https://github.com/cran/pracma
Tip revision: c1688b374d201c13fb40b4dda2d2a89e34b94ec6 authored by Hans W. Borchers on 23 January 2021, 09:10:02 UTC
version 2.3.3
version 2.3.3
Tip revision: c1688b3
midpoint.R
.midp <- function(f, x0, xfinal, y0, nsteps) {
# stopifnot(nsteps >= 2, x0 < xfinal, f(...) column vector)
h <- (xfinal - x0) / nsteps
x <- x0
y1 <- y0
y2 <- y1 + h * f(x, y1)
for (i in 1:(nsteps-1)) {
x <- x + h
yy <- y1 + 2.0 * h * f(x, y2)
y1 <- y2
y2 <- yy
}
0.5*(y1 + y2 + h * f(x, yy))
}
midpoint <- function(f, t0, tfinal, y0, tol = 1e-07, kmax = 25) {
stopifnot(is.numeric(y0), is.numeric(t0), length(t0) == 1,
is.numeric(tfinal), length(tfinal) == 1)
n <- length(y0)
# Richardson extrapolation
nsteps <- 2
r <- matrix(NA, kmax, n)
r[1, ] <- .midp(f, t0, tfinal, y0, nsteps)
rold <- r[1, ]
for (k in 2:kmax) {
nsteps <- 2*nsteps
r[k, 1:n] <- .midp(f, t0, tfinal, y0, nsteps)
for (j in (k-1):1) {
cc <- 4^(k-j)
r[j, ] <- (cc * r[j+1, ] - r[j, ]) / (cc - 1.0)
}
dr <- r[1, ] - rold
err <- sqrt(dot(dr, dr)/n)
if (err < tol) break
rold <- r[1, ]
}
return(r[1, ])
}
bulirsch_stoer <- function(f, t, y0, ..., tol = 1e-07) {
stopifnot(is.numeric(t), is.numeric(y0))
fun <- match.fun(f)
f <- function(t, y) fun(t, y, ...) # as row vector
x <- t
D <- length(f(x[1], y0)) # image of f
if (length(y0) != D)
stop("Argument 'y0' and 'f(..)' must be of the same length.")
N <- length(x) # no. of grid points
if (N < 2)
stop("Argument 't' must be a vector of length at least 2.")
mtol <- tol / (N-1)
z <- matrix(0, N, D)
z[1, ] <- y0
for (i in 2:N) {
z[i, ] <- midpoint(f, x[i-1], x[i], z[i-1, ], tol = mtol)
}
return(z)
}