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DifferentialAlgebra[Tools]

 Coeffs
 extracts the coefficients of a rational differential fraction

 Calling Sequence Coeffs(q,v,R,opts)

Parameters

 q - a rational differential fraction v - a variable R - a differential polynomial ring or ideal opts (optional) - a sequence of options

Options

 • The opts arguments may contain one or more of the options below.
 • notation = jet, tjet, diff or Diff. Specifies the notation used for the result of the function call. If not specified, the notation of the input rational differential fraction q is used.
 • memout = nonnegative. Specifies a memory limit, in MB, for the computation. Default is zero (no memory out).

Description

 • The function call Coeffs(q,v,R) decomposes q as a sum of coefficients multiplied by monomials. The subexpressions of q which only involve derivatives greater than or equal to v appear among the monomials. The ones which only involve derivatives lower than v appear among the coefficients. The variable v is regarded as a derivative of R, or a derivative of its embedding polynomial ring, if R is an ideal.
 • The denominator of q appears either in each coefficient, if all the derivatives it depends on are lower than v, or, in each monomial, if all the derivatives it depends on, are greater than or equal to v. The function does not apply if the denominator of q involves derivatives lower than v and derivatives greater than or equal to v.
 • The function call Coeffs(q,v,R) returns a sequence of two lists. The first list is the list of coefficients. The second one is the list of monomials (in the same order). Monomials are sorted by decreasing rank of their numerators.
 • This command is part of the DifferentialAlgebra:-Tools package. It can be called using the form Coeffs(...) after executing the command with(DifferentialAlgebra:-Tools). It can also be directly called using the form DifferentialAlgebra[Tools][Coeffs](...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right):$$\mathrm{with}\left(\mathrm{Tools}\right):$
 > $R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{blocks}=\left[\left[\mathrm{dv},\mathrm{du}\right],\left[v,u\right]\right]\right)$
 ${R}{:=}{\mathrm{differential_ring}}$ (1)

The denominator only depends on derivatives greater than or equal to u. Thus it appears among the monomials.

 > $\mathrm{Tools}:-\mathrm{Coeffs}\left(\frac{1\left({u}_{x}{u}_{y}u-{u}_{x}{u}_{y}\right)}{4u},u,R\right)$
 $\left[\frac{{1}}{{4}}{,}{-}\frac{{1}}{{4}}\right]{,}\left[{{u}}_{{x}}{}{{u}}_{{y}}{,}\frac{{{u}}_{{x}}{}{{u}}_{{y}}}{{u}}\right]$ (2)

All the derivatives are less than ${u}_{x,x}$

 > $\mathrm{Tools}:-\mathrm{Coeffs}\left(\frac{1\left({u}_{x}{u}_{y}u-{u}_{x}{u}_{y}\right)}{4u},{u}_{x,x},R\right)$
 $\left[\frac{{1}}{{4}}{}\frac{{u}{}{{u}}_{{x}}{}{{u}}_{{y}}{-}{{u}}_{{x}}{}{{u}}_{{y}}}{{u}}\right]{,}\left[{1}\right]$ (3)
 > $\mathrm{nf}≔\frac{1\left({\mathrm{du}}_{y,y}{u}_{x}{u}^{2}-{\mathrm{du}}_{y,y}{u}_{x}u-{\mathrm{du}}_{y}{u}_{x}{u}_{y}u+{\mathrm{du}}_{y}{u}_{x}{u}_{y}-\mathrm{du}{u}_{x}\right)}{4{u}^{2}}$
 ${\mathrm{nf}}{:=}\frac{{1}}{{4}}{}\frac{{{u}}^{{2}}{}{{\mathrm{du}}}_{{y}{,}{y}}{}{{u}}_{{x}}{-}{u}{}{{\mathrm{du}}}_{{y}}{}{{u}}_{{x}}{}{{u}}_{{y}}{-}{u}{}{{\mathrm{du}}}_{{y}{,}{y}}{}{{u}}_{{x}}{+}{{\mathrm{du}}}_{{y}}{}{{u}}_{{x}}{}{{u}}_{{y}}{-}{\mathrm{du}}{}{{u}}_{{x}}}{{{u}}^{{2}}}$ (4)

A differential of the differential rational fraction is considered. Its coefficients are extracted.

 > $\mathrm{Tools}:-\mathrm{Coeffs}\left(\mathrm{nf},\mathrm{du},R\right)$
 $\left[\frac{{1}}{{4}}{}\frac{{u}{}{{u}}_{{x}}{-}{{u}}_{{x}}}{{u}}{,}\frac{{1}}{{4}}{}\frac{{-}{u}{}{{u}}_{{x}}{}{{u}}_{{y}}{+}{{u}}_{{x}}{}{{u}}_{{y}}}{{{u}}^{{2}}}{,}{-}\frac{{1}}{{4}}{}\frac{{{u}}_{{x}}}{{{u}}^{{2}}}\right]{,}\left[{{\mathrm{du}}}_{{y}{,}{y}}{,}{{\mathrm{du}}}_{{y}}{,}{\mathrm{du}}\right]$ (5)