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Non-linear interpolation is a method used to estimate values between two known data points by fitting a curve, rather than a straight line, through the data points. This approach is particularly useful when the data exhibits a non-linear trend, allowing for more accurate predictions in complex datasets.
Polynomial interpolation is a method of estimating values between known data points by fitting a polynomial that passes through all the given points. It is widely used in numerical analysis and computer graphics for constructing new data points within the range of a discrete set of known data points.
Spline interpolation is a mathematical method used to construct a smooth curve through a set of data points. It leverages piecewise polynomial functions, known as splines, to achieve a balance between flexibility and smoothness, minimizing oscillations that can occur with higher-degree polynomials.
A cubic spline is a piecewise polynomial function used to interpolate or approximate data points with a high degree of smoothness at the boundaries of the pieces. It ensures continuity and smoothness up to the second derivative, making it ideal for applications requiring smooth transitions between data points.
Rational interpolation is a form of interpolation where the interpolant is a ratio of two polynomials, offering greater flexibility and accuracy for approximating functions with poles or rapid oscillations compared to polynomial interpolation. This method is particularly useful in numerical analysis for functions that exhibit singularities or steep gradients, as it can provide better convergence properties and reduced error rates.
A Bézier curve is a parametric curve frequently used in computer graphics and related fields to model smooth curves that can be scaled indefinitely. It is defined by a set of control points, and its shape is determined by a linear combination of these points, offering a versatile way to represent complex shapes and paths.
Runge's Phenomenon describes the large oscillations that occur when using high-degree polynomial interpolation over equidistant points, particularly noticeable near the endpoints of the interval. This effect highlights the limitations of polynomial interpolation for certain functions, emphasizing the need for alternative approaches like spline interpolation or using Chebyshev nodes for more accurate results.
Piecewise interpolation involves constructing a function that passes through a given set of points by defining different polynomial functions for each interval between points. This method is particularly useful for creating smooth curves that accurately represent data points, especially when the data exhibits varying trends or rates of change across different intervals.
Exponential interpolation is a mathematical technique used to estimate values between two known points by assuming the rate of change follows an exponential trend. This method is particularly useful in fields where growth or decay processes are observed, allowing for predictions that align with the natural progression of such phenomena.
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