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<?xml version="1.0" encoding="iso-8859-1"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<link rel="stylesheet" type="text/css" href="html.css" />
<title>JAS - Algorithms for Computer Algebra book</title>
</head>
<body class="main">
<h1>Algorithms for Computer Algebra book and JAS methods</h1>
<p>
Summary of algorithms from the
<a href="http://www.springer.com/computer/theoretical+computer+science/book/978-0-7923-9259-0"
target="gbb">Algorithms for Computer Algebra</a>
book and corresponding JAS classes and methods.
</p>
<h2>Algorithms for Computer Algebra book</h2>
<p>
The JAS base package <code>edu.jas</code> name is omitted in the
following table.
JAS also contains improved versions of the algorithms which may be located through the links.
A short explanation of code organization with interfaces and several implementing classes
can be found in the <a href="design.html">API guide</a>.
</p>
<table border="1" cellpadding="3" summary="Algo CA book to JAS summary" >
<tr>
<td>Algorithms for Computer Algebra</td>
<td>JAS interfaces, classes and methods</td>
<td>remarks</td>
</tr>
<tr>
<td>2.1 Euclidean Algorithm, <code>Euclid</code></td>
<td><a href="api/edu/jas/structure/RingElem.html#gcd(C)" target="classFrame"><code>structure.RingElem.gcd</code></a>
</td>
<td>all classes which implement this interface
</td>
</tr>
<tr>
<td>2.2 Extended Euclidean Algorithm, <code>EEA</code></td>
<td><a href="api/edu/jas/structure/RingElem.html#egcd(C)" target="classFrame"><code>structure.RingElem.egcd</code></a>
</td>
<td>all classes which implement this interface
</td>
</tr>
<tr>
<td>2.3 Primitive Euclidean Algorithm, <code>PrimitiveEuclidean</code></td>
<td><a href="api/edu/jas/ufd/GreatestCommonDivisorPrimitive.html" target="classFrame"><code>ufd.GreatestCommonDivisorPrimitive</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td colspan="3"> </td>
</tr>
<tr>
<td>4.1 Multiprecision Integer Multiplication, <code>BigIntegerMultiply</code></td>
<td><a href="api/edu/jas/arith/BigInteger.html#multiply(edu.jas.arith.BigInteger)" target="classFrame"><code>BigInteger.multiply</code></a>
</td>
<td>adapter for native Java implementation in <code>java.math.BigInteger.multiply</code>
</td>
</tr>
<tr>
<td>4.2 Karatsuba's Multiplication Algorithm, <code>Karatsuba</code></td>
<td><a href="api/edu/jas/arith/" target="classFrame"><code></code></a>
</td>
<td>implemented in <code>java.math.BigInteger.multiply</code>
</td>
</tr>
<tr>
<td>4.3 Polynomial Trial Division Algorithm, <code>TrialDivision</code></td>
<td>not implemented
</td>
<td>see
<a href="api/edu/jas/poly/GenPolynomial.html#divide(edu.jas.poly.GenPolynomial)" target="classFrame"><code>GenPolynomial.divide</code></a>
and
<a href="api/edu/jas/poly/PolyUtil.html#basePseudoDivide(edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial)" target="classFrame"><code>PolyUtil.basePseudoDivide</code></a>
</td>
</tr>
<tr>
<td>4.4 Fast Fourier Transform, <code>FFT</code></td>
<td><a href="api/edu/jas/" target="classFrame"><code></code></a>
not implemented
</td>
<td>
</td>
</tr>
<tr>
<td>4.5 Fast Fourier Polynomial Multiplication, <code>FFT_Multiply</code></td>
<td><a href="api/edu/jas/" target="classFrame"><code></code></a>
not implemented
</td>
<td>
</td>
</tr>
<tr>
<td>4.6 Newtons's Method for Power Series Inversion, <code>FastNewtonInversion</code></td>
<td>not implemented
</td>
<td>see
