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module Algebra.
PrincipalIdealDomain No documentation available.
module MathObj.
PartialFraction Implementation of partial fractions. Useful e.g. for fractions of integers and fractions of polynomials. For the considered ring the prime factorization must be unique.
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Routines and abstractions for permutations of Integers.
- ** Seems to be a candidate for Algebra directory. Algebra.PermutationGroup ?
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Polynomials and rational functions in a single indeterminate. Polynomials are represented by a list of coefficients. All non-zero coefficients are listed, but there may be extra '0's at the end. Usage: Say you have the ring of Integer numbers and you want to add a transcendental element x, that is an element, which does not allow for simplifications. More precisely, for all positive integer exponents n the power x^n cannot be rewritten as a sum of powers with smaller exponents. The element x must be represented by the polynomial [0,1]. In principle, you can have more than one transcendental element by using polynomials whose coefficients are polynomials as well. However, most algorithms on multi-variate polynomials prefer a different (sparse) representation, where the ordering of elements is not so fixed. If you want division, you need Number.Ratios of polynomials with coefficients from a Algebra.Field. You can also compute with an algebraic element, that is an element which satisfies an algebraic equation like x^3-x-1==0. Actually, powers of x with exponents above 3 can be simplified, since it holds x^3==x+1. You can perform these computations with Number.ResidueClass of polynomials, where the divisor is the polynomial equation that determines x. If the polynomial is irreducible (in our case x^3-x-1 cannot be written as a non-trivial product) then the residue classes also allow unrestricted division (except by zero, of course). That is, using residue classes of polynomials you can work with roots of polynomial equations without representing them by radicals (powers with fractional exponents). It is well-known, that roots of polynomials of degree above 4 may not be representable by radicals.
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Power series, either finite or unbounded. (zipWith does exactly the right thing to make it work almost transparently.)
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Two-variate power series.
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For a multi-set of numbers, we describe a sequence of the sums of powers of the numbers in the set. These can be easily converted to polynomials and back. Thus they provide an easy way for computations on the roots of a polynomial.
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numeric-prelude Number.Complex We like to build the Complex Algebraic instance on top of the Algebraic instance of the scalar type. This poses no problem to sqrt. However, root requires computing the complex argument which is a transcendent operation. In order to keep the type class dependencies clean for more sophisticated algebraic number types, we introduce a type class which actually performs the radix operation.
module Number.
PartiallyTranscendental Define Transcendental functions on arbitrary fields. These functions are defined for only a few (in most cases only one) arguments, that's why we discourage making these types instances of C. But instances of C can be useful when working with power series. If you intend to work with power series with Rational coefficients, you might consider using MathObj.PowerSeries.T (Number.PartiallyTranscendental.T Rational) instead of MathObj.PowerSeries.T Rational.
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Lazy Peano numbers represent natural numbers inclusive infinity. Since they are lazily evaluated, they are optimally for use as number type of genericLength et.al.