The SampleSolutions command computes all real solutions of the system

when the parameters are evaluated at the given point.

Solutions are returned as a list of lists of equations of the form variable=number, or variable=[number,number] when the output=interval option is specified.

The point can be specified in three different formats:

as a list s of equations of the form parameter=rational number,

as a list p of rational numbers, in which case the $i$th parameter in m:-Parameters gets replaced by $p_i$ for all $i$, or

as a cell index k, in which case the point is taken to be the $k$th sample point in m:-SamplePoints.

Any optional arguments are passed directly to RootFinding[Isolate].

This command is part of the RootFinding[Parametric] package, so it can be used in the form SampleSolutions(..) only after executing the command with(RootFinding[Parametric]). However, it can always be accessed through the long form of the command by using RootFinding[Parametric][SampleSolutions](..).

$\mathrm{with}\left(\mathrm{RootFinding}\left[\mathrm{Parametric}\right]\right)\:$

$m≔\mathrm{CellDecomposition}\left(\left[x^2+y^2=a\&comma;x-y=b\&comma;0<a\right]\&comma;\left[x\&comma;y\right]\right)\&colon;$

$m:-\mathrm{SamplePoints}$

$\left[\left[a=\frac{302231454903657293676531}{1208925819614629174706176}\&comma;b=\mathrm{-1}\right]\&comma;\left[a=1\&comma;b=\mathrm{-1}\right]\&comma;\left[a=\frac{302231454903657293676531}{1208925819614629174706176}\&comma;b=1\right]\&comma;\left[a=1\&comma;b=1\right]\right]$

$\mathrm{SampleSolutions}\left(m\&comma;\left[a=1\&comma;b=-1\right]\right)$

$\left[\left[x=\mathrm{-1.}\&comma;y=0.\right]\&comma;\left[x=0.\&comma;y=1.\right]\right]$

$\mathrm{SampleSolutions}\left(m\&comma;\left[1\&comma;-1\right]\right)$

$\left[\left[x=\mathrm{-1.}\&comma;y=0.\right]\&comma;\left[x=0.\&comma;y=1.\right]\right]$

$\mathrm{SampleSolutions}\left(m\&comma;2\right)$

$\left[\left[x=\mathrm{-1.}\&comma;y=0.\right]\&comma;\left[x=0.\&comma;y=1.\right]\right]$

$\mathrm{SampleSolutions}\left(m\&comma;\left[a=1\&comma;b=-1\right]\&comma;\mathrm{output}=\mathrm{interval}\right)$

$\left[\left[x=\left[\mathrm{-1}\&comma;\mathrm{-1}\right]\&comma;y=\left[0\&comma;0\right]\right]\&comma;\left[x=\left[0\&comma;0\right]\&comma;y=\left[1\&comma;1\right]\right]\right]$

$\mathrm{SampleSolutions}\left(m\&comma;\left[1\&comma;\frac{1}{2}\right]\&comma;\mathrm{digits}=15\right)$

$\left[\left[x=\mathrm{-0.411437827766148}\&comma;y=\mathrm{-0.911437827766148}\right]\&comma;\left[x=0.911437827766148\&comma;y=0.411437827766148\right]\right]$