From ab8cc85adde879fb963c94d15675783f2cf4b183 Mon Sep 17 00:00:00 2001 From: dos-reis Date: Tue, 14 Aug 2007 05:14:52 +0000 Subject: Initial population. --- src/algebra/real0q.spad.pamphlet | 129 +++++++++++++++++++++++++++++++++++++++ 1 file changed, 129 insertions(+) create mode 100644 src/algebra/real0q.spad.pamphlet (limited to 'src/algebra/real0q.spad.pamphlet') diff --git a/src/algebra/real0q.spad.pamphlet b/src/algebra/real0q.spad.pamphlet new file mode 100644 index 00000000..5a891c92 --- /dev/null +++ b/src/algebra/real0q.spad.pamphlet @@ -0,0 +1,129 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/algebra real0q.spad} +\author{Andy Neff, Barry Trager} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\section{package REAL0Q RealZeroPackageQ} +<>= +)abbrev package REAL0Q RealZeroPackageQ +++ Author: Andy Neff, Barry Trager +++ Date Created: +++ Date Last Updated: 7 April 1991 +++ Basic Functions: +++ Related Constructors: UnivariatePolynomial, RealZeroPackageQ +++ Also See: +++ AMS Classifications: +++ Keywords: +++ References: +++ Description: +++ This package provides functions for finding the real zeros +++ of univariate polynomials over the rational numbers to arbitrary user-specified +++ precision. The results are returned as a list of +++ isolating intervals, expressed as records with "left" and "right" rational number components. + +RealZeroPackageQ(Pol): T == C where + RN ==> Fraction Integer + I ==> Integer + SUP ==> SparseUnivariatePolynomial + Pol: UnivariatePolynomialCategory RN + Interval ==> Record(left : RN, right : RN) + isoList ==> List(Interval) + ApproxInfo ==> Record(approx : RN, exFlag : Boolean) + T == with + -- next two functions find isolating intervals + realZeros: (Pol) -> isoList + ++ realZeros(pol) returns a list of isolating intervals for + ++ all the real zeros of the univariate polynomial pol. + realZeros: (Pol, Interval) -> isoList + ++ realZeros(pol, range) returns a list of isolating intervals + ++ for all the real zeros of the univariate polynomial pol which + ++ lie in the interval expressed by the record range. + -- next two functions return intervals smaller then tolerence + realZeros: (Pol, RN) -> isoList + ++ realZeros(pol, eps) returns a list of intervals of length less + ++ than the rational number eps for all the real roots of the + ++ polynomial pol. + realZeros: (Pol, Interval, RN) -> isoList + ++ realZeros(pol, int, eps) returns a list of intervals of length + ++ less than the rational number eps for all the real roots of the + ++ polynomial pol which lie in the interval expressed by the + ++ record int. + refine: (Pol, Interval, RN) -> Interval + ++ refine(pol, int, eps) refines the interval int containing + ++ exactly one root of the univariate polynomial pol to size less + ++ than the rational number eps. + refine: (Pol, Interval, Interval) -> Union(Interval,"failed") + ++ refine(pol, int, range) takes a univariate polynomial pol and + ++ and isolating interval int which must contain exactly one real + ++ root of pol, and returns an isolating interval which + ++ is contained within range, or "failed" if no such isolating interval exists. + C == add + import RealZeroPackage SparseUnivariatePolynomial Integer + + convert2PolInt: Pol -> SparseUnivariatePolynomial Integer + + convert2PolInt(f : Pol) == + pden:I :=lcm([denom c for c in coefficients f]) + map(numer,pden * f)$UnivariatePolynomialCategoryFunctions2(RN,Pol,I,SUP I) + + realZeros(f : Pol) == realZeros(convert2PolInt f) + realZeros(f : Pol, rn : RN) == realZeros(convert2PolInt f, rn) + realZeros(f : Pol, bounds : Interval) == + realZeros(convert2PolInt f, bounds) + realZeros(f : Pol, bounds : Interval, rn : RN) == + realZeros(convert2PolInt f, bounds, rn) + refine(f : Pol, int : Interval, eps : RN) == + refine(convert2PolInt f, int, eps) + refine(f : Pol, int : Interval, bounds : Interval) == + refine(convert2PolInt f, int, bounds) + +@ +\section{License} +<>= +--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +--All rights reserved. +-- +--Redistribution and use in source and binary forms, with or without +--modification, are permitted provided that the following conditions are +--met: +-- +-- - Redistributions of source code must retain the above copyright +-- notice, this list of conditions and the following disclaimer. +-- +-- - Redistributions in binary form must reproduce the above copyright +-- notice, this list of conditions and the following disclaimer in +-- the documentation and/or other materials provided with the +-- distribution. +-- +-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the +-- names of its contributors may be used to endorse or promote products +-- derived from this software without specific prior written permission. +-- +--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. +@ +<<*>>= +<> + +<> +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} -- cgit v1.2.3