intquad1#

Purpose#

Integrates a specified function using Gauss-Legendre quadrature. A suite of upper and lower bounds may be calculated in one procedure call.

Format#

y = intquad1(&f, lims[, ...])#

Parameters:

&f (scalar) – pointer to the procedure containing the function to be integrated. This must be a function of x.
lims (2xN matrix) – the limits of x. The first row is the upper limit and the second row is the lower limit. N integrations are computed.
... (any) – Optional. Extra scalar arguments to pass to the user function. These arguments will be passed to the user function untouched.

Returns:

y (Nx1 vector) – the estimated integral(s) of \(f(x)\) evaluated between the limits given by lims.

Global Input#

_intord#

scalar, the order of the integration. The larger _intord, the more precise the final result will be. _intord may be set to 2, 3, 4, 6, 8, 12, 16, 20, 24, 32, 40.

Default = 12.

Examples#

Basic example#

This will integrate the function \(f(x) = x*sin(x)\) between 0 and 1. Note the use of the .* instead of *.

// Define function to be integrated
proc f(x);
   retp(x.*sin(x));
endp;

// Limits of integration
lims = { 1, 0 };

// Calculate integral
y = intquad1(&f, lims);

After the code above, y should equal:

0.30116868

Passing in additional arguments#

/*
** Define function to be integrated
** with a second input
*/
proc f(x, a);
   retp(x.*sin(x .* a));
endp;

// Create additional scalar argument 'a'
a = 3.14;

// Limits of integration
lims = { 1, 0 };

/*
** Calculate integral, passing in extra input
** 'a' as the final input to 'intquad1'
*/
y = intquad1(&f, lims, a);

After the code above, y should equal:

0.31863247

Remarks#

The user-defined function f must return a vector of function values. intquad1() will pass to the user-defined function a vector or matrix for x and expect a vector or matrix to be returned. Use the .* and ./ instead of * and /.

Source#

integral.src

Globals#

_intord, _intq12, _intq16, _intq2, _intq20, _intq24, _intq3, _intq32, _intq4, _intq40, _intq6, _intq8