<a href="api/edu/jas/ps/UnivPowerSeries.html#inverse()" target="classFrame"><code>UnivPowerSeries.inverse()</code></a>
and
<a href="api/edu/jas/ps/MultiVarPowerSeries.html#inverse()" target="classFrame"><code>MultiVarPowerSeries.inverse()</code></a>
</td>
</tr>
<tr>
<td>4.7 Newtons's Method for Solving P(y) = 0, <code>NewtonSolve</code></td>
<td>not implemented
</td>
<td>see
<a href="api/edu/jas/ps/UnivPowerSeriesRing.html#solveODE(edu.jas.ps.UnivPowerSeries,%20C)" target="classFrame"><code>UnivPowerSeriesRing.solveODE()</code></a>
</td>
</tr>
<tr>
<td colspan="3"> </td>
</tr>
<tr>
<td>5.1 Garner's Chinese Remainder Algorithm, <code>IntegerCRA</code></td>
<td><a href="api/edu/jas/arith/ModIntegerRing.html#chineseRemainder(edu.jas.arith.ModInteger,%20edu.jas.arith.ModInteger,%20edu.jas.arith.ModInteger)" target="classFrame"><code>ModIntegerRing.chineseRemainder()</code></a>
</td>
<td>only for two moduli
</td>
</tr>
<tr>
<td>5.2 Newtons Interpolation Algorithm, <code>NewtonInterp</code></td>
<td>not implemented
</td>
<td>see
<a href="api/edu/jas/poly/PolyUtil.html#chineseRemainder(edu.jas.poly.GenPolynomialRing,%20edu.jas.poly.GenPolynomial,%20C,%20edu.jas.poly.GenPolynomial)" target="classFrame"><code>PolyUtil.chineseRemainder()</code></a>
and
<a href="api/edu/jas/poly/PolyUtil.html#interpolate(edu.jas.poly.GenPolynomialRing,%20edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial,%20C,%20C,%20C)" target="classFrame"><code>PolyUtil.interpolate()</code></a>
</td>
</tr>
<tr>
<td colspan="3"> </td>
</tr>
<tr>
<td>6.1 Univariate Hensel Lifting Algorithm, <code>UnivariateHensel</code></td>
<td><a href="api/edu/jas/ufd/HenselUtil.html#liftHensel(edu.jas.poly.GenPolynomial,%20java.util.List,%20long,%20edu.jas.arith.BigInteger)" target="classFrame"><code>HenselUtil.liftHensel()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td>6.2 Multivariate Polynomial Diophantine Equantions, <code>MultivariateDiophant</code></td>
<td><a href="api/edu/jas/ufd/HenselMultUtil.html#liftDiophant(java.util.List,%20edu.jas.poly.GenPolynomial,%20java.util.List,%20long,%20long)" target="classFrame"><code>HenselMultUtil.liftDiophant()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td>6.3 Univariate Polynomial Diophantine Equantions, <code>UnivariateDiophant</code></td>
<td><a href="api/edu/jas/ufd/HenselUtil.html#liftDiophant(edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial,%20long,%20long)" target="classFrame"><code>HenselUtil.liftDiophant()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td>6.4 Multivariate Hensel Lifting Algorithm, <code>MultivariateHensel</code></td>
<td><a href="api/edu/jas/ufd/HenselMultUtil.html#liftHensel(edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial,%20java.util.List,%20java.util.List,%20long,%20java.util.List)" target="classFrame"><code>HenselMultUtil.liftHensel()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td colspan="3"> </td>
</tr>
<tr>
<td>7.1 Modular GCD Algorithm, <code>MGCD</code></td>
<td><a href="api/edu/jas/ufd/GreatestCommonDivisorModular.html#baseGcd(edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial)" target="classFrame"><code>GreatestCommonDivisorModular.</code> <code>baseGcd()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td>7.2 Multivariate GCD Reduction Algorithm, <code>PGCD</code></td>
<td><a href="api/edu/jas/ufd/GreatestCommonDivisorModEval.html#gcd(edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial)" target="classFrame"><code>GreatestCommonDivisorModEval.gcd()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td></td>
