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Concept
Method of steepest descent
Summary
In mathematics, the method of steepest descent 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
where C is a contour, and λ is large. One version of the method of steepest descent deforms the contour of integration C into a new path integration C′ so that the following conditions hold:
C′ passes through one or more zeros of the derivative g′(z),
the imaginary part of g(z) is constant on C′.
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descent was first published by , who used it to estimate Bessel functions and pointed out that it occurred in the unpublished note by about hypergeometric functions.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 described some other unpublished notes of Riemann, where he used this method to derive the Riemann–Siegel formula.
The method of steepest descent is a method to approximate a complex integral of the formfor large , where and are analytic functions of . Because the integrand is analytic, the contour can be deformed into a new contour without changing the integral. In particular, one seeks a new contour on which the imaginary part of is constant. Then and the remaining integral can be approximated with other methods like Laplace's method.
The method is called the method of steepest descent because for analytic , constant phase contours are equivalent to steepest descent contours.
If is an analytic function of , it satisfies the Cauchy–Riemann equationsThen so contours of constant phase are also contours of steepest descent.
Let f, S : Cn → C and C ⊂ Cn. If
where denotes the real part, and there exists a positive real number λ0 such that
then the following estimate holds:
Proof of the simple estimate:
Let x be a complex n-dimensional vector, and
denote the Hessian matrix for a function S(x).
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