And indeed one of the great achievements of our civilization over the past several centuries has been to build up the paradigms of mathematics, the exact sciences-and, most importantly, now computation-and to create a tower of capabilities quite different from what pure human-like thinking can achieve. Some of it is instead more formal and structured. And, as I discuss elsewhere, I think its success is probably telling us some very fundamental things about the nature of human thinking.īut while ChatGPT is a remarkable achievement in automating the doing of major human-like things, not everything that’s useful to do is quite so “human like”. Finally, and suddenly, here’s a system that can successfully generate text about almost anything-that’s very comparable to what humans might write. And even having watched developments in the past few years I find the performance of ChatGPT thoroughly remarkable. I’ve been tracking neural net technology for a long time (about 43 years, actually). And it’s happening now with OpenAI’s ChatGPT. It happened with our Physics Project in 2020. It happened to us with Wolfram|Alpha back in 2009. It’s always amazing when things suddenly “just work”. "Numerical Integration."įrom MathWorld-A Wolfram Web Resource.“What Is ChatGPT Doing … and Why Does It Work?” » Referenced on Wolfram|Alpha Numerical Integration Cite this as: "Numerical Integration and Summation." Ch. 7Ĭalculus of Observations: A Treatise on Numerical Mathematics, 4th ed. "Numerical Integration."Ĭomputation 2: Methods, Software, and Analysis. Cambridge, England:Ĭambridge University Press, 1992.
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Princeton, NJ: Princeton University Press, 1949. Milne,Ĭalculus: Approximations, Interpolation, Finite Differences, Numerical IntegrationĪnd Curve Fitting. Integration on Advanced Computer Systems. "Numerical Integration: From Trapezoids to RMS: Object-Oriented Numerical Integration." Dr. Numerical integration of x^3/(e^x-1) from 0 to pi Numerical integration of x lnx from 1 to 2 Methods (which are based on approximation theory) Systems, communication systems, and control systems since in these cases, the classical Modern numerical integrations methods based on information theory have been developed to simulate information systems such as computer controlled Twice as many points on a regular grid (which also permits the already computed values Quadrature formalism often makes it less desirable than simply brute-force calculating Speed of modern computers, the additional complication of the Gaussian Quadrature produces the most accurate approximations possible. By picking the abscissas at which to evaluate the function, Gaussian If the functions are known analytically instead of being tabulated at equally spaced intervals, the best numerical method of integration is called Gaussian Integration, which can yield accurate results for many fewer function evaluations. The 5-pointįormula is called Boole's rule. Then the 2- and 3-point formulas are called the trapezoidal The most straightforward numerical integration technique uses the Newton-Cotes formulas (also called quadrature formulas), which approximate a function tabulatedĪt a sequence of regularly spaced intervals by variousĭegree polynomials. In the Wolfram Language as NIntegrate[ f, A good source for such techniques is Press et al.
There are a wide range of methods available for numerical integration. Mean numerical computation of a univariate integral,Īnd " cubature" to mean numerical computation Numerical integration is the approximate computation of an integral using numerical techniques.
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