MultivariatePowerSeries/UpdatePrecision - Maple Help

MultivariatePowerSeries

 UpdatePrecision
 Update the precision to which a power series or univariate polynomial over power series is known

 Calling Sequence UpdatePrecision(p,d) UpdatePrecision(u,d)

Parameters

 p - power series generated by this package u - univariate polynomial over power series generated by this package d - non-negative integer

Description

 • The command UpdatePrecision(p,d) increases the precision of p so that this precision equals at least d. This is achieved by repeated calls to the generator of p.
 • The command UpdatePrecision(u,d) increases the precision of each power series coefficient of u so that this precision equals at least d.
 • When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series and univariate polynomials over power series. If you do, you may see invalid results.

Examples

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

We create a new power series object. Initially, it is known only to low precision.

 > $p≔\frac{1}{\mathrm{PowerSeries}\left(1+x\right)}$
 ${p}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{+}{x}}{:}{1}{+}{\dots }\right]$ (1)
 > $\mathrm{Precision}\left(p\right)$
 ${0}$ (2)

If we update the precision, more terms are known.

 > $\mathrm{UpdatePrecision}\left(p,10\right)$
 $\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{+}{x}}{:}{1}{-}{x}{+}{{x}}^{{2}}{-}{{x}}^{{3}}{+}{{x}}^{{4}}{-}{{x}}^{{5}}{+}{{x}}^{{6}}{-}{{x}}^{{7}}{+}{{x}}^{{8}}{-}{{x}}^{{9}}{+}{{x}}^{{10}}{+}{\dots }\right]$ (3)
 > $\mathrm{Precision}\left(p\right)$
 ${10}$ (4)

We create a new polynomial over power series involving the power series defined above, $p$, and a  new power series, $q$.

 > $q≔\mathrm{GeometricSeries}\left(\left[x,y\right]\right)$
 ${q}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{:}{1}{+}{x}{+}{y}{+}{\dots }\right]$ (5)
 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[q,p\right],z\right)$
 ${f}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}{+}{x}{+}{y}{+}{\dots }\right){+}\left({1}{-}{x}{+}{{x}}^{{2}}{-}{{x}}^{{3}}{+}{{x}}^{{4}}{-}{{x}}^{{5}}{+}{{x}}^{{6}}{-}{{x}}^{{7}}{+}{{x}}^{{8}}{-}{{x}}^{{9}}{+}{{x}}^{{10}}{+}{\dots }\right){}{z}\right]$ (6)

Because $q$ is new, it is again known only to low precision.

 > $\mathrm{Precision}\left(q\right)$
 ${1}$ (7)

Updating the precision of $f$ updates the precision of each of its coefficients, $p$ and $q$, if necessary. In this case, $p$ was already known to higher precision, but $q$'s precision is updated.

 > $\mathrm{UpdatePrecision}\left(f,8\right)$
 $\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}{+}{x}{+}{y}{+}{{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{{x}}^{{3}}{+}{3}{}{{x}}^{{2}}{}{y}{+}{3}{}{x}{}{{y}}^{{2}}{+}{{y}}^{{3}}{+}{{x}}^{{4}}{+}{4}{}{{x}}^{{3}}{}{y}{+}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{4}{}{x}{}{{y}}^{{3}}{+}{{y}}^{{4}}{+}{{x}}^{{5}}{+}{5}{}{{x}}^{{4}}{}{y}{+}{10}{}{{x}}^{{3}}{}{{y}}^{{2}}{+}{10}{}{{x}}^{{2}}{}{{y}}^{{3}}{+}{5}{}{x}{}{{y}}^{{4}}{+}{{y}}^{{5}}{+}{{x}}^{{6}}{+}{6}{}{{x}}^{{5}}{}{y}{+}{15}{}{{x}}^{{4}}{}{{y}}^{{2}}{+}{20}{}{{x}}^{{3}}{}{{y}}^{{3}}{+}{15}{}{{x}}^{{2}}{}{{y}}^{{4}}{+}{6}{}{x}{}{{y}}^{{5}}{+}{{y}}^{{6}}{+}{{x}}^{{7}}{+}{7}{}{{x}}^{{6}}{}{y}{+}{21}{}{{x}}^{{5}}{}{{y}}^{{2}}{+}{35}{}{{x}}^{{4}}{}{{y}}^{{3}}{+}{35}{}{{x}}^{{3}}{}{{y}}^{{4}}{+}{21}{}{{x}}^{{2}}{}{{y}}^{{5}}{+}{7}{}{x}{}{{y}}^{{6}}{+}{{y}}^{{7}}{+}{{x}}^{{8}}{+}{8}{}{{x}}^{{7}}{}{y}{+}{28}{}{{x}}^{{6}}{}{{y}}^{{2}}{+}{56}{}{{x}}^{{5}}{}{{y}}^{{3}}{+}{70}{}{{x}}^{{4}}{}{{y}}^{{4}}{+}{56}{}{{x}}^{{3}}{}{{y}}^{{5}}{+}{28}{}{{x}}^{{2}}{}{{y}}^{{6}}{+}{8}{}{x}{}{{y}}^{{7}}{+}{{y}}^{{8}}{+}{\dots }\right){+}\left({1}{-}{x}{+}{{x}}^{{2}}{-}{{x}}^{{3}}{+}{{x}}^{{4}}{+}{\dots }\right){}{z}\right]$ (8)
 > $\mathrm{Precision}\left(q\right)$
 ${8}$ (9)
 > $\mathrm{Precision}\left(p\right)$
 ${10}$ (10)

Compatibility

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