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MultivariatePowerSeries

 ApproximatelyZero
 Determine equality with zero up to some precision

 Calling Sequence ApproximatelyZero(p, deg) ApproximatelyZero(u, deg)

Parameters

 p - power series generated by this package u - univariate polynomial over power series generated by this package deg - (optional) the precision up to which to compare

Description

 • Two power series p and q are said to be equal up to a degree deg, called the precision, if for each degree d <= deg, the homogeneous components of p and q of degree d are equal.
 • The command ApproximatelyZero(p) returns true if the power series is equal to zero up to its currently computed precision, otherwise false.
 • The command ApproximatelyZero(p,deg) returns true,  if the power series is equal to zero up to precision deg, otherwise false.
 • The command ApproximatelyZero(u) returns true if each coefficient of the univariate polynomial over power series is equal to zero up to its currently computed precision, otherwise false.
 • The command ApproximatelyZero(u,deg) returns true if each coefficient of the univariate polynomial over power series is equal to zero up to precision deg, otherwise false.
 • 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 two power series and compute their difference.

 > $a≔\mathrm{Inverse}\left(\mathrm{PowerSeries}\left(1-x-y\right)\right)$
 ${a}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{:}{1}{+}{\dots }\right]$ (1)
 > $b≔\mathrm{GeometricSeries}\left(\left[x,y\right]\right)$
 ${b}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{:}{1}{+}{x}{+}{y}{+}{\dots }\right]$ (2)
 > $c≔a-b$
 ${c}{≔}\left[{PowⅇrSⅇriⅇs:}{0}\right]$ (3)

The power series $a$ is known to have nonzero terms.

 > $\mathrm{ApproximatelyZero}\left(a\right)$
 ${\mathrm{false}}$ (4)

In order to test whether $c$ has any nonzero terms of homogeneous degree 10 or less, we can issue the following command. We see that $a$ and $b$ are the same up to homogeneous degree 10 (and indeed, from the analytic expression we can see that they are exactly equal).

 > $\mathrm{ApproximatelyZero}\left(c,10\right)$
 ${\mathrm{true}}$ (5)

Because $c$ is defined as a difference, we can test the same thing using the ApproximatelyEqual command.

 > $\mathrm{ApproximatelyEqual}\left(a,b,10\right)$
 ${\mathrm{true}}$ (6)

The following two univariate polynomials over power series are exactly equal to the polynomial $z\left(1+x+y\right)$. Hence, their difference is equal to 0 at any precision.

 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(0\right),\mathrm{Inverse}\left(\mathrm{GeometricSeries}\left(\left[x,y\right]\right)\right)\right],z\right)$
 ${f}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}\right){+}\left({1}{+}{\dots }\right){}{z}\right]$ (7)
 > $g≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(0\right),\mathrm{PowerSeries}\left(1-x-y\right)\right],z\right)$
 ${g}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}\right){+}\left({1}{-}{x}{-}{y}\right){}{z}\right]$ (8)
 > $h≔f-g$
 ${h}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}\right){+}\left({0}\right){}{z}\right]$ (9)
 > $\mathrm{ApproximatelyZero}\left(h\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{ApproximatelyZero}\left(h,10\right)$
 ${\mathrm{true}}$ (11)

Compatibility

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