QSIMP

The QSIMP function performs numerical integration of a function over the closed interval [ A, B ] using Simpson's rule. The result will have the same structure as the smaller of A and B , and the resulting type will be single- or double-precision floating, depending on the input types.

QSIMP is based on the routine


qsimp

described in section 4.2 of Numerical Recipes in C: The Art of Scientific Computing (Second Edition), published by Cambridge University Press, and is used by permission.

Arguments

Func

A scalar string specifying the name of a user-supplied IDL function to be integrated. This function must accept a single scalar argument X and return a scalar result. It must be defined over the closed interval [ A, B ].

For example, if we wish to integrate the fourth-order polynomial

y = ( x ⁴ - 2 x ² ) sin( x )

we define a function SIMPSON to express this relationship in the IDL language:

FUNCTION simpson, X

RETURN, (X^4 - 2.0 * X^2) * SIN(X)

END

A

The lower limit of the integration. A can be either a scalar or an array.

B

The upper limit of the integration. B can be either a scalar or an array.

Note: If arrays are specified for A and B , then QSIMP integrates the user-supplied function over the interval [ A _i , B _i ] for each i . If either A or B is a scalar and the other an array, the scalar is paired with each array element in turn.

Keywords

DOUBLE

Set this keyword to force the computation to be done in double-precision arithmetic.

EPS

The desired fractional accuracy. For single-precision calculations, the default value is 1.0 ¥ 10^-6 . For double-precision calculations, the default value is 1.0 ¥ 10^-12 .

JMAX

2^{(JMAX - 1)} is the maximum allowed number of steps. If not specified, a default of 20 is used.

Example

To integrate the SIMPSON function (listed above) over the interval [0, p /2] and print the result:

A = 0.0 ; Define lower limit of integration.

B = !PI/2.0 ; Define upper limit of integration.

PRINT, QSIMP('simpson', A, B)

IDL prints:

-0.479158

The exact solution can be found using the integration-by-parts formula:

FB = 4.*B*(B^2-7.)*SIN(B) - (B^4-14.*B^2+28.)*COS(B)

FA = 4.*A*(A^2-7.)*SIN(A) - (A^4-14.*A^2+28.)*COS(A)

exact = FB - FA

PRINT, exact