help links subscribe changes refresh edit

	ToDelete
SandBoxTest1 ←	↑	→ Test14

SparseMultivariateTaylorSeries

Edit detail for SparseMultivariateTaylorSeries revision 2 of 3

	1 2 3
Editor: Bill Page Time: 2014/08/22 21:19:20 GMT+0
Note: Naive solution

added:

From BillPage Fri Aug 22 21:19:19 +0000 2014
From: Bill Page
Date: Fri, 22 Aug 2014 21:19:19 +0000
Subject: Naive solution
Message-ID: <20140822211919+0000@axiom-wiki.newsynthesis.org>

This is not what I expected:
\begin{axiom}
x:=taylor 'x
y:=taylor 'y
sinh(x)*cosh(y)
\end{axiom}

Can it be converted somehow to the solution above?

Example for multivariate Taylor series expansion

In order to work with multivariate Taylor series one first has to do a few preparation steps in order to create an appropriate domain.

fricas

Z==>Integer

Type: Void

fricas

Q==>Fraction Z

Type: Void

fricas

vl: List Symbol := [x,y]

$\label{eq1}\left[ x , \: y \right]$

(1)

Type: List(Symbol)

fricas

V==>OrderedVariableList vl

Type: Void

fricas

E ==> DirectProduct(2, NonNegativeInteger)

Type: Void

fricas

P==>SparseMultivariatePolynomial(Q, V)

Type: Void

fricas

M==>SparseMultivariateTaylorSeries(Q,V,P)

Type: Void

fricas

X:=monomial(1$M,x,1)

$\label{eq2}x$

(2)

Type: SparseMultivariateTaylorSeries?(Fraction(Integer),OrderedVariableList?([x,y]),SparseMultivariatePolynomial?(Fraction(Integer),OrderedVariableList?([x,y])))

fricas

Y:=monomial(1$M,y,1)

$\label{eq3}y$

(3)

Type: SparseMultivariateTaylorSeries?(Fraction(Integer),OrderedVariableList?([x,y]),SparseMultivariatePolynomial?(Fraction(Integer),OrderedVariableList?([x,y])))

fricas

sinh(X)*cosh(Y)

$\label{eq4}\begin{array}{@{}l} \displaystyle x +{\left({{1 \over 6}\ {{x}^{3}}}+{{1 \over 2}\ {{y}^{2}}\ x}\right)}+{\left({{1 \over{120}}\ {{x}^{5}}}+{{1 \over{12}}\ {{y}^{2}}\ {{x}^{3}}}+{{1 \over{24}}\ {{y}^{4}}\ x}\right)}+ \ \ \displaystyle {\left({{1 \over{5040}}\ {{x}^{7}}}+{{1 \over{240}}\ {{y}^{2}}\ {{x}^{5}}}+{{1 \over{144}}\ {{y}^{4}}\ {{x}^{3}}}+{{1 \over{720}}\ {{y}^{6}}\ x}\right)}+ \ \ \displaystyle {\left({ \begin{array}{@{}l} \displaystyle {{1 \over{362880}}\ {{x}^{9}}}+{{1 \over{10080}}\ {{y}^{2}}\ {{x}^{7}}}+{{1 \over{2880}}\ {{y}^{4}}\ {{x}^{5}}}+ \ \ \displaystyle {{1 \over{4320}}\ {{y}^{6}}\ {{x}^{3}}}+{{1 \over{40320}}\ {{y}^{8}}\ x}$

(4)

Type: SparseMultivariateTaylorSeries?(Fraction(Integer),OrderedVariableList?([x,y]),SparseMultivariatePolynomial?(Fraction(Integer),OrderedVariableList?([x,y])))

Naive solution --Bill Page, Fri, 22 Aug 2014 21:19:19 +0000 reply

This is not what I expected:

fricas

x:=taylor 'x

$\label{eq5}x$

(5)

Type: UnivariateTaylorSeries?(Expression(Integer),x,0)

fricas

y:=taylor 'y

$\label{eq6}y$

(6)

Type: UnivariateTaylorSeries?(Expression(Integer),y,0)

fricas

sinh(x)*cosh(y)

$\label{eq7}\begin{array}{@{}l} \displaystyle x +{{1 \over 6}\ {{x}^{3}}}+{{1 \over{120}}\ {{x}^{5}}}+{{1 \over{5 040}}\ {{x}^{7}}}+{{1 \over{362880}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{1 \over 2}\ x}+{{1 \over{12}}\ {{x}^{3}}}+{{1 \over{240}}\ {{x}^{5}}}+{{1 \over{10080}}\ {{x}^{7}}}+{{1 \over{725760}}\ {{x}^{9}}}+ \ \ \displaystyle {O \left({{x}^{11}}\right)}$

(7)

Type: UnivariateTaylorSeries?(UnivariateTaylorSeries?(Expression(Integer),x,0),y,0)

Can it be converted somehow to the solution above?