# integral: Adaptive Numerical Integration In pracma: Practical Numerical Math Functions

## Description

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

## Usage

 ```1 2 3 4``` ```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)

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.

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

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42``` ```## 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) if (x<=0 || x>=pi) 0 else sin(x) Fun <- Vectorize(fun) integral(fun, -100, 100, no_intervals = 1) # 0 integral(Fun, -100, 100, no_intervals = 1) # 0 integral(fun, -100, 100, random=FALSE) # 2.00000000371071 integral(fun, -100, 100, random=TRUE) # 2.00000000340142 integral(Fun, -1000, 1000, random=FALSE) # 2.00000000655435 integral(Fun, -1000, 1000, random=TRUE) # 2.00000001157690 (sometimes 0 !) ```

pracma documentation built on Jan. 30, 2018, 3:01 a.m.