Search results
Results from the WOW.Com Content Network
Consider a quadratic form given by f(x,y) = ax 2 + bxy + cy 2 and suppose that its discriminant is fixed, say equal to −1/4. In other words, b 2 − 4ac = 1. One can ask for the minimal value achieved by | (,) | when it is evaluated at non-zero vectors of the grid , and if this minimum does not exist, for the infimum.
This can be seen in the following tables, the left of which shows Newton's method applied to the above f(x) = x + x 4/3 and the right of which shows Newton's method applied to f(x) = x + x 2. The quadratic convergence in iteration shown on the right is illustrated by the orders of magnitude in the distance from the iterate to the true root (0,1 ...
of an infinitely many times differentiable function f : R → R as its "infinite order Taylor polynomial" at a. Now the estimates for the remainder imply that if, for any r, the derivatives of f are known to be bounded over (a − r, a + r), then for any order k and for any r > 0 there exists a constant M k,r > 0 such that
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
This is the quadratic function whose first and second derivatives are the same as those of f at a given point. The formula for the best quadratic approximation to a function f around the point x = a is () + ′ () + ″ (). This quadratic approximation is the second-order Taylor polynomial for the function centered at x = a.
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 study of quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is Fermat's theorem on sums of two squares, which determines when an integer may be expressed in the form x 2 + y 2, where x, y are integers.
Since one knows the first and second derivatives of P(x) − f(x), one can calculate approximately how far a test point has to be moved so that the derivative will be zero. Calculating the derivatives of a polynomial is straightforward. One must also be able to calculate the first and second derivatives of f(x).