<td><a href="api/edu/jas/ufd/GreatestCommonDivisorSubres.html#gcd(edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial)" target="classFrame"><code>GreatestCommonDivisorSubres.gcd()</code></a>
</td>
<td>many more algorithms, for example using polynomial remainder sequences
(PRS), in particular a sub-resultant PRS
</td>
</tr>
<tr>
<td>7.3 Extended Zassenhaus GCD Algorithm, <code>EZ-GCD</code></td>
<td><a href="api/edu/jas/ufd/GreatestCommonDivisorHensel.html#recursiveUnivariateGcd(edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial)" target="classFrame"><code>GreatestCommonDivisorHensel.</code> <code>recursiveUnivariateGcd()</code></a>
</td>
<td>not complete in all cases
</td>
</tr>
<tr>
<td>7.4 GCD Heuristic Algorithm, <code>GCDHEU</code></td>
<td>not implemented
</td>
<td>
</td>
</tr>
<tr>
<td colspan="3"> </td>
</tr>
<tr>
<td>8.1 Square-Free Factorization, <code>SquareFree</code></td>
<td><a href="api/edu/jas/ufd/SquarefreeFieldChar0.html#squarefreeFactors(edu.jas.poly.GenPolynomial)" target="classFrame"><code>SquarefreeFieldChar0.</code> <code>squarefreeFactors()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td>8.2 Yun's Square-Free Factorization, <code>SquareFree2</code></td>
<td><a href="api/edu/jas/ufd/SquarefreeFieldChar0Yun.html#squarefreeFactors(edu.jas.poly.GenPolynomial)" target="classFrame"><code>SquarefreeFieldChar0Yun.</code> <code>squarefreeFactors()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td>8.3 Finite Field Square-Free Factorization, <code>SquareFreeFF</code></td>
<td><a href="api/edu/jas/ufd/SquarefreeFiniteFieldCharP.html#squarefreeFactors(edu.jas.poly.GenPolynomial)" target="classFrame"><code>SquarefreeFiniteFieldCharP.</code> <code>squarefreeFactors()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td></td>
<td><a href="api/edu/jas/ufd/SquarefreeInfiniteFieldCharP.html#squarefreeFactors(edu.jas.ufd.Quotient)" target="classFrame"><code>SquarefreeInfiniteFieldCharP.</code> <code>squarefreeFactors()</code></a>
</td>
<td>Algorithm for infinite fields of characteristic p, not in the book.
</td>
</tr>
<tr>
<td>8.4 Berlekamp's Factorization Algorithm, <code>Berlekamp</code></td>
<td><a href="api/edu/jas/ufd/FactorModularBerlekamp.html#baseFactorsSquarefree(edu.jas.poly.GenPolynomial)" target="classFrame"><code>FactorModularBerlekamp.</code> <code>baseFactorsSquarefree()</code></a>
</td>
<td>The method <code>baseFactorsSquarefreeSmallPrime()</code> contains the implementation.
</td>
</tr>
<tr>
<td>8.5 Form Q Matrix, <code>FormMatrixQ</code></td>
<td><a href="api/edu/jas/ufd/PolyUfdUtil.html#constructQmatrix(edu.jas.poly.GenPolynomial)" target="classFrame"><code>PolyUfdUtil.constructQmatrix()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td>8.6 Null Space Basis Algorithm, <code>NullSpaceBasis</code></td>
<td><a href="api/edu/jas/vector/LinAlg.html#nullSpaceBasis(edu.jas.vector.GenMatrix)" target="classFrame"><code>LinAlg.nullSpaceBasis()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td>8.7 Big Prime Berlekamp Factoring Algorithm, <code>BigPrimeBerlekamp</code></td>
<td><a href="api/edu/jas/ufd/FactorModularBerlekamp.html#baseFactorsSquarefree(edu.jas.poly.GenPolynomial)" target="classFrame"><code>FactorModularBerlekamp.</code> <code>baseFactorsSquarefree()</code></a>
</td>
<td>The method <code>baseFactorsSquarefreeBigPrime()</code> contains the implementation.
