interp cubic spline evaluation function [ yp [,yp1 [,yp2 [,yp3]]]] = interp(xp, x, y, d [, out_mode]) 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 [x_1,\ 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). The cubic spline function s(X) interpolating the (x,y) set of given points is a continuous and derivable piece-wise function defined over [x_1,\ 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\ \in\ [x_k,\ x_{k+1}],\ s(X) = p_k(X), such that s(x_i) = y_i,\quad \mbox{and}\quad 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 [x_1,\ x_n] : "by_zero" an extrapolation by zero is done "by_nan" extrapolation by Nan (%nan) "C0" the extrapolation is defined as follows : x_n \Rightarrow yp_i = y_n ]]> "natural" the extrapolation is defined as follows (p_i(x) being the polynomial defining s(X) on [x_i,\ x_{i+1}]) x_n \Rightarrow yp_i = p_{n-1}(xp_i) ]]> "linear" the extrapolation is defined as follows : x_n \Rightarrow yp_i = y_n + d_n.(xp_i - x_n) ]]> "periodic" s(X) is extended by periodicity: 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)") 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") splin lsq_splin 5.4.0 previously, imaginary part of input arguments were implicitly ignored.