evalp
Evaluate a polynomial at a point
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evalp(p
,x = v, …
) evalp(p
,[x = v, …]
) evalp(f
, <vars
>,x = v, …
) evalp(f
, <vars
>,[x = v, …]
)
evalp(p, x = v)
evaluates the polynomial p
in
the variable x
at the point v
.
An error occurs if x
is not an indeterminate
of p
. The value v
may be any
object that could also be used as coefficient. The result is an element
of the coefficient ring of p
if p
is
univariate. If p
is multivariate, the result is
a polynomial in the remaining variables.
If several evaluation points are given, the evaluations take place in succession from left to right. Each evaluation follows the rules above.
For a polynomial p
in the variables x1,x2,...
,
the syntax p(v1,v2,...)
can be used instead of evalp(p,
x1 = v1, x2 = v2, ...)
.
evalp(f, vars, x = v, ...)
first converts
the polynomial expression f
to a polynomial with
the variables given by vars
. If no variables are
given, they are searched for in f
. See poly
about details of
the conversion. FAIL
is returned if f
cannot
be converted to a polynomial. A successfully converted polynomial
is evaluated as above. The result is converted to an expression.
Horner's rule is used to evaluate the polynomial. The evaluation
of variables at the point 0
is most efficient and
should take place first. After that, the remaining main variable should
be evaluated first.
The result of evalp
is not evaluated further.
One may use eval
to
fully evaluate the result.
Instead of evalp(p, x1 = v1, x2 = v2, ...)
one
may also use the equivalent form evalp(p, [x1 = v1, x2 =
v2, ...])
.
evalp
is used to evaluate the polynomial
expression x^{2} +
2 x + 3 at the point x = a +
2. The form of the resulting expression reflects
the fact that Horner's rule was used:
evalp(x^2 + 2*x + 3, x = a + 2)
evalp
is used to evaluate a polynomial in
the indeterminates x and y at
the point x = 3.
The result is a polynomial in the remaining indeterminate y:
p := poly(x^2 + x*y + 2, [x, y]): evalp(p, x = 3)
delete p:
Polynomials may be called like functions in order to evaluate all variables:
p := poly(x^2 + x*y, [x, y]): evalp(p, x = 3, y = 2) = p(3, 2)
delete p:
If not all variables are replaced by values, the result is a polynomial in the remaining variables:
evalp(poly(x*y*z + x^2 + y^2 + z^2, [x, y, z]), x = 1, y = 1)
The result of evalp
is not evaluated further.
We first define a polynomial p
with coefficient a
and
then change the value of a
. The change is not reflected
by p
, because polynomials do not evaluate their
coefficients implicitly. One must map the function eval
onto the coefficients
in order to enforce evaluation:
p := poly(x^2 + a*y + 1, [x,y]): a := 2: p, mapcoeffs(p, eval)
If we use evalp
to evaluate p
at
the point x = 1,
the result is not fully evaluated. One must use eval
to get fully evaluated coefficients:
r := evalp(p, x = 1): r, mapcoeffs(r, eval)
delete p, a, r:

A polynomial of
type 

An indeterminate 

The value for 
 

A list of indeterminates of the polynomial: typically, identifiers or indexed identifiers 
Element of the coefficient ring, or a polynomial, or a polynomial
expression, or FAIL
f
, p