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 ApproximatelyEqual(p,q) returns true if the two power series are equal up to the minimum of their currently computed precisions, otherwise false.

The command ApproximatelyEqual(p,q,deg) returns true if the two power series are equal up to precision deg, otherwise false. This calling sequence computes any coefficients needed that have not been computed so far.

The command ApproximatelyEqual(u,v) returns true if the two univariate polynomials over power series are equal up to the currently computed precision of each coefficient power series, otherwise false.

The command ApproximatelyEqual(u,v,deg) returns true if the two univariate polynomials over power series are equal by comparing each power series coefficient up to precision deg, otherwise false.

By default, if p and q know their analytic expressions and these analytic expressions are equal, the ApproximatelyEqual command will immediately return true. If you pass the force or force = true option, then the ApproximatelyEqual command will ignore the analytic expressions. The default behavior can be explicitly selected by passing the option force = false.

Any form of the force option passed to the ApproximatelyEqual command when comparing univariate polynomials over power series are used as described above when comparing the power series coefficients.

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.

$\mathrm{with}\left(\mathrm{MultivariatePowerSeries}\right)\&colon;$

$a≔\mathrm{Inverse}\left(\mathrm{PowerSeries}\left(1+x+y\right)\right)$

$a≔\left[\mathrm{Pow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s\; of}\frac{1}{1+x+y}\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(a\&comma;1\right)$

$1-x-y$

$b≔\mathrm{GeometricSeries}\left(\left[x\&comma;y\right]\right)$

$b≔\left[\mathrm{Pow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s\; of}\frac{1}{1-x-y}\mathrm{:}1+x+y+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(b\&comma;1\right)$

$1+x+y$

$c≔\mathrm{Inverse}\left(\mathrm{SumOfAllMonomials}\left(\left[x\&comma;y\right]\right)\right)$

$c≔\left[\mathrm{Pow\&ExponentialE;rS\&ExponentialE;ri\&ExponentialE;s\; of}\left(1-x\right)\left(1-y\right)\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(c\&comma;1\right)$

$1-x-y$

$\mathrm{ApproximatelyEqual}\left(a\&comma;b\right)$

$\mathrm{false}$

$\mathrm{ApproximatelyEqual}\left(a\&comma;c\right)$

$\mathrm{true}$

$\mathrm{ApproximatelyEqual}\left(a\&comma;c\&comma;2\right)$

$\mathrm{false}$

$\mathrm{Truncate}\left(a\right)$

$x^2+2xy+y^2-x-y+1$

$\mathrm{Truncate}\left(c\right)$

$xy-x-y+1$

$f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(0\right)\&comma;\mathrm{GeometricSeries}\left(\left[x\&comma;y\right]\right)\right]\&comma;z\right)\&colon;$

$g≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(0\right)\&comma;\mathrm{Inverse}\left(\mathrm{PowerSeries}\left(1-x-y\right)\right)\right]\&comma;z\right)\&colon;$

$\mathrm{ApproximatelyEqual}\left(f\&comma;g\right)$

$\mathrm{true}$

$\mathrm{ApproximatelyEqual}\left(f\&comma;g\&comma;10\right)$

$\mathrm{true}$

$\mathrm{ApproximatelyEqual}\left(f\&comma;g\&comma;10\&comma;\mathrm{force}\right)$

$\mathrm{true}$

The MultivariatePowerSeries[ApproximatelyEqual] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.

The MultivariatePowerSeries[ApproximatelyEqual] command was updated in Maple 2022.

The force option was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.