ROBERT F.H.CHAO
The gradient of a scalar function is a vector pointing to the directionof steepest ascent of the function and its magnitude is the rate of ascentin that direction.In this paper,this property is used to devise a methodof successive approximations for solving a system of simultaneous equa-tions of any type.This method is therefore called the gradient method.Let the given simultaneous equations bew_i=w_i(x_1,x_2,……,x_n)=0,i=1,2,……,n,and let Q be the magnitude of the vector W=(W_1,W_2,……,W_n),then theformula of successive approximations obtained is(?)where X_1 is the trial vector,its magnitude being Q_1,and X_2 is the nextapproximation reached.When used to solve a system of n linear equations,it is proved thatsuccessive approximations by the gradient method converge monotonicallyto the solution.Each approximation requires only about 2/n!of the totalwork of solving the same problem by Cramer's Rule.Hence for large n,this method can be used with advantage.When used to find conjugate complex roots of an algebraic equation,the gradient method is Newton's method.When used to find the coef-ficients of a quadratic factor of a quartic form,the gradient method hasbeen found to be also successful.
