pracma (version 1.8.8)

integral: Adaptive Numerical Integration

Description

Combines several approaches to adaptive numerical integration of functions of one variable.

Usage

integral(fun, xmin, xmax,
            method = c("Kronrod","Richardson","Clenshaw","Simpson","Romberg"),
            vectorized = TRUE, arrayValued = FALSE,
            reltol = 1e-8, abstol = 0, ...)

Arguments

fun

integrand, univariate (vectorized) function.

xmin,xmax

endpoints of the integration interval.

method

integration procedure, see below.

vectorized

logical; is the integrand a vectorized function; not used.

arrayValued

logical; is the integrand array-valued.

reltol

relative tolerance.

abstol

absolute tolerance; not used.

...

additional parameters to be passed to the function.

Value

• Returns the integral, no error terms given.

Details

integral combines the following methods for adaptive numerical integration (also available as separate functions):

• Kronrod (Gauss-Kronrod)
• Richardson (Gauss-Richardson)
• Clenshaw (Clenshaw-Curtis; not yet made adaptive)
• Simpson (adaptive Simpson)
• Romberg

Recommended default method is Gauss-Kronrod. Most methods require that function f is vectorized.

If the interval is infinite, quadinf will be called that accepts the same methods as well. If the function is array-valued, quadv is called that applies an adaptive Simpson procedure -- other methods are ignored.

References

Davis, Ph. J., and Ph. Rabinowitz (1984). Methods of Numerical Integration. Dover Publications, New York.

See Also

quadgk, quadgr, quadcc, simpadpt, romberg, quadv, quadinf

Examples

##  Very smooth function
fun <- function(x) 1/(x^4+x^2+0.9)
val <- 1.582232963729353
for (m in c("Kron", "Rich", "Clen", "Simp", "Romb")) {
    Q <- integral(fun, -1, 1, reltol = 1e-12, method = m)
    cat(m, Q, abs(Q-val), "")}
# Kron 1.582233 3.197442e-13 
# Rich 1.582233 3.197442e-13 
# Clen 1.582233 3.199663e-13 
# Simp 1.582233 3.241851e-13 
# Romb 1.582233 2.555733e-13 

##  Highly oscillating function
fun <- function(x) sin(100*pi*x)/(pi*x)
val <- 0.4989868086930458
for (m in c("Kron", "Rich", "Clen", "Simp", "Romb")) {
    Q <- integral(fun, 0, 1, reltol = 1e-12, method = m)
    cat(m, Q, abs(Q-val), "")}
# Kron 0.4989868 2.775558e-16 
# Rich 0.4989868 4.440892e-16 
# Clen 0.4989868 2.231548e-14
# Simp 0.4989868 6.328271e-15 
# Romb 0.4989868 1.508793e-13

## Evaluate improper integral
fun <- function(x) log(x)^2 * exp(-x^2)
val <- 1.9475221803007815976
for (m in c("Kron", "Rich", "Clen", "Simp", "Romb")) {
    Q <- integral(fun, 0, Inf, reltol = 1e-12, method = m)
    cat(m, Q, abs(Q-val), "")}
# Kron 1.947522 1.101341e-13 
# Rich 1.947522 2.928655e-10 
# Clen 1.948016 1.960654e-13 
# Simp 1.947522 9.416912e-12 
# Romb 1.952683 0.00516102

## Array-valued function
log1 <- function(x) log(1+x)
fun <- function(x) c(exp(x), log1(x))
Q <- integral(fun, 0, 1, reltol = 1e-12, arrayValued = TRUE, method = "Simpson")
abs(Q - c(exp(1)-1, log(4)-1))          # 2.220446e-16 4.607426e-15