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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{\OrderlyDifferentialPolynomialXmpTitle}{OrderlyDifferentialPolynomial}
+\newcommand{\OrderlyDifferentialPolynomialXmpNumber}{9.60}
+%
+% =====================================================================
+\begin{page}{OrderlyDifferentialPolynomialXmpPage}{9.60 OrderlyDifferentialPolynomial}
+% =====================================================================
+\beginscroll
+
+Many systems of differential equations may be transformed to equivalent
+%-% \HDindex{differential equation}{OrderlyDifferentialPolynomialXmpPage}{9.60}{OrderlyDifferentialPolynomial}
+systems of ordinary differential equations where the equations are
+%-% \HDindex{equation!differential}{OrderlyDifferentialPolynomialXmpPage}{9.60}{OrderlyDifferentialPolynomial}
+expressed polynomially in terms of the unknown functions.
+%-% \HDindex{polynomial!differential polynomial}{OrderlyDifferentialPolynomialXmpPage}{9.60}{OrderlyDifferentialPolynomial}
+In \Language{}, the domain constructors
+\spadtype{OrderlyDifferentialPolynomial}
+%-% \HDindex{differential polynomial}{OrderlyDifferentialPolynomialXmpPage}{9.60}{OrderlyDifferentialPolynomial}
+(abbreviated \spadtype{ODPOL}) and
+\spadtype{SequentialDifferentialPolynomial} (abbreviation
+\spadtype{SDPOL}) implement two domains of ordinary differential
+polynomials over any differential ring.
+In the simplest case, this differential ring is usually either the ring of
+integers, or the field of rational numbers.
+However, \Language{} can handle ordinary differential polynomials over a
+field of rational functions in a single indeterminate.
+%-% \HDexptypeindex{OrderlyDifferentialPolynomial}{OrderlyDifferentialPolynomialXmpPage}{9.60}{OrderlyDifferentialPolynomial}
+%-% \HDexptypeindex{SequentialDifferentialPolynomial}{OrderlyDifferentialPolynomialXmpPage}{9.60}{OrderlyDifferentialPolynomial}
+
+The two domains \spadtype{ODPOL} and \spadtype{SDPOL} are almost
+identical, the only difference being the choice of a different ranking,
+which is an ordering of the derivatives of the indeterminates.
+The first domain uses an orderly ranking, that is, derivatives of higher
+order are ranked higher, and derivatives of the same order are ranked
+alphabetically.
+The second domain uses a sequential ranking, where derivatives are ordered
+first alphabetically by the differential indeterminates, and then by
+order.
+A more general domain constructor,
+\spadtype{DifferentialSparseMultivariatePolynomial} (abbreviation
+\spadtype{DSMP}) allows both a user-provided list of differential
+indeterminates as well as a user-defined ranking.
+We shall illustrate \spadtype{ODPOL(FRAC INT)}, which constructs a domain
+of ordinary differential polynomials in an arbitrary number of
+differential indeterminates with rational numbers as coefficients.
+\xtc{
+}{
+\spadpaste{dpol:= ODPOL(FRAC INT) \bound{dpol}}
+}
+
+\xtc{
+A differential indeterminate \spad{w} may be viewed as an infinite
+sequence of algebraic indeterminates, which are the derivatives of
+\spad{w}.
+To facilitate referencing these, \Language{} provides the operation
+\spadfunFrom{makeVariable}{OrderlyDifferentialPolynomial} to convert an
+element of type \spadtype{Symbol} to a map from the natural numbers to the
+differential polynomial ring.
+}{
+\spadpaste{w := makeVariable('w)\$dpol \free{dpol}\bound{w}}
+}
+\xtc{
+}{
+\spadpaste{z := makeVariable('z)\$dpol \free{dpol}\bound{z}}
+}
+\xtc{
+The fifth derivative of \spad{w} can be obtained by applying the map
+\spad{w} to the number \spad{5.}
+Note that the order of differentiation is given as a subscript (except
+when the order is 0).
+}{
+\spadpaste{w.5 \free{w}}
+}
+\xtc{
+}{
+\spadpaste{w 0 \free{w}}
+}
+\xtc{
+The first five derivatives of \spad{z} can be generated by a list.
+}{
+\spadpaste{[z.i for i in 1..5] \free{z}}
+}
+\xtc{
+The usual arithmetic can be used to form a differential polynomial from
+the derivatives.
+}{
+\spadpaste{f:= w.4 - w.1 * w.1 * z.3 \free{w}\free{z}\bound{f}}
+}
+\xtc{
+}{
+\spadpaste{g:=(z.1)**3 * (z.2)**2 - w.2 \free{z}\free{w}\bound{g}}
+}
+\xtc{
+The operation \spadfunFrom{D}{OrderlyDifferentialPolynomial}
+computes the derivative of any differential polynomial.
+}{
+\spadpaste{D(f) \free{f}}
+}
+\xtc{
+The same operation can compute higher derivatives, like the
+fourth derivative.
+}{
+\spadpaste{D(f,4) \free{f}}
+}
+\xtc{
+The operation \spadfunFrom{makeVariable}{OrderlyDifferentialPolynomial}
+creates a map to facilitate referencing the derivatives of \spad{f},
+similar to the map \spad{w}.
+}{
+\spadpaste{df:=makeVariable(f)\$dpol \free{f}\bound{df}}
+}
+\xtc{
+The fourth derivative of f may be referenced easily.
+}{
+\spadpaste{df.4 \free{df}}
+}
+\xtc{
+The operation \spadfunFrom{order}{OrderlyDifferentialPolynomial}
+returns the order of a differential polynomial, or the order
+in a specified differential indeterminate.
