# Operations on polynomials¶

This chapter contains commands to manipulate polynomials. This includes functions for constructing and evaluating orthogonal polynomials.

Expand(expr)
Expand(expr, var)
Expand(expr, varlist)

transform a polynomial to an expanded form

Param expr

a polynomial expression

Param var

a variable

Param varlist

a list of variables

This command brings a polynomial in expanded form, in which polynomials are represented in the form $$c_0 + c_1x + c_2x^2 + ... + c_nx^n$$. In this form, it is easier to test whether a polynomial is zero, namely by testing whether all coefficients are zero. If the polynomial {expr} contains only one variable, the first calling sequence can be used. Otherwise, the second form should be used which explicitly mentions that {expr} should be considered as a polynomial in the variable {var}. The third calling form can be used for multivariate polynomials. Firstly, the polynomial {expr} is expanded with respect to the first variable in {varlist}. Then the coefficients are all expanded with respect to the second variable, and so on.

Example

In> Expand((1+x)^5)
Out> x^5+5*x^4+10*x^3+10*x^2+5*x+1
In> Expand((1+x-y)^2, x);
Out> x^2+2*(1-y)*x+(1-y)^2
In> Expand((1+x-y)^2, {x,y})
Out> x^2+((-2)*y+2)*x+y^2-2*y+1

Degree(expr[, var])

degree of a polynomial

Param expr

a polynomial

Param var

a variable occurring in {expr}

This command returns the degree of the polynomial expr with respect to the variable var. If only one variable occurs in expr, the first calling sequence can be used. Otherwise the user should use the second form in which the variable is explicitly mentioned.

Example

In> Degree(x^5+x-1);
Out> 5;
In> Degree(a+b*x^3, a);
Out> 1;
In> Degree(a+b*x^3, x);
Out> 3;


Coef(expr, var, order)

coefficient of a polynomial

Param expr

a polynomial

Param var

a variable occurring in {expr}

Param order

integer or list of integers

This command returns the coefficient of {var} to the power {order} in the polynomial {expr}. The parameter {order} can also be a list of integers, in which case this function returns a list of coefficients.

Example

In> e := Expand((a+x)^4,x)
Out> x^4+4*a*x^3+(a^2+(2*a)^2+a^2)*x^2+
(a^2*2*a+2*a^3)*x+a^4;
In> Coef(e,a,2)
Out> 6*x^2;
In> Coef(e,a,0 .. 4)
Out> {x^4,4*x^3,6*x^2,4*x,1};

Content(expr)

content of a univariate polynomial

Param expr

univariate polynomial

This command determines the content of a univariate polynomial.

Example

In> poly := 2*x^2 + 4*x;
Out> 2*x^2+4*x;
In> c := Content(poly);
Out> 2*x;
In> pp := PrimitivePart(poly);
Out> x+2;
In> Expand(pp*c);
Out> 2*x^2+4*x;


PrimitivePart(expr)

primitive part of a univariate polynomial

Param expr

univariate polynomial

This command determines the primitive part of a univariate polynomial. The primitive part is what remains after the content is divided out. So the product of the content and the primitive part equals the original polynomial.

Example

In> poly := 2*x^2 + 4*x;
Out> 2*x^2+4*x;
In> c := Content(poly);
Out> 2*x;
In> pp := PrimitivePart(poly);
Out> x+2;
In> Expand(pp*c);
Out> 2*x^2+4*x;

LeadingCoef(poly)
LeadingCoef(poly, var)

This function returns the leading coefficient of poly, regarded as a polynomial in the variable var. The leading coefficient is the coefficient of the term of highest degree. If only one variable appears in the expression poly, it is obvious that it should be regarded as a polynomial in this variable and the first calling sequence may be used.

Example

In> poly := 2*x^2 + 4*x;
Out> 2*x^2+4*x;
Out> 2;
In> m := Monic(poly);
Out> x^2+2*x;
In> Expand(lc*m);
Out> 2*x^2+4*x;
In> LeadingCoef(2*a^2 + 3*a*b^2 + 5, a);
Out> 2;
In> LeadingCoef(2*a^2 + 3*a*b^2 + 5, b);
Out> 3*a;


Monic(poly)
Monic(poly, var)

monic part of a polynomial

This function returns the monic part of poly, regarded as a polynomial in the variable var. The monic part of a polynomial is the quotient of this polynomial by its leading coefficient. So the leading coefficient of the monic part is always one. If only one variable appears in the expression poly, it is obvious that it should be regarded as a polynomial in this variable and the first calling sequence may be used.

