### A Note on the Downhill MethodA Note on the Downhill Method Access Restriction
Subscribed

 Author Caldwell, George C. Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©1959 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Abstract In  Ward described the “downhill” method for determining roots of $\textit{f}(\textit{z})$ = 0, whqere $\textit{f}(\textit{z})$ is analytic. He denoted by $\textit{R}(\textit{x,$ y) and $\textit{J}(\textit{x,$ y) the real and imaginary parts, respectively, of $\textit{f}(\textit{z})$ and defined the surface $\textit{W}$ by (1) $\textit{W}(\textit{x,$ y) = | $\textit{R}(\textit{x,$ y) | + | $\textit{J}$ (x, y) |. He proved that $\textit{W}$ (x, y) is a minimum if and only if $\textit{x}$ + $\textit{iy}$ is a zero of $ƒ(\textit{z}),$ i.e., $\textit{W}$ is a minimum if and only if $\textit{W}$ = 0. The application of this method to the solution of $\textit{f}(\textit{z})$ = 0 is conceptually simple in view of the results in . A starting value, $\textit{x}0$ + $\textit{iy}0,$ is chosen (at random, if it is desired so), and the corresponding $\textit{W}$ $(\textit{x}0,$ $\textit{y}0)$ is computed. If $\textit{W}$ $(\textit{x}0,$ $\textit{y}0)$ > 0, new values of $\textit{x}$ and $\textit{y},$ $\textit{x}1$ = $\textit{x}0$ + $\textit{h}1,$ $\textit{y}1$ = $\textit{y}0$ + $\textit{k}1,$ are chosen, the $\textit{h}1$ and $\textit{k}1$ being subject only to the restriction that $\textit{W}(\textit{x}1,$ $\textit{y}1)$ < $\textit{W}$ $(\textit{x}0,$ $\textit{y}0).$ Therefore, the actual problem consists of determining suitable values of $\textit{h}1$ and $\textit{k}1.$ It is the purpose of this note to indicate a method for the determination of these increments. Let (2) $ƒ(\textit{z})$ = $\textit{c}0$ + $∑∞\textit{j}=\textit{p}$ $\textit{cj}$ $(\textit{z}$ - $\textit{z})\textit{j}$ be analytic in whatever region about $\textit{z}$ is being considered. It is assumed that $\textit{c}0$ ≠ 0, that $\textit{cp}$ ≠ 0, and the $\textit{p}$ ≥ 1. The $\textit{cj}$ are complex numbers. Although the final result will be given in rectangular coordinates, it is easier to work here with the polar form. Accordingly, let $\textit{cj}$ = $\textit{α}\textit{j$ $ei}&psgr;\textit{j}$ and $\textit{z}$ - $\textit{z}$ = $\textit{re}\textit{i}&thgr;.$ Then $ƒ(\textit{z})$ = $\textit{α}0\textit{e}\textit{i}&psgr;0$ + $∑∞\textit{j}=\textit{p}$ $\textit{α}\textit{j}$ $\textit{rj}$ $\textit{ei}(\textit{j}&thgr;+&psgr;$ $\textit{j}),$ from which it follows that (3) $\textit{R}(\textit{r},$ &thgr;) = $\textit{α}0$ cos &psgr;0 + $∑∞\textit{j}=\textit{p}$ $\textit{α}\textit{j}$ $\textit{rj}$ cos $(\textit{j&thgr;}$ + $\textit{&psgr;j}),$ (4) $\textit{J}$ (r, &thgr;) = $\textit{α}0$ sin &psgr;0 + $∑∞\textit{j}=\textit{p}$ $\textit{α}\textit{j}\textit{rj}$ sin $(\textit{j&thgr;}$ + $&psgr;\textit{j}),$ and (5) $\textit{W}(\textit{r,$ &thgr;) = | $\textit{R}(\textit{r,$ &thgr;) | + | $\textit{J}$ (r, &thgr;) |. In particular, $\textit{W}(0,$ 0) = $\textit{α}0$ {| cos &psgr;0 | + | sin &psgr;0 |}. If the angle @@@@ is defined by the equation (6) @@@@ = $1/\textit{p}$ (π + &psgr;0 - $&psgr;\textit{p}),$ then $\textit{W}$ (δ, @@@@) ∼ | $\textit{α}0$ cos &psgr;0 - $\textit{α}\textit{p}δ\textit{p}$ cos &psgr;0 | + | $\textit{α}0$ sin &psgr;0 - $\textit{α}\textit{p}δ\textit{p}$ sin &psgr;0 | = | $\textit{α}0$ - $\textit{α}\textit{p}δ\textit{p}$ | {| cos &psgr;0 | + | sin &psgr;0|} < $\textit{W}$ (0, 0) for δ sufficiently small. To put these results in rectangular form, suitable for computation, let $\textit{x}$ + $\textit{i}\textit{y}$ = $\textit{z};$ $\textit{cj}$ = $\textit{aj}$ + $\textit{ibj},$ with $\textit{aj}$ and $\textit{bj}$ real; and $\textit{h}$ and $\textit{k}$ denote the increments in $\textit{x}$ and $\textit{y},$ respectively. It follows from equation (6) that (7) $\textit{k}/\textit{h}$ = tan $[1/\textit{p}$ (π + tan-1 $\textit{b}0/\textit{a}0$ - tan $-1\textit{bp}/\textit{ap})],$ and the following theorem has been proved.THEOREM. If $\textit{c}0$ ≠ 0, $\textit{c}1$ = $\textit{c}2$ = … = $\textit{cp}-1$ = 0, $\textit{cp}$ ≠ 0, $\textit{c}\textit{p}+1,$ … are complex coefficients in (1), then $\textit{W}$ $(\textit{x}$ + $\textit{h},$ $\textit{y}$ + $\textit{k})$ < $\textit{W}$ (x, y, for $\textit{h}$ and $\textit{k}$ sufficiently small, if $\textit{h}$ and $\textit{k}$ satisfy (7). As an example of the application of the theorem, consider the polynomial $\textit{f}(\textit{z})$ = 1 + $\textit{z}4.$ This polynomial was discussed in , and it was pointed out that the usual starting technique failed. Let $\textit{x}0$ + $\textit{iy}0$ = 0. Then $\textit{x}1$ = $\textit{h}1$ and $\textit{y}1$ = $\textit{k}1,$ where (8) $\textit{k}1$ = $\textit{h}1$ tan 1/4 (π + 0 - 0) = $\textit{h}1.$ Now, $\textit{W}$ (x, y) = | $\textit{x}4$ - $6x2\textit{y}2$ + $\textit{y}4$ + 1 | + | $4\textit{xy}$ | · | $\textit{x}2$ - $\textit{y}2$ |, so that $\textit{W}$ (0, 0) = 1. Taking $\textit{h}1$ = $\textit{k}1$ = 1/2 yields $\textit{W}$ (1/2, 1/2) = 3/4 < 1. Therefore, $\textit{z}1$ = $\textit{x}1$ + $\textit{iy}1$ = 1/2 (1 + $\textit{i})$ is permissible. For this example, the roots of $\textit{f}(\textit{z})$ = 0 are ± (√2/2) (1 ± $\textit{i}).$ Therefore, one could adjust $\textit{h}1$ (and $\textit{k}1)$ in accordance with (8) to obtain one of these roots. However, if we take $\textit{z}1$ = 1/2 (1 + $\textit{i}),$ then $\textit{f}(\textit{z})$ = 3/4 + $∑4\textit{j}=1$ $\textit{cj}$ $(\textit{z}$ - $\textit{z}1),$ where $\textit{c}1$ = -1 + $\textit{i}.$ Using (7) again, $\textit{k}2$ = $\textit{h}2$ tan (π + tan-1 0/4 - tan-1 1/-1) = $\textit{h}2.$ This procedure is followed step by step until $\textit{W}$ (x, y) differs from zero by no more than an initially prescribed amount. In this example, it can be seen easily that $\textit{k}3$ = $\textit{h}3,$ $\textit{k}4$ = $\textit{h}4,$ …. The only problem remaining is that of choosing the $\textit{hj}$ properly, and this is done readily with simple tests. The downhill method has the following principal advantages for use on an automatic digital computer:The method always converges (theoretically).The method is definitive, i.e., it can be programmed for use in the general case.On the other hand, there are some apparent disadvantages:In order to utilize effectively the result of the theorem, it is necessary at each step to compute $\textit{c}0$ and all of the other coefficients in (2) up through the first non-vanishing one. Subroutines for tan $\textit{&thgr;}$ and tan-1 $\textit{u}$ must be available. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 1959-04-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 6 Issue Number 2 Page Count 3 Starting Page 223 Ending Page 225

#### Open content in new tab

Source: ACM Digital Library