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Piecewise Cubic Hermite Interpolating Polynomial (PCHIP)

`yi = pchip(x,y,xi)pp = pchip(x,y)`

`yi = pchip(x,y,xi)` returns
vector `yi` containing elements corresponding to
the elements of `xi` and determined by piecewise
cubic interpolation within vectors `x` and `y`.
The vector `x` specifies the points at which the
data `y` is given. If `y` is a matrix,
then the interpolation is performed for each column of `y` and `yi` is `length(xi)`-by-`size(y,2)`.

`pp = pchip(x,y)` returns
a piecewise polynomial structure for use by `ppval`. `x` can be a row or column vector. `y` is
a row or column vector of the same length as `x`,
or a matrix with `length(x)` columns.

`pchip` finds values of an underlying interpolating
function
at intermediate points, such
that:

On each subinterval , is the cubic Hermite interpolant to the given values and certain slopes at the two endpoints.

interpolates

*y*, i.e., , and the first derivative is continuous. is probably not continuous; there may be jumps at the .The slopes at the are chosen in such a way that preserves the shape of the data and respects monotonicity. This means that, on intervals where the data are monotonic, so is ; at points where the data has a local extremum, so does .

[1] Fritsch, F. N. and R. E. Carlson, "Monotone
Piecewise Cubic Interpolation," *SIAM J. Numerical Analysis*,
Vol. 17, 1980, pp.238-246.

[2] Kahaner, David, Cleve Moler, Stephen
Nash, *Numerical Methods and Software*, Prentice
Hall, 1988.

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