The Down-Hill Method of Solving f(z) = 0

The purpose of this paper is to show a numerical method of determining to any desired accuracy the complex roots of f ( z ) = O, where f ( z ) is any analytic function. The technique of using this method on a small digital computer is also given in the case for which f ( z ) is a polynomial with real coefficients, Let f ( x + iy) = R(x , y) + /J(x, y) where R and J are real and let W ( z , y ) = I R I + I J I . Then we may think of W as a surface over the XY-plane. Since W is nonnegative, the surface never extends below the XY-plane. The surface WIU intersect the XY-plane at (a, b) if, and only if, a + b / is a root of f ( z ) = O, for only at a root of f ( z ) = 0 are R and J both zero. In theorem 1 we will prove that there are no minimum points on the surface W such that W ~ 0. Hence, the only minimum points of W are those where the surface intersects the XY-plane. Let (ao, b0, W0) be any point on the surface W. If W0 = 0, ao + boi is a root off(z) = 0. If W0 ~ 0, there is a nearby point (a l , b l , W1) such that Wt < Wo. Such a point is said to be " d o w n hill" from (ao, bo, Wo). If Wt ~ 0, there is a nearby point (a2, b2, W~) which is "down hill" from (a~, bl , W~), etc. The general procedure is to start at any point (ao, bo, W0) on the surface W and proceed "down hill" to (a~, b . , W,) such that W~ < e. In this case, a,, + ibn is sufficiently near the root a + bi of f ( z ) = O. The procedure is exactly the same whether the root is real or imaginary. To determine a point "down hill" from (ak, bk, Wk) we start on the X Y plane and compute Wk • Then with an initial step size h we compute Q ( ± h ) = W(ak ± h, bk) and Q ( ± i h ) = W ( a k , bk ± h). If any Q is less than W, take (ak+~, bk+l) as the point which gives a minimum Q and continue the procedure. If no Q is less than W, reduce the size of h and re-compute the Q's. In this way the sequence (ak, bk, Wk) moves always "down hill" on the surface W and converges to a root a + bi of f ( z ) = O. In the case f ( z ) is a polynomial with real coefficients we can avoid multiplying complex numbers by computing R and J from the formulas