interp

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, x_n]. 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, x_n]. It consists of a set of cubic polynomials, each one p_k(X) being defined on [x_k, x_k+1] and connected in values and slopes to both its neighbours. Thus, we can state that for each X ∈ [x_k, x_k+1], s(X) = p_k(X), such that s(x_i) = y_i,  and   s'(x_i) = d_i. 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 [x₁, x_n] :

"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 (p_i(x) being the polynomial defining s(X) on [x_i, x_i+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)")

// 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")

See also

• splin — cubic spline interpolation
• lsq_splin — weighted least squares cubic spline fitting