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calculates the value of all possibly nonzero b-splines at |
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x of order
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jout = max( jhigh , (j+1)*(index-1) )
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****** |
i n p u t ******
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t.....knot sequence, of length | | |
left + jout , assumed to be nonde-
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creasing. | | |
a s s u m p t i o n . . . .
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t(left) |
[char46]lt. t(left + 1) .
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d i v i s i o n | | |
b y z e r o will result if t(left) = t(left+1)
jhigh,
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index.....integers which determine the order | | |
jout = max(jhigh,
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(j+1)*(index-1)) | | |
of the b-splines whose values at x are to
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be returned. | | |
index is used to avoid recalculations when seve-
ral columns of the triangular array of b-spline values are nee-
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ded (e.g., in | | |
bsplpp or in bsplvd ). precisely,
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if |
index = 1 ,
the calculation starts from scratch and the entire triangular
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array of b-spline values of orders 1,2,...,jhigh | | |
is generated
order by order , i.e., column by column .
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only the b-spline values of order | | |
j+1, j+2, ..., jout are ge-
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nerated, the assumption being that | | |
biatx , j , deltal , deltar
are, on entry, as they were on exit at the previous call.
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in particular, if | | |
jhigh = 0, then jout = j+1, i.e., just
the next column of b-spline values is generated.
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w a r n i n g . . . | | |
the restriction jout .le. jmax (= 20) is im-
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posed arbitrarily by the dimension statement for | | |
deltal and
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deltar |
below, but is n o w h e r e c h e c k e d for .
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x.....the point at which the b-splines are to be evaluated.
left.....an integer chosen (usually) so that
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t(left) .le. x .le. t(left+1) | | |
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****** |
o u t p u t ******
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biatx.....array of length | | |
jout , with biatx(i) containing the val-
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ue at |
x of the polynomial of order jout which agrees with
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the b-spline | | |
b(left-jout+i,jout,t) on the interval (t(left),
t(left+1)) .
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****** |
m e t h o d ******
the recurrence relation
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x - t(i) t(i+j+1) - x
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b(i,j+1)(x) | | |
= -----------b(i,j)(x) + ---------------b(i+1,j)(x)
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t(i+j)-t(i) | | |
t(i+j+1)-t(i+1)
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is used (repeatedly) to generate the (j+1)-vector |
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b(left-j,j+1)(x),
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...,b(left,j+1)(x) | | |
from the j-vector b(left-j+1,j)(x),...,
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b(left,j)(x), storing the new values in | | |
biatx over the old. the
facts that
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b(i,1) = 1 | | |
if t(i) .le. x .lt. t(i+1)
and that
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b(i,j)(x) = 0 | | |
unless t(i) .le. x .lt. t(i+j)
are used. the particular organization of the calculations follows al-
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gorithm | | |
(8) in chapter x of the text.
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real biatx(jhigh),t(*),x, deltal(20),deltar(20),saved,term
dimension biatx(jout), t(left+jout)
current fortran standard makes it impossible to specify the length of
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t |
and of biatx precisely without the introduction of otherwise
superfluous additional arguments.
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