A method for finding a minimum of a multivariate function with applications to the reduction of missile and satellite data

Abstract

Let (XI, X2, . . e , Xn) be an approximation to the location of a minim~ of the continuous function f (Xl, x2, . . . , Xn). Let h i be a number associated with x i, We define the following abbreviations: ~o ~ f(x~, x2,. o °, 4), f(X i + h i ) = f(Xl, X 2, ° ~ • ~ Xi + hi, . . . , Xn), z(x i h~, xj + hi) ~ f(Xl, . . . , X~ hi, o . ., Xj ÷ hi, . ° . , Xn), e~. Thus any variables not appearing in parentheses will be assumed to have the values at the point of approximation to the minimum. The method consists of fitting a complete second-degree polynomial in the n variables to the f~mction in the neighborhood of a minim~mLo Let this polynomial be n n n n-! (I) ~ fo ~ ~ bi ~i + 1⁄2 ~ a~ xi 2 ~ ~ T aij x~ ~j i=l i=l j =i+l i=l i To fit this polynomial to the function, we force the polynomial and the function to correspond at 1⁄2n(n + 3) points in addition to the point (Xl, . ° . , Xn, fo) ° The resulting set of simultaneous equations is solved for the coefficients b i and aij in (I). Then setting the partial derivatives of (I) equal to zero, we obtain a set of n simultaneous linear equations. The solution of this set gives a new approximation to the location of a minimum of the function.