Content deleted Content added
m Dated {{Cite check}}. (Build p612) |
Repairing links to disambiguation pages - You can help! - Multiplicity |
||
Line 1:
In [[mathematics]], the '''Lehmer–Schur algorithm''' (named after [[Derrick Henry Lehmer]] and [[Issai Schur]]) is a [[root-finding algorithm]] extending the one-dimensional bracketing used by the [[bisection method]] to find the roots of a function of one complex variable inside any rectangular region of the function's [[Holomorphic function|holomorphicity]] (''i.e.'', [[Analytic function|analyticity]]).
The rectangle in question is quadrisected into four, [[Congruence (geometry)|congruent]] quarter rectangles. The [[argument principle]] is then applied to the boundary of each quarter to find the [[winding number]] (about the origin) for the boundary. Given that the function is [[Analytic function|analytic]] within each of these quarters, a nonzero [[winding number]] ''N'' (always an integer) identifies ''N'' zeros of the function inside the quarter in question, each zero counted as many times as its [[Multiplicity (mathematics)|multiplicity]].
Analogously with the bisection method, the algorithm is then applied recursively to any quarter whose boundary has nonzero winding number to further refine the estimates of the zeros. The recursion is repeated until the zero-containing rectangles are either small enough that their centres give sufficiently accurate zero estimates or, alternatively, that another root-finding algorithm can be applied to the estimates to further refine them.
| |||