coeff
Coefficients of a polynomial
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coeff(p
, <All>) coeff(p
, <x
>,n
, <All>) coeff(p
, <[x, …]
>,[n, …]
, <All>) coeff(f
, <vars
>, <All>) coeff(f
, <vars
>, <x
>,n
, <All>) coeff(f
, <vars
>, <[x, …]
>,[n, …]
, <All>)
coeff(p)
returns a sequence of all nonzero
coefficients of the polynomial p
.
coeff(p, x, n)
regards p
as
a univariate polynomial in x
and returns the coefficient
of the term x^n
.
coeff(p, [x,...], [n,...])
regards p
as
a multivariate polynomial in x,...
and returns
the coefficient of the term x^n,...
.
If the first argument f
is not element of
a polynomial domain, then coeff
converts the expression
internally to a polynomial of type DOM_POLY
via poly
(f)
. If a list
of indeterminates is specified, the polynomial poly
(f, vars)
is considered.
Coefficients of polynomial expressions f
are
returned as arithmetical
expressions.
There are various ways to call coeff
with
a polynomial p
of type DOM_POLY
:
coeff(p)
returns a sequence of
all nonzero coefficients of p
. They are ordered
according to the lexicographical term ordering. The order in descending.
The returned coefficients are elements of the coefficient ring
of p
.
coeff(p, x, n)
regards p
as
a univariate polynomial in the variable x
and returns
the coefficient of the term x^n
.
For univariate polynomials, the returned coefficients are elements
of the coefficient ring of p
.
For multivariate polynomials, the coefficients are returned
as polynomials of type DOM_POLY
in the “remaining”
variables.
coeff(p, n)
is equivalent to coeff(p,
x, n)
, where x
is the “main variable”
of p
. This variable is the first element of the
list of indeterminates op(p, 2)
.
coeff(p, [x1,x2,...], [n1,n2,...])
regards p
as
a multivariate polynomial in the variables x1,x2,...
and
returns the coefficient of the term x1^n1*x2^n2...
.
Variable and exponent lists must have the same length.
The returned coefficients are either elements of the coefficient
ring of p
or polynomials of type DOM_POLY
in the “remaining”
variables.
coeff(p, [n1,n2,...])
is equivalent
to coeff(p, [x1,x2,...], [n1,n2,...])
, where the
variables x1,x2,...
are the “main variables”
of p
, i.e., the leading elements of the list of
indeterminates op(p, 2)
.
coeff(p, All)
returns a sequence
of coefficients of p
including those equal to zero.
The function returns the result in ascending lexicographical order.
For univariate polynomial p
, the call coeff(p,
All)
is equivalent to coeff(p, i) $ i = 0 .. degree(p)
.
coeff
returns 0 or
a zero polynomial if the polynomial does not contain a term corresponding
to the specified powers. In particular, this happens for a univariate
polynomial if n
is larger than the degree of the
polynomial.
coeff
returns FAIL
if
an expression cannot be regarded as a polynomial.
The result of coeff
is not fully evaluated.
Evaluation can be enforced by the function eval
. See Example 5.
coeff(f)
returns a sequence of all nonzero
coefficients:
f := 10*x^10 + 5*x^5 + 2*x^2: coeff(f)
coeff(f, i)
returns a single coefficient:
coeff(f, i) $ i = 0..15
delete f:
We demonstrate how the indeterminates influence the result:
f := 3*x^3 + x^2*y^2 + 17*x + 23*y + 2
coeff(f); coeff(f, [x, y]); coeff(f, [y, x])
delete f:
The coefficients of f
are selected with respect
to the main variable x
which is the first entry
of the list of indeterminates:
f := 3*x^3 + x^2*y^2 + 2: coeff(f, [x, y], i) $ i = 0..3
The coefficients of f
can be selected with
respect to another main variable (in this case, y
):
coeff(f, [y, x], i) $ i = 0..2
Alternatively:
coeff(f, y, i) $ i = 0..2
The coefficients of f
can also be selected
with respect to a multivariate term:
coeff(f, [x,y], [3,0]), coeff(f, [x,y], [2,2]), coeff(f, [x,y], [0,0])
delete f:
In the same way, coeff
can be applied to
polynomials of type DOM_POLY
:
p := poly(3*x^3 + x, [x], Dom::IntegerMod(7)): coeff(p)
coeff(p, i) $ i = 0..3
For multivariate polynomials, the coefficients with respect to an indeterminate are polynomials in the other indeterminates:
p := poly(3*x^3 + x^2*y^2 + 2, [x, y]):
coeff(p, y, 0), coeff(p, y, 1), coeff(p, y, 2);
coeff(p, x, 0), coeff(p, x, 1), coeff(p, x, 2)
Note that the indeterminates passed to coeff
will
be used, even if the polynomial provides different indeterminates
:
coeff(p, z, 0), coeff(p, z, 1), coeff(p, z, 2)
delete p:
The result of coeff
is not fully evaluated:
p := poly(27*x^2 + a*x, [x]): a := 5: coeff(p, x, 1), eval(coeff(p, x, 1))
delete p, a:
To return all coefficients of a polynomial, use the All
option:
p := poly(a*x^3 + b*x^2 + c*x + d, [x, y]): coefficients := coeff(p, All)
To revert the order of the resulting sequence, use the revert
function. This
function does not operate on sequences. To convert a sequence to a
list, call revert
for
this list, and convert the result back to a sequence:
op(revert([coefficients]))
The All
option also works for polynomial
expressions:
p_expr := 2*x^5 + 5*x^2 + 10*x + 3: coeff(p_expr, All)
You can use the coeff
function with the All
option
to compute scalar products of polynomials. For example, the following
procedure computes a scalar product of two polynomials in an orthonormal
basis. The coeff
function extracts the coefficients
of the polynomials and returns two lists of coefficients. The zip
function multiplies
the entries of these lists pairwise and returns a list. The op
function accesses
the entries of that list. Finally, the _plus
function computes
the sum of all products:
scalarProduct := proc(p, q) local lp, lq; begin lp := [coeff(p, All)]; lq := [coeff(conjugate(q), All)]; _plus(op(zip(lp, lq, _mult))); end_proc:
The following polynomials are orthogonal:
scalarProduct(poly(x^2 + 2), poly(x^3 + 2*x^2  1))
coeff(p, All)
also works for multivariate
polynomials and polynomial expressions:
p := poly(2*x^2*y + PI*x + y^2  2, [x, y]): coeff(p, All)
For a multivariate polynomial or polynomial expression, the
order in which coeff
returns the coefficients is
such that the coefficient of the exponent vector [e_{1}, e_{2},
…] appears at position e_{1} d_{1} + e_{2} d_{2} +
… + 1. For example, represent the coefficients
returned for bivariate polynomial as a matrix:
A := matrix(degree(p, x) + 1, degree(p, y) + 1, [coeff(p, All)])

A polynomial of
type 

An indeterminate 

A power: a nonnegative integer 
 

A list of indeterminates of the polynomial: typically, identifiers or indexed identifiers 

The 
One or more coefficients of the coefficient
ring of the polynomial, or a polynomial, or FAIL
.
f
, p