CommonPolynomials: Common Polynomials

In polynomial: Handling Univariate Polynomials

Description

Determines the polynomial coefficients of some common n-th order polynomials.

Usage

Abel(n, a = 1)

Bessel(n)
rBessel(n)

Chebyshev1(n)
Chebyshev2(n)
ChebyshevT(n)
ChebyshevU(n)

Cyclotomic(n)

Hermite(n)

Arguments

n	a non-negative integer; the order of the polynomial. For Cyclotomic, n must be positive.
a	a real or complex number; most commonly a positive integer.

Details

The n-th order Abel polynomial is defined as

p[n](x) = x (x - a n)^{n - 1}

The n-th order Bessel polynomial (y[n]) and reverse Bessel polynomial (θ[n]) are defined as

y[n](x) = ∑ (n + k)!/((n - k)! k!) (x/2)^k for k = 0, …, n

θ[n](x) = x^n y[n](1/x) = ∑ (n + k)!/((n - k)! k!) x^(n - k)/2^k for k = 0, …, n

The n-th order Chebyshev polynomials of the first kind (T[n]) and the second kind (U[n]) are defined as

T[n](cos(θ)) = cos(n θ)

U[n](cos(θ)) sin(θ) = sin((n + 1) θ)

where upon defining x = cos(θ) and using trig identities to replace cos(n θ) and sin((n + 1) θ), we find two polynomials in terms of x.

The n-th order Cyclotomic polynomial is defined as

Φ[n](x) = ∏ (x - e^(2 i π k/n)) for k = 1, …, n and k is coprime with n

The n-th order Hermite polynomial is defined as

He(x) = (-1)^n e^(x^2/2) d^n/dx^n e^(-x^2/2)

Value

For Cyclotomic, a numeric vector of length sum(coprime(1:n, n)) + 1; the coefficients of the n-th order Cyclotomic polynomial.

For all others, a numeric vector of length n + 1; the coefficients of the n-th order polynomial.

Examples

colours <- grDevices::palette.colors()[c("vermillion", "bluishgreen",
    "blue", "skyblue", "reddishpurple", "orange")]


## Abel polynomials
n <- 0:4
plot(
    xlim = c(-3, 6), ylim = c(-20, 30),
    panel.first = graphics::grid(col = "gray69"),
    x = NA_real_, y = NA_real_,
    xlab = "x", ylab = as.call(list(call("[", as.symbol("p"),
        as.symbol("n")), as.symbol("x"))),
    main = "Abel Polynomials"
)
for (i in seq_along(n)) {
    graphics::lines(as.polynomial(Abel(n[[i]])),
        col = colours[[i]], n = 1001, lwd = 2,
        xlim = graphics::par("usr")[1:2])
}
graphics::box()
graphics::legend(
    x        = "bottomright",
    legend   = as.expression(lapply(X = n, FUN = function(n1) {
        call("==", as.symbol("n"), n1)
    })),
    col      = colours[seq_along(n)],
    lwd      = 2,
    bty      = "n"
)
graphics::title(sub = ~"All with" ~ list(a == 1), adj = 1)


## Bessel polynomials
n <- 0:5
plot(
    xlim = c(-3, 6), ylim = c(-10, 20),
    panel.first = graphics::grid(col = "gray69"),
    x = NA_real_, y = NA_real_,
    xlab = "x", ylab = as.call(list(call("[", as.symbol("y"),
        as.symbol("n")), as.symbol("x"))),
    main = "Bessel Polynomials"
)
for (i in seq_along(n)) {
    graphics::lines(as.polynomial(Bessel(n[[i]])),
        col = colours[[i]], n = 1001, lwd = 2,
        xlim = graphics::par("usr")[1:2])
}
graphics::box()
graphics::legend(
    x        = "bottomright",
    legend   = as.expression(lapply(X = n, FUN = function(n1) {
        call("==", as.symbol("n"), n1)
    })),
    col      = colours[seq_along(n)],
    lwd      = 2,
    bty      = "n"
)


