Title

integrate-- Numerical integration for one dimensional functions

integrate[,options]

optionsDescription ------------------------------------------------------------------------- Mainfunction(string)specifies the function to be integrated, this must be a function of x.lower(#)specifies the lower limit of the integral; the default is -1.upper(#)specifies the upper limit of the integral; the default is 1.quadpts(#)specifies the number of quadrature points to use in the numerical integration; the default is 100.vectorisespecifies that the function to be integrated is not defined in terms of vector operators. -------------------------------------------------------------------------

integrateis an implementation of three numerical integration algorithms: Gauss-Legendre quadrature; Gauss-Hermite quadrature ; and Gauss-Laguerre. Gauss-Legendre quadrature is used for the definite integrals, Gauss-Hermite quadrature is used for the indefinite integral between -infinity and +infinity; and Gauss-Laguerre quadrature is used for the indefinite integral is between 0 and +infinity. Any limits can be chosen and the command will select a combination of quadrature techniques to calculate the result. The current command does not attempt to look at numerical errors but the user can alter the number of quadrature points to inspect any numerical instabilities.This command has been primarily written in the MATA language but is a Stata command. The function can be any single line expression and the integration is with respect to x. The text from the option function() will be used to create a new function in Mata which is then passed to the integration algorithm.

The number of quadrature points can be chosen to be any number above 1 but the larger this number the slower the algorithm. There is no upper limit because the quadrature points are chosen by calculating the eigenvalues and eigenvectors of a companion matrix.

Updating this command using SSCTo obtain the latest version click the following to install the new version

ssc install integrate,replace

+------+ ----+ Main +-------------------------------------------------------------

function(string)specifies the function to be integrated. This function needs to be defined in terms of x. If the function contains any other unknowns then it will crash. The command is much quicker if this function is written in terms of vector operations. If the function is written without vector operations then the vectorise option needs to be specified and another function is constructed that is the vector equivalent of the function (this is slower given the extra calculations).

lower(#)specifies the lower limit of the integral; the default is -1. To specify that the lower limit is -infinity just specify the missing value . in this option.

upper(#)specifies the upper limit of the integral; the default is +1. To specify that the upper limit is +infinity just specify the missing value . in this option.

quadpts(#)specifies the number of quadrature points to use in the numerical integration; the default is 100. The numerical integration function can allow any number of quadrature points but if too many are specified then the program will become slow.

vectorisespecifies that the function specified in the function() option is not defined in terms of vector operators. The code will generate an additional step of creating a new function that allows x as a vector. This involves looping over the elements of the rowvector x so will be considerably slower but does allow flexibility in the specification of the function.The distribution functions in Mata already accept vectors as arguments so can be used in the function directly. The following examples are all standard results that can be obtained using the cumulative distribution functions.

integrate, f(normalden(x)) l(-1) u(1) integrate, f(normalden(x)) l(-1.96) u(1.96) integrate, f(normalden(x)) l(-1.96) u(.) integrate, f(normalden(x)) l(.) u(1.96) integrate, f(normalden(x)) l(.) u(.)

An example of a user-defined function would be the polynomial x+x^2+x^3 note that because this function is not defined by the appropriate vector operations then the option vectorise needs to be used. integrate, f(x+x^2+x^3) v l(-10) u(10)

A quicker implementation of the same function would be integrate, f(x:+x:^2+x:^3) l(-10) u(10)

AuthorAdrian Mander, MRC Biostatistics Unit, Cambridge, UK.

Email adrian.mander@mrc-bsu.cam.ac.uk