Maxima Function
cspline (points)
cspline(points,option1,option2,...)
Computes the polynomial interpolation by the cubic splines method. Argument points must be either:
a two column matrix, p:matrix([2,4],[5,6],[9,3])
,
a list of pairs, p: [[2,4],[5,6],[9,3]]
,
a list of numbers, p: [4,6,3]
, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
There are three options to fit specific needs:
'd1
, default 'unknown
, is the first derivative at x_1; if it is 'unknown
, the second derivative at x_1 is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.
'dn
, default 'unknown
, is the first derivative at x_n; if it is 'unknown
, the second derivative at x_n is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.
'varname
, default 'x
, is the name of the independent variable.
Examples:
(%i1) load(interpol)$ (%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$ (%i3) /* Unknown first derivatives at the extremes is equivalent to natural cubic splines */ cspline(p); 3 2 1159 x 1159 x 6091 x 8283 (%o3) (------- - ------- - ------ + ----) charfun2(x, minf, 3) 3288 1096 3288 1096 3 2 2587 x 5174 x 494117 x 108928 + (- ------- + ------- - -------- + ------) charfun2(x, 7, inf) 1644 137 1644 137 3 2 4715 x 15209 x 579277 x 199575 + (------- - -------- + -------- - ------) charfun2(x, 6, 7) 1644 274 1644 274 3 2 3287 x 2223 x 48275 x 9609 + (- ------- + ------- - ------- + ----) charfun2(x, 3, 6) 4932 274 1644 274 (%i4) f(x):=''%$ (%i5) /* Some evaluations */ map(f,[2.3,5/7,%pi]), numer; (%o5) [1.991460766423356, 5.823200187269903, 2.227405312429507] (%i6) load(draw)$ /* load draw package */ (%i7) /* Plotting interpolating function */ draw2d( color = red, key = "Cubic splines", explicit(f(x),x,0,10), point_size = 3, color = blue, key = "Sample points", points(p))$ (%i8) /* New call, but giving values at the derivatives */ cspline(p,d1=0,dn=0); 3 2 1949 x 11437 x 17027 x 1247 (%o8) (------- - -------- + ------- + ----) charfun2(x, minf, 3) 2256 2256 2256 752 3 2 1547 x 35581 x 68068 x 173546 + (- ------- + -------- - ------- + ------) charfun2(x, 7, inf) 564 564 141 141 3 2 607 x 35147 x 55706 x 38420 + (------ - -------- + ------- - -----) charfun2(x, 6, 7) 188 564 141 47 3 2 3895 x 1807 x 5146 x 2148 + (- ------- + ------- - ------ + ----) charfun2(x, 3, 6) 5076 188 141 47 (%i8) /* Defining new interpolating function */ g(x):=''%$ (%i9) /* Plotting both functions together */ draw2d( color = black, key = "Cubic splines (default)", explicit(f(x),x,0,10), color = red, key = "Cubic splines (d1=0,dn=0)", explicit(g(x),x,0,10), point_size = 3, color = blue, key = "Sample points", points(p))$