## Reverse Bessel polynomials
n <- 0:5
plot(
    xlim = c(-10, 10), ylim = c(-10, 20),
    panel.first = graphics::grid(col = "gray69"),
    x = NA_real_, y = NA_real_,
    xlab = "x", ylab = as.call(list(call("[", as.symbol("theta"),
        as.symbol("n")), as.symbol("x"))),
    main = "Reverse Bessel Polynomials"
)
for (i in seq_along(n)) {
    graphics::lines(as.polynomial(rBessel(n[[i]])),
        col = colours[[i]], n = 1001, lwd = 2,
        xlim = graphics::par("usr")[1:2])
}
graphics::box()
graphics::legend(
    x        = "bottomright",
    legend   = as.expression(lapply(X = n, FUN = function(n1) {
        call("==", as.symbol("n"), n1)
    })),
    col      = colours[seq_along(n)],
    lwd      = 2,
    bty      = "n"
)


## Chebyshev polynomials of the first kind
n <- 0:4
plot(
    xlim = c(-1, 1), ylim = c(-1, 1),
    panel.first = graphics::grid(col = "gray69"),
    x = NA_real_, y = NA_real_,
    xlab = "x", ylab = as.call(list(call("[", as.symbol("T"),
        as.symbol("n")), as.symbol("x"))),
    main = "Chebyshev polynomials of the first kind"
)
for (i in seq_along(n)) {
    graphics::lines(as.polynomial(Chebyshev1(n[[i]])),
        col = colours[[i]], n = 1001, lwd = 2,
        xlim = graphics::par("usr")[1:2]
    )
}
graphics::box()
graphics::legend(
    x        = "bottomright",
    legend   = as.expression(lapply(X = n, FUN = function(n1) {
        call("==", as.symbol("n"), n1)
    })),
    col      = colours[seq_along(n)],
    lwd      = 2,
    bty      = "n"
)


## Chebyshev polynomials of the second kind
n <- 0:4
plot(
    xlim = c(-1, 1), ylim = c(-4, 5),
    panel.first = graphics::grid(col = "gray69"),
    x = NA_real_, y = NA_real_,
    xlab = "x", ylab = as.call(list(call("[", as.symbol("U"),
        as.symbol("n")), as.symbol("x"))),
    main = "Chebyshev polynomials of the second kind"
)
for (i in seq_along(n)) {
    graphics::lines(as.polynomial(Chebyshev2(n[[i]])),
        col = colours[[i]], n = 1001, lwd = 2,
        xlim = graphics::par("usr")[1:2]
    )
}
graphics::box()
graphics::legend(
    x        = "bottomright",
    legend   = as.expression(lapply(X = n, FUN = function(n1) {
        call("==", as.symbol("n"), n1)
    })),
    col      = colours[seq_along(n)],
    lwd      = 2,
    bty      = "n"
)


## Cyclotomic polynomials
n <- 1:5
plot(
    xlim = c(-3, 3), ylim = c(-10, 10),
    panel.first = graphics::grid(col = "gray69"),
    x = NA_real_, y = NA_real_,
    xlab = "x", ylab = as.call(list(call("[", as.symbol("Phi"),
        as.symbol("n")), as.symbol("x"))),
    main = "Cyclotomic Polynomials"
)
for (i in seq_along(n)) {
    graphics::lines(as.polynomial(Cyclotomic(n[[i]])),
        col = colours[[i]], n = 1001, lwd = 2,
        xlim = graphics::par("usr")[1:2]
    )
}
graphics::box()
graphics::legend(
    x        = "bottomright",
    legend   = as.expression(lapply(X = n, FUN = function(n1) {
        call("==", as.symbol("n"), n1)
    })),
    col      = colours[seq_along(n)],
    lwd      = 2,
    bty      = "n"
)


## Hermite polynomials
n <- 0:5
plot(
    xlim = c(-3, 6), ylim = c(-10, 20),
    panel.first = graphics::grid(col = "gray69"),
    x = NA_real_, y = NA_real_,
    xlab = "x", ylab = as.call(list(call("[", as.symbol("H"),
        as.symbol("n")), as.symbol("x"))),
    main = "Hermite Polynomials"
)
for (i in seq_along(n)) {
    graphics::lines(as.polynomial(Hermite(n[[i]])),
        col = colours[[i]], n = 1001, lwd = 2,
        xlim = graphics::par("usr")[1:2]
    )
}
graphics::box()
graphics::legend(
    x        = "bottomright",
    legend   = as.expression(lapply(X = n, FUN = function(n1) {
        call("==", as.symbol("n"), n1)
    })),
    col      = colours[seq_along(n)],
    lwd      = 2,
    bty      = "n"
)

polynomial documentation built on Feb. 22, 2021, 5:08 p.m.