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from |
* a practical guide to splines * by c. de boor
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input ***************************
n = number of data points. assumed to be .ge. 2.
(tau(i), c(1,i), i=1,...,n) = abscissae and ordinates of the
data points. tau is assumed to be strictly increasing.
ibcbeg, ibcend = boundary condition indicators, and
c(2,1), c(2,n) = boundary condition information. specifically,
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ibcbeg = 0 | | |
means no boundary condition at tau(1) is given.
in this case, the not-a-knot condition is used, i.e. the
jump in the third derivative across tau(2) is forced to
zero, thus the first and the second cubic polynomial pieces
are made to coincide.)
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ibcbeg = 1 | | |
means that the slope at tau(1) is made to equal
c(2,1), supplied by input.
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ibcbeg = 2 | | |
means that the second derivative at tau(1) is
made to equal c(2,1), supplied by input.
ibcend = 0, 1, or 2 has analogous meaning concerning the
boundary condition at tau(n), with the additional infor-
mation taken from c(2,n).
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output **************************
c(j,i), j=1,...,4; i=1,...,l (= n-1) = the polynomial coefficients
of the cubic interpolating spline with interior knots (or
joints) tau(2), ..., tau(n-1). precisely, in the interval
(tau(i), tau(i+1)), the spline f is given by
f(x) = c(1,i)+h*(c(2,i)+h*(c(3,i)+h*c(4,i)/3.)/2.)
where h = x - tau(i). the function program *ppvalu* may be
used to evaluate f or its derivatives from tau,c, l = n-1,
and k=4.
****** a tridiagonal linear system for the unknown slopes s(i) of
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f |
at tau(i), i=1,...,n, is generated and then solved by gauss elim-
ination, with s(i) ending up in c(2,i), all i.
c(3,.) and c(4,.) are used initially for temporary storage.
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