linear function


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linear function

[′lin·ē·ər ′fəŋk·shən]
(mathematics)

Linear Function

 

a function of the form y = kx + b. The basic property of a linear function is that an increment in the function is proportional to the corresponding increment in the independent variable. The graph of a linear function is a straight line. If we use the same units on the coordinate axes, then the coefficient k (the slope) is equal to the tangent of the angle formed by the line with the Ox axis (k = tan ϕ; see Figure 1) and b is the segment cut off by the line on the Oy axis. When b = 0, the linear function is called homogeneous; its graph depicts the direct proportion: y = kx.

Figure 1

Linear functions are widely used in physics and engineering in order to represent (often approximately) the dependence of various quantities. Linear functions of several variables are also studied. Homogeneous linear functions of several variables are called linear forms. Homogeneous linear functions which send vectors to vectors are called linear transformations.

linear function

A recursive function is linear if it is of the form

f x = if p x then q x else h f x

where h is a "linear functional" which means that

(1) for all functions, a, b c and some function ht

h (if a then b else c) = if ht a then h b else h c

Function ht is known as the "predicate transformer" of h.

(2) If for some x,

h (\ y . bottom) x /= bottom

then

for all g, ht g x = True.

I.e. if h g x terminates despite g x not terminating then ht g x doesn't depend on g.

See also linear argument.
References in periodicals archive ?
A statement that linear function should be used to shift the last experimental graph [3] is omitted.
That is, the first difference is a linear function and the second difference is a constant.
Aggregate by Linear Function. In this approach of ensemble method, the set of predicted values using the training dataset are mapped to the desired output values using a linear function f(*).
In this way we can express the objective function of problem (8) as a linear function in {[z.sub.l,n] | l [member of] L}.
The first method utilizes a constant value for the latent heat of vaporization ([l.sub.[upsilon]]) in the Clausius-Clapeyron equation, and the second method uses a previously published temperature-dependent linear function for [l.sub.[upsilon]].
In this paper, the cubic function is adopted to build the BDLM, the prediction precision of which is better than the BDLM built based on linear function model and quadratic function model about the SHM data.
When the point is in an EPI with a fixed [y.sup.*], its depth can be expressed by a linear function only on x (Figure 1),
If the general recurrent neural network is activated by the linear function, the exponential convergence can be achieved.
Transformation of original variables in new abstract variables that form a canonical discriminant linear function is the principle of the technique.
And the dynamic power can be modeled as the linear function of the frequency [12, 16].
In this extension question, the motif is slightly modified and the parent function is changed to a linear function. Students were then asked to choose one of [h.sub.1], [h.sub.2], [h.sub.3], [h.sub.4], [h.sub.5], or [h.sub.6] in Figure 6 and plot that function on their graphing calculators according to the parent functions y = 2x or y = -2x.
Exponential function has much higher slope with respect to linear function and thus we also examined the results of the automatic speaker recognition when auditory critical bands are based on exponential function.