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author | dos-reis <gdr@axiomatics.org> | 2007-08-14 05:14:52 +0000 |
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committer | dos-reis <gdr@axiomatics.org> | 2007-08-14 05:14:52 +0000 |
commit | ab8cc85adde879fb963c94d15675783f2cf4b183 (patch) | |
tree | c202482327f474583b750b2c45dedfc4e4312b1d /src/hyper/pages/LODO.ht | |
download | open-axiom-ab8cc85adde879fb963c94d15675783f2cf4b183.tar.gz |
Initial population.
Diffstat (limited to 'src/hyper/pages/LODO.ht')
-rw-r--r-- | src/hyper/pages/LODO.ht | 118 |
1 files changed, 118 insertions, 0 deletions
diff --git a/src/hyper/pages/LODO.ht b/src/hyper/pages/LODO.ht new file mode 100644 index 00000000..02174a75 --- /dev/null +++ b/src/hyper/pages/LODO.ht @@ -0,0 +1,118 @@ +% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved. +% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk. +\newcommand{\LinearOrdinaryDifferentialOperatorXmpTitle}{LinearOrdinaryDifferentialOperator} +\newcommand{\LinearOrdinaryDifferentialOperatorXmpNumber}{9.44} +% +% ===================================================================== +\begin{page}{LinearOrdinaryDifferentialOperatorXmpPage}{9.44 LinearOrdinaryDifferentialOperator} +% ===================================================================== +\beginscroll + +\spadtype{LinearOrdinaryDifferentialOperator(A, diff)} is the domain of linear +%-% \HDindex{operator!linear ordinary differential}{LinearOrdinaryDifferentialOperatorXmpPage}{9.44}{LinearOrdinaryDifferentialOperator} +ordinary differential operators with coefficients in a ring +\spad{A} with a given derivation. +%This includes the cases of operators which are polynomials in \spad{D} +%acting upon scalar or vector expressions of a single variable. +%The coefficients of the operator polynomials can be integers, rational +%functions, matrices or elements of other domains. +\showBlurb{LinearOrdinaryDifferentialOperator} + +\beginmenu + \menudownlink{{9.44.1. Differential Operators with Series Coefficients}}{ugxLinearOrdinaryDifferentialOperatorSeriesPage} +\endmenu +\endscroll +\autobuttons +\end{page} +% +% +\newcommand{\ugxLinearOrdinaryDifferentialOperatorSeriesTitle}{Differential Operators with Series Coefficients} +\newcommand{\ugxLinearOrdinaryDifferentialOperatorSeriesNumber}{9.44.1.} +% +% ===================================================================== +\begin{page}{ugxLinearOrdinaryDifferentialOperatorSeriesPage}{9.44.1. Differential Operators with Series Coefficients} +% ===================================================================== +\beginscroll + +\noindent +{\bf Problem:} +Find the first few coefficients of \spad{exp(x)/x**i} of \spad{Dop phi} where +\begin{verbatim} +Dop := D**3 + G/x**2 * D + H/x**3 - 1 +phi := sum(s[i]*exp(x)/x**i, i = 0..) +\end{verbatim} + +\noindent +{\bf Solution:} +\xtc{ +Define the differential. +}{ +\spadpaste{Dx: LODO(EXPR INT, f +-> D(f, x)) \bound{Dxd}} +} +\xtc{ +}{ +\spadpaste{Dx := D() \free{Dxd}\bound{Dx}} +} +\xtc{ +Now define the differential operator \spad{Dop}. +}{ +\spadpaste{Dop:= Dx**3 + G/x**2*Dx + H/x**3 - 1 \free{Dx}\bound{Dop}} +} +\xtc{ +}{ +\spadpaste{n == 3 \bound{n3}} +} +\xtc{ +}{ +\spadpaste{phi == reduce(+,[subscript(s,[i])*exp(x)/x**i for i in 0..n]) \bound{phi}} +} +\xtc{ +}{ +\spadpaste{phi1 == Dop(phi) / exp x \bound{phi1}\free{Dop phi}} +} +\xtc{ +}{ +\spadpaste{phi2 == phi1 *x**(n+3) \bound{phi2}\free{phi1}} +} +\xtc{ +}{ +\spadpaste{phi3 == retract(phi2)@(POLY INT) \bound{phi3}\free{phi2}} +} +\xtc{ +}{ +\spadpaste{pans == phi3 ::UP(x,POLY INT) \free{phi3}\bound{pans}} +} +\xtc{ +}{ +\spadpaste{pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1] \bound{pans1}\free{pans}} +} +\xtc{ +}{ +\spadpaste{leq == solve(pans1,[subscript(s,[i]) for i in 1..n]) \bound{leq}\free{pans1}} +} +\xtc{ +Evaluate this for several values of \spad{n}. +}{ +\spadpaste{leq \free{n3 leq}} +} +\xtc{ +}{ +\spadpaste{n==4 \bound{n4}} +} +\xtc{ +}{ +\spadpaste{leq \free{n4 leq}} +} +\xtc{ +}{ +\spadpaste{n==7 \bound{n7}} +} +\xtc{ +}{ +\spadpaste{leq \free{n7 leq}} +} + +\endscroll +\autobuttons +\end{page} +% |