An approximation technique for finding the value of a function or a measurement that lies within known values. If the values f(x0), f(x1), …, f(xn) of a function f of a variable x are known in the interval [x0,xn], the value of f(x) for a value of x inside the interval [x0,xn] can be found by interpolation. One method of interpolation, called linear interpolation for x0 < x < x1, gives:
which is derived using the assumption that between the points x0 and x1, the graph of the function f(x) can be regarded as a straight line. More complicated methods of interpolation exist, using more than two values for the function. The techniques used for interpolation are usually much better than the techniques used in extrapolation.
Suppose that the values f(x0), f(x1),…, f(xn) of a certain function f are known for the particular values x0, x1,…, xn. A method of finding an approximation for f(x), for a given value of x somewhere between these particular values, is called interpolation. If x0<x<x1, the method known as linear interpolation gives
This is obtained by supposing, as an approximation, that between x0 and x1 the graph of the function is a straight line joining the points (x0, f(x0)) and (x1, f(x1)). More complicated methods of interpolation use the values of the function at more than two values. See Lagrange interpolation, spline.
The use of a formula to estimate an intermediate data value. Compare extrapolation. For example, if we discover that 1 kg of fertilizer per hectare results in a yield of 50 kg of produce per hectare, and that 2 kg gives 70 kg, then we might reasonably interpolate between these values to deduce that 1.5 kg would give a yield of about 60 kg ha−1.
In general, if x1<x0<x2 and the values of y corresponding to x1 and x2 are y1 and y2, respectively, then the estimate of y0, the value corresponding to x0, is given by linear interpolation as
A method of obtaining data that falls between two known items of data. It is used in numerical analysis in which new data can be obtained using techniques such as curve fitting.
A simple means of approximating a function f(x) in which the approximation, say p(x), is constructed by requiring that
Here f(xi) are given values p(xi) that fit exactly at the distinct points xi (compare smoothing). The value of f can be approximated by p(x) for x ≠ xi. In practice p is often a polynomial, linear and quadratic polynomials providing the simplest examples. In addition the idea can be extended to include matching of p′(xi) with f′(xi); this is Hermite interpolation. The process is also widely used in the construction of many numerical methods, for example in numerical integration and ordinary differential equations. The interpolating polynomial can be represented in many equivalent forms. For example, when the xi are equally spaced, the forward and backward difference forms (see difference equation) are convenient. More commonly, nonequally spaced xi give rise to the divided difference form, which incorporates successive differencesThese are the first divided differences; second divided differences are obtained by a similar differencing process and so on for higher order differences.
A method that enables additional values to be obtained between sampled values of a signal.
Inserting missing data in a sample, usually by calculating the prediction based on the available data.
- waterspout
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- Watson-Crick model
- Watson, Geoffrey Stuart (1922–98)
- Watson, Henry William (1827–1903)
- Watson, James Dewey
- Watson, James Dewey (1928–)
- Watson, James Dewey (1928– )
- Watson, John Broadus (1878–1958)
- Watson, John Christian (1867–1941)
- Watson, Sir William
- Watson's test
- Watson-Watt, Sir Robert Alexander
- Watsuji Tetsuro (1889–1960)
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