splin

cubic spline interpolation

Calling Sequence

d = splin(x, y [,spline_type [, der]])

Arguments

:x a strictly increasing (row or column) vector (x must have at least

2 components)

: :y a vector of same format than x : :spline_type (optional) a string selecting the kind of spline to

compute

: :der (optional) a vector with 2 components, with the end points

derivatives (to provide when spline_type=”clamped”)

: :d vector of the same format than x ( di is the derivative of

the spline at xi)

:

Description

This function computes a cubic spline or sub-spline s which interpolates the (xi,yi) points, ie, we have s(xi)=yi for all i=1,..,n. The resulting spline s is completely defined by the triplet (x,y,d) where d is the vector with the derivatives at the xi: s’(xi)=di (this is called the Hermite form). The evaluation of the spline at some points must be done by the `interp`_ function. Several kind of splines may be computed by selecting the appropriate spline_type parameter:

:”not_a_knot” this is the default case, the cubic spline is computed

by using the following conditions (considering n points x1,...,xn):

: :”clamped” in this case the cubic spline is computed by using the

end points derivatives which must be provided as the last argument der:

: :”natural” the cubic spline is computed by using the conditions: : :”periodic” a periodic cubic spline is computed ( y must verify

y1=yn) by using the conditions:

: :”monotone” in this case a sub-spline ( s is only one continuously

differentiable) is computed by using a local scheme for the di such that s is monotone on each interval:

: :”fast” in this case a sub-spline is also computed by using a simple

local scheme for the di : d(i) is the derivative at x(i) of the interpolation polynomial of (x(i-1),y(i-1)), (x(i),y(i)),(x(i+1),y(i+1)), except for the end points (d1 being computed from the 3 left most points and dn from the 3 right most points).

: :”fast_periodic” same as before but use also a centered formula for

d1 = s’(x1) = dn = s’(xn) by using the periodicity of the underlying function ( y must verify y1=yn).

:

Remarks

From an accuracy point of view use essentially the clamped type if you know the end point derivatives, else use not_a_knot. But if the underlying approximated function is periodic use the periodic type. Under the good assumptions these kind of splines got an O(h^4) asymptotic behavior of the error. Don’t use the natural type unless the underlying function have zero second end points derivatives.

The monotone, fast (or fast_periodic) type may be useful in some cases, for instance to limit oscillations (these kind of sub- splines have an O(h^3) asymptotic behavior of the error).

If n=2 (and spline_type is not clamped) linear interpolation is used. If n=3 and spline_type is not_a_knot, then a fast sub-spline type is in fact computed.

Examples

// example 1
`deff`_("y=runge(x)","y=1 ./(1 + x.^2)")
a = -5; b = 5; n = 11; m = 400;
x = `linspace`_(a, b, n)';
y = runge(x);
d = splin(x, y);
xx = `linspace`_(a, b, m)';
yyi = `interp`_(xx, x, y, d);
yye = runge(xx);
`clf`_()
`plot2d`_(xx, [yyi yye], style=[2 5], leg="interpolation spline@exact function")
`plot2d`_(x, y, -9)
`xtitle`_("interpolation of the Runge function")

// example 2 : show behavior of different splines on random data
a = 0; b = 1;        // interval of interpolation
n = 10;              // nb of interpolation  points
m = 800;             // discretisation for evaluation
x = `linspace`_(a,b,n)'; // abscissae of interpolation points
y = `rand`_(x);          // ordinates of interpolation points
xx = `linspace`_(a,b,m)';
yk = `interp`_(xx, x, y, splin(x,y,"not_a_knot"));
yf = `interp`_(xx, x, y, splin(x,y,"fast"));
ym = `interp`_(xx, x, y, splin(x,y,"monotone"));
`clf`_()
`plot2d`_(xx, [yf ym yk], style=[5 2 3], strf="121", ...
       leg="fast@monotone@not a knot spline")
`plot2d`_(x,y,-9, strf="000")  // to show interpolation points
`xtitle`_("Various spline and sub-splines on random data")
`show_window`_()

See Also

• `interp`_ cubic spline evaluation function
• `lsq_splin`_ weighted least squares cubic spline fitting

History

Version Description 5.4.0 previously, imaginary part of input arguments were implicitly ignored. .. _interp: interp.html .. _lsq_splin: lsq_splin.html