interp cubic spline evaluation function

## Syntax

                [yp, yp1, yp2, yp3] = interp(xp, x, y, d)
                [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 [x1, xn]. 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 [x1, xn]. It consists of a set of cubic polynomials, each one pk(X) being defined on [xk, xk+1] and connected in values and slopes to both its neighbours. Thus, we can state that for each X ∈ [xk, xk+1], s(X) = pk(X) , such that s(xi) = yi, and s'(xi) = di . Then, interp() evaluates `s(X)` (and `s'(X), s''(X), s'''(X)` if needed) at `xp(i)`, such that

ypi = s(xpi) or ypij = s(xpij)
yp1i = s'(xpi) or yp1ij = s'(xpij)
yp2i = s''(xpi) or yp2ij = s''(xpij)
yp3i = s'''(xpi) or yp3ij = s'''(xpij)

The `out_mode` parameter set the evaluation rule for extrapolation, i.e. for `xp(i)` outside [x1, xn] :

"by_zero"

  an extrapolation by zero is done

"by_nan"

  extrapolation by Nan (%nan)

"C0"

  the extrapolation is defined as follows :

  xpi < x1 ⇒ ypi = y1
  xpi > xn ⇒ ypi = yn

"natural"

  the extrapolation is defined as follows (pi(x) being the polynomial defining `s(X)` on [xi, xi+1]):

  xpi < x1 ⇒ ypi = p1(xpi)
  xpi > xn ⇒ ypi = pn-1(xpi)

"linear"

  the extrapolation is defined as follows :

  xpi < x1 ⇒ ypi = y1 + d1.(xpi - x1)
  xpi > xn ⇒ ypi = yn + dn.(xpi - xn)

"periodic"

  `s(X)` is extended by periodicity:

  ypi = s( x1 + (xpi - x1) modulo (xn-x1) )

## 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)")

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")

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")

## See also

- splin
- lsq_splin