Greatest common divisor

Description

Find a monic polynomial of maximal degree that divides each of a set of polynomials exactly

Usage

GCD(...)

greatest_common_divisor(...)

## S3 method for class 'polynom'
GCD(...)

## S3 method for class 'polylist'
GCD(...)

Arguments

Value

A polynomial giving the greatest common divisor, as defined above

Examples

p <- poly_calc(0:5)
r <- poly_calc(1:6)
greatest_common_divisor(p, r)
solve(greatest_common_divisor(p, r))
lowest_common_multiple(p, r)
solve(lowest_common_multiple(p, r))

Summary and Math methods for polynomials

Description

These provide methods for the generic function Summary and Math for polynomial and polylist objects. For Summary only sum and prod members are implemented

Usage

## S3 method for class 'polynom'
Summary(..., na.rm = FALSE)

## S3 method for class 'polylist'
Summary(..., na.rm = FALSE)

## S3 method for class 'polynom'
Math(x, ...)

## S3 method for class 'polylist'
Math(x, ...)

Arguments

Value

The result of the group generic operation

Examples

lis <- as_polylist(lapply(-2:3, function(x) polynomial() - x))
prod(lis)
sum(lis)
solve(prod(lis))
solve(sum(lis))

Lowest Common Multiple

Description

For a list of polynomials, find the lowest degree monic polynomial into which each divides exactly

Usage

LCM(...)

lowest_common_multiple(...)

## S3 method for class 'polynom'
LCM(...)

## S3 method for class 'polylist'
LCM(...)

Arguments

Value

A polynomial giving the lowest common multiple

Examples

p <- poly_calc(0:5)
r <- poly_calc(1:6)
greatest_common_divisor(p, r)
solve(greatest_common_divisor(p, r))
lowest_common_multiple(p, r)
solve(lowest_common_multiple(p, r))

Polynomial arithmetic

Description

Group generic function to implement arithmetic operations on polynomial objects

Usage

## S3 method for class 'polynom'
Ops(e1, e2)

## S3 method for class 'polylist'
Ops(e1, e2)

Arguments

Value

A polynomial or polylist object representing the result of the operation.

Examples

x <- polynomial()
(p <- (x-1)^5 - 1)
(p1 <- (p + 1)/(x - 1)^2 - 1)
for(i in 0:10) cat(coef((x+1)^i), "\n")

Extract components of a list of polynomials

Description

Extract components of a list of polynomials

Usage

## S3 method for class 'polylist'
x[i]

Arguments

Value

A polylist object of the components

Polynomial coercion to character

Description

Produce a text representation of a polynomial object

Usage

## S3 method for class 'polynom'
as.character(x, variable = "x", decreasing = FALSE, ...)

Arguments

Value

A character string representation of the polynomial

Examples

p <- poly_from_zeros(-2:3)
as.character(p, "z", FALSE)
as.character(p, "z", TRUE)
parse(text = as.character(p, "z", TRUE))[[1]]

Coercion to function

Description

PolynomF objects ARE functions, but this coercion method creates from a polynomial object a pure function with the coefficients fully exposed in the code and which evaluates the polynomial more efficiently.

Usage

## S3 method for class 'polynom'
as.function(x, variable = "x", ...)

## S3 method for class 'polylist'
as.function(x, ...)

Arguments

Value

An explicit R function evaluating the polynomial

Examples

p <- poly_from_zeros(-2:3)
p
as.function(p)

Concatenation of polynomial objects into lists

Description

Concatenation of polynomial objects into lists

Usage

## S3 method for class 'polynom'
c(..., recursive = FALSE)

## S3 method for class 'polylist'
c(..., recursive = FALSE)

Arguments

Value

A polylist object with all argumets included

Change origin of a polynomial

Description

Given a polynomial P(x) and a new origin o, find the polynomial Q(x) = P(x + o). I.e. Q(0) = P(o)

Usage

change_origin(p, o, ...)

## Default S3 method:
change_origin(p, o, ...)

## S3 method for class 'polynom'
change_origin(p, o, ...)

## S3 method for class 'polylist'
change_origin(p, o, ...)

Arguments

Value

A polynom or polylist object with x measured from the new origin

Polynomial coefficients

Description

Extract polynomial coefficients

Usage

## S3 method for class 'polynom'
coef(object, ...)

## S3 method for class 'polylist'
coef(object, ...)

Arguments

Value

A numeric vector of coefficients

Examples

p <- polynomial(1:3)*polynomial(5:1)
coef(p)

Polynomial Calculus

Description

Find the derivative or indefinite integral of a polynomial object, or list thereof.

Usage

## S3 method for class 'polynom'
deriv(expr, ...)

integral(expr, ...)

## Default S3 method:
integral(expr, ...)

## S3 method for class 'polynom'
integral(expr, limits = NULL, ...)

## S3 method for class 'polylist'
deriv(expr, ...)

## S3 method for class 'polylist'
integral(expr, ...)

Arguments

Value

A coeffieient vector, or list thereof

Examples

p <- poly_from_roots(-2:3)
p
deriv(p)
integral(p)

Lagrange Interpolation Polynomials

Description

Compute the Lagrange Interpolation Polynomial from a given set of x- and y-values, or, alterntively, compute the interpolated values at a set of given x-values. Two algorithms are provided, namely Neville's algorithm, or a more direct version based on the usual Lagrange formula. The latter is generally faster but the former can be more accurate numerically.

