Laplace's method

In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the formLower bound: Let ε > 0 {displaystyle varepsilon >0} . Since f ″ {displaystyle f''} is continuous there exists δ > 0 {displaystyle delta >0} such that if | x 0 − c | < δ {displaystyle |x_{0}-c|<delta } then f ″ ( c ) ≥ f ″ ( x 0 ) − ε . {displaystyle f''(c)geq f''(x_{0})-varepsilon .} By Taylor's Theorem, for any x ∈ ( x 0 − δ , x 0 + δ ) , {displaystyle xin (x_{0}-delta ,x_{0}+delta ),} The “approximation” in this method is related to the relative error and not the absolute error. Therefore, if we setFirst of all, let me set the global maximum is located at x 0 = 0 {displaystyle x_{0}=0} which can simplify the derivation and does not lost any important information; therefore, all the derivation inside this sub-section is under this assumption. Besides, what we want is the relative error | R | {displaystyle |R|} as shown below In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form where f ( x ) {displaystyle f(x)} is a twice-differentiable function, M is a large number, and the endpoints a and b could possibly be infinite. This technique was originally presented in Laplace (1774, pp. 366–367). Suppose the function f ( x ) {displaystyle f(x)} has a unique global maximum at x0. Let M be a constant and consider the following two functions: Note that x0 will be the global maximum of g {displaystyle g} and h {displaystyle h} as well. Now observe: As M increases the ratio for h {displaystyle h} will grow exponentially while the ratio for g {displaystyle g} does not change. Thus, significant contributions to the integral of this function will come only from points x in a neighbourhood of x0, which can then be estimated. To state and motivate the method, we need several assumptions. We will assume that x0 is not an endpoint of the interval of integration, that the values f ( x ) {displaystyle f(x)} cannot be very close to f ( x 0 ) {displaystyle f(x_{0})} unless x is close to x0, and that f ″ ( x 0 ) < 0. {displaystyle f''(x_{0})<0.} We can expand f ( x ) {displaystyle f(x)} around x0 by Taylor's theorem, where R = O ( ( x − x 0 ) 3 ) {displaystyle R=Oleft((x-x_{0})^{3} ight)} (see: big O notation). Since f {displaystyle f} has a global maximum at x0, and since x0 is not an endpoint, it is a stationary point, so the derivative of f {displaystyle f} vanishes at x0. Therefore, the function f ( x ) {displaystyle f(x)} may be approximated to quadratic order