+}{
+\spadpaste{order(g)  \free{g}}
+}
+\xtc{
+}{
+\spadpaste{order(g, 'w)  \free{g}}
+}
+\xtc{
+The operation
+\spadfunFrom{differentialVariables}{OrderlyDifferentialPolynomial} returns
+a list of differential indeterminates occurring in a differential
+polynomial.
+}{
+\spadpaste{differentialVariables(g)  \free{g}}
+}
+\xtc{
+The operation \spadfunFrom{degree}{OrderlyDifferentialPolynomial} returns
+the degree, or the degree in the differential indeterminate specified.
+}{
+\spadpaste{degree(g) \free{g}}
+}
+\xtc{
+}{
+\spadpaste{degree(g, 'w)  \free{g}}
+}
+\xtc{
+The operation \spadfunFrom{weights}{OrderlyDifferentialPolynomial} returns
+a list of weights of differential monomials appearing in differential
+polynomial, or a list of weights in a specified differential
+indeterminate.
+}{
+\spadpaste{weights(g)  \free{g}}
+}
+\xtc{
+}{
+\spadpaste{weights(g,'w) \free{g}}
+}
+\xtc{
+The operation \spadfunFrom{weight}{OrderlyDifferentialPolynomial} returns
+the maximum weight of all differential monomials appearing in the
+differential polynomial.
+}{
+\spadpaste{weight(g)  \free{g}}
+}
+\xtc{
+A differential polynomial is {\em isobaric} if the weights of all
+differential monomials appearing in it are equal.
+}{
+\spadpaste{isobaric?(g) \free{g}}
+}
+\xtc{
+To substitute {\em differentially}, use
+\spadfunFrom{eval}{OrderlyDifferentialPolynomial}.
+Note that we must coerce \spad{'w} to \spadtype{Symbol}, since in
+\spadtype{ODPOL}, differential indeterminates belong to the domain
+\spadtype{Symbol}.
+Compare this result to the next, which substitutes {\em algebraically} (no
+substitution is done since \spad{w.0} does not appear in \spad{g}).
+}{
+\spadpaste{eval(g,['w::Symbol],[f]) \free{f}\free{g}}
+}
+\xtc{
+}{
+\spadpaste{eval(g,variables(w.0),[f]) \free{f}\free{g}}
+}
+\xtc{
+Since \spadtype{OrderlyDifferentialPolynomial} belongs to
+\spadtype{PolynomialCategory}, all the operations defined in the latter
+category, or in packages for the latter category, are available.
+}{
+\spadpaste{monomials(g) \free{g}}
+}
+\xtc{
+}{
+\spadpaste{variables(g) \free{g}}
+}
+\xtc{
+}{
+\spadpaste{gcd(f,g) \free{f}\free{g}}
+}
+\xtc{
+}{
+\spadpaste{groebner([f,g]) \free{f}\free{g}}
+}
+\xtc{
+The next three operations are essential for elimination procedures in
+differential polynomial rings.
+The operation \spadfunFrom{leader}{OrderlyDifferentialPolynomial} returns
+the leader of a differential polynomial, which is the highest ranked
+derivative of the differential indeterminates that occurs.
+}{
+\spadpaste{lg:=leader(g)  \free{g}\bound{lg}}
+}
+\xtc{
+The operation \spadfunFrom{separant}{OrderlyDifferentialPolynomial} returns
+the separant of a differential polynomial, which is the partial derivative
+with respect to the leader.
+}{
+\spadpaste{sg:=separant(g)  \free{g}\bound{sg}}
+}
+\xtc{
+The operation \spadfunFrom{initial}{OrderlyDifferentialPolynomial} returns
+the initial, which is the leading coefficient when the given differential
+polynomial is expressed as a polynomial in the leader.
+}{
+\spadpaste{ig:=initial(g)  \free{g}\bound{ig}}
+}
+\xtc{
+Using these three operations, it is possible to reduce \spad{f} modulo the
+differential ideal generated by \spad{g}.
+The general scheme is to first reduce the order, then reduce the degree in
+the leader.
+First, eliminate \spad{z.3} using the derivative of \spad{g}.
+}{
+\spadpaste{g1 := D g \free{g}\bound{g1}}
+}
+\xtc{
+Find its leader.
+}{
+\spadpaste{lg1:= leader g1 \free{g1}\bound{lg1}}
+}
+\xtc{
+Differentiate \spad{f} partially with respect to this leader.
+}{
+\spadpaste{pdf:=D(f, lg1) \free{f}\free{lg1}\bound{pdf}}
+}
+\xtc{
+Compute the partial remainder of \spad{f} with respect to \spad{g}.
+}{
+\spadpaste{prf:=sg * f- pdf * g1 \free{f}\free{sg}\free{pdf}\free{g1}\bound{prf}}
+}
+\xtc{
+Note that high powers of \spad{lg} still appear in \spad{prf}.
+Compute the leading coefficient of \spad{prf}
+as a polynomial in the leader of \spad{g}.
+}{
+\spadpaste{lcf:=leadingCoefficient univariate(prf, lg) \free{prf}\free{lg}\bound{lcf}}
+}
+\xtc{
+Finally, continue eliminating the high powers of \spad{lg} appearing in
+\spad{prf} to obtain the (pseudo) remainder of \spad{f} modulo \spad{g}
+and its derivatives.
+}{
+\spadpaste{ig * prf - lcf * g * lg \free{ig}\free{prf}\free{lcf}\free{g}\free{lg}}
+}
+\showBlurb{OrderlyDifferentialPolyomial}
+\showBlurb{SequentialDifferentialPolynomial}
+\endscroll
+\autobuttons
+\end{page}
+%