# splin

cubic spline interpolation

### Syntax

d = splin(x, y) d = splin(x, y, spline_type) 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`

:*s'(x*_{1}) = der(1)*s'(x*_{n}) = der(2)- "natural"
the cubic spline is computed by using the conditions:

*s''(x*_{1}) = 0*s''(x*_{n}) = 0- "periodic"
a periodic cubic spline is computed (

`y`

must verify*y1=yn*) by using the conditions:*s'(x*_{1}) = s'(x_{n})*s''(x*_{1}) = s''(x_{n})- "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:- If y(i) ≤ y(i+1), s is increasing on
`[x(i), x(i+1)]`

. - If y(i) ≥ y(i+1), s is increasing on
`[x(i), x(i+1)]`

.

- If y(i) ≤ y(i+1), s is increasing on
- "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; // discretization 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()

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