interp - cubic spline evaluation function
Given three vectors (x,y,d) defining a spline or sub-spline function (see splin ) with yi=s(xi), di = s'(xi) this function evaluates s (and s', s'', s''' if needed) at xp(i) :
yp(i) = s(xp(i)) or yp(i,j) = s(xp(i,j)) yp1(i) = s'(xp(i)) or yp1(i,j) = s'(xp(i,j)) yp2(i) = s''(xp(i)) or yp2(i,j) = s''(xp(i,j)) yp3(i) = s'''(xp(i)) or yp3(i,j) = s'''(xp(i,j))
The out_mode parameter set the evaluation rule for extrapolation, i.e. for xp(i) not in [x1,xn] :
s(x) = y1 for x < x1 s(x) = yn for x > xn
s(x) = p_1(x) for x < x1 s(x) = p_{n-1}(x) for x > xn
s(x) = y1 + s'(x1)(x-x1) for x < x1 s(x) = yn + s'(xn)(x-xn) for x > xn
// 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); xbasc() 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"); xbasc() 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")
B. Pincon