quad, quad8

Synopsis

a = quad('function',a,b)
a = quad('function',a,b,tol)
a = quad('function',a,b,tol,trace)
a = quad8('function',a,b)
a = quad8('function',a,b,tol)
a = quad8('function',a,b,tol,trace)

Description

Quadrature is a numerical method of finding the area under the graph of a function, that is, computing a definite integral.

quad and quad8 implement two different quadrature algorithms. quad implements a low order method using an adaptive recursive Simpson's rule. quad8 implements a higher order method using an adaptive recursive Newton-Cotes 8 panel rule.

q = quad('function',a,b) returns the result of numerically integrating the function fun(x) between the limits a and b. The function function must return a vector of output values when given a vector of input values.

q = quad('function',a,b,tol) iterates until the relative error is less than tol. The default value for tol is 1e-3.

If the final argument trace is nonzero, quad plots a graph showing the progress of the integration.

quad8 has the same calling sequence as quad.

Examples

Integrate the sine function from 0 to ś:

a = quad('sin',0,pi)
          
a =
    2.0000

Algorithm

quad uses an adaptive recursive Simpson's rule. quad8 uses an adaptive recursive Newton-Cotes 8 panel rule. quad8 is better than quad at handling functions with soft singularities:

Diagnostics

quad and quad8 have recursion level limits of 10 to prevent infinite recursion for a singular integral. Reaching this limit in one of the integration intervals produces the warning message:

Recursion level limit reached in quad. Singularity likely.

The computation continues using the best value available in that interval.

Limitations

Neither quad nor quad8 is set up to handle integrable singularities:

If you need to evaluate an integral with such a singularity, recast the problem by transforming the problem into one in which you can explicitly evaluate the integrable singularities and let quad or quad8 take care of the remainder.

References

[1] G.E. Forsythe, M.A. Malcolm and C.B. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, 1977.