Example

In> poly := 2*x^2 + 4*x;
Out> 2*x^2+4*x;
Out> 2;
In> m := Monic(poly);
Out> x^2+2*x;
In> Expand(lc*m);
Out> 2*x^2+4*x;
In> Monic(2*a^2 + 3*a*b^2 + 5, a);
Out> a^2+(a*3*b^2)/2+5/2;
In> Monic(2*a^2 + 3*a*b^2 + 5, b);
Out> b^2+(2*a^2+5)/(3*a);

SquareFree(p)

return the square-free part of polynomial

Param p

a polynomial in {x}

Given a polynomial $$p = p_1^{n_1}\ldots p_m^{n_m}$$ with irreducible polynomials $$p_i$$, return the square-free version part (with all the factors having multiplicity 1): $$p_1\ldots p_m$$

Example

In> Expand((x+1)^5)
Out> x^5+5*x^4+10*x^3+10*x^2+5*x+1;
In> SquareFree(%)
Out> (x+1)/5;
In> Monic(%)
Out> x+1;

SquareFreeFactorize(p, x)

return square-free decomposition of polynomial

Param p

a polynomial in {x}

Given a polynomial $$p$$ having square-free decomposition $$p = p_1^{n_1}\ldots p_m^{n_m}$$ where $$p_i$$ are square-free and $$n_{i+1}>n_i$$, return the list of pairs ($$p_i$$, $$n_i$$)

Example

In> Expand((x+1)^5)
Out> x^5+5*x^4+10*x^3+10*x^2+5*x+1
In> SquareFreeFactorize(%,x)
Out> {{x+1,5}}

Horner(expr, var)

convert a polynomial into the Horner form

This command turns the polynomial expr, considered as a univariate polynomial in var, into the Horner form. A polynomial in normal form is an expression such as $$c_0 + c_1x + \ldots + c_nx^n$$. If one converts this polynomial into Horner form, one gets the equivalent expression $$(\ldots( c_nx + c_{n-1}) x + \ldots + c_1)x + c_0$$. Both expression are equal, but the latter form gives a more efficient way to evaluate the polynomial as the powers have disappeared.

Example

In> expr1:=Expand((1+x)^4)
Out> x^4+4*x^3+6*x^2+4*x+1;
In> Horner(expr1,x)
Out> (((x+4)*x+6)*x+4)*x+1;

ExpandBrackets(expr)

expand all brackets

This command tries to expand all the brackets by repeatedly using the distributive laws $$a(b+c) = ab + ac$$ and $$(a+b)c = ac + bc$$. It goes further than Expand(), in that it expands all brackets.

Example

In> Expand((a-x)*(b-x),x)
Out> x^2-(b+a)*x+a*b;
In> Expand((a-x)*(b-x),{x,a,b})
Out> x^2-(b+a)*x+b*a;
In> ExpandBrackets((a-x)*(b-x))
Out> a*b-x*b+x^2-a*x;

EvaluateHornerScheme(coeffs, x)

fast evaluation of polynomials

This function evaluates a polynomial given as a list of its coefficients, using the Horner scheme. The list of coefficients starts with the $$0$$-th power.

OrthoP(n, x)
OrthoP(n, a, b, x)

Legendre and Jacobi orthogonal polynomials

The first calling format with two arguments evaluates the Legendre polynomial of degree n at the point x. The second form does the same for the Jacobi polynomial with parameters a and b, which should be both greater than $$-1$$.

The Jacobi polynomials are orthogonal with respect to the weight function $$(1-x)^a(1+x)^b$$ on the interval $$[-1,1]$$. They satisfy the recurrence relation $$P(n,a,b,x) = \frac{2n+a+b-1}{2n+a+b-2}\frac{a^2-b^2+x(2n+a+b-2)(n+a+b)}{2n(n+a+b)}P(n-1,a,b,x) - \frac{(n+a-1)(n+b-1)(2n+a+b)}{n(n+a+b)(2n+a+b-2)}P(n-2,a,b,x)$$ for $$n > 1$$, with $$P(0,a,b,x) = 1$$, $$P(1,a,b,x) = \frac{a-b}{2}+x(1+\frac{a+b}{2})$$.

OrthoH(n, x)

Hermite orthogonal polynomials

This function evaluates the Hermite polynomial of degree n at the point x.

The Hermite polynomials are orthogonal with respect to the weight function $$\exp(\frac{-x^2}{2})$$ on the entire real axis. They satisfy the recurrence relation $$H(n,x) = 2xH(n-1,x) - 2(n-1)H(n-2,x)$$ for $$n > 1$$, with $$H(0,x) = 1$$, $$H(1,x) = 2x$$.

Example

In> OrthoH(3, x);
Out> x*(8*x^2-12);
In> OrthoH(6, 0.5);
Out> 31;

OrthoG(n, a, x)

Gegenbauer orthogonal polynomials

This function evaluates the Gegenbauer (or ultraspherical) polynomial with parameter a and degree n at the point x. The parameter a should be greater than $$-\frac{1}{2}$$.

The Gegenbauer polynomials are orthogonal with respect to the weight function $$(1-x^2)^{a-\frac{1}{2}}$$ on the interval $$[-1,1]$$. Hence they are connected to the Jacobi polynomials via $$G(n, a, x) = P(n, a-\frac{1}{2}, a-\frac{1}{2}, x)$$. They satisfy the recurrence relation $$G(n,a,x) = 2(1+\frac{a-1}{n})xG(n-1,a,x)-(1+2\frac{a-2}{n})G(n-2,a,x)$$ for $$n>1$$, with $$G(0,a,x) = 1$$, $$G(1,a,x) = 2x$$.

