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diff --git a/src/hyper/pages/UP.ht b/src/hyper/pages/UP.ht new file mode 100644 index 00000000..b3afe6e0 --- /dev/null +++ b/src/hyper/pages/UP.ht @@ -0,0 +1,274 @@ +% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved. +% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk. +\newcommand{\UnivariatePolynomialXmpTitle}{UnivariatePolynomial} +\newcommand{\UnivariatePolynomialXmpNumber}{9.83} +% +% ===================================================================== +\begin{page}{UnivariatePolynomialXmpPage}{9.83 UnivariatePolynomial} +% ===================================================================== +\beginscroll + +The domain constructor \spadtype{UnivariatePolynomial} +%-% \HDindex{polynomial!one variable}{UnivariatePolynomialXmpPage}{9.83}{UnivariatePolynomial} +(abbreviated \spadtype{UP}) +creates domains of univariate polynomials in a specified variable. +For example, the domain +\spadtype{UP(a1,POLY FRAC INT)} provides polynomials in the single variable +\spad{a1} whose coefficients are general polynomials with rational +number coefficients. + +\beginImportant +\noindent {\bf Restriction:} +\texht{\begin{quotation}\noindent}{\newline\indent{5}} +\Language{} does not allow you to create types where +\spadtype{UnivariatePolynomial} is contained in the coefficient type of +\spadtype{Polynomial}. Therefore, +\spadtype{UP(x,POLY INT)} is legal but \spadtype{POLY UP(x,INT)} is not. +\texht{\end{quotation}}{\indent{0}} +\endImportant + +\xtc{ +\spadtype{UP(x,INT)} is the domain of polynomials in the single +variable \spad{x} with integer coefficients. +}{ +\spadpaste{(p,q) : UP(x,INT) \bound{pdec}\bound{qdec}} +} +\xtc{ +}{ +\spadpaste{p := (3*x-1)**2 * (2*x + 8) \free{pdec}\bound{p}} +} +\xtc{ +}{ +\spadpaste{q := (1 - 6*x + 9*x**2)**2 \free{qdec}\bound{q}} +} +\xtc{ +The usual arithmetic operations are available for univariate +polynomials. +}{ +\spadpaste{p**2 + p*q \free{p q}} +} +\xtc{ +The operation \spadfunFrom{leadingCoefficient}{UnivariatePolynomial} +extracts the coefficient of the term of highest degree. +}{ +\spadpaste{leadingCoefficient p \free{p}} +} +\xtc{ +The operation \spadfunFrom{degree}{UnivariatePolynomial} returns +the degree of the polynomial. +Since the polynomial has only one variable, the variable is not supplied +to operations like \spadfunFrom{degree}{UnivariatePolynomial}. +}{ +\spadpaste{degree p \free{p}} +} +\xtc{ +The reductum of the polynomial, the polynomial obtained by +subtracting the term of highest order, is returned by +\spadfunFrom{reductum}{UnivariatePolynomial}. +}{ +\spadpaste{reductum p \free{p}} +} +\xtc{ +The operation \spadfunFrom{gcd}{UnivariatePolynomial} computes the +greatest common divisor of two polynomials. +}{ +\spadpaste{gcd(p,q) \free{p q}} +} +\xtc{ +The operation \spadfunFrom{lcm}{UnivariatePolynomial} computes the +least common multiple. +}{ +\spadpaste{lcm(p,q) \free{p q}} +} +\xtc{ +The operation \spadfunFrom{resultant}{UnivariatePolynomial} +computes the resultant of two univariate polynomials. +In the case of \spad{p} and \spad{q}, the resultant is \spad{0} because they +share a common root. +}{ +\spadpaste{resultant(p,q) \free{p q}} +} +\xtc{ +To compute the derivative of a univariate polynomial with respect to its +variable, use \spadfunFrom{D}{UnivariatePolynomial}. +}{ +\spadpaste{D p \free{p}} +} +\xtc{ +Univariate polynomials can also be used as if they were functions. +To evaluate a univariate polynomial at some point, apply +the polynomial to the point. +}{ +\spadpaste{p(2) \free{p}} +} +\xtc{ +The same syntax is used for composing two univariate polynomials, i.e. +substituting one polynomial for the variable in another. +This substitutes \spad{q} for the variable in \spad{p}. +}{ +\spadpaste{p(q) \free{p q}} +} +\xtc{ +This substitutes \spad{p} for the variable in \spad{q}. +}{ +\spadpaste{q(p) \free{p q}} +} +\xtc{ +To obtain a list of coefficients of the polynomial, use +\spadfunFrom{coefficients}{UnivariatePolynomial}. +}{ +\spadpaste{l := coefficients p \free{p}\bound{l}} +} +\xtc{ +From this you can use \spadfunFrom{gcd}{UnivariatePolynomial} +and \spadfunFrom{reduce}{List} +to compute the content of the polynomial. +}{ +\spadpaste{reduce(gcd,l) \free{l}} +} +\xtc{ +Alternatively (and more easily), +you can just call \spadfunFrom{content}{UnivariatePolynomial}. +}{ +\spadpaste{content p \free{p}} +} + +Note that the operation \spadfunFrom{coefficients}{UnivariatePolynomial} +omits the zero coefficients from the list. +Sometimes it is useful to convert a univariate polynomial +to a vector whose \eth{\spad{i }} position contains the degree \spad{i-1} +coefficient of the polynomial. +\xtc{ +}{ +\spadpaste{ux := (x**4+2*x+3)::UP(x,INT) \bound{ux}} +} +\xtc{ +To get a complete vector of coefficients, use the operation +\spadfunFrom{vectorise}{UnivariatePolynomial}, which takes a +univariate polynomial and an integer denoting the length of the +desired vector. +}{ +\spadpaste{vectorise(ux,5) \free{ux}} +} + +It is common to want to do something to every term of a polynomial, +creating a new polynomial in the process. +\xtc{ +This is a function for iterating across the terms of a polynomial, +squaring each term. +}{ +\begin{spadsrc}[\bound{squareTerms}] +squareTerms(p) == + reduce(+,[t**2 for t in monomials p]) +\end{spadsrc} +} +\xtc{ +Recall what \spad{p} looked like. +}{ +\spadpaste{p \free{p}} +} +\xtc{ +We can demonstrate \userfun{squareTerms} on \spad{p}. +}{ +\spadpaste{squareTerms p \free{p}\free{squareTerms}} +} + +When the coefficients of the univariate polynomial belong to a +field,\footnote{For example, when the coefficients are rational +numbers, as opposed to integers. The important property of +a field is that non-zero elements can be divided and produce +another element. The quotient of the integers 2 and 3 is not +another integer.} +it is possible to compute quotients and remainders. +\xtc{ +}{ +\spadpaste{(r,s) : UP(a1,FRAC INT) \bound{rdec}\bound{sdec}} +} +\xtc{ +}{ +\spadpaste{r := a1**2 - 2/3 \free{rdec}\bound{r}} +} +\xtc{ +}{ +\spadpaste{s := a1 + 4 \free{sdec}\bound{s}} +} +\xtc{ +When the coefficients are rational numbers or rational expressions, the +operation \spadfunFrom{quo}{UnivariatePolynomial} computes the quotient +of two polynomials. +}{ +\spadpaste{r quo s \free{r s}} +} +\xtc{ +The operation +\spadfunFrom{rem}{UnivariatePolynomial} computes the remainder. +}{ +\spadpaste{r rem s \free{r s}} +} +\xtc{ +The operation \spadfunFrom{divide}{UnivariatePolynomial} can be used to +return a record of both components. +}{ +\spadpaste{d := divide(r, s) \free{r s}\bound{d}} +} +\xtc{ +Now we check the arithmetic! +}{ +\spadpaste{r - (d.quotient * s + d.remainder) \free{r s d}} +} +\xtc{ +It is also possible to integrate univariate polynomials when the +coefficients belong to a field. +}{ +\spadpaste{integrate r \free{r}} +} +\xtc{ +}{ +\spadpaste{integrate s \free{s}} +} + +One application of univariate polynomials is to see expressions in terms +of a specific variable. +% +\xtc{ +We start with a polynomial in \spad{a1} whose coefficients +are quotients of polynomials in \spad{b1} and \spad{b2}. +}{ +\spadpaste{t : UP(a1,FRAC POLY INT) \bound{tdec}} +} +\xtc{ +Since in this case we are not talking about using multivariate +polynomials in only two variables, we use \spadtype{Polynomial}. +We also use \spadtype{Fraction} because we want fractions. +}{ +\spadpaste{t := a1**2 - a1/b2 + (b1**2-b1)/(b2+3) \free{tdec}\bound{t}} +} +\xtc{ +We push all the variables into a single quotient of polynomials. +}{ +\spadpaste{u : FRAC POLY INT := t \bound{u}\free{t}} +} +\xtc{ +Alternatively, we can view this as a polynomial in the variable +This is a {\it mode-directed} conversion: you indicate +as much of the structure as you care about and let \Language{} +decide on the full type and how to do the transformation. +}{ +\spadpaste{u :: UP(b1,?) \free{u}} +} + +See \downlink{``\ugProblemFactorTitle''}{ugProblemFactorPage} in Section \ugProblemFactorNumber\ignore{ugProblemFactor} +for a discussion of the factorization facilities +in \Language{} for univariate polynomials. +For more information on related topics, see +\downlink{``\ugIntroVariablesTitle''}{ugIntroVariablesPage} in Section \ugIntroVariablesNumber\ignore{ugIntroVariables}, +\downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert}, +\downlink{`Polynomial'}{PolynomialXmpPage}\ignore{Polynomial}, +\downlink{`MultivariatePolynomial'}{MultivariatePolynomialXmpPage}\ignore{MultivariatePolynomial}, and +\downlink{`DistributedMultivariatePolynomial'}{DistributedMultivariatePolynomialXmpPage}\ignore{DistributedMultivariatePolynomial}. +% +\showBlurb{UnivariatePolynomial} +\endscroll +\autobuttons +\end{page} +% |