Graphics Programs Reference
In-Depth Information
=
Similarly,
x
x
i
+
2
yields
P
2
[
x
i
,
x
i
+
1
,
=
P
1
[
x
i
+
1
,
=
x
i
+
2
]
x
i
+
2
]
y
i
+
2
=
Finally, when
x
x
i
+
1
wehave
P
1
[
x
i
,
=
P
1
[
x
i
+
1
,
=
x
i
+
1
]
x
i
+
2
]
y
i
+
1
so that
(
x
i
+
1
−
x
i
+
2
)
y
i
+
1
+
(
x
i
−
x
i
+
1
)
y
i
+
1
P
2
[
x
i
,
x
i
+
1
,
=
=
x
i
+
2
]
y
i
+
1
x
i
−
x
i
+
2
Having established the pattern, wecan nowdeduce
the general recursive
formula:
P
k
[
x
i
,
x
i
+
1
,...,
x
i
+
k
]
(3.8)
(
x
−
x
i
+
k
)
P
k
−
1
[
x
i
,
x
i
+
1
,...,
x
i
+
k
−
1
]
+
(
x
i
−
x
)
P
k
−
1
[
x
i
+
1
,
x
i
+
2
,...,
x
i
+
k
]
=
x
i
+
k
Given the valueof
x
, the computationscan becarried out in the following tabular
format (shown for four datapoints):
x
i
−
k
=
0
k
=
1
k
=
2
k
=
3
x
1
P
0
[
x
1
]
=
y
1
P
1
[
x
1
,
x
2
]
P
2
[
x
1
,
x
2
,
x
3
]
P
3
[
x
1
,
x
2
,
x
3
,
x
4
]
x
2
P
0
[
x
2
]
=
y
2
P
1
[
x
2
,
x
3
]
P
2
[
x
2
,
x
3
,
x
4
]
x
3
P
0
[
x
3
]
=
y
3
P
1
[
x
3
,
x
4
]
x
4
P
0
[
x
4
]
=
y
4
Table 3.2
neville
This algorithmworks with the one-dimensional array
y
, which initially contains the
y
-values of the data (the second column in Table 3.2).Each pass through the for-
loop computes the terms in next column of the table, which overwrite the previous
elements of
y
.At the end of the procedure,
y
contains the diagonalterms of the table.
The valueoftheinterpolant(evaluatedat
x
)that passes through all the datapoints is
y
1
, the first elementof
y
.
function yInterp = neville(xData,yData,x)
% Neville's polynomial interpolation;
% returns the value of the interpolant at x.
Search WWH ::
Custom Search