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An illustration of the five-point stencil in one and two dimensions (top, and bottom, respectively). In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors".
The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
The four roots of the depressed quartic x 4 + px 2 + qx + r = 0 may also be expressed as the x coordinates of the intersections of the two quadratic equations y 2 + py + qx + r = 0 and y − x 2 = 0 i.e., using the substitution y = x 2 that two quadratics intersect in four points is an instance of Bézout's theorem.
If the constant term a 4 = 0, then one of the roots is x = 0, and the other roots can be found by dividing by x, and solving the resulting cubic equation, a 0 x 3 + a 1 x 2 + a 2 x + a 3 = 0. {\displaystyle a_{0}x^{3}+a_{1}x^{2}+a_{2}x+a_{3}=0.\,}
A nonchaotic case Schröder also illustrated with his method, f(x) = 2x(1 − x), yielded Ψ(x) = − 1 / 2 ln(1 − 2x), and hence f n (x) = − 1 / 2 ((1 − 2x) 2 n − 1). If f is the action of a group element on a set, then the iterated function corresponds to a free group .
In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in.
When k is the field of rational numbers, K is generally assumed to be the field of complex numbers, because each solution belongs to a field extension of k, which is isomorphic to a subfield of the complex numbers. This article is about the methods for solving, that is, finding all solutions or describing them.
In a typical backtracking solution to this problem, one could define a partial candidate as a list of integers c = (c[1], c[2], …, c[k]), for any k between 0 and n, that are to be assigned to the first k variables x[1], x[2], …, x[k]. The root candidate would then be the empty list (). The first and next procedures would then be