</td>
</tr>
<tr>
<td>8.8 Distinct Degree Factorization I, <code>PartialFactorDD</code></td>
<td><a href="api/edu/jas/ufd/FactorModular.html#baseDistinctDegreeFactors(edu.jas.poly.GenPolynomial)" target="classFrame"><code>FactorModular.</code> <code>baseDistinctDegreeFactors()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td>8.9 Distinct Degree Factorization II, <code>SplitDD</code></td>
<td><a href="api/edu/jas/ufd/FactorModular.html#baseEqualDegreeFactors(edu.jas.poly.GenPolynomial)" target="classFrame"><code>FactorModular.</code> <code>baseEqualDegreeFactors()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td></td>
<td><a href="api/edu/jas/ufd/FactorInteger.html#factorsSquarefree(edu.jas.poly.GenPolynomial)" target="classFrame"><code>FactorInteger.factorsSquarefree()</code></a>
</td>
<td>Algorithm of P. Wang, not presented in the book.
</td>
</tr>
<tr>
<td>8.10 Factorization over Algebraic Number Fields, <code>AlgebraicFactorization</code></td>
<td><a href="api/edu/jas/ufd/FactorAlgebraic.html#baseFactorsSquarefree(edu.jas.poly.GenPolynomial)" target="classFrame"><code>FactorAlgebraic.</code> <code>baseFactorsSquarefree()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td colspan="3"> </td>
</tr>
<tr>
<td>9.1 Fraction-Free Gaussian Elimination, <code>FractionFreeElim</code></td>
<td><a href="api/edu/jas/vector/LinAlg.html#fractionfreeGaussElimination(edu.jas.vector.GenMatrix)" target="classFrame"><code>LinAlg.</code> <code>fractionfreeGaussElimination()</code></a>
</td>
<td>see also
<a href="api/edu/jas/gbufd/GroebnerBasePseudoSeq.html#GB(int,%20java.util.List)" target="classFrame"><code>GroebnerBasePseudoSeq.GB()</code></a>
</td>
</tr>
<tr>
<td>9.2 Nonlinear Elimination Algorithm, <code>NonlinearElim</code></td>
<td><a href="api/edu/jas/" target="classFrame"><code></code></a>
not implemented
</td>
<td>Based on iterated resultant computations.
See also the characteristic set method
<a href="api/edu/jas/gbufd/CharacteristicSetSimple.html#characteristicSet(java.util.List)" target="classFrame"><code>CharacteristicSetSimple.characteristicSet()</code></a>
</td>
</tr>
<tr>
<td>9.3 Solution of Nonlinear System of Equations, <code>NonlinearSolve</code></td>
<td><a href="api/edu/jas/" target="classFrame"><code></code></a>
not implemented
</td>
<td>Based on resultant computations and algebraic root substitution.
See also the ideal complex and real root computation and decomposition methods
<a href="api/edu/jas/application/PolyUtilApp.html#complexAlgebraicRoots(edu.jas.application.Ideal)" target="classFrame"><code>PolyUtilApp.complexAlgebraicRoots()</code></a>
</td>
</tr>
<tr>
<td colspan="3"> </td>
</tr>
<tr>
<td>10.1 Full Reduction Algorithm, <code>Reduce</code></td>
<td><a href="api/edu/jas/gb/Reduction.html#normalform(java.util.List,%20edu.jas.poly.GenPolynomial)" target="classFrame"><code>Reduction.normalform()</code></a>
</td>
<td>all classes which implement this interface
</td>
</tr>
<tr>
<td>10.2 Buchbergers's Algorithm for Gröbner Bases, <code>Gbasis</code></td>
<td><a href="api/edu/jas/" target="classFrame"><code></code></a>
not implemented
</td>
<td>
</td>
</tr>
<tr>
<td>10.3 Construction of a Reduced Ideal Basis, <code>ReduceSet</code></td>
<td><a href="api/edu/jas/gb/GroebnerBase.html#minimalGB(java.util.List)" target="classFrame"><code>GroebnerBase.minimalGB()</code></a>
</td>
<td>all classes which implement this interface
</td>
</tr>
<tr>
<td>10.4 Improved Construction of a Reduced Gröbner Basis, <code>Gbasis</code></td>
<td><a href="api/edu/jas/gb/GroebnerBaseSeq.html#GB(int,%20java.util.List)" target="classFrame"><code>GroebnerBaseSeq.GB()</code></a>
</td>
<td>can be parametrized also with different strategies, e.g. Gebauer & Möller
