Mathematical Conventions for Data Fitting
Functions
This section explains the notation used for Data
Fitting function descriptions. Spline notations are based on the terminology
and definitions of [
deBoor2001
].
The Subbotin quadratic spline definition follows the conventions of [StechSub76
].
The quasi-uniform partition definition is based on [Schumaker2007
].
Concept
| Mathematical Notation
|
---|---|
Partition of interpolation interval [ a ,
b ] , where
| { x i i =1,...,n a =
x 1 <
x 2 <...
<x n b |
Quasi-uniform partition of interpolation interval
[ a ,
b ]
| Partition { x i i =1,...,n C defined as
1
≤ M /
m ≤ C ,
where
|
Vector-valued function of dimension
p being fit
| ƒ (x ) = (ƒ 1 (x ),...,
ƒ p x ))
|
Piecewise polynomial (PP) function
ƒ of order
k +1 | ƒ (x ) ≔
P i x ), if
x ∈ [
x i x i +1i = 1,...,
n -1
where
|
Function
p agrees
with function
ƒ at the points {x i i =1,...,n | For every point ζ in sequence { x i i =1,...,n m times,
the equality
p ( (ζ) =
i -1)ƒ ( (ζ) holds for
all
i -1)i = 1,...,m , where
p ( (i )t ) is the derivative of the
i -th
order.
|
The
k -th
divided difference of function
ƒ at points
x i x i +
k k +1 that
agrees with
ƒ at
x i x i +
k | [
x i x i +
k ƒ In particular,
|
A
k -order
derivative of interpolant
ƒ (x ) at interpolation site
![]() | ![]() |
Concept
| Mathematical Notation
|
---|---|
Linear interpolant
| P i x ) =
c 1,
+
i c 2,
(i x -
x i where
|
Piecewise parabolic interpolant
| P i x ) =
c 1,
+
i c 2,
(i x -
x i c 3,
(i x -
x i 2 ,
x ∈ [
x i x i +1 Coefficients
c 1,
,
i c 2,
, and
i c 3,
depend on the conditions:
i
where parameter
v i +1P i -1(1) (x i P i (1) (x i |
Piecewise parabolic Subbotin interpolant
| P (x ) =
P i x ) =
c 1, +i c 2, (i x -x i c 3, (i x -x i 2 +d 3, ((i x -t i + )2 ,
where
Coefficients
c 1, ,
i c 2, ,
i c 3, , and
i d 3, depend on the
following conditions:
i
|
Piecewise cubic Hermite interpolant
| P i x ) =
c 1, +
i c 2, (i x
-
x i c 3, (i x
-
x i 2
+
c 4, (i x
-
x i 3 ,
where
|
Piecewise cubic Bessel interpolant
| P i x ) =
c 1, +
i c 2, (i x
-
x i c 3, (i x
-
x i 2
+
c 4, (i x
-
x i 3 ,
where
|