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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 :
Such complex logarithm functions are analogous to the real logarithm function: >, which is the inverse of the real exponential function and hence satisfies e ln x = x for all positive real numbers x. Complex logarithm functions can be constructed by explicit formulas involving real-valued functions, by integration of 1 / z {\displaystyle 1/z ...
The backward Euler method is an implicit method, meaning that the formula for the backward Euler method has + on both sides, so when applying the backward Euler method we have to solve an equation. This makes the implementation more costly.
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
The complex logarithm is the complex number analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However, a multivalued function can be defined which satisfies most of the identities. It is usual to consider this as a function defined on a Riemann surface.
The natural logarithm function, if considered as a real-valued function of a positive real variable, is the inverse function of the exponential function, leading to the identities: = + = Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition: [ 5 ] ln ( x ⋅ y ) = ln x + ln y ...
Root-finding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is differentiable and the derivative is known, then Newton's method is a popular choice. [16] [17] Linearization is another technique for solving nonlinear equations.
Solving an equation f(x) = g(x) is the same as finding the roots of the function h(x) = f(x) – g(x). Thus root-finding algorithms can be used to solve any equation of continuous functions. 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 ...