Developer Reference for Intel® oneAPI Math Kernel Library - C

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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