  GNU-Darwin Web
i/spline Home Manual Packages Global Index Keywords Quick Reference ``` /* SPLINE.I Cubic spline interpolator. \$Id\$ */ /* Copyright (c) 1994. The Regents of the University of California. All rights reserved. */ func spline (dydx, y, x, xp, dydx1=, dydx0=) /* DOCUMENT dydx= spline(y, x) -or- yp= spline(dydx, y, x, xp) -or- yp= spline(y, x, xp) computes the cubic spline curve passing through the points (X, Y). With two arguments, Y and X, spline returns the derivatives DYDX at the points, an array of the same length as X and Y. The DYDX values are chosen so that the piecewise cubic function returned by the four argument call will have a continuous second derivative. The X array must be strictly monotonic; it may either increase or decrease. The values Y and the derivatives DYDX uniquely determine a piecewise cubic function, whose value is returned in the four argument form. In this form, spline is analogous to the piecewise linear interpolator interp; usually you will regard it as a continuous function of its fourth argument, XP. The first argument, DYDX, will normally have been computed by a previous call to the two argument spline function. However, this need not be the case; another DYDX will generate a piecewise cubic function with continuous first derivative, but a discontinuous second derivative. For XP outside the extreme values of X, spline is linear (if DYDX1 or DYDX0 keywords were specified, the function will NOT have continuous second derivative at the endpoint). The XP array may have any dimensionality; the result YP will have the same dimensions as XP. If you only want the spline evaluated at a single set of XP, use the three argument form. This is equivalent to: yp= spline(spline(y,x), y, x, xp) The keywords DYDX1 and DYDX0 can be used to set the values of the returned DYDX(1) and DYDX(0) -- the first and last values of the slope, respectively. If either is not specified or nil, the slope at that end will be chosen so that the second derivative is zero there. The function tspline (tensioned spline) gives an interpolation function which lies between spline and interp, at the cost of requiring you to specify another parameter (the tension). SEE ALSO: interp, tspline */ { if (is_void(x)) { /* spline(y,x) form */ x= y; y= dydx; dx= x(dif); dy= y(dif); diag= (2./dx)(pcen); if (numberof(x)>2) diag(2:-1)*= 2.; rhs= (3.*dy/(dx*dx))(pcen); if (numberof(x)>2) rhs(2:-1)*= 2.; dx= 1./dx; if (is_void(dydx1)) { if (is_void(dydx0)) { return TDsolve(dx, diag, dx, rhs); /* simple natural spline */ } else { dx= dx(1:-1); rhs= rhs(1:-1); rhs(0)-= dydx0*dx(0); return grow(TDsolve(dx, diag(1:-1), dx, rhs), dydx0); } } else { dydx1= double(dydx1); if (is_void(dydx0)) { dx= dx(2:0); rhs= rhs(2:0); rhs(1)-= dydx1*dx(1); return grow(dydx1, TDsolve(dx, diag(2:0), dx, rhs)); } else { if (numberof(x)==2) return double([dydx1, dydx0]); dx= dx(2:-1); rhs= rhs(2:-1); rhs(1)-= dydx1*dx(1); rhs(0)-= dydx0*dx(0); return grow(dydx1, TDsolve(dx, diag(2:-1), dx, rhs), dydx0); } } } if (is_void(xp)) { /* spline(y,x,xp) form */ xp= x; x= y; y= dydx; dydx= spline(y,x,dydx1=dydx1,dydx0=dydx0); } /* spline(dydx,y,x,xp) form */ l= digitize(xp, x); /* index of lower boundary of interval containing xp */ u= l+1; /* extend x, y, dydx so that l and u can be used as index lists */ dx= x(0)-x(1); x= grow(x(1)-dx, x, x(0)+dx); y= grow(y(1)-dydx(1)*dx, y, y(0)+dydx(0)*dx); dydx= grow(dydx(1), dydx, dydx(0)); xl= x(l); dx= double(x(u)-xl); yl= y(l); dy= y(u)-yl; dl= dydx(l); du= dydx(u); dydx= dy/dx; return poly(xp-xl, yl, dl, (3.