integral: Adaptive Numerical Integration

In pracma: Practical Numerical Math Functions

Adaptive Numerical Integration

Description

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

Usage

integral(fun, xmin, xmax,
         method = c("Kronrod", "Clenshaw","Simpson"),
         no_intervals = 8, random = FALSE,
         reltol = 1e-8, abstol = 0, ...)

Arguments

fun	integrand, univariate (vectorized) function.
xmin,xmax	endpoints of the integration interval.
method	integration procedure, see below.
no_intervals	number of subdivisions at at start.
random	logical; shall the length of subdivisions be random.
reltol	relative tolerance.
abstol	absolute tolerance; not used.
...	additional parameters to be passed to the function.

Details

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

• Kronrod (Gauss-Kronrod)

• Clenshaw (Clenshaw-Curtis; not yet made adaptive)

• Simpson (adaptive Simpson)

Recommended default method is Gauss-Kronrod. Also try Clenshaw-Curtis that may be faster at times.

Most methods require that function f is vectorized. This will be checked and the function vectorized if necessary.

By default, the integration domain is subdivided into no_intervals subdomains to avoid 0 results if the support of the integrand function is small compared to the whole domain. If random is true, nodes will be picked randomly, otherwise forming a regular division.

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 – not true anymore.]

Value

Returns the integral, no error terms given.

Note

integral does not provide ‘new’ functionality, everything is already contained in the functions called here. Other interesting alternatives are Gauss-Richardson (quadgr) and Romberg (romberg) integration.

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", "Clen", "Simp")) {
    Q <- integral(fun, -1, 1, reltol = 1e-12, method = m)
    cat(m, Q, abs(Q-val), "\n")}
# Kron 1.582233 3.197442e-13 
# Rich 1.582233 3.197442e-13  # use quadgr()
# Clen 1.582233 3.199663e-13 
# Simp 1.582233 3.241851e-13 
# Romb 1.582233 2.555733e-13  # use romberg()

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

## Evaluate improper integral
fun <- function(x) log(x)^2 * exp(-x^2)
val <- 1.9475221803007815976
Q <- integral(fun, 0, Inf, reltol = 1e-12)
# For infinite domains Gauss integration is applied!
cat(m, Q, abs(Q-val), "\n")
# Kron 1.94752218028062 2.01587635473288e-11 

## Example with small function support
fun <- function(x)
            ifelse (x <= 0 | x >= pi, 0, sin(x))
integral(fun, -100, 100, no_intervals = 1)      # 0
integral(fun, -100, 100, no_intervals = 10)     # 1.99999999723
integral(fun, -100, 100, random=FALSE)          # 2
integral(fun, -100, 100, random=TRUE)           # 2 (sometimes 0 !)
integral(fun, -1000, 10000, random=FALSE)       # 0
integral(fun, -1000, 10000, random=TRUE)        # 0 (sometimes 2 !)

pracma documentation built on Nov. 10, 2023, 1:14 a.m.