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An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments. [ 1 ] : 204–206 For example, the equation x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} of the unit circle defines y as an implicit function ...
The unit circle can be specified as the level curve f(x, y) = 1 of the function f(x, y) = x 2 + y 2.Around point A, y can be expressed as a function y(x).In this example this function can be written explicitly as () =; in many cases no such explicit expression exists, but one can still refer to the implicit function y(x).
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f.For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value.
In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly x and y. For example, the unit circle is defined by the implicit equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} .
An implicit function is a function that is defined implicitly by an implicit equation, by associating one of the variables (the value) with the others (the arguments). [ 56 ] : 204–206 Thus, an implicit function for y {\displaystyle y} in the context of the unit circle is defined implicitly by x 2 + f ( x ) 2 − 1 = 0 {\displaystyle x^{2}+f ...
In calculus, a real-valued function of a real variable or real function is a partial function from the set of the real numbers to itself. Given a real function f : x ↦ f ( x ) {\displaystyle f:x\mapsto f(x)} its multiplicative inverse x ↦ 1 / f ( x ) {\displaystyle x\mapsto 1/f(x)} is also a real function.
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.
The stability function of an explicit Runge–Kutta method is a polynomial, so explicit Runge–Kutta methods can never be A-stable. [32] If the method has order p, then the stability function satisfies () = + (+) as . Thus, it is of interest to study quotients of polynomials of given degrees that approximate the exponential function the best.