Definite integrals arise in many different areas and the Fundamental Theorem of Calculus is a powerful tool for evaluating definite integrals. However, it cannot always be applied. There are some functions which do not have an anti-derivative which can be expressed in terms of familiar functions such as polynomials, exponentials and trigonometric functions. One such example is E(-X2). Of course, this is an important function since it is the probability density function for the normal distribution. Moreover, we sometimes only have information about a function by making observations at a certain number of points. In that case, we do not have a nice formula for the function we are integrating, but only some data points. One of the current solution to the above problem is the Trapezoidal Rule.


The trapezoidal rule uses trapezoids instead of rectangles to approximate the definite interval over a closed bounded interval. By using points on the graph of the function determined by a uniform width partition of the interval the upper boundary of the trapezoid is formed. Of course the more sub-intervals, (or said another way: the more trapezoids) the more accuracy of the estimation. And here lies the biggest challenge in the implementation of the Trapezoidal Rule - the sheer computational complexity involved - particularly when high levels of accuracy are required.


We have proposed a parallel algorithm for the Trapezoidal Rule, which exploits the power of CUDA. Running 4 blocks of 256 threads each, per call - subject to a maximum limit of 2^27 calls (after this the function starts making approximations).


On execution, the user is asked to choose a mode for computation - Quick, Standard or Extended - depending on which the relevant function is called. In the Quick or the Default Mode, the Integration is performed over <X^2> from 0 to 1. The accuracy is two decimal places. In the Standard and Extended modes, the user gets to choose one out of the 5 common types of functions: Inverse, Logarithmic, Algebraic, Trigonometric and Exponential. The accuracy is three decimal places in Standard, while it increases to 6 decimal places in Extended Mode. In addition, the Extended Mode also allows the user to control the main kernel function. He can specify the Depth of Recursion at which the function should start making serial calls, as well as the Depth of recursion at which it should quit

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Last edited Jul 7, 2009 at 8:07 AM by thebetaguy, version 2