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
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
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).
interpreted function, defined at i/spline.i line 122