*dydx-du-2.*dl)/dx, (du+dl-2.*dydx)/(dx*dx)); } func tspline (tension, d2ydx2, y, x, xp, dydx1=, dydx0=) /* DOCUMENT d2ydx2= tspline(tension, y, x) -or- yp= tspline(tension, d2ydx2, y, x, xp) -or- yp= tspline(tension, y, x, xp) computes a tensioned spline curve passing through the points (X, Y). The first argument, TENSION, is a positive number which determines the "tension" in the spline. In a cubic spline, the second derivative of the spline function varies linearly between the points X. In the tensioned spline, the curvature is concentrated near the points X, falling off at a rate proportional to the tension. Between the points of X, the function varies as: y= C1*exp(k*x) + C2*exp(-k*x) + C3*x + C4 The parameter k is proportional to the TENSION; for k->0, the function reduces to the cubic spline (a piecewise cubic function), while for k->infinity, the function reduces to the piecewise linear function connecting the points. The TENSION argument may either be a scalar value, in which case, k will be TENSION*(numberof(X)-1)/(max(X)-min(X)) in every interval of X, or TENSION may be an array of length one less than the length of X, in which case the parameter k will be abs(TENSION/X(dif)), possibly varying from one interval to the next. You can use a variable tension to flatten "bumps" in one interval without affecting nearby intervals. Internally, tspline forces k*X(dif) to lie between 0.01 and 100.0 in every interval, independent of the value of TENSION. Typically, the most dramatic variation occurs between TENSION of 1.0 and 10.0. With three arguments, Y and X, spline returns the derivatives D2YDX2 at the points, an array of the same length as X and Y. The D2YDX2 values are chosen so that the tensioned spline function returned by the five argument call will have a continuous first derivative. The X array must be strictly monotonic; it may either increase or decrease. The values Y and the derivatives D2YDX2 uniquely determine a tensioned spline function, whose value is returned in the five argument form. In this form, tspline is analogous to the piecewise linear interpolator interp; usually you will regard it as a continuous function of its fifth (or fourth) argument, XP. The XP array may have any dimensionality; the result YP will have the same dimensions as XP. The D2YDX2 argument will normally have been computed by a previous call to the three argument tspline function. If you will be computing the values of the spline function for many sets of XP, use this five argument form. If you only want the tspline evaluated at a single set of XP, use the four argument form. This is equivalent to: yp= tspline(tension, tspline(tension,y,x), y, x, xp) The keywords DYDX1 and DYDX0 can be used to set the values of the returned DYDX(1) and DYDX(0) -- the first and last values of the slope, respectively. If either is not specified or nil, the slope at that end will be chosen so that the second derivative is zero there. The function tspline (tensioned spline) gives an interpolation function which lies between spline and interp, at the cost of requiring you to specify another parameter (the tension). SEE ALSO: interp, tspline */ { if (is_void(x)) { /* tspline(tension, y,x) form */ x= double(y); y= d2ydx2; dx= x(dif); dy= y(dif); if (numberof(tension)==numberof(dx)) k= tension/abs(dx); else k= tension*numberof(dx)/(max(x)-min(x)); k= max(min(k, 100./abs(dx)), 