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MultivariatePowerSeries

 Truncate
 Truncate a power series or univariate polynomial over power series

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

Parameters

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

Description

 • The command Truncate(p, d) returns the sum of all homogeneous parts of p of degree at most d. In other words, Truncate(p, d) returns  the natural image of the power series p modulo  the ${\left(d+1\right)}^{\mathrm{th}}$ power of the maximal ideal of the ring of power series.
 • If d is greater than the current precision of p, then the necessary extra terms are computed.
 • If d is not specified, all currently computed terms are used and no extra ones are computed. In other words, the default value of d is the precision of p.
 • The command  Truncate(u, d) returns the polynomial obtained from u by replacing each coefficient p with the sum of the homogeneous parts of p of degree at most d. In other words, each coefficient is replaced by its image modulo the ${\left(d+1\right)}^{\mathrm{th}}$ power of the maximal ideal of the ring of power series.
 • If d is not specified, the precision of each coefficient is used. If the coefficients are currently known to different precisions, they will consequently be truncated at different degrees.
 • 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 define the geometric power series in the variables $x$ and $y$.

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

It is initially computed only to low precision.

 > $\mathrm{Truncate}\left(a\right)$
 ${1}{+}{x}{+}{y}$ (2)

If we update its precision, then the Truncate command returns more terms.

 > $\mathrm{UpdatePrecision}\left(a,3\right)$
 $\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{:}{1}{+}{x}{+}{y}{+}{{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{{x}}^{{3}}{+}{3}{}{{x}}^{{2}}{}{y}{+}{3}{}{x}{}{{y}}^{{2}}{+}{{y}}^{{3}}{+}{\dots }\right]$ (3)
 > $\mathrm{Truncate}\left(a\right)$
 ${{x}}^{{3}}{+}{3}{}{{x}}^{{2}}{}{y}{+}{3}{}{x}{}{{y}}^{{2}}{+}{{y}}^{{3}}{+}{{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{x}{+}{y}{+}{1}$ (4)

We can get lower precision by specifying the truncation degree.

 > $\mathrm{Truncate}\left(a,2\right)$
 ${{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{x}{+}{y}{+}{1}$ (5)

We define a univariate polynomial over power series involving $a$.

 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{GeometricSeries}\left(x\right),\mathrm{GeometricSeries}\left(y\right),a\right],z\right)$
 ${f}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}{+}{x}{+}{\dots }\right){+}\left({1}{+}{y}{+}{\dots }\right){}{z}{+}\left({1}{+}{x}{+}{y}{+}{{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{{x}}^{{3}}{+}{3}{}{{x}}^{{2}}{}{y}{+}{3}{}{x}{}{{y}}^{{2}}{+}{{y}}^{{3}}{+}{\dots }\right){}{{z}}^{{2}}\right]$ (6)

The constant and linear coefficients of $z$ are known to a different precision than the quadratic coefficient. By default, the Truncate command will return all known coefficients regardless of degree.

 > $\mathrm{Truncate}\left(f\right)$
 $\left({{x}}^{{3}}{+}{3}{}{{x}}^{{2}}{}{y}{+}{3}{}{x}{}{{y}}^{{2}}{+}{{y}}^{{3}}{+}{{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{x}{+}{y}{+}{1}\right){}{{z}}^{{2}}{+}\left({1}{+}{y}\right){}{z}{+}{1}{+}{x}$ (7)

If we specify the truncation degree as 2, then more terms of the constant and linear coefficient are computed and some terms of the quadratic coefficient are omitted.

 > $\mathrm{Truncate}\left(f,2\right)$
 $\left({{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{x}{+}{y}{+}{1}\right){}{{z}}^{{2}}{+}\left({{y}}^{{2}}{+}{y}{+}{1}\right){}{z}{+}{{x}}^{{2}}{+}{x}{+}{1}$ (8)

Compatibility

 • The MultivariatePowerSeries[Truncate] command was introduced in Maple 2021.
 • For more information on Maple 2021 changes, see Updates in Maple 2021.

 See Also