Usage

neville(x, y, x0 = polynomial())

lagrange(x, y, x0 = polynomial())

Arguments

Value

Either an interpolation polynomial object or a vector of interpolated y-values

Examples

set.seed(123)
x <- 1:5
y <- rnorm(x)
xout <- 0.5 + 1:4

p1 <- neville(x, y)
plot(p1, xlim = range(x), ylim = extendrange(y, f = 1), panel.first = grid())
points(x, y, col = 4)
points(xout, lagrange(x, y, xout), col = 2)

Plot method for polynomials

Description

Plot methods for polynom or polylist objects

Usage

## S3 method for class 'polylist'
plot(
  x,
  xlim = 0:1,
  ylim = range(Px),
  type = "l",
  xlab = "x",
  ylab = "P(x)",
  ...,
  col = seq_along(x),
  lty = if (length(col) == 1) seq_along(x) else "solid",
  len = 1000,
  legend = FALSE
)

## S3 method for class 'polynom'
plot(
  x,
  xlim = 0:1,
  ylim = range(Px),
  type = "l",
  xlab = "x",
  ylab = "p(x)",
  ...,
  len = 1000,
  limits = pu[1:2]
)

## S3 method for class 'polynom'
lines(x, ..., len = 1000, limits = pu[1:2])

## S3 method for class 'polynom'
points(x, ..., len = 100, limits = pu[1:2])

## S3 method for class 'polylist'
lines(
  x,
  ...,
  len = 1000,
  limits = pu[1:2],
  col = seq_along(x),
  lty = if (length(col) == 1) seq_along(x) else "solid"
)

## S3 method for class 'polylist'
points(x, ..., len = 100)

Arguments

Value

Nothing of interest, invisibly

Examples

p <- poly_from_zeros((-3):4)
plot(p)
lines(deriv(p), col = "red")

Lagrange interpolation polynomial

Description

Calculate the Lagrange interpolation polynomial, or list of polynomials, given a set of (x, y) points to fit

Usage

poly_calc(x, y, tol = sqrt(.Machine$double.eps), lab = dimnames(y)[[2]])

poly_from_zeros(...)

poly_from_roots(...)

poly_from_roots(...)

poly_from_values(x, y, tol = sqrt(.Machine$double.eps), lab = dimnames(y)[[2]])

Arguments

Value

An interpolation polynomial, or list of interpolating polynomials.

Examples

(p <- poly_calc(0:5)) ## same as poly_from_zeros(0:5)
(p <- poly_calc(0:5, exp(0:5)))
plot(p)
curve(exp, add = TRUE, col = "red")

Simpl orthogonal polynomials

Description

Generate a list of polynomials up to a specified degree, orthogonal with respect to the natural inner product on a discrete, finite set of x-values with equal weights.

Usage

poly_orth(x, degree = length(unique(x)) - 1, norm = TRUE)

Arguments

Value

A list of orthogonal polynomials as a polylist object

Examples

x <- c(0:3, 5)
P <- poly_orth(x)
plot(P, lty = "solid")
Pf <- as.function(P)
zap(crossprod(Pf(x)))

General Orthogonal Polynomials

Description

Generate sets of polynomials orthogonal with respect to a general inner product. The inner product is specified by an R function of (at least) two polynomial arguments.

Usage

poly_orth_general(inner_product, degree, norm = FALSE, ...)

Hermite(p, q = p)

Legendre(p, q = p)

ChebyshevT(p, q = p)

ChebyshevU(p, q = p)

Jacobi(p, q = p, alpha = -0.5, beta = alpha)

Discrete(p, q = p, x, w = function(x, ...) 1, ...)

Arguments

Details

Discrete orthogonal polynomials, equally or unequally weighted, are included as special cases. See the Discrete inner product function.

Computations are done using the recurrence relation with computed coefficients. If the algebraic expressions for these recurrence relation coefficients are known the computation can be made much more efficient.

Value

A "polylist" object containing the orthogonal set

Examples

(P0 <- poly_orth(0:5, norm = FALSE))
(P1 <- poly_orth_general(Discrete, degree = 5, x = 0:5, norm = FALSE))
sapply(P0-P1, function(x) max(abs(coef(x))))  ## visual check for equality
(P0 <- poly_orth_general(Legendre, 5))
   ### should be same as P0, up to roundoff
(P1 <- poly_orth_general(Jacobi, 5, alpha = 0, beta = 0))
                ### check
sapply(P0-P1, function(x) max(abs(coef(x))))

Polynomial construction

Description

Functions to construct polynomial objects and check class membership

Usage

polynom(a = c(0, 1), ..., eps = 0)

polynomial(a = c(0, 1), ..., eps = 0)

as_polynom(a)

is_polynom(a)

polylist(...)

is_polylist(x)

as_polylist(x)

Arguments

Value

A polynomial object.

