piecewise linear function

The notion of a piecewise linear function makes sense in several different contexts.

These may be defined as indeed higher dimensional piecewise linear function (see second figure below).

The air quality index is a piecewise linear function of the pollutant concentration.

In mathematics, a piecewise linear function is a function composed of straight-line sections.

This is done with a "transfer function" which can be a simple ramp, a piecewise linear function or an arbitrary table.

In general, for every n dimensional continuous piecewise linear function , there is a such that:

As we refine the triangulation, the space of piecewise linear functions must also change with , hence the notation .

Triangulated irregular network-based linear interpolation (a type of piecewise linear function)

Other examples of piecewise linear functions include the absolute value function, the square wave, the sawtooth function, and the floor function.

Therefore, the complete equation will be a piecewise linear function which has a separate influence-line equation for each linear section of the influence line.

The exponential cent scale can therefore be accurately approximated as a piecewise linear function which is numerically correct at semitones.

The Franklin system is therefore an orthonormal basis for L([0, 1]), consisting of continuous piecewise linear functions.

The transfer functions usually have a sigmoid shape, but they may also take the form of other non-linear functions, piecewise linear functions, or step functions.

Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions.

Mathematically, apart from discretization (whole euros both for income and for tax), the tax is a continuous, convex, piecewise linear function of income.

Since the graph of a linear function is a line, the graph of a piecewise linear function consists of line segments and rays.

It is also known that for the piecewise linear functions there are no obstructions, i.e. the climbers can always coordinate their movements to get to the top (see Whittaker).

Splines generalize piecewise linear functions to higher-order polynomials, which are in turn contained in the category of piecewise-differentiable functions, PDIFF .

A monomial in this semiring is a linear map, and a polynomial is the minimum of a finite number of such functions, and therefore a concave, continuous, piecewise linear function.

It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation.

If this condition is not satisfied, we obtain a nonconforming element method, an example of which is the space of piecewise linear functions over the mesh which are continuous at each edge midpoint.

Important sub-classes of piecewise linear functions include the continuous piecewise linear functions and the convex piecewise linear functions.

In the original Chua-Yang CNN (CY-CNN) processor, the state of the cell was a weighted sum of the inputs and the output was a piecewise linear function.

Piecewise linear functions may be defined on n-dimensional Euclidean space, or more generally any vector space or affine space, as well as on piecewise linear manifolds, simplicial complexes, and so forth.

The phase transforms are all assembled from piecewise linear functions under binary logic control and shows characteristic sharp knees (and for some transforms, even sudden jumps) as they move from minimum to maximum, where the frequency counters accumulator wraps around and starts over.