Negate - Maple Help

MultivariatePowerSeries

 Negate
 negate a power series or a Puiseux series or a univariate polynomial over power series or over Puiseux series

 Calling Sequence -p Negate(p) -s Negate(s) -u Negate(u)

Parameters

 p - power series generated by this package s - Puiseux series generated by this package u - univariate polynomial over power series or over Puiseux series generated by this package

Description

 • The commands -p and Negate(p) return the additive inverse of the power series p.
 • The commands -s and Negate(s) return the additive inverse of the Puiseux series s.
 • The commands -u and Negate(u) return the additive inverse of the univariate polynomial over power series u.
 • When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series, Puiseux series, and univariate polynomials over these series. If you do, you may see invalid results.

Examples

 > $\mathrm{with}\left(\mathrm{MultivariatePowerSeries}\right):$

We create a power series and form its additive inverse in two ways.

 > $a≔\mathrm{GeometricSeries}\left(\left[x,y,z\right]\right)$
 ${a}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}{-}{z}}{:}{1}{+}{x}{+}{y}{+}{z}{+}{\dots }\right]$ (1)
 > $b≔-a$
 ${b}{≔}\left[{PowⅇrSⅇriⅇs of}{-}\frac{{1}}{{1}{-}{x}{-}{y}{-}{z}}{:}{-1}{+}{\dots }\right]$ (2)
 > $c≔\mathrm{Negate}\left(a\right)$
 ${c}{≔}\left[{PowⅇrSⅇriⅇs of}{-}\frac{{1}}{{1}{-}{x}{-}{y}{-}{z}}{:}{-1}{-}{x}{-}{y}{-}{z}{+}{\dots }\right]$ (3)

We verify that the results are the same up to homogeneous degree 10.

 > $\mathrm{ApproximatelyEqual}\left(b,c,10\right)$
 ${\mathrm{true}}$ (4)

We create a univariate polynomial over power series and form its additive inverse.

 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left({z}^{3}-{z}^{2}x-2xy,z\right)$
 ${f}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({-}{2}{}{x}{}{y}\right){+}\left({0}\right){}{z}{+}\left({-}{x}\right){}{{z}}^{{2}}{+}\left({1}\right){}{{z}}^{{3}}\right]$ (5)
 > $g≔-f$
 ${g}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}{+}{\dots }\right){+}\left({0}\right){}{z}{+}\left({0}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({-1}\right){}{{z}}^{{3}}\right]$ (6)

The additive inverse of $g$ should be equal to $f$.

 > $\mathrm{ApproximatelyEqual}\left(f,\mathrm{Negate}\left(g\right),10\right)$
 ${\mathrm{true}}$ (7)

Now we define a Puiseux series s and compute its additive inverse.

 > $s≔\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y},\left[x=x{y}^{\frac{1}{2}},y=x{y}^{-1}\right]\right)$
 ${s}{≔}\left[{PuisⅇuxSⅇriⅇs of}\frac{{x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}}{{1}{+}{x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}}{:}{0}{+}{\dots }\right]$ (8)
 > $\mathrm{ns}≔\mathrm{Negate}\left(s\right)$
 ${\mathrm{ns}}{≔}\left[{PuisⅇuxSⅇriⅇs of}{-}\frac{{x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}}{{1}{+}{x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}}{:}{0}{+}{\dots }\right]$ (9)
 > $\mathrm{Truncate}\left(s+\mathrm{ns},20\right)$
 ${0}$ (10)

Finally, we create a univariate polynomial over power series from a list of Puiseux series.

 > $h≔\mathrm{UnivariatePolynomialOverPuiseuxSeries}\left(\left[\mathrm{PuiseuxSeries}\left(1\right),\mathrm{PuiseuxSeries}\left(0\right),\mathrm{PuiseuxSeries}\left(x,\left[x={x}^{\frac{1}{3}}\right]\right),\mathrm{PuiseuxSeries}\left(y,\left[y={y}^{\frac{1}{2}}\right]\right),\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y},\left[x=x{y}^{\frac{1}{2}},y=x{y}^{-1}\right]\right)\right],z\right)$
 ${h}{≔}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}\right){+}\left({0}\right){}{z}{+}\left({{x}}^{{1}}{{3}}}\right){}{{z}}^{{2}}{+}\left(\sqrt{{y}}\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}\right]$ (11)

We compute -h.

 > $\mathrm{nh}≔\mathrm{Negate}\left(h\right)$
 ${\mathrm{nh}}{≔}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({-1}\right){+}\left({0}\right){}{z}{+}\left({-}{{x}}^{{1}}{{3}}}\right){}{{z}}^{{2}}{+}\left({-}\sqrt{{y}}\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}\right]$ (12)
 > $\mathrm{Truncate}\left(h+\mathrm{nh},20\right)$
 ${0}$ (13)

Compatibility

 • The MultivariatePowerSeries[Negate] command was introduced in Maple 2021.