Steepest-Descent Integral Approximation Algorithm

See: Steepest/Gradient-Descent Algorithm, Numerical Integration Algorithm, Steepest-Descent Optimization Algorithm, Laplace's Method, Saddle Point.

References

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Method_of_steepest_descent Retrieved:2015-6-24.
- In mathematics, the 'method of steepest descent or stationary phase method or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals.
  The integral to be estimated is often of the form : [math]\displaystyle{ \int_Cf(z)e^{\lambda g(z)}dz }[/math] where C is a contour and λ is large. One version of the method of steepest descent deforms the contour of integration
  so that it passes through a zero of the derivative g′(z) in such a way that on the contour g is (approximately) real and has a maximum at the zero.
  The method of steepest descent was first published by , who used it to estimate Bessel functions and pointed out that it occurred in the unpublished note about hypergeometric functions. The contour of steepest descent has a minimax property, see . described some other unpublished notes of Riemann, where he used this method to derive the Riemann-Siegel formula.