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  2. Implicit function - Wikipedia

    en.wikipedia.org/wiki/Implicit_function

    In calculus, a method called implicit differentiation makes use of the chain rule to differentiate implicitly defined functions. To differentiate an implicit function y ( x ) , defined by an equation R ( x , y ) = 0 , it is not generally possible to solve it explicitly for y and then differentiate.

  3. Explicit and implicit methods - Wikipedia

    en.wikipedia.org/wiki/Explicit_and_implicit_methods

    For such problems, to achieve given accuracy, it takes much less computational time to use an implicit method with larger time steps, even taking into account that one needs to solve an equation of the form (1) at each time step. That said, whether one should use an explicit or implicit method depends upon the problem to be solved.

  4. Numerical methods for ordinary differential equations - Wikipedia

    en.wikipedia.org/wiki/Numerical_methods_for...

    It costs more time to solve this equation than explicit methods; this cost must be taken into consideration when one selects the method to use. The advantage of implicit methods such as is that they are usually more stable for solving a stiff equation, meaning that a larger step size h can be used.

  5. Backward Euler method - Wikipedia

    en.wikipedia.org/wiki/Backward_Euler_method

    The backward Euler method is an implicit method: the new approximation + appears on both sides of the equation, and thus the method needs to solve an algebraic equation for the unknown +. For non-stiff problems, this can be done with fixed-point iteration:

  6. Trapezoidal rule (differential equations) - Wikipedia

    en.wikipedia.org/wiki/Trapezoidal_rule...

    In numerical analysis and scientific computing, the trapezoidal rule is a numerical method to solve ordinary differential equations derived from the trapezoidal rule for computing integrals. The trapezoidal rule is an implicit second-order method, which can be considered as both a Runge–Kutta method and a linear multistep method.

  7. Backward differentiation formula - Wikipedia

    en.wikipedia.org/wiki/Backward_differentiation...

    The backward differentiation formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations.They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed time points, thereby increasing the accuracy of the approximation.

  8. Euler method - Wikipedia

    en.wikipedia.org/wiki/Euler_method

    In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta ...

  9. Linear multistep method - Wikipedia

    en.wikipedia.org/wiki/Linear_multistep_method

    Iterative methods such as Newton's method are often used to solve the implicit formula. Sometimes an explicit multistep method is used to "predict" the value of +. That value is then used in an implicit formula to "correct" the value. The result is a predictor–corrector method.