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With low-order polynomials, the curve is more likely to fall near the midpoint (it's even guaranteed to exactly run through the midpoint on a first degree polynomial). Low-order polynomials tend to be smooth and high order polynomial curves tend to be "lumpy". To define this more precisely, the maximum number of inflection points possible in a ...
A ninth order polynomial interpolation (exact replication of the red curve at 10 points) In the mathematical field of numerical analysis, Runge's phenomenon (German:) is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points.
hp-FEM is a generalization of the finite element method (FEM) for solving partial differential equations numerically based on piecewise-polynomial approximations. hp-FEM originates from the discovery by Barna A. Szabó and Ivo Babuška that the finite element method converges exponentially fast when the mesh is refined using a suitable combination of h-refinements (dividing elements into ...
A polynomial function is one that has the form = + + + + + where n is a non-negative integer that defines the degree of the polynomial. A polynomial with a degree of 0 is simply a constant function; with a degree of 1 is a line; with a degree of 2 is a quadratic; with a degree of 3 is a cubic, and so on.
In contrast, the p-version finite element method spans a space of high order polynomials by nodeless basis functions, chosen approximately orthogonal for numerical stability. Since not all interior basis functions need to be present, the p-version finite element method can create a space that contains all polynomials up to a given degree with ...
The determination of the upper bound for the number of limit cycles in two-dimensional polynomial vector fields of degree n and an investigation of their relative positions. The first problem is yet unsolved for n = 8. Therefore, this problem is what usually is meant when talking about Hilbert's sixteenth problem in real algebraic geometry.
Trapezoidal rule — second-order method, based on (piecewise) linear approximation; Simpson's rule — fourth-order method, based on (piecewise) quadratic approximation Adaptive Simpson's method; Boole's rule — sixth-order method, based on the values at five equidistant points; Newton–Cotes formulas — generalizes the above methods
In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function , the Taylor polynomial is the truncation at the order k {\textstyle k} of the Taylor series of the function.