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+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\FactoredXmpTitle}{Factored}
+\newcommand{\FactoredXmpNumber}{9.22}
+%
+% =====================================================================
+\begin{page}{FactoredXmpPage}{9.22 Factored}
+% =====================================================================
+\beginscroll
+
+\spadtype{Factored} creates a domain whose objects are kept in factored
+%-% \HDindex{factorization}{FactoredXmpPage}{9.22}{Factored}
+form as long as possible.
+Thus certain operations like \spadopFrom{*}{Factored}
+(multiplication) and \spadfunFrom{gcd}{Factored} are relatively
+easy to do.
+Others, such as addition, require somewhat more work, and
+the result may not be completely factored
+unless the argument domain \spad{R} provides a
+\spadfunFrom{factor}{Factored} operation.
+Each object consists of a unit and a list of factors, where each
+factor consists of a member of \spad{R} (the {\em base}), an
+exponent, and a flag indicating what is known about the base.
+A flag may be one of \spad{"nil"}, \spad{"sqfr"}, \spad{"irred"}
+or \spad{"prime"}, which mean that nothing is known about the
+base, it is square-free, it is irreducible, or it is prime,
+respectively.
+The current restriction to factored objects of integral domains
+allows simplification to be performed without worrying about
+multiplication order.
+
+\beginmenu
+    \menudownlink{{9.22.1. Decomposing Factored Objects}}{ugxFactoredDecompPage}
+    \menudownlink{{9.22.2. Expanding Factored Objects}}{ugxFactoredExpandPage}
+    \menudownlink{{9.22.3. Arithmetic with Factored Objects}}{ugxFactoredArithPage}
+    \menudownlink{{9.22.4. Creating New Factored Objects}}{ugxFactoredNewPage}
+    \menudownlink{{9.22.5. Factored Objects with Variables}}{ugxFactoredVarPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFactoredDecompTitle}{Decomposing Factored Objects}
+\newcommand{\ugxFactoredDecompNumber}{9.22.1.}
+%
+% =====================================================================
+\begin{page}{ugxFactoredDecompPage}{9.22.1. Decomposing Factored Objects}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+\xtc{
+In this section we will work with a factored integer.
+}{
+\spadpaste{g := factor(4312) \bound{g}}
+}
+\xtc{
+Let's begin by decomposing \spad{g} into pieces.
+The only possible units for integers are \spad{1} and \spad{-1}.
+}{
+\spadpaste{unit(g) \free{g}}
+}
+\xtc{
+There are three factors.
+}{
+\spadpaste{numberOfFactors(g) \free{g}}
+}
+\xtc{
+We can make a list of the bases, \ldots
+}{
+\spadpaste{[nthFactor(g,i) for i in 1..numberOfFactors(g)] \free{g}}
+}
+\xtc{
+and the exponents, \ldots
+}{
+\spadpaste{[nthExponent(g,i) for i in 1..numberOfFactors(g)] \free{g}}
+}
+\xtc{
+and the flags.
+You can see that all the bases (factors) are prime.
+}{
+\spadpaste{[nthFlag(g,i) for i in 1..numberOfFactors(g)] \free{g}}
+}
+\xtc{
+A useful operation for pulling apart a factored object into a list
+of records of the components is \spadfunFrom{factorList}{Factored}.
+}{
+\spadpaste{factorList(g) \free{g}}
+}
+\xtc{
+If you don't care about the flags, use \spadfunFrom{factors}{Factored}.
+}{
+\spadpaste{factors(g) \free{g}\bound{prev1}}
+}
+\xtc{
+Neither of these operations returns the unit.
+}{
+\spadpaste{first(\%).factor \free{prev1}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFactoredExpandTitle}{Expanding Factored Objects}
+\newcommand{\ugxFactoredExpandNumber}{9.22.2.}
+%
+% =====================================================================
+\begin{page}{ugxFactoredExpandPage}{9.22.2. Expanding Factored Objects}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+\xtc{
+Recall that we are working with this factored integer.
+}{
+\spadpaste{g := factor(4312) \bound{g}}
+}
+\xtc{
+To multiply out the factors with their multiplicities, use
+\spadfunFrom{expand}{Factored}.
+}{
+\spadpaste{expand(g) \free{g}}
+}
+\xtc{
+If you would like, say, the distinct factors multiplied together
+but with multiplicity one, you could do it this way.
+}{
+\spadpaste{reduce(*,[t.factor for t in factors(g)]) \free{g}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFactoredArithTitle}{Arithmetic with Factored Objects}
+\newcommand{\ugxFactoredArithNumber}{9.22.3.}
+%
+% =====================================================================
+\begin{page}{ugxFactoredArithPage}{9.22.3. Arithmetic with Factored Objects}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+\xtc{
+We're still working with this factored integer.
