mapcoeffs -- apply a function
to the coefficients of a polynomial
Introductionmapcoeffs(p, F, a1, a2...) applies the
function F to the polynomial p by replacing
each coefficient c in p by F(c, a1,
a2...).
Call(s)mapcoeffs(p, F <, a1, a2...>)
mapcoeffs(f, <vars, > F <, a1, a2...>)
Parametersp |
- | a polynomial of type
DOM_POLY |
F |
- | a procedure |
a1, a2... |
- | additional parameters for the function
F |
f |
- | a polynomial expression |
vars |
- | a list of indeterminates of the polynomial: typically, identifiers or indexed identifiers |
Returnsa polynomial of type DOM_POLY, or a polynomial
expression, or FAIL.
p, f
Related
Functionscoeff, degree, degreevec, lcoeff, ldegree, lterm, map, nterms, nthcoeff, nthmonomial, nthterm, poly, tcoeff
Detailsp of type DOM_POLY generated by poly, the function F
must accept arguments from the coefficient ring of p and
must produce corresponding results.f is first converted 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. After
applying the function F, the result is converted to an
expression.mapcoeffs evaluates its arguments. Note, however, that
polynomials of type DOM_POLY do not evaluate their
coefficients for efficiency reasons. Cf. example 4.mapcoeffs is a function of the system kernel.
Example
1The function sin is mapped to the coefficients of a
polynomial expression in the indeterminates x and
y:
>> mapcoeffs(3*x^3 + x^2*y^2 + 2, sin)
3 2 2
sin(2) + x sin(3) + x y sin(1)
The following call makes mapcoeffs regard
this expression as a polynomial in x. Consequently,
y is regarded as a parameter that becomes part of the
coefficients:
>> mapcoeffs(3*x^3 + x^2*y^2 + 2, [x], sin)
3 2 2
sin(2) + x sin(3) + x sin(y )
The system function _plus adds its arguments. In the
following call, it is used to add 2 to all coefficients by
providing this shift as an additional argument:
>> mapcoeffs(c1*x^3 + c2*x^2*y^2 + c3, [x, y], _plus, 2)
3 2 2
c3 + x (c1 + 2) + x y (c2 + 2) + 2
Example
2The function sin is mapped to the coefficients of a
polynomial in the indeterminates x and y:
>> mapcoeffs(poly(3*x^3 + x^2*y^2 + 2, [x, y]), sin)
3 2 2
poly(sin(3) x + sin(1) x y + sin(2), [x, y])
In the following call, the polynomial has the
indeterminate x. Consequently, y is regarded
as a parameter that becomes part of the coefficients:
>> mapcoeffs(poly(3*x^3 + x^2*y^2 + 2, [x]), sin)
3 2 2
poly(sin(3) x + sin(y ) x + sin(2), [x])
A user-defined function is mapped to a polynomial:
>> F := (c, a1, a2) -> exp(c + a1 + a2): mapcoeffs(poly(x^3 + c*x, [x]), F, a1, a2)
3
poly(exp(a1 + a2 + 1) x + exp(c + a1 + a2) x, [x])
>> delete F:
Example
3We consider a polynomial over the integers modulo 7:
>> p := poly(x^3 + 2*x*y, [x, y], Dom::IntegerMod(7)):
A function to be applied to the coefficients must produce values in the coefficient ring of the polynomial:
>> mapcoeffs(p, c -> c^2)
3
poly(x + 4 x y, [x, y], Dom::IntegerMod(7))
The following call maps a function which converts its
argument to an integer modulo 3. Such a return value is not
a valid coefficient of p:
>> mapcoeffs(p, c -> Dom::IntegerMod(3)(expr(c)))
FAIL
>> delete p:
Example
4Note that polynomials of type DOM_POLY do not evaluate their
arguments:
>> delete a, x: p := poly(a*x, [x]): a := PI: p
poly(a x, [x])
Evaluation can be enforced by the function eval:
>> mapcoeffs(p, eval)
poly(PI x, [x])
We map the sine function to the
coefficients of p. The polynomial does not evaluate its
coefficient sin(a) to 0:
>> mapcoeffs(p, sin)
poly(sin(a) x, [x])
The composition of sin and eval is mapped to the coefficients of
the polynomial:
>> mapcoeffs(p, eval@sin)
poly(0, [x])
>> delete p, a: