Documentation |
Create a polynomial
This functionality does not run in MATLAB.
poly(f, <[x_{1}, x_{2}, …]>, <ring>) poly(p, <[x_{1}, x_{2}, …]>, <ring>) poly(list, [x_{1}, x_{2}, …], <ring>) poly(coeffs, [x], <ring>)
poly(f) converts a polynomial expression f to a polynomial of the kernel domain DOM_POLY.
The kernel domain DOM_POLY represents polynomials. The arithmetic for this data structure is more efficient than the arithmetic for polynomial expressions. Moreover, this domain allows you to use special coefficient rings that cannot be represented by expressions. The function poly is the tool for generating polynomials of this type.
poly(f, [x1, x2, ...], ring) converts the expression f to a polynomial in the indeterminates x1, x2, ... over the specified coefficient ring. The poly function does not require an expanded form of the expression f. The function internally expands expressions.
If you do not specify indeterminates, MuPAD^{®} searches for them internally. If MuPAD cannot identify indeterminates, it returns FAIL.
By default, the poly function uses the coefficient ring of arbitrary MuPAD expressions. In this case, you can use arbitrary MuPAD expressions as coefficients.
If the poly function cannot convert an expression to a polynomial, the function returns FAIL. See Example 10.
If f is a domain element, the system calls f::dom::poly for the conversion into a polynomial. If f contains domain elements, the system recursively calls f::dom::poly for domain elements inside f. See Example 11.
poly(p, [x1, x2, ...], ring) converts a polynomial p of the type DOM_POLY to a polynomial in the indeterminates x1, x2, ... over the specified coefficient ring. The indeterminates and the coefficient ring are part of the data structure DOM_POLY. Using this function call, you can change the indeterminates and the coefficient ring of a polynomial.
If you do not specify indeterminates, poly uses the indeterminates of the original polynomial p.
If you do not specify a coefficient ring, poly uses the ring of the original polynomial p.
poly(list, [x]) converts a list of coefficients [a_{0}, a_{1}, a_{2}, …] to a univariate polynomial a_{0} + a_{1} x + a_{2} x^{2} + …. See Example 3.
For a univariate polynomial p, the call poly(list, [x]) converts the result of the call coeff(p, All) back to a polynomial.
poly(list, [x1, x2, ...], ring) converts a list of coefficients and exponents to a polynomial in the indeterminates x1, x2, ... over the specified coefficient ring. See Example 4 and Example 7.
This call is the fastest method to create polynomials of the type DOM_POLY because the input already has the form that MuPAD uses internally.
The list must contain an element for each nonzero monomial of the polynomial. Therefore, you must use sparse input involving only nonzero terms. In particular, an empty list results in the zero polynomial.
Each element of the list must be a list with two elements: the coefficient of the monomial and the exponent (or exponent vector). For a univariate polynomial in the variable x, the list
corresponds to c_{1} x^{e1} + c_{2} x^{e2} + …. For a multivariate polynomial, the exponent vectors are lists containing the exponents of all indeterminates of the polynomial. The order of the exponents must be the same as the order given by the list of indeterminates. For a multivariate polynomial in the variables x_{1}, x_{2}, the term list
corresponds to c_{1} x_{1}^{e11} x_{2}^{e12} + c_{2} x_{1}^{e21} x_{2}^{e22} + ….
The order of the elements of the term list does not affect the resulting polynomial. If you provide multiple entries corresponding to the same term, poly adds the coefficients.
This call lets you restore polynomials from the term lists returned by poly2list.
The position of the indeterminates in the input list [x1, x2, ...] determines their order in the resulting polynomial. If you do not specify indeterminates, MuPAD searches the expression f for possible indeterminates and determines their order. See Example 2.
You can perform arithmetical operations on polynomials that have the same indeterminates and the same coefficient ring. Also, you can perform arithmetical operations on polynomials and arithmetical expressions. When you operate on a polynomial and an arithmetical expression, MuPAD internally converts that arithmetical expression to a polynomial and performs the calculation. See Example 1.
The poly function does not limit acceptable indeterminates to identifiers or indexed identifiers. You can use any expression (except for rational expressions) as an indeterminate. For example, poly accepts the expressions sin(x) and f(x) as indeterminates. See Example 5.
After creating a polynomial, the poly function does not evaluate the coefficients of the polynomial. If the coefficients contain free identifiers, poly does not replace these identifiers with their values. See Example 12.
