Modifications and alternatives to the cubic interpolation process for one-dimensional search
Gonzalez Castro, Salvador
Master of Science
In this thesis, the numerical solution of the problem of minimizing a unimodal function f(a) is considered, where a is a scalar. Two modifications of the cubic interpolation process are presented, so as to improve the robustness of the method and force the process to converge in a reasonable number of iterations, even in pathological cases. Modification Ml includes the ncnoptional bisection of the interval of interpolation at each iteration of the process. Modification M2 includes the optional bisection of the interval of interpolation: this depends on whether the slopes f(a) and f(aQ) at the terminal points agand a of two consecutive iterations have the same sign or opposite sign. An alternative to the cubic interpolation process is also presented. This is a Lagrange interpolation scheme in which the quadratic approximation to the derivative of the function is considered. The coefficients of the quadratic are determined from the values of the slope at three points: aj, a2f. and CX3 = (aj + a2l/2, where a! and a2 are the endpoints of the interval of interpolation. The proposed alternative is investigated in two versions, Version Al and Version A2. They differ in the way in which the next interval of interpolation is chosen; for Version Al, the choice depends on the sign of the slope fa(a); for Version A2, the choice depends on the signs of the slopes fa(cto) and ^(13). Twenty-nine numerical examples are presented. The numerical results show that both modifications of the cubic interpolation process improve the robustness of the process. They also show the promising characteristics of Version A2 of the proposed alternative. Therefore, the one-dimensional search schemes described here have potential interest for those minimization algorithms which depend critically of the precise selection of the stepsize, namely, the conjugate gradient method and the variable metric method.