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In numerical analysis, multivariate interpolation or multidimensional interpolation is interpolation on multivariate functions, having more than one variable or defined over a multi-dimensional domain. [1] A common special case is bivariate interpolation or two-dimensional interpolation, based on two variables or two dimensions.
Multivariate interpolation is the interpolation of functions of more than one variable. Methods include nearest-neighbor interpolation, bilinear interpolation and bicubic interpolation in two dimensions, and trilinear interpolation in three dimensions. They can be applied to gridded or scattered data.
In mathematics, bilinear interpolation is a method for interpolating functions of two variables (e.g., x and y) using repeated linear interpolation. It is usually applied to functions sampled on a 2D rectilinear grid , though it can be generalized to functions defined on the vertices of (a mesh of) arbitrary convex quadrilaterals .
We fix the interpolation nodes x 0, ..., x n and an interval [a, b] containing all the interpolation nodes. The process of interpolation maps the function f to a polynomial p. This defines a mapping X from the space C([a, b]) of all continuous functions on [a, b] to itself.
Quadratic profile. For the one-dimensional domain shown in the figure the Φ value at a control volume face is approximated using three-point quadratic function passing through the two bracketing or surrounding nodes and one other node on upstream side. [4]
Slerp (spherical linear interpolation) — interpolation between two points on a sphere Generalized quaternion interpolation — generalizes slerp for interpolation between more than two quaternions; Irrational base discrete weighted transform; Nevanlinna–Pick interpolation — interpolation by analytic functions in the unit disc subject to a ...
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".
Simpson's 3/8 rule, also called Simpson's second rule, is another method for numerical integration proposed by Thomas Simpson. It is based upon a cubic interpolation rather than a quadratic interpolation.