</td>
</tr>
<tr>
<td>10.5 Solution of System P in Variable x, <code>Solve1</code></td>
<td><a href="api/edu/jas/application/Ideal.html#constructUnivariate(int)" target="classFrame"><code>Ideal.constructUnivariate()</code></a>
</td>
<td>univariate polynomials of minimal degree in the ideal
</td>
</tr>
<tr>
<td>10.6 Complete Solution of System P, <code>GröbnerSolve</code></td>
<td><a href="api/edu/jas/application/Ideal.html#zeroDimDecomposition()" target="classFrame"><code>Ideal.zeroDimDecomposition()</code></a>,
</td>
<td>univariate polynomials in the ideal are irreducible
</td>
</tr>
<tr>
<td>10.7 Solution of P using Lexicographic Gröbner Basis, <code>LexSolve</code></td>
<td><a href="api/edu/jas/application/Ideal.html#zeroDimRootDecomposition()" target="classFrame"><code>Ideal.zeroDimRootDecomposition()</code></a>
</td>
<td>additionally to 10.6, the ideal basis consists of maximally bi-variate polynomials
</td>
</tr>
<tr>
<td colspan="3"> </td>
</tr>
<tr>
<td>11.1 Hermite's Method for Rational Functions, <code>HermiteReduction</code></td>
<td><a href="api/edu/jas/integrate/ElementaryIntegration.html#integrateHermite(edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial)" target="classFrame"><code>ElementaryIntegration.</code> <code>integrateHermite()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td>11.2 Horowitz's Reduction for Rational Functions, <code>HorowitzReduction</code></td>
<td><a href="api/edu/jas/integrate/ElementaryIntegration.html#integrate(edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial)" target="classFrame"><code>ElementaryIntegration.integrate()</code></a>
</td>
<td>
</td>
</tr>
<tr>
<td>11.3 Rothstein/Trager method, <code>LogarithmicPartIntegral</code></td>
<td><a href="api/edu/jas/integrate/ElementaryIntegration.html#integrateLogPart(edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial)" target="classFrame"><code>ElementaryIntegration.</code> <code>integrateLogPart()</code></a>
</td>
<td>using resultants
</td>
</tr>
<tr>
<td>11.4 Lazard/Rioboo/Trager improvement, <code>LogarithmicPartIntegral</code></td>
<td><a href="api/edu/jas/integrate/ElementaryIntegrationLazard.html#integrateLogPart(edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial)" target="classFrame"><code>ElementaryIntegrationLazard.</code> <code>integrateLogPart()</code></a>
</td>
<td>using sub-resultants
</td>
</tr>
<tr>
<td>11.x Czichowski variant, <code>LogarithmicPartIntegral</code></td>
<td><a href="api/edu/jas/integrate/ElementaryIntegrationCzichowski.html#integrateLogPart(edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial)" target="classFrame"><code>ElementaryIntegrationCzichowski.</code> <code>integrateLogPart()</code></a>
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<td>using Gröbner bases
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<tr>
<td>11.y Bernoulli variant, <code>LogarithmicPartIntegral</code></td>
<td><a href="api/edu/jas/integrate/ElementaryIntegrationBernoulli.html#integrateLogPart(edu.jas.poly.GenPolynomial,%20edu.jas.poly.GenPolynomial)" target="classFrame"><code>ElementaryIntegrationBernoulli.</code> <code>integrateLogPart()</code></a>
</td>
<td>using absolute factorization into linear factors
</td>
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</table>
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<p><!--a href="README" target="readme" >README</a-->
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<address><a href="mailto:kredel at rz.uni-mannheim.de">Heinz Kredel</a></address>
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Last modified: Sat Aug 7 19:41:34 CEST 2021
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