cubic spline evaluation function
[ yp [,yp1 [,yp2 [,yp3]]]] = interp(xp, x, y, d [, out_mode])
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.
real vector of size(x): The derivative s'(x). Most often, s'(x) will be priorly estimated through the function splin(x, y,..)
(optional) string defining s(X)
for X
outside
.
Possible values: "by_zero" | "by_nan" | "C0" | "natural" | "linear" | "periodic"
real vector or matrix: abscissae at which Y
is unknown and must be estimated with s(xp)
vector or matrix of size(xp): yp(i) = s(xp(i))
or yp(i,j) = s(xp(i,j))
vectors (or matrices) of size(x): elementwise evaluation of the derivatives s'(xp)
, s''(xp)
and s'''(xp)
.
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
:
an extrapolation by zero is done
extrapolation by Nan (%nan)
the extrapolation is defined as follows :
the extrapolation is defined as follows (
being the polynomial defining
s(X)
on
)
the extrapolation is defined as follows :
s(X)
is extended by periodicity:
// 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") | ![]() | ![]() |
Version | Description |
5.4.0 | previously, imaginary part of input arguments were implicitly ignored. |