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# interp

cubic spline evaluation function

### Syntax

[ yp [,yp1 [,yp2 [,yp3]]]] = interp(xp, x, y, d [, out_mode])

### Arguments

- x,y
real vectors of same size

`n`

: Coordinates of data points on which the interpolation and the related cubic spline (called`s(X)`

in the following) or sub-spline function is based and built.- d
real vector of size(x): The derivative s'(x). Most often, s'(x) will be priorly estimated through the function splin(x, y,..)

- out_mode
(optional) string defining

`s(X)`

for`X`

outside . Possible values: "by_zero" | "by_nan" | "C0" | "natural" | "linear" | "periodic"- xp
real vector or matrix: abscissae at which

`Y`

is unknown and must be estimated with`s(xp)`

- yp
vector or matrix of size(xp):

`yp(i) = s(xp(i))`

or`yp(i,j) = s(xp(i,j))`

- yp1, yp2, yp3
vectors (or matrices) of size(x): elementwise evaluation of the derivatives

`s'(xp)`

,`s''(xp)`

and`s'''(xp)`

.

### Description

The cubic spline function `s(X)`

interpolating the `(x,y)`

set of given points is a continuous and derivable piece-wise function defined over . It consists of a set of cubic polynomials, each one being defined on and connected in values and slopes to both its neighbours. Thus, we can state that for each , such that
. Then, interp() evaluates `s(X)`

(and `s'(X), s''(X), s'''(X)`

if needed) at `xp(i)`

, such that

The `out_mode`

parameter set the evaluation rule
for extrapolation, i.e. for `xp(i)`

outside :

- "by_zero"
an extrapolation by zero is done

- "by_nan"
extrapolation by Nan (%nan)

- "C0"
the extrapolation is defined as follows :

- "natural"
the extrapolation is defined as follows ( being the polynomial defining

`s(X)`

on )- "linear"
the extrapolation is defined as follows :

- "periodic"
`s(X)`

is extended by periodicity:

### Examples

// see the examples of splin and lsq_splin // an example showing C2 and C1 continuity of spline and subspline a = -8; b = 8; x = linspace(a,b,20)'; y = sinc(x); dk = splin(x,y); // not_a_knot df = splin(x,y, "fast"); xx = linspace(a,b,800)'; [yyk, yy1k, yy2k] = interp(xx, x, y, dk); [yyf, yy1f, yy2f] = interp(xx, x, y, df); clf() subplot(3,1,1) plot2d(xx, [yyk yyf]) plot2d(x, y, style=-9) legends(["not_a_knot spline","fast sub-spline","interpolation points"],... [1 2 -9], "ur",%f) xtitle("spline interpolation") subplot(3,1,2) plot2d(xx, [yy1k yy1f]) legends(["not_a_knot spline","fast sub-spline"], [1 2], "ur",%f) xtitle("spline interpolation (derivatives)") subplot(3,1,3) plot2d(xx, [yy2k yy2f]) legends(["not_a_knot spline","fast sub-spline"], [1 2], "lr",%f) xtitle("spline interpolation (second derivatives)")

// here is an example showing the different extrapolation possibilities x = linspace(0,1,11)'; y = cosh(x-0.5); d = splin(x,y); xx = linspace(-0.5,1.5,401)'; yy0 = interp(xx,x,y,d,"C0"); yy1 = interp(xx,x,y,d,"linear"); yy2 = interp(xx,x,y,d,"natural"); yy3 = interp(xx,x,y,d,"periodic"); clf() plot2d(xx,[yy0 yy1 yy2 yy3],style=2:5,frameflag=2,leg="C0@linear@natural@periodic") xtitle(" different way to evaluate a spline outside its domain")

### History

Version | Description |

5.4.0 | previously, imaginary part of input arguments were implicitly ignored. |

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