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last edited 10 years ago by hemmecke |
Edit detail for SparseMultivariateTaylorSeries revision 1 of 3
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Editor: hemmecke
Time: 2014/08/22 18:22:23 GMT+0 |
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changed: - 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. \begin{axiom} Z==>Integer Q==>Fraction Z vl: List Symbol := [x,y] V==>OrderedVariableList vl E ==> DirectProduct(2, NonNegativeInteger) P==>SparseMultivariatePolynomial(Q, V) M==>SparseMultivariateTaylorSeries(Q,V,P) X:=monomial(1$M,x,1) Y:=monomial(1$M,y,1) sinh(X)*cosh(Y) \end{axiom}
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.
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Z==>Integer
Type: Void
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Q==>Fraction Z
Type: Void
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vl: List Symbol := [x,y]
![\label{eq1}\left[ x , \: y \right]](images/3908585630146559634-16.0px.png "\label{eq1}\left[ x , \: y \right]") | (1) |
Type: List(Symbol)
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V==>OrderedVariableList vl
Type: Void
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E ==> DirectProduct(2,NonNegativeInteger)
Type: Void
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P==>SparseMultivariatePolynomial(Q,V)
Type: Void
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M==>SparseMultivariateTaylorSeries(Q,V, P)
Type: Void
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X:=monomial(1$M,x, 1)
 | (2) |
Type: SparseMultivariateTaylorSeries?(Fraction(Integer),
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Y:=monomial(1$M,y, 1)
 | (3) |
Type: SparseMultivariateTaylorSeries?(Fraction(Integer),
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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}](images/7647231367283329238-16.0px.png "\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) |