What is Quadrature for unequal subdivision of the range of integration?

Question

What is Quadrature for unequal subdivision of the range of integration? please explain

Answer 1

Quadrature is the term that is used to refer to the process of numerically estimating a definite integral. Instead of numerical integration, the term quadrature is used.

Now, I'm not completely sure what you mean by "...quadrature for unequal subdivision of the range of integration"

There is a concept known as "adaptive quadrature." This is when the step size in the numerical quadrature method adapts to the function. For instance, if the fuction is 'changing' a lot on a certain interval, the method will adapt the step size to be 'smaller' and if the function is not changing much on an interval, the method will adapt the step size to be bigger.

Hope this helps.

Answer 2

As you know, many of the most commonly-used integration schemes (trapezoidal rule, Simpson's rule, etc.) use function values at equally-spaced values of the variable of integration. This is convenient, and fairly easy to code for use in a computer program.

As a "side benefit", those function evaluations can be re-used if one decides, for example, to halve the subdivison and recalculate the integral using the finer grid; convergence is often detected by comparing results on the coarser and finer grid (and in some cases, those two results can be used to extrapolate to an estimate that is better yet). This re-use doesn't matter so much now, with the powerful computers commonly available; it did matter, not so many years ago. And it was critical in pre-computer and pre-calculator days!

For a given equal-subdivision size (which dictates the number of function evaluations required), the various schemes generally are capable of a certain amount of accuracy. However, if you are willing to forgo the equal-subdivision attribute, you can get more accuracy from a given number of function evaluations. Integration schemes like Gauss integration in its several varieties can achieve impressive accuracy as compared with equal-subdivision schemes like Simpson's rule. Some of these schemes are designed for integrands of a specific form, and some of those are designed for integrals on an infinite interval (for which, obviously, schemes like Simpson's rule can't be applied).

There is a large body of work on this topic, to which the little I've written above does not do justice. I offer a few links in the source section below.