are computed, i .e.: while every run should give

the same four solutions back, the order of solutions could be permuted.

The output we see on screen of the sequence above is

ans =

' x l ** 2 + 4*x2**2 - 4 '

' - 2*x2**2 + x l - 2 '

ns =

4

s2 =

t i m e : 1


e r r : 4 . 3 7 3 0 0 0 0 0 0 0 0 0 0 0 0 e - 0 7

r c o : 3 . i 4 7 0 0 0 0 0 0 0 0 0 0 0 0 e - 0 7

r e s :

7 .235000000000000e- 13

x l :

2 .0000000000000006+00 - 5 .04 8709 793 414 480 e-29i

x 2 : - 2 .49333914 60120 10e- 07 - 1 .879166705634450e- 07 i

Recall th at we const ructed the equat ions in our second sys tem / ( x ) = 0 so

that there is a double solution at (2,0). However, since we are not yet close

enough to the actual double solution (2,0) , the magnitude of the condit ion


29/180

20 YUN GUAN AND JAN VERSCHELDE

number is about 10^, below the threshold of 10^, so phc does not recognize

th e solution as a dou ble roo t. We will ne xt see how to get closer to th e

actual solution.

4 .3 .

Re f i n i n g a n d r e c o n d i t i o n i n g s i n g u l a r s o l u t i o n s . To refine

the solution we save in s2, we execute 10 addit ion Newton steps, applying

ref inejsols to the second solut ion s2:

r 2 = r e f i n e _ s o l s ( d o u b l e , s 2 , 1 . 0 e - 1 6 , 1 . 0 e - 0 8 , 1 . 0 e - 1 6 , 1 0 )

We allow 10 i terat io ns ( last param eter of refine_sols) of Newton's method,

requir ing that ei ther the magnitude of the correct ion vector (err) or the

residual (res) is less or equal than 10~^®, as specified respectively by the

third and f if th parameter of reflne_sols.

Below, on the output we see the est imate for the inverse condit ion

number has decreased, along with the value for x2:

r2 =

time:

multiplicity:

err:

rco:

res:

xl:

x2:

1

1

3.3000000000000006-09

3.8850000000000006-09

6.4279999999999996-17

2.OOOOOOOOOOOOOOOe+00

-2.9990621833465416-0

4.309100000000000e-41i

-

3.017695139191104e-10i

Now that the est imate for the inverse condit ion number has dropped

from 10~^ to 10 "^, below the thresho ld of 10~^, we expe ct this solution to

be singular. To deflate the multiplicity [9] and recondition the solution, we

execute

d e f . s o l s = d e f l a t i o n ( d o u b l e , s o l s ) ;

d e f _ s o l s { 4 , l }

and then we see on screen

ans =

t i m e : 1


e r r : 2 .1860000000000006- 07

r c o : 1 .2420000000000006-01

r e s :

1 .0030000000000006-13

x l : 2 .0000000000000106+00 + 1 .92928 62559 18420 e-14i

x 2 : - 1 .7426214785 217806 - 14 + 8 .26617945 7715231 e- 15 i

l m_l _ l : 3 .0779398018996406- 01 + 6 .678691166401400e- 01 i

l m _ l _ 2 :

- 6 .73752454 60803 006- 01 - 2 .9469292 68111 410e- 01 i

Not ice the value rc o which has increased dramat ical ly f irom 3. 88 5e -0 9 to

1.242e-01, as a clear indicat ion that the solution returned by deflat ion is

well con ditioned . Yet th e m ultiplicity is two as a solution of th e original

system. The deflat ion procedure has constructed an augmented system for

which the dou ble solution of th e original system is a regulax roo t. T he


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PHCLAB: A MATLAB/OCTAVE INTERFACE TO PHCPACK 21

values for lm_l_l and lm_l_2 are the values of the multipliers Ai,i and Ai^2

used in th e first deflation of th e system . T he nu m ber of m ultipliers used

equals the one plus the numerical rank of the given approximate solution

evaluated at the Jacobian matrix of the original system. The augmented

system is returned in def _ s o l s { 3 , 1 } .

4 . 4 .

M i x e d v o l u m e s a n d r a n d o m c o ef fi ci en t s y s t e m s . In o rd er

to solve the system (4.2) we used the output of the blackbox solver on a

more general system . Th e blackbox solver uses polyhedral hom otopies [5] to

solve a system with th e same spa rse structu re b ut w ith ran do m coefficients .

Such random coefficient system has exactly as many isolated solutions as its

mixed volume [10] . The funct ion mixed_volume in PHClab gives access

to the code MixedVol [3] as it is available as translated form in PHCpack.

If we continue our session with in double the tableau input for the

system (4.2) , then we can compute i ts mixed volume and solve a random

coefficients start system via the following sequence of commands:

[ v , g , s ] = m i x e d _ v o l u m e ( d o u b l e , l ) ; % c om p u te m i xe d v o lu m e

v 7. che ck th e mixed volume

US = s i z e ( s , 2 ) 7. c h e c k n um b er of s o l u t i o n s

g '/. ramdom c o e f f i c i e n t sys tem

The output to these command is

V = 4

ns = 4

[ 1 ,1 ] =

+( 9.51900029533701E-01 + 3.06408769087537E-01*i)*xl~2

+( 9.94012861166580E-01 + 1.09263131180786E-01*i)

[2.1] =

+( 6 .10442645118414E- 01 - 7 .920604 62982 993E - 01* i ) *x2- 2

+( - 5 .76175858933 274E- 01 - 8 .17325748757804E- 01* i )

5 . So l v i n g m a n y s y s t e m s . Us ing PHCpack from wi th i n a MATLAB

or O ctave session provides novel opp ortun it ies to solve polynom ial system s.

In this sect ion we show how the script ing environments can help to control

th e quali ty of the d eveloped software. T he high level paral lel p rogram m ing

capabil i t ies of MPITB will speed up this process in a convenient manner.

5 . 1 . A u t o m a t i c t e s t i n g a n d b e n c h m a r k

software for algebraic geometry

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