Polynomials

A set of polynomials can be regarded as the elements of a vector space. As an example we consider the set of all real polynomials of degree $d$,

\[P_α(x) = α_0 + α_1 x + ⋯ + α_d x^d.\]

These are maps $P_α:\mathbb{\mathbb{R\rightarrow\mathbb{R}}}$ that satisfy the group operation 'addition of polynomials' because the sum of two polynomials of degree $d$ is again a polynomial of degree $d$,

\[(P_α + P_β)(x) ≡ (α_0 + β_{0})+(α_1 + β_1)x + ⋯ + (α_d + β_d) x^d = P_α(x) + P_β(x),\]

and remains a polynomial of degree $d$ under 'scalar multiplication',

\[(λ P_α)(x) ≡ P_{\lambdaα}(x)=\lambdaα_0+\lambdaα_1 x + ⋯ + \lambdaα_d x^d = λ P_α(x).\]

The 'zero element' of the vector space is the polynomial $P_α(x)=0$ and the 'inverse element' of the element $P_α(x)$ is the polynomial $-P_α(x)$. Also the 'associative' and 'distributive' properties are easily verified.

Hence, the set of all real polynomials of degree $d$ defines a vector space (of dimension $d + 1$) over the field $\mathbb{R}$ and the polynomials $1,x,x^{2},\cdots x^d$ represent a basis. The coefficients $α_0, ⋯, α_d$, represent the coordinates of the vector $P_α$ with respect to this basis.

In CamiMath we define a polynomial by specifying the coordinate vector polynom.

polynom

CamiMath.polynom — Type

polynom

The set of coordinates defining the vector or tuple representation of a polynomial.

Example:

julia> polynom = (1, 1, 1, 1, 1) 
(1, 1, 1, 1, 1)

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polynom power

CamiMath.polynom_power — Method

polynom_power(polynom, p)

The polynom of a polynomial of degree $d$ raised to the power p, which define a polynomial in a vector space of dimension $p d + 1$.

Examples:

julia> polynom = (1,1,1)    # coordinates of polynomial vector of degree d = 2
(1, 1, 1)

julia> polynom = (1,1,1);

julia> polynom_power(polynom, 3)
7-element Vector{Int64}:
 1
 3
 6
 7
 6
 3
 1

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polynom product

CamiMath.polynom_product — Method

polynom_product(polynom1, polynom2)

The coefficients of the product of two polynomials, a = polynom1 and b = polynom2 of degree $m$ and $n$, which represents a polynomial in a vector space of dimension $d=m+n+1$,

\[ P(x)=a_0b_0 + (a_0b_1 + b_0a_1)x + ⋯ + a_n b_m x^{n+m}.\]

Examples:

julia> polynom_product((1, 1), (1, -1, 2))
4-element Vector{Int64}:
 1
 0
 1
 2

julia> polynom_product((1, 1), (1, -1.0, 2))
4-element Vector{Real}:
 1
 0.0
 1.0
 2

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polynom product expansion

CamiMath.polynom_product_expansion — Method

polynom_product_expansion(polynom1, polynom2, p::Int)

The coefficients of the product of two polynomials, a = polynom1 (of degree $n$) and b = polynom2 (of degree $m$), truncated at the order p, which represents a polynomial in a vector space of dimension $d=p+1$

Examples:

julia> a = (1,-1,1);

julia> b = (1,1,-1,1,1,1);

julia> o = polynom_product(a, b); println(o)
[1, 0, -1, 3, -1, 1, 0, 1]
 
julia> o = polynom_product_expansion(a, b, 4); println(o)
[1, 0, -1, 3, -1]

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polynomial

CamiMath.polynomial — Method

polynomial(polynom, x::T [; deriv=0]) where T<:Real

Polynomial of degree $d$,

\[ P(x)=c_0 + c_1 x + ⋯ + c_d x^d,\]

where the coefficients polynom = $(c_0,⋯\ c_d)$ define the vector representation of the polynomial in a vector space of dimension $d+1$.

Examples:

julia> polynom = (1, 1, 1, 1, 1);
           
julia> P(x) = polynomial(polynom,x);

julia> P(1)
5

julia> polynomial(polynom, 1; deriv=1)     # P′(1)
10

julia> polynomial(polynom, 2; deriv=2)     # P″(1)
20

julia> polynomial(polynom, 1; deriv=-1)   # primitive (zero integration constant)
137 // 60

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