If any domain of type DOM_DOMAIN provides arithmetical operations, you can use that domain as a coefficient ring. See the "Background" section for details.
If you specify a coefficient domain, MuPAD accepts only the elements of that domain as coefficients of the polynomial. On input, poly tries to convert a polynomial expression f to a polynomial over the coefficient ring. For some coefficient rings, you cannot use arithmetical expressions to represent a polynomial. The reason is that multiplication with the indeterminates can be an invalid operation in the ring. In such cases, you can define the polynomial by using a term list. See Example 7.
The poly function creates a polynomial from a polynomial expression:
p := poly(2*x*(x + 3))
The operators *, +, - and ^ work on polynomials:
p^2 - p*(p + 1)
You can multiply a polynomial by an arithmetical expression. MuPAD internally converts the expression to a polynomial of the appropriate type, and then multiplies polynomials. For example, multiply the polynomial p by the constant 5:
p*5
Now, multiply the polynomial p by x - 1:
p*(x - 1)
If MuPAD cannot convert the expression to a polynomial of the appropriate type, the arithmetical operation between a polynomial and this expression fails:
p*(1/x - 1)
Error: The argument is invalid. [_mult]
delete p:
You can create a polynomial with parameters. In the following call, y is a parameter (not an indeterminate):
poly((x*(y + 1))^2, [x])
If you do not specify indeterminates, MuPAD tries to find indeterminates automatically. The following call converts a multivariate expression to a multivariate polynomial:
poly((x*(y + 1))^2)
Now, specify the order of the indeterminates explicitly:
poly((x*(y + 1))^2, [y, x])
Use the poly function to convert the following list of coefficients to a univariate polynomial in x. The first entry of the list produces the term with the zero exponent. The last entry produces the term with the highest exponent:
p := poly([1, 2, 3, 4, 5], [x])
To revert the ordering of the coefficients in a polynomial, use the revert function:
revert(p)
Create the following polynomials by term lists:
poly([[c2, 3], [c1, 7], [c3, 0]], [x])
If you provide multiple coefficients corresponding to the same exponent, poly adds those coefficients:
poly([[c2, 3], [c1, 7], [c3, 0], [a, 3]], [x])
For multivariate polynomials, specify exponent vectors by lists:
poly([[c1, [2, 2]], [c2, [2, 1]], [c3, [2, 0]]], [x, y])
You can use expressions as indeterminates:
poly(f(x)*(f(x) + x^2))
The residue class ring IntMod(7) is a valid coefficient ring:
p := poly(9*x^3 + 4*x - 7, [x], IntMod(7))
For computations that involve polynomials over this ring, MuPAD uses modular arithmetic:
p^3
However, MuPAD does not return coefficients as elements of a special domain. Instead, it returns coefficients as plain integers of the type DOM_INT:
coeff(p)
delete p:
To create the following polynomial, combine the input syntax that uses term lists with a specified coefficient ring:
poly([[9, 3], [4, 1], [-2, 0]], [x], IntMod(7))
MuPAD interprets the input coefficients as elements of the coefficient domain. For example, conversions such as 9 mod 7 = 2 mod 7 occur on input. You also can use the domain Dom::IntegerMod(7) to define an equivalent polynomial. If you use IntMod(7), MuPAD uses the symmetric modulo function mods and represents the coefficients by the numbers - 3, …, 3. If you use Dom::IntegerMod(7), MuPAD uses the positive modulo function modp and represents the coefficients by the numbers 0, …, 6:
poly([[9, 3], [4, 1], [-2, 0]], [x], Dom::IntegerMod(7))
The domain Dom::IntegerMod(7) does not allow multiplication with identifiers:
c := Dom::IntegerMod(7)(3)
poly(c*x^2, [x], Dom::IntegerMod(7))
Instead, use the term list to specify the polynomial:
poly([[c, 2]], [x], Dom::IntegerMod(7))
delete c:
Change the indeterminates in a polynomial:
p:= poly(((a + b)*x - a^2)*x, [x]): p, poly(p, [a, b])
Change the coefficient ring of a polynomial:
p := poly(-4*x + 5*y - 5, [x, y], IntMod(7)): p, poly(p, IntMod(3))
Create a polynomial over the coefficient ring Dom::Float:
poly(3*x - y, Dom::Float)
The identifier y cannot appear in coefficients from this ring because it cannot be converted to a floating-point number:
poly(3*x - y, [x], Dom::Float)
You can overload poly by its first operand. For example, create a domain polyInX that represents polynomials in x:
domain polyInX new := () -> new(dom, poly(args(), [x])); print := p -> expr(extop(p, 1)); poly := p -> if args(0) = 1 then print(Unquoted, "polyInX::poly called with 1 argument"); extop(p, 1); else print(Unquoted, "polyInX::poly called with more than 1 argument"); poly(extop(p, 1), args(2..args(0))); end; end_domain: p := polyInX(3*x^2-2)
