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A value c that satisfies this equation, that is, f (c) = 0, is called a root or zero of the function f and is a solution of the original equation. If f is a continuous function and there exist two points a 0 and b 0 such that f ( a 0 ) and f ( b 0 ) are of opposite signs, then, by the intermediate value theorem , the function f has a root in ...
In this case a and b are said to bracket a root since, by the intermediate value theorem, the continuous function f must have at least one root in the interval (a, b). At each step the method divides the interval in two parts/halves by computing the midpoint c = (a+b) / 2 of the interval and the value of the function f(c) at that point.
However, most root-finding algorithms do not guarantee that they will find all roots of a function, and if such an algorithm does not find any root, that does not necessarily mean that no root exists. Most numerical root-finding methods are iterative methods, producing a sequence of numbers that ideally converges towards a root as a limit.
Costate equations — equation for the "Lagrange multipliers" in Pontryagin's minimum principle; Hamiltonian (control theory) — minimum principle says that this function should be minimized; Types of problems: Linear-quadratic regulator — system dynamics is a linear differential equation, objective is quadratic
In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method , so it is considered a quasi-Newton method .
Finding roots in a specific region of the complex plane, typically the real roots or the real roots in a given interval (for example, when roots represents a physical quantity, only the real positive ones are interesting). For finding one root, Newton's method and other general iterative methods work generally well.
The idea to combine the bisection method with the secant method goes back to Dekker (1969).. Suppose that we want to solve the equation f(x) = 0.As with the bisection method, we need to initialize Dekker's method with two points, say a 0 and b 0, such that f(a 0) and f(b 0) have opposite signs.
In the figure, Excel is used to find the smallest root of the quadratic equation x 2 + bx + c = 0 for c = 4 and c = 4 × 10 5. The difference between direct evaluation using the quadratic formula and the approximation described above for widely spaced roots is plotted vs. b.