Spline Toolbox | ![]() ![]() |
Linear System to Be Solved
Since , we convert this last system into a system for the B-spline coefficients of
. This requires the values, first, and second derivatives at every
and for all the relevant B-splines. The command
spcol
was expressly written for this purpose.
We use spcol
to supply the matrix
From this, we get the collocation matrix by combining the row triple of colmat
for using the weights
to get the row for
of the actual matrix. For this, we need a current approximation
. Initially, we get it by interpolating some reasonable initial guess from our piecewise-polynomial space at the
sites
. We use the parabola (i.e., the function
) which satisfies the end conditions as the initial guess, and pick the matrix from the full matrix
colmat
. Here it is, in several cautious steps:
intmat = colmat([2 1+[1:8]*3,1+9*3],:); coefs = intmat\[0 colpnts.*colpnts-1 0].'
; y = spmak(knots,coefs.'
);
![]() | Linearization | Iteration | ![]() |