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  2. Newton's method - Wikipedia

    en.wikipedia.org/wiki/Newton's_method

    An illustration of Newton's method. In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function.

  3. Newton's method in optimization - Wikipedia

    en.wikipedia.org/wiki/Newton's_method_in...

    Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function, which are solutions to the equation =.

  4. Root-finding algorithm - Wikipedia

    en.wikipedia.org/wiki/Root-finding_algorithm

    Newton's method may not converge if started too far away from a root. However, when it does converge, it is faster than the bisection method; its order of convergence is usually quadratic whereas the bisection method's is linear. Newton's method is also important because it readily generalizes to higher-dimensional problems.

  5. Polynomial root-finding - Wikipedia

    en.wikipedia.org/wiki/Polynomial_root-finding...

    For finding one root, Newton's method and other general iterative methods work generally well. For finding all the roots, arguably the most reliable method is the Francis QR algorithm computing the eigenvalues of the companion matrix corresponding to the polynomial, implemented as the standard method [ 1 ] in MATLAB .

  6. Broyden's method - Wikipedia

    en.wikipedia.org/wiki/Broyden's_method

    Anderson's iterative method, which uses a least squares approach to the Jacobian. [9] Schubert's or sparse Broyden algorithm – a modification for sparse Jacobian matrices. [10] The Pulay approach, often used in density functional theory. [11] [12] A limited memory method by Srivastava for the root finding problem which only uses a few recent ...

  7. Explicit and implicit methods - Wikipedia

    en.wikipedia.org/wiki/Explicit_and_implicit_methods

    In the vast majority of cases, the equation to be solved when using an implicit scheme is much more complicated than a quadratic equation, and no analytical solution exists. Then one uses root-finding algorithms, such as Newton's method, to find the numerical solution. Crank-Nicolson method. With the Crank-Nicolson method

  8. Line search - Wikipedia

    en.wikipedia.org/wiki/Line_search

    The line-search method first finds a descent direction along which the objective function will be reduced, and then computes a step size that determines how far should move along that direction. The descent direction can be computed by various methods, such as gradient descent or quasi-Newton method. The step size can be determined either ...

  9. Quasi-Newton method - Wikipedia

    en.wikipedia.org/wiki/Quasi-Newton_method

    Quasi-Newton methods are a generalization of the secant method to find the root of the first derivative for multidimensional problems. In multiple dimensions the secant equation is under-determined , and quasi-Newton methods differ in how they constrain the solution, typically by adding a simple low-rank update to the current estimate of the ...