Examples

(s <- polynomial())
(p <- polynomial(c(1, 5, 4, 1)/11))
oldPar <- par(mar = c(5,5,2,2)+0.1)
plot(p, xlim = 0:1, ylim = 0:1, type = "n", bty="n",
     xlab = "s", ylab = expression({P^(n)}(s)))
lines(s, limits = 0:1)
P <- p
for(j in 1:7) {
  lines(P, col = j+1, limits = 0:1)
  P <- p(P)
}
lines(P, limits = 0:1, col = 9)
(r <- Re(solve((p-s)/(1-s))))
arrows(r, p(r), r, par("usr")[3], lwd = 0.5,
       length = 0.125, angle = 15)
text(r, 0.025, paste("r =", format(r, digits = 3)))
leg <- sapply(0:8, function(x) bquote({P^(.(x))}(s)))
legend("topleft", legend = as.expression(leg),
       lty = "solid", col = 1:9, bty = "n", ncol=3)
par(oldPar)
rm(leg, oldPar, p, P, r, s, j)

Evaluate a polynomial

Description

Evaluate a polynomial, or polylist object components.

Usage

## S3 method for class 'polynom'
predict(object, newdata, ...)

## S3 method for class 'polylist'
predict(object, newdata, ...)

Arguments

Value

If newdata is a numeric vector, a numeric vector of results. If newdata is a polynomial, then the composition is returned as a polynomial, or polylist object.

Print method for polynomial objects

Description

Print method for polynomial objects

Usage

## S3 method for class 'polylist'
print(x, ...)

Arguments

Value

The original object, invisibly.

Print method for polynomial objects

Description

Standard method for printing polynomial objects

Usage

## S3 method for class 'polynom'
print(x, variable = "x", digits = getOption("digits"), decreasing = FALSE, ...)

Arguments

Value

The original object x, invisibly

Component repition

Description

Repeat components of a polylist object

Usage

## S3 method for class 'polylist'
rep(x, times, ...)

## S3 method for class 'polynom'
rep(x, times, ...)

Arguments

Value

The resulting polylist object.

Find Polynomial Zeros

Description

Solve polynomial equations, a(x) = b(x), or alternatively find the zeros of the polynomial a(x) - b(x)

Usage

## S3 method for class 'polynom'
solve(a, b, ...)

## S3 method for class 'polylist'
solve(a, b, ...)

Arguments

Value

A vector of roots, usually complex

Examples

p <- poly_calc(0:5)
solve(p)
solve(p, 1)

Polynomial summary

Description

Provide a succinct summary of the critical points of a polynomial, or list thereof

Usage

## S3 method for class 'polynom'
summary(object, ...)

## S3 method for class 'polylist'
summary(object, ...)

## S3 method for class 'summary.polynom'
print(x, ...)

Arguments

Value

A list giving the zeros, stationary points and points of inflexion of the polynomial(s)

Examples

p <- poly_calc(0:5)
summary(p)

Tangent lines

Description

Find the tangent line to a polynomial at one or more x-points

Usage

tangent(p, x0)

Arguments

Value

A linear polynomial giving the tangent line, or a list of such polynomials

Examples

p <- poly_from_zeros(c(0, 0:5, 4))
plot(p, xlab = expression(italic(x)), ylab = expression(italic(P(x))),
  main = parse(text = paste("italic(P(x) ==",
                             as.character(p, decreasing = TRUE),")")))
x0 <- solve(deriv(p))        ## stationary points
lines(tangent(p, x0), col = "dark green", lty = "solid",
      limits = cbind(x0-1/4, x0+1/4))
points(x0, p(x0), col = "dark green")

x0 <- solve(deriv(deriv(p))) ## points of inflexion
lines(tangent(p, x0), col = "red", lty = "solid", lwd = 2,
      limits = cbind(x0-1/4, x0+1/4))
points(x0, p(x0), col = "red")
legend("bottomleft", c("Stationary points", "Points of inflexion"),
       pch = 19, col = c("dark green", "red"), lty = "solid",
       cex = 0.7, bg = "beige", box.lwd = 0.25)

Unique components

Description

Remove duplicated polynomials in a polylist object

Usage

## S3 method for class 'polylist'
unique(x, incomparables = FALSE, ...)

Arguments

Value

A polylist object with no duplicated components

Remove minuscule coefficients

Description

A convenience function for setting polynomial coefficients likely to be entirely round-off error to zero. The decision is relegated to the function base::zapsmall, to which this is a front-end.

Usage

zap(x, digits = getOption("digits"))

## Default S3 method:
zap(x, digits = getOption("digits"))

## S3 method for class 'polynom'
zap(x, digits = getOption("digits"))

## S3 method for class 'polylist'
zap(x, digits = getOption("digits"))

## S3 method for class 'list'
zap(x, digits = getOption("digits"))

Arguments

Value

A polynomial or polylist object with minuscule coefficients set to zero.

Examples

(P <- poly_orth(-2:2, norm = FALSE))
zap(35*P)