You can convert the elements of polyInX into polynomials of the type DOM_POLY. The poly function calls the poly method of the domain:
poly(p)
polyInX::poly called with 1 argument
By reacting to additional arguments, the overloading defined above also allows you to create polynomials over other coefficient rings:
poly(p, [x], IntMod(2))
polyInX::poly called with more than 1 argument
Create a polynomial with coefficients containing the identifier y. Although you assign the value 1 to y, MuPAD does not substitute the new value into the polynomial:
f := poly(x^2 - y, [x]): y := 1: eval(f)
You can evaluate the coefficients explicitly. Use the mapcoeffs function to apply eval to the coefficients of the polynomial:
f := mapcoeffs(f, eval)
f | |
x_{1}, x_{2}, … |
The indeterminates of the polynomial: typically, identifiers or indexed identifiers. |
ring |
The coefficient ring: either Expr, or IntMod(n) with some integer n greater than 1, or a domain of type DOM_DOMAIN. The default is the ring Expr of arbitrary MuPAD expressions. |
p |
A polynomial of type DOM_POLY generated by poly |
list |
A list containing coefficients and exponents |
coeffs |
A list containing coefficients of a univariate polynomial |
x |
The indeterminate of a univariate polynomial |
Expr |
The default ring Expr represents arbitrary MuPAD expressions. Mathematically, this ring coincides with Dom::ExpressionField(). However, MuPAD operates differently on the polynomials created over Expr and the polynomials created over Dom::ExpressionField(). In particular, MuPAD performs arithmetic operations for polynomials over the ring Expr faster. |
IntMod |
The ring IntMod(n) represents the residue class ring ℤ_{n}, using the symmetrical representation. Here, n is an integer greater than 1. Mathematically, this ring coincides with Dom::IntegerMod(n). However, MuPAD operates differently on the polynomials created over IntMod(n) and the polynomials created over Dom::IntegerMod(n). In particular, MuPAD performs arithmetic operations for polynomials over the ring IntMod faster. Also, for polynomials over IntMod, coeff and similar functions return requested coefficients as integers of the type DOM_INT. See Example 6, Example 7, and Example 9. |
Polynomial of the domain type DOM_POLY. If conversion to a polynomial is not possible, the return value is FAIL.
To use a domain as a coefficient, the domain must contain the following:
The entry "zero" that provides the neutral element with respect to addition.
The entry "one" that provides the neutral element with respect to multiplication.
The method "_plus" that adds domain elements.
The method "_negate" that returns the inverse with respect to addition.
The method "_mult" that multiplies domain elements.
The method "_power" that computes integer powers of a domain element. Call this method with the domain element as the first argument and an integer as the second argument.
In addition, you must define the following methods. Functions (such as gcd, diff, divide, norm and so on) call these methods:
The method "gcd" that returns the greatest common divisor of domain elements.
The method "diff" that differentiates a domain element with respect to a variable.
The method "_divide" that divides two domain elements. It must return FAIL if division is not possible.
The method "norm" that computes the norm of a domain element and returns it as a number.
The method "convert" that converts an expression to a domain element. The method must return FAIL if such conversion is not possible.
The system calls this method to convert the coefficients of polynomial expressions to coefficients of the specified domain. If this method does not exist, you can specify the coefficients only by using domain elements.
The method "expr" that converts a domain element to an expression.
The system function expr calls this method to convert a polynomial over the coefficient domain to a polynomial expression. If this method does not exist, expr inserts domain elements into the expression.
You can convert a polynomial over a certain coefficient domain into a polynomial over the same domain, but a different set of indeterminates. This conversion is much more efficient when the domain has the axiom Ax::indetElements. MuPAD implicitly assumes that this axiom holds for the domain IntMod(n), but not for Expr.
Internally, MuPAD stores polynomials of the type DOM_POLY in a sparse representation and uses machine integers for the exponents. This method implies that in a 32-bit environment, the exponent of each variable in each monomial cannot exceed 2^{31} - 1.