cics_explicit {ICSsmoothing}

Construct the explicit form of non-uniform clamped interpolating cubic spline (NcICS).

Description

cics_explicit constructs the explicit form of non-uniform clamped interpolating cubic spline (via Hermite cubic spline) for nodes uu, function values yy and exterior-node derivatives d.

Usage

cics_explicit(
  uu,
  yy,
  d,
  clrs = c("blue", "red"),
  xlab = NULL,
  ylab = NULL,
  title = NULL
)

Arguments

uu

a vector of arbitrary nodes (ordered ascendingly), with magnitude n+2, n\ge1.

yy

a vector of function values pertaining to nodes in uu.

d

a vector of two values of derivative, in the first and the last node of uu.

clrs

a vector of colours that are used alternately to plot the graph of spline's components.

xlab

a title (optional parameter) for the x axis.

ylab

a title (optional parameter) for the y axis.

title

a title (optional parameter) for the plot.

Value

a list with components

spline_coeffs

matrix, whose i-th row contains coefficients of non-uniform ICS's i-th component.

spline_polynomials

list of NcICS's components string representations.

B

4-element array of (n+1)x(n+4) matrices, whereas element in i-th row and j-th column of l-th matrix contains coefficient by x^{l-1} of cubic polynomial that is in i-th row and j-th column of matrix B from spline's explicit form

S=B.\gamma.

gamma

\gamma= vector of spline coefficients - function values and exterior-node derivatives that takes part in the explicit form S=B.\gamma.

aux_BF

A basis function of the spline

aux_tridiag_inverse

An inverse of the tridiagonal matrix used for spline derivatives construction

Examples

cics_explicit(
 uu = c(1, 2.2, 3, 3.8, 7),
 CERN$y[1:5],
 d=c(0,-2),
 xlab="X axis",
 ylab="Y axis"
)

uu <- c(0, 1, 4, 6);
yy <- c(4, 5, 2, 1.8);
sp <- cics_explicit(uu, yy, c(1,0))
sp$spline_polynomials
### <~~>
### Spline components' coefficients
explicit_spline(sp$B, sp$gamma)
sp$spline_coeffs == .Last.value