Contents

# 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
].
Mathematical Notation in the Data Fitting Component
Concept
Mathematical Notation
Partition of interpolation interval [
a
,
b
] , where
• x
i
denotes breakpoints.
• [
x
i
,
x
i
+1
) denotes a sub-interval (cell) of size
Δ
i
=
x
i
+1
-
x
i
.
{
x
i
}
i
=1,...,
n
, where
a
=
x
1
<
x
2
<... <
x
n
=
b
Quasi-uniform partition of interpolation interval [
a
,
b
]
Partition {
x
i
}
i
=1,...,
n
which meets the constraint with a constant
C
defined as
1
M
/
m
C
,
where
• M
= max
i
=1,...,
n
-1
(
Δ
i
)
• m
= min
i
=1,...,
n
-1
(
Δ
i
)
• Δ
i
=
x
i
+1
-
x
i
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
+1
),
i
= 1,...,
n
-1
where
• {
x
i
}
i
= 1,...,
n
is a strictly increasing sequence of breakpoints.
• P
i
(
x
) =
c
i
,0
+
c
i
,1
(
x
-
x
i
) + ... +
c
i
,
k
(
x
-
x
i
)
k
is a polynomial of degree
k
(order
k
+1) over the interval
x
∈ [
x
i
,
x
i
+1
).
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
that occurs
m
times, the equality
p
(
i
-1)
(ζ) =
ƒ
(
i
-1)
(ζ) holds for all
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
. This difference is the leading coefficient of the polynomial of order
k
+1 that agrees with
ƒ
at
x
i
,...,
x
i
+
k
.
[
x
i
,...,
x
i
+
k
]
ƒ
In particular,
• [
x
1
]
ƒ
=
ƒ
(
x
1
)
• [
x
1
,
x
2
]
ƒ
= (
ƒ
(
x
1
) -
ƒ
(
x
2
)) / (
x
1
-
x
2
)
A
k
-order derivative of interpolant
ƒ
(
x
) at interpolation site .
Interpolants to the Function
ƒ
at
x
1
,...,
x
n
and Boundary Conditions
Concept
Mathematical Notation
Linear interpolant
P
i
(
x
) =
c
1,
i
+
c
2,
i
(
x
-
x
i
),
where
• x
∈ [
x
i
,
x
i
+1
)
• c
1,
i
=
ƒ
(
x
i
)
• c
2,
i
= [
x
i
,
x
i
+1
]
ƒ
• i
= 1,...,
n
-1
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,
i
, and
c
3,
i
depend on the conditions:
• P
i
(
x
i
) =
ƒ
(
x
i
)
• P
i
(
x
i
+1
) =
ƒ
(
x
i
+1
)
• P
i
((
x
i
+1
+
x
i
) / 2) =
v
i
+1
where parameter
v
i
+1
depends on the interpolant being continuously differentiable:
P
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
• x
∈ [
t
i
,
t
i
+1
)
• {
t
i
}
i
=1,...,
n
+1
is a sequence of knots such that
• t
1
=
x
1
,
t
n
+1
=
x
n
• t
i
∈ (
x
i
-1
,
x
i
),
i
= 2,...,
n
Coefficients
c
1,
i
,
c
2,
i
,
c
3,
i
, and
d
3,
i
depend on the following conditions:
• P
i
(
x
i
) =
ƒ
(
x
i
),
P
i
(
x
i
+1
) =
ƒ
(
x
i
+1
)
• P
(
x
) is a continuously differentiable polynomial of the second degree on [
t
i
,
t
i
+1
),
i
= 1,...,
n
.
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
• x
∈ [
x
i
,
x
i
+1
)
• c
1,
i
=
ƒ
(
x
i
)
• c
2,
i
=
s
i
• c
3,
i
= ([
x
i
,
x
i
+1
]
ƒ
-
s
i
) / (
Δ
x
i
) -
c
4,
i
(
Δ
x
i
)
• c
4,
i
= (
s
i
+
s
i
+1
- 2[
x
i
,
x
i
+1
]
ƒ
) / (
Δ
x
i
)
2
• i
= 1,...,
n
-1
• s
i
=
ƒ
(1)
(
x
i
)
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
• x
∈ [