+}{
+\spadpaste{g := factor(4312) \bound{g}}
+}
+\xtc{
+We'll also define this factored integer.
+}{
+\spadpaste{f := factor(246960) \bound{f}}
+}
+\xtc{
+Operations involving multiplication and division are particularly
+easy with factored objects.
+}{
+\spadpaste{f * g \free{f g}}
+}
+\xtc{
+}{
+\spadpaste{f**500 \free{f}}
+}
+\xtc{
+}{
+\spadpaste{gcd(f,g) \free{f g}}
+}
+\xtc{
+}{
+\spadpaste{lcm(f,g) \free{f g}}
+}
+\xtc{
+If we use addition and subtraction things can slow down because
+we may need to compute greatest common divisors.
+}{
+\spadpaste{f + g \free{f g}}
+}
+\xtc{
+}{
+\spadpaste{f - g \free{f g}}
+}
+\xtc{
+Test for equality with \spad{0} and \spad{1} by using
+\spadfunFrom{zero?}{Factored} and \spadfunFrom{one?}{Factored},
+respectively.
+}{
+\spadpaste{zero?(factor(0))}
+}
+\xtc{
+}{
+\spadpaste{zero?(g) \free{g}}
+}
+\xtc{
+}{
+\spadpaste{one?(factor(1))}
+}
+\xtc{
+}{
+\spadpaste{one?(f) \free{f}}
+}
+\xtc{
+Another way to get the zero and one factored objects is to use
+package calling (see \downlink{``\ugTypesPkgCallTitle''}{ugTypesPkgCallPage} in Section \ugTypesPkgCallNumber\ignore{ugTypesPkgCall}).
+}{
+\spadpaste{0\$Factored(Integer)}
+}
+\xtc{
+}{
+\spadpaste{1\$Factored(Integer)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFactoredNewTitle}{Creating New Factored Objects}
+\newcommand{\ugxFactoredNewNumber}{9.22.4.}
+%
+% =====================================================================
+\begin{page}{ugxFactoredNewPage}{9.22.4. Creating New Factored Objects}
+% =====================================================================
+\beginscroll
+
+The \spadfunFrom{map}{Factored} operation is used to iterate across the
+unit and bases of a factored object.
+See \downlink{`FactoredFunctions2'}{FactoredFunctionsTwoXmpPage}\ignore{FactoredFunctions2} for a discussion of
+\spadfunFrom{map}{Factored}.
+
+\labelSpace{1pc}
+\xtc{
+The following four operations take a base and an exponent and create
+a factored object.
+They differ in handling the flag component.
+}{
+\spadpaste{nilFactor(24,2) \bound{prev2}}
+}
+\xtc{
+This factor has no associated information.
+}{
+\spadpaste{nthFlag(\%,1) \free{prev2}}
+}
+\xtc{
+This factor is asserted to be square-free.
+}{
+\spadpaste{sqfrFactor(30,2) \bound{prev3}}
+}
+\xtc{
+This factor is asserted to be irreducible.
+}{
+\spadpaste{irreducibleFactor(13,10) \bound{prev4}}
+}
+\xtc{
+This factor is asserted to be prime.
+}{
+\spadpaste{primeFactor(11,5) \bound{prev5}}
+}
+
+\xtc{
+A partial inverse to \spadfunFrom{factorList}{Factored} is
+\spadfunFrom{makeFR}{Factored}.
+}{
+\spadpaste{h := factor(-720) \bound{h}}
+}
+\xtc{
+The first argument is the unit and the second is a list of records as
+returned by \spadfunFrom{factorList}{Factored}.
+}{
+\spadpaste{h - makeFR(unit(h),factorList(h)) \free{h}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFactoredVarTitle}{Factored Objects with Variables}
+\newcommand{\ugxFactoredVarNumber}{9.22.5.}
+%
+% =====================================================================
+\begin{page}{ugxFactoredVarPage}{9.22.5. Factored Objects with Variables}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+\xtc{
+Some of the operations available for polynomials are also available
+for factored polynomials.
+}{
+\spadpaste{p := (4*x*x-12*x+9)*y*y + (4*x*x-12*x+9)*y + 28*x*x - 84*x + 63 \bound{p}}
+}
+\xtc{
+}{
+\spadpaste{fp := factor(p) \free{p}\bound{fp}}
+}
+\xtc{
+You can differentiate with respect to a variable.
+}{
+\spadpaste{D(p,x) \free{p}}
+}
+\xtc{
+}{
+\spadpaste{D(fp,x) \free{fp}\bound{prev1}}
+}
+\xtc{
+}{
+\spadpaste{numberOfFactors(\%) \free{prev1}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%