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The imaginary unit or unit imaginary number (i) is a mathematical constant that is a solution to the quadratic equation x 2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers , using addition and multiplication .
The algorithm is easy to understand and explain, but is often competitive with much more complex approaches. With the use of a priority queue, the algorithm is performant on large inputs, since the importance of each point can be computed using only its neighbors, and removing a point only requires recomputing the importance of two other points.
An imaginary number is the product of a real number and the imaginary unit i, [note 1] which is defined by its property i 2 = −1. [1] [2] The square of an imaginary number bi is −b 2. For example, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary. [3]
The computation of (1 + iπ / N ) N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 + iπ / N ) N. It can be seen that as N gets larger (1 + iπ / N ) N approaches a limit of −1. Euler's identity asserts that is
Once the iterations are complete, we have () =,, meaning that , is the desired result. De Boor's algorithm is more efficient than an explicit calculation of B-splines B i , p ( x ) {\displaystyle B_{i,p}(x)} with the Cox-de Boor recursion formula, because it does not compute terms which are guaranteed to be multiplied by zero.
A simple predictor–corrector method (known as Heun's method) can be constructed from the Euler method (an explicit method) and the trapezoidal rule (an implicit method). ...
CORDIC (coordinate rotation digital computer), Volder's algorithm, Digit-by-digit method, Circular CORDIC (Jack E. Volder), [1] [2] Linear CORDIC, Hyperbolic CORDIC (John Stephen Walther), [3] [4] and Generalized Hyperbolic CORDIC (GH CORDIC) (Yuanyong Luo et al.), [5] [6] is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots ...
In computational physics, the term advection scheme refers to a class of numerical discretization methods for solving hyperbolic partial differential equations.In the so-called upwind schemes typically, the so-called upstream variables are used to calculate the derivatives in a flow field.