C Library Functions - splint (3)

from

* a practical guide to splines * by c. de boor

calls bsplvb, banfac/slv

   splint  produces the b-spline coeff.s  bcoef  of the spline of order
   k  with knots  t(i), i=1,..., n + k , which takes on the value
   gtau(i) at  tau(i), i=1,..., n .

******

i n p u t ******

tau.....array of length

n , containing data point abscissae.

a s s u m p t i o n . . .
	tau is strictly increasing

gtau.....corresponding array of length

n , containing data point or- dinates

t.....knot sequence, of length

n+k

n.....number of data points and dimension of spline space

s(k,t) k.....order of spline

******

o u t p u t ******

q.....array of size

(2*k-1)*n , containing the triangular factoriz- ation of the coefficient matrix of the linear system for the b- coefficients of the spline interpolant. the b-coeffs for the interpolant of an additional data set

(tau(i),htau(i)), i=1,...,n
	with the same data abscissae can be obtained without going through all the calculations in this
routine, simply by loading
	htau into bcoef and then execut-
ing the
	call banslv ( q, 2*k-1, n, k-1, k-1, bcoef )

bcoef.....the b-coefficients of the interpolant, of length
iflag.....an integer indicating success (= 1)

or failure (= 2) the linear system to be solved is (theoretically) invertible if and only if

t(i) .lt. tau(i) .lt. t(i+k),
	all i.

violation of this condition is certain to lead to

iflag = 2 .

******

m e t h o d ******

the i-th equation of the linear system
	a*bcoef = b for the b-co-

effs of the interpolant enforces interpolation at

tau(i), i=1,...,n.

hence,

b(i) = gtau(i), all i, and a is a band matrix with 2k-1 bands (if it is invertible).

the matrix
	a is generated row by row and stored, diagonal by di-

agonal, in the

r o w s of the array q , with the main diagonal go-

ing into row

k . see comments in the program below.

the banded system is then solved by a call to
	banfac (which con-

structs the triangular factorization for

a and stores it again in

q ), followed by a call to

banslv (which then obtains the solution

bcoef

by substitution).

banfac

does no pivoting, since the total positivity of the matrix

makes this unnecessary.

     dimension q(2*k-1,n), t(n+k)

current fortran standard makes it impossible to specify precisely the

dimension of

q and t without the introduction of otherwise super- fluous additional arguments.

Nemo Release 3.1

splint (3)

June 29, 2025

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