ROOTS WITH MULTIPLICITY
|
Author : | Jean Michel Ferrard |
Syntax : | MZEROS ( expr , var ) |
Require : | ****** |
It is said that
is a root of the polynomial "P" if P(
)=0.
It is said that
is root of P with the multiplicity m if (X-
)m
divides P, but not (X-
)^(m+
).
One speaks about simple' root if m=1, double if m=2, triple if m=3, ... ...
. One speaks of multiple root if m>1.
It is shown that the multiplicity m of a root ""
of P is characterized by:
P()=p
' (
)=p "
(
)=... =P^(m-
)
(
)=0, and P(m)^(
)/=0
(the multiplicity appears as the order of the first derivative which is not
cancelled at
).
The integrated functions "zeros()" and "czeros()" only give
the different zero from a polynomial. The function "mzeros()" corrects
this problem while returning the list with all the zeros.
Syntax is mzeros(P, x), where P is an polynomial expression (and not a list! compared to the variable "X" (x or another name of variable).
Exemple :
![]() |
Here : P(x)=(x-1)^3*(x-)^2*(x+1) : roots
(and multiplicities) are obvious. |