pracma (version 1.9.9)

bulirsch-stoer: Bulirsch-Stoer Algorithm

Description

Bulirsch-Stoer algorithm for solving Ordinary Differential Equations (ODEs) very accurately.

Usage

bulirsch_stoer(f, t, y0, ..., tol = 1e-07)
midpoint(f, t0, tfinal, y0, tol = 1e-07, kmax = 25)

Arguments

f

function describing the differential equation $y' = f(t, y)$.

t

vector of x-values where the values of the ODE function will be computed; needs to be increasingly sorted.

y0

starting values as column vector.

...

additional parameters to be passed to the function.

tol

relative tolerance in the Ricardson extrapolation.

t0, tfinal

start and end point of the interval.

kmax

maximal number of steps in the Richardson extrapolation.

Value

bulirsch_stoer returns a list with x the grid points input, and y a vector of function values at the se points.

Details

The Bulirsch-Stoer algorithm is a well-known method to obtain high-accuracy solutions to ordinary differential equations with reasonable computational efforts. It exploits the midpoint method to get good accuracy in each step. The (modified) midpoint method computes the values of the dependent variable y(t) from t0 to tfinal by a sequence of substeps, applying Richardson extrapolation in each step.

Bulirsch-Stoer and midpoint shall not be used with non-smooth functions or singularities inside the interval. The best way to get intermediate points t = (t[1], ..., t[n]) may be to call ode23 or ode23s first and use the x-values returned to start bulirsch_stoer on.

References

J. Stoer and R. Bulirsch (2002). Introduction to Numerical Analysis. Third Edition, Texts in Applied Mathematics 12, Springer Science + Business, LCC, New York.

See Also

ode23, ode23s

Examples

## Example: y'' = -y
f1 <- function(t, y) as.matrix(c(y[2], -y[1]))
y0 <- as.matrix(c(0.0, 1.0))
tt <- linspace(0, pi, 13)
yy <- bulirsch_stoer(f1, tt, c(0.0, 1.0))   # 13 equally-spaced grid points
yy[nrow(yy), 1]                             # 1.1e-11

## Not run: 
# S  <- ode23(f1, 0, pi, c(0.0, 1.0))
# yy <- bulirsch_stoer(f1, S$t, c(0.0, 1.0))  # S$x 13 irregular grid points
# yy[nrow(yy), 1]                             #  2.5e-11
# S$y[nrow(S$y), 1]                           # -7.1e-04
# 
# ## Example: y' = -200 x y^2                 # y(x) = 1 / (1 + 100 x^2)
# f2 <- function(t, y) -200 * t * y^2
# y0 < 1
# tic(); S <- ode23(f2, 0, 1, y0); toc()            # 0.002 sec
# tic(); yy <- bulirsch_stoer(f2, S$t, y0); toc()   # 0.013 sec## End(Not run)