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In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable [1] (multivariate functions); when the variates are spatial coordinates, it is also known as spatial interpolation. The function to be interpolated is known at given points (,,, …
The last two examples illustrate what happens if x is a rather small number. In the second from last example, x = 1.110111⋯111 × 2 −50 ; 15 bits altogether. The binary is replaced very crudely by a single power of 2 (in this example, 2 −49 ) and its decimal equivalent is used.
Consider the above example of estimating f(2.5). Since 2.5 is midway between 2 and 3, it is reasonable to take f(2.5) midway between f(2) = 0.9093 and f(3) = 0.1411, which yields 0.5252. Generally, linear interpolation takes two data points, say (x a,y a) and (x b,y b), and the interpolant is given by:
Even for proper assumptions about the function, the extrapolation can diverge severely from the function. The classic example is truncated power series representations of sin(x) and related trigonometric functions. For instance, taking only data from near the x = 0, we may estimate that the function behaves as sin(x) ~ x.
To make comparisons based on dates (e.g., if the current date and time is after some other date and time), first convert the time(s) to the number of seconds after January 1, 1970, using the function {{#time: U }}, then compare (or add, subtract, etc.) those numerical values.
For example, given a = f(x) = a 0 x 0 + a 1 x 1 + ··· and b = g(x) = b 0 x 0 + b 1 x 1 + ···, the product ab is a specific value of W(x) = f(x)g(x). One may easily find points along W(x) at small values of x, and interpolation based on those points will yield the terms of W(x) and the specific product ab. As fomulated in Karatsuba ...
Fitted curves can be used as an aid for data visualization, [12] [13] to infer values of a function where no data are available, [14] and to summarize the relationships among two or more variables. [15] Extrapolation refers to the use of a fitted curve beyond the range of the observed data, [16] and is subject to a degree of uncertainty [17 ...
Example of bilinear interpolation on the unit square with the z values 0, 1, 1 and 0.5 as indicated. Interpolated values in between represented by color. In mathematics, bilinear interpolation is a method for interpolating functions of two variables (e.g., x and y) using repeated linear interpolation.
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