0.01/abs(dx)); kdx= k*dx; skdx= sinh(kdx); diag= (cosh(kdx)/skdx-1./kdx)/k; diag= diag(pcen); if (numberof(x)>2) diag(2:-1)*= 2.; offd= (1./kdx-1./skdx)/k; ddydx= (dy/dx)(dif); if (is_void(dydx1)) { if (is_void(dydx0)) { if (numberof(x)==2) return [0., 0.]; diag= diag(2:-1); offd= offd(2:-1); return grow(0., TDsolve(offd, diag, offd, ddydx), 0.); } else { dydx0-= dy(0)/dx(0); if (numberof(x)==2) return [0., dydx0/diag(0)]; diag= diag(2:0); offd= offd(2:0); ddydx= grow(ddydx, dydx0); return grow(0., TDsolve(offd, diag, offd, ddydx)); } } else { dydx1= dy(1)/dx(1) - dydx1; if (is_void(dydx0)) { if (numberof(x)==2) return [dydx1/diag(1), 0.]; diag= diag(1:-1); offd= offd(1:-1); ddydx= grow(dydx1, ddydx); return grow(TDsolve(offd, diag, offd, ddydx), 0.); } else { dydx0-= dy(0)/dx(0); if (numberof(x)==2) return [dydx1/diag(1), dydx0/diag(0)]; ddydx= grow(dydx1, ddydx, dydx0); return TDsolve(offd, diag, offd, ddydx); } } } if (is_void(xp)) { /* tspline(tension, y,x,xp) form */ xp= x; x= y; y= d2ydx2; d2ydx2= tspline(tension, y,x,dydx1=dydx1,dydx0=dydx0); } /* tspline(tension, d2ydx2,y,x,xp) form */ l= digitize(xp, x); /* index of lower boundary of interval containing xp */ u= l+1; /* extend x so that l and u can be used as index lists -- be careful not to make new intervals larger than necessary */ n= numberof(x)-1; /* number of original intervals */ dxavg= (x(0)-x(1))/n; if (dxavg>0.) { dx0= max(max(xp)-x(0), dxavg); dx1= max(x(1)-min(xp), dxavg); } else { dx0= min(min(xp)-x(0), dxavg); dx1= min(x(1)-max(xp), dxavg); } x= grow(x(1)-dx1, x, x(0)+dx0); /* compute k so that sinh(k*dx) is safe to compute */ dx= x(dif); if (numberof(tension)==n) { k= grow(0., tension, 0.)/abs(dx); } else { k= tension/abs(dxavg); } k= max(min(k, 100./abs(dx)), 0.01/abs(dx)); /* extend y carefully so that linear extrapolation happens automatically */ k1= k(2); k0= k(-1); dydx1= (y(2)-y(1))/dx1; kdx= k1*dx1; d2u= d2ydx2(2); d2l= d2ydx2(1); dydx1+= ((d2u-d2l*cosh(kdx))/sinh(kdx) - (d2u-d2l)/kdx)/k1; dydx0= (y(0)-y(-1))/dx0; kdx= k0*dx0; d2u= d2ydx2(0); d2l= d2ydx2(-1); dydx0+= ((d2u*cosh(kdx)-d2l)/sinh(kdx) - (d2u-d2l)/kdx)/k0; y= grow(y(1)-dydx1*dx1, y, y(0)+dydx0*dx0); d2ydx2= grow(0., d2ydx2, 0.); /* begin interpolation */ xu= x(u); xl= x(l); dx= xu-xl; dxl= xp-xl; yl= y(l); dydx= (y(u)-yl)/dx; km2= 1./(k*k); km2(1)= 0.; km2(0)= 0.; km2= km2(l); k= k(l); skdx= sinh(k*dx); d2u= d2ydx2(u); d2l= d2ydx2(l); d3= km2*(d2u-d2l)/dx; d2ydx2= d2u*sinh(k*dxl)/skdx + d2l*(sinh(k*(xu-xp))/skdx-1.); return yl + km2*d2ydx2 + (dydx-d3)*dxl; } func sprime (dydx, y, x, xp) /* DOCUMENT ypprime= sprime(dydx, y, x, xp) computes the derivative of the cubic spline curve passing through the points (X, Y) at the points XP. The DYDX values will have been computed by a previous call to SPLINE, and are chosen so that the piecewise cubic function returned by the four argument call will have a continuous second derivative. The X array must be strictly monotonic; it may either increase or decrease. */ { /* spline(dydx,y,x,xp) form */ l= digitize(xp, x); /* index of lower boundary of interval containing xp */ u= l+1; /* extend x, y, dydx so that l and u can be used as index lists */ dx= x(0)-x(1); x= grow(x(1)-dx, x, x(0)+dx); y= grow(y(1)-dydx(1)*dx, y, y(0)+dydx(0)*dx); dydx= grow(dydx(1), dydx, dydx(0)); xl= x(l); dx= double(x(u)-xl); yl= y(l); dy= y(u)-yl; dl= dydx(l); du= dydx(u); dydx= dy/dx; return poly(xp-xl, dl, 2.*(3.*dydx-du-2.*dl)/dx, 3.*(du+dl-2.*dydx)/(dx*dx)); } ```