bsplvb Subroutine

BSPLVB evaluates all possibly nonzero B-splines at X of order

  JOUT = MAX ( JHIGH, (J+1)*(INDEX-1) )

with knot sequence T.

The recurrence relation

                 X - T(I)               T(I+J+1) - X
B(I,J+1)(X) = ----------- * B(I,J)(X) + --------------- * B(I+1,J)(X)
              T(I+J)-T(I)               T(I+J+1)-T(I+1)

is used to generate B(LEFT-J:LEFT,J+1)(X) from B(LEFT-J+1:LEFT,J)(X)
storing the new values in BIATX over the old.

The facts that

  B(I,1)(X) = 1  if  T(I) <= X < T(I+1)

and that

  B(I,J)(X) = 0  unless  T(I) <= X < T(I+J)

are used.

The particular organization of the calculations follows 
algorithm 8 in chapter X of the text.

14 February 2007

Carl DeBoor

Carl DeBoor,
A Practical Guide to Splines,
Springer, 2001,
ISBN: 0387953663,
LC: QA1.A647.v27.

Input, real ( kind = 8 ) T(LEFT+JOUT), the knot sequence.  T is assumed to 
be nondecreasing, and also, T(LEFT) must be strictly less than 
T(LEFT+1).

Input, integer ( kind = 4 ) JHIGH, INDEX, determine the order 
JOUT = max ( JHIGH, (J+1)*(INDEX-1) )  
of the B-splines whose values at X are to be returned.  
INDEX is used to avoid recalculations when several 
columns of the triangular array of B-spline values are
needed, for example, in BVALUE or in BSPLVD.
If INDEX = 1, the calculation starts from scratch and the entire 
triangular array of B-spline values of orders
1, 2, ...,JHIGH is generated order by order, that is, 
column by column.
If INDEX = 2, only the B-spline values of order J+1, J+2, ..., JOUT  
are generated, the assumption being that BIATX, J, 
DELTAL, DELTAR are, on entry, as they were on exit 
at the previous call.  In particular, if JHIGH = 0, 
then JOUT = J+1, that is, just the next column of B-spline 
values is generated.
Warning: the restriction  JOUT <= JMAX (= 20) is
imposed arbitrarily by the dimension statement for DELTAL
and DELTAR, but is nowhere checked for.

Input, real ( kind = 8 ) X, the point at which the B-splines 
are to be evaluated.

Input, integer ( kind = 4 ) LEFT, an integer chosen so that 
T(LEFT) <= X <= T(LEFT+1).

Output, real ( kind = 8 ) BIATX(JOUT), with BIATX(I) containing the
value at X of the polynomial of order JOUT which agrees 
with the B-spline B(LEFT-JOUT+I,JOUT,T) on the interval 
(T(LEFT),T(LEFT+1)).