OrthoL(n, a, x)

Laguerre orthogonal polynomials

This function evaluates the Laguerre polynomial with parameter a and degree n at the point x. The parameter a should be greater than $$-1$$.

The Laguerre polynomials are orthogonal with respect to the weight function $$x^a\exp(-x)$$ on the positive real axis. They satisfy the recurrence relation $$L(n,a,x) = (2+\frac{a-1-x}{n})L(n-1,a,x) -(1-\frac{a-1}{n})L(n-2,a,x)$$ for $$n>1$$, with $$L(0,a,x) = 1$$, $$L(1,a,x) = a + 1 - x$$.

OrthoT(n, x)
OrthoU(n, x)

Chebyshev polynomials

These functions evaluate the Chebyshev polynomials of the first kind $$T(n,x)$$ and of the second kind $$U(n,x)$$, of degree n at the point x. (The name of this Russian mathematician is also sometimes spelled Tschebyscheff.)

The Chebyshev polynomials are orthogonal with respect to the weight function $$(1-x^2)^{-\frac{1}{2}}$$. Hence they are a special case of the Gegenbauer polynomials $$G(n,a,x)$$, with $$a=0$$. They satisfy the recurrence relations $$T(n,x) = 2xT(n-1,x) - T(n-2,x)$$, $$U(n,x) = 2xU(n-1,x) - U(n-2,x)$$ for $$n > 1$$, with $$T(0,x) = 1$$, $$T(1,x) = x$$, $$U(0,x) = 1$$, $$U(1,x) = 2x$$.

Example

In> OrthoT(3, x);
Out> 2*x*(2*x^2-1)-x;
In> OrthoT(10, 0.9);
Out> -0.2007474688;
In> OrthoU(3, x);
Out> 4*x*(2*x^2-1);
In> OrthoU(10, 0.9);
Out> -2.2234571776;

OrthoPSum(c, x)
OrthoPSum(c, a, b, x)
OrthoGSum(c, a, x)
OrthoHSum(c, x)
OrthoLSum(c, a, x)
OrthoTSum(c, x)
OrthoUSum(c, x)

sums of series of orthogonal polynomials

These functions evaluate the sum of series of orthogonal polynomials at the point x, with given list of coefficients c of the series and fixed polynomial parameters a, b (if applicable). The list of coefficients starts with the lowest order, so that for example OrthoLSum(c, a, x) = c L(a,x) + c L(a,x) + ... + c[N] L[N-1](a,x).

See pages for specific orthogonal polynomials for more details on the parameters of the polynomials. Most of the work is performed by the internal function OrthoPolySum(). The individual polynomials entering the series are not computed, only the sum of the series.

Example

In> Expand(OrthoPSum({1,0,0,1/7,1/8}, 3/2, 2/3, x));
Out> (7068985*x^4)/3981312+(1648577*x^3)/995328+
(-3502049*x^2)/4644864+(-4372969*x)/6967296
+28292143/27869184

OrthoPoly(name, n, params, x)

internal function for constructing orthogonal polynomials

This function is used internally to construct orthogonal polynomials. It returns the n-th polynomial from the family name with parameters params at the point x. All known families are stored in the association list returned by the function KnownOrthoPoly(). The name serves as key.

At the moment the following names are known to yacas: "Jacobi", "Gegenbauer", "Laguerre", "Hermite", "Tscheb1", and "Tscheb2". The value associated to the key is a pure function that takes two arguments: the order n and the extra parameters p, and returns a list of two lists: the first list contains the coefficients {A,B} of the n=1 polynomial, i.e. $$A+Bx$$; the second list contains the coefficients {A,B,C} in the recurrence relation, i.e. $$P_n = (A+Bx)P_{n-1}+CP_{n-2}$$.

Note

There are only 3 coefficients in the second list, because none of the considered polynomials use $$C+Dx$$ instead of $$C$$ in the recurrence relation. This is assumed in the implementation.

If the argument x is numerical, the function OrthoPolyNumeric() is called. Otherwise, the function OrthoPolyCoeffs() computes a list of coefficients, and EvaluateHornerScheme() converts this list into a polynomial expression.

OrthoPolySum(name, c, params, x)

internal function for computing series of orthogonal polynomials

This function is used internally to compute series of orthogonal polynomials. It is similar to the function OrthoPoly() and returns the result of the summation of series of polynomials from the family name with parameters params at the point x, where c is the list of coefficients of the series. The algorithm used to compute the series without first computing the individual polynomials is the Clenshaw-Smith recurrence scheme. (See the algorithms book for explanations.)

If the argument x is numerical, the function OrthoPolySumNumeric() is called. Otherwise, the function OrthoPolySumCoeffs() computes the list of coefficients of the resulting polynomial, and EvaluateHornerScheme() converts this list into a polynomial expression.