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n (x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − x 2) α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer.
Bernstein polynomials can be generalized to k dimensions – the resulting polynomials have the form B i 1 (x 1) B i 2 (x 2) ... B i k (x k). [1] In the simplest case only products of the unit interval [0,1] are considered; but, using affine transformations of the line, Bernstein polynomials can also be defined for products [a 1, b 1] × [a 2 ...
[1] [2] Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE. For the sake of simplicity, we confine our attention to the case of a function of two independent variables x and y for the moment. Consider a quasilinear PDE of the form [3]
Most but not all overdetermined systems, when constructed with random coefficients, are inconsistent. For example, the system x 3 – 1 = 0, x 2 – 1 = 0 is overdetermined (having two equations but only one unknown), but it is not inconsistent since it has the solution x = 1.
One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.
Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations A x = b, where b is not an element of the column space of the matrix A. The approximate solution is realized as an exact solution to A x = b', where b' is the projection of b onto the column space of A. The best ...