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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.
Brahmagupta's interpolation formula — seventh-century formula for quadratic interpolation; Extensions to multiple dimensions: Bilinear interpolation; Trilinear interpolation; Bicubic interpolation; Tricubic interpolation; Padua points — set of points in R 2 with unique polynomial interpolant and minimal growth of Lebesgue constant; Hermite ...
A Lozenge diagram is a diagram that is used to describe different interpolation formulas that can be constructed for a given data set. A line starting on the left edge and tracing across the diagram to the right can be used to represent an interpolation formula if the following rules are followed: [5]
In the figure, in order to calculate the value of the property at the face, we should have three nodes i.e. two bracketing or surrounding nodes and one upstream node. Φ w when u w > 0 and u e > 0 a quadratic fit through WW, W and P is used, Φ e when u w > 0 and u e > 0 a quadratic fit through W, P and E is used,
Modern improvements on Brent's method include Chandrupatla's method, which is simpler and faster for functions that are flat around their roots; [3] [4] Ridders' method, which performs exponential interpolations instead of quadratic providing a simpler closed formula for the iterations; and the ITP method which is a hybrid between regula-falsi ...
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 .
The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method is unlikely to be used, as linear interpolation (see below) is almost as easy, but in higher-dimensional multivariate interpolation, this could be a favourable choice for its speed and simplicity.
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".