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CE 341/441 - Lecture 8 - Fall 2004
p. 8.
LECTURE 8INTERPOLATION USING CHEBYSHEV ROOTS • For both Lagrange and Newton interpolation through
data points (
degree
polynomial which fits through all
data points).
⇒
- Typically over a small interval,
won’t change dramatically (although strictly
speaking
does
depend on
portion of
since we don’t know
and we can’t
specify
N
N
th
N
e x
x
x
o
x
x
1
x
x
2
x
x
N
N
f
N
1
(
)
ξ (
x
o
ξ
x
N
e x
N
x
x
i
f
N
1
(
)
ξ (
i
0 N =
x
o
ξ
x
N
f
N
1
(
)
ξ (
ξ
x
f
N
1
(
)
ξ (
f
N
1
(
)
x
m
(
where
x
m
x
o
x
N
+^2
f
N
1
(
)
ξ (
e x
f
ξ
CE 341/441 - Lecture 8 - Fall 2004
p. 8.
- The error is for the most part controlled by•
is small in the center of the interval
, but large within the end zones.
N=
, we examine a plot of
- Our objective is to minimize
by minimizing
by selecting a “special”
set of non-equispaced interpolating points
x
x
i
i^
1 N =
x
x
i
i^
0
N =
x
o
x
N
[
]
x
x
i
i^
0 (^5) =
1
4
0
2
3
5
Error small Error large
e x
x
x
i
i^
1 N =
CE 341/441 - Lecture 8 - Fall 2004
p. 8.
- Second Degree Chebyshev Polynomial:
- From the CRC Math Handbook: Double Angle Relation
⇒
⇒
- Third Degree Chebyshev Polynomial:
- From the CRC Math Handbook: Multiple Angle Relation
⇒
T
^2
x (
cos
x
cos
α
cos
cos
α
T
^2
x (
cos
x
cos
T
^2
x (
cos
x
cos [
]
cos
x
cos [
]
T
^2
x (
x
2
T
^3
cos
x
cos
α
cos
cos
α
α
cos
T
^3
x (
cos
x
cos [
]
3
cos
x
cos
T
^3
x (
x
3
x
CE 341/441 - Lecture 8 - Fall 2004
p. 8.
- In general recursive relationship can be developed• Chebyshev polynomials can be normalized
, the coefficient of the highest degree polynomial term equals
unity. For example:
T
j
x (
xT
j^
1
x (
T
j^
2
x (
ψ
j
x (
T
j
x (
j
1
ψ
j
x (
ψ
3
x (
T
^3
x (
(^2)
x
3
x
CE 341/441 - Lecture 8 - Fall 2004
p. 8.
where the
n
values are selected such that all roots falling within the range
are
defined. Extending the range for
n
will only lead to repeated roots. Thus
are the roots of
we may write the following
⇒
−
cos
π--- 2
π 2
π 2
N
cos
x
n
1
c
N
n
π
n
N
x
x
n
1
c
N
n
π
N
cos
n
N
x
c o
x
c 1
x
c 2
ψ
N
1
x (
ψ
N
1
x (
x
x
c o
x
x
c 1
x
x
c N
ψ
N
1 +
x (
x
x
c i
i^
0
N =
CE 341/441 - Lecture 8 - Fall 2004
p. 8.
Example • Find the Chebyshev roots for the
degree Chebyshev polynomial
- The roots are computed as:
N
ψ
3
x (
x
3
x
x
c o
π ⋅
cos
π
cos
x
c 1
π ⋅
cos
π--- 2
cos
x
c 2
π ⋅
cos
CE 341/441 - Lecture 8 - Fall 2004
p. 8.
Application of Chebyshev Roots as Interpolation Points •
In general if
are the roots of
, then
- For Lagrange interpolation through
data points, the error function is expressed as:
Thus if we select the roots of the
degree Chebyshev polynomial as our interpo-
lation (or data) points for Lagrange Interpolation (or any
degree polynomial inter-
polation scheme with variably spaced data points)
x
c o
x
c 1
x
c 2
ψ
N
1
x (
ψ
N
1 +
x (
x
x
c i
i
0
N =
N
e x
x
x
o
x
x
1
x
x
2
x
x
N
N
f
N
1
(
)
ξ (
N
N
e
c
x (
N
ψ
N
1 +
x (
f
N
1 +
(
)
ξ (
CE 341/441 - Lecture 8 - Fall 2004
p. 8.
- Notes:
- This error estimate is only good for the case where the roots of the Chebyshev poly-
nomial are used as interpolation points and the interval is
(the normalized Chebyshev polynomial) is mini-
mized to
we have effectively minimized the maximum error
over the interval (as far as
we can)!
- The distribution of error is now more even on the interval.•
We haven’t entirely minimized
since
depends on
and we can’t do much
about this.
x
ψ
N
1
x (
N
ψ
N
1
x (
N
e x
e x
ξ
x
CE 341/441 - Lecture 8 - Fall 2004
p. 8.
- Substituting in values for • Note that for
and
we are strictly speaking extrapolating, not
interpolating.
- However since we have carefully placed the points, we will not incur excessive
errors in these extrapolated ranges.
x
c o
x
c 1
and
x
c 2
g x
f
o
x
x
f
^1
x
x
f
^2
x
x
-
+
f
1
f
3
f
0
0
[
]
[
]
CE 341/441 - Lecture 8 - Fall 2004
p. 8.
- The maximum error over the interval may be estimated as (
⇒
⇒
- However we noted that• Thus over the interval
⇒
can be estimated using a forward/backward difference formula
N
e
c
x (
ψ
3
x (
f
3 (
)
ξ (
ξ
e
c
x (
ψ
3
x (
f
3 (
)
x
m
(
x
m
max e
c
x (
max
ψ
3
x (
f
(^3)
x
m
(
N
ψ
N
1
x (
N
max e
c
x (
2
f
3 (
)
x
m
(
max e
c
x (
f
3 (
)
x
m
f
3 (
)
x
m
(
