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CE 341/441 - Lecture 5 -Fall 2004
p. 5.
LECTURE 5INTRODUCTION TO INTERPOLATION • Interpolation function: a function that passes exactly through a set of data points. • Interpolating functions to interpolate values in tables
• In tables, the function is only specified at a limited number or discrete set of indepen-
dent variable values (as opposed to a continuum function).
• We can use interpolation to find functional values at other values of the independent
variable, e.g. sin(0.63253)
x^
sin(
x )
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
• In numerical methods, like tables, the values of the function are only specified at a
discrete number of points! Using interpolation, we can describe or at least approximatethe function at every point in space.
• For numerical methods, we use interpolation to
• Interpolate values from computations• Develop numerical integration schemes• Develop numerical differentiation schemes• Develop finite element methods
• Interpolation is typically not used to obtain a functional description of measured data
since errors in the data may lead to a poor representation.
• Curve fitting to data is handled with a separate set of techniques
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
• If
is a linear function then
(1)
where
and
are unknown coefficients
• To pass through points
and
we must have:
⇒
(2)
⇒
(3)
• 2 unknowns and 2 equations
⇒
solve for
• Using (2)
Substituting into (3)
g x
(^
g x
(^
)^
Ax
B
A
B
x^ o
f^
x^ ( o
(^
)^
x^1
f^
x ( 1
(^
g x
o (^
)^
f^
x^ ( o
Ax
o^
B
f^
x^ ( o
g x
1 (^
)^
f^
x ( 1
Ax
1
B
f^
x ( 1
A B
B
f^
x^ ( o
)^
Ax
o
A
x^1
f^
x^ ( o
)^
Ax
o
f^
x ( 1
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
• Substituting for
and
into equation (1)
This is the formula for linear interpolation
A
f^
x ( 1
)^
f^
x^ ( o
x^1
x^ o
B
f^
x^ ( o
) x
1
f^
x ( 1
) x
o
x^1
x^ o
A
B^ g x
(^
)^
f^
x^ o (^
)^
x 1
x
(^
x 1
x^ o
(^
-^
f^
x (^1
)^
x^
x^ o
(^
x 1
x^ o
(^
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
Error for Linear Interpolating Functions • Error is defined as:•^
represents the difference between the exact function
and the interpolating or
approximating function
• We note that at the interpolating points
and
• This is because at the interpolating point we have by definition
e x (
)^
f^
x (
)^
g x
(^
e x (
)^
f^
x ( )
g x
(^
x^ o
x^1
e x
o (^
)^
e x
1 (^
)^
g x
o (^
)^
f^
x^ ( o
g x
1 (^
)^
f^
x ( 1
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
Derivation of
e(x)
Step 1 • Expand
in Taylor Series (T.S.) about
where
(4)
• The third term is the actual remainder term and represents all other terms in the series
since
it is evaluated at
x
f(x
f(x
x^0
x^1
f(x)
g(x)
f(x)
e x (
)^
f^
x (
)^
g x
(^
f^
x ( )
x^ o
f^
x ( )
f^
x^ ( o
)^
x^
x^ o
(^
df ) ----- dx
x^
x^ o
x^
x^ o
(^
d - 2
f d x
2
x^
ξ
x^ o
ξ^
x
x^
ξ
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
Step 3 • Substitute Equation 7 into T.S. form of
, Equation (4).
• This gives us an expression for
in terms of the discrete values
and
(8)
⇒
(9)
⇒
(10)
⇒
(11)
f^
x ( )
f^
x (
)^
f^
x^ ( o
)^
f^
x ( 1
f^
x ( )
f^
x^ ( o
)^
x^
x^ o
(^
f^
x ( 1
x^1
x^ o
(^
-^
f^
x^ ( o
x^1
x^ o
(^
–^
x^1
x^ o
(^
d - 2 f d x
2
x^
ξ
–^
x^
x^ o
(^
d - 2 f d x
2
x^
ξ
f^
x ( )
f^
x^ ( o
)^
x^
x^ o
(^
x^1
x^ o
(^
-^
f^
x ( 1
)^
x^
x^ o
(^
x^1
x^ o
(^
-^
f^
x^ ( o
–^
x^
x^ o
(^
)^
x
-^
1
x^ o
(^
-^
x^
x^ o
(^
d 2 f d x
2
x^
ξ
f^
x ( )
x^1
x^ o
-^
x
-^
x^ o
(^
)^
f^
x^ ( o
x^1
x^ o
(^
-^
x^
x^ o
(^
)^
f^
x ( 1
x^1
x^ o
(^
-^
x^1
-^
x^ o
x^
x^ o
(^
)^
x^
x^ o
(^
d - 2 f d x
2
x^
ξ
f^
x (
)^
f^
x^ ( o
)^
x 1
x
x^ o
----------------
-^
f^
x (^1
)^
x^
x^ o
x^ o
----------------
-^
x^
x^ o
(^
)^
x^
x 1
(^
d - 2 f d x
2 --------
x^
ξ =
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
• The first part of Equation (11) is simply the linear interpolation formula. The second
part is in fact the error. Thus:
⇒ ⇒
• If we assume that the interval
is small, then the second derivative won’t change
dramatically in the interval!
where
e x (
)^
f^
x ( )
g x
(^
e x (
)^
f^
x^ ( o
)^
x^1
x
x^ o
-^
f^
x ( 1
)^
x^
x^ o
x^ o
-^
x^
x^ o
(^
)^
x^
x^1
(^
d -
2 f d x
2
x^
ξ
f^
x^ ( o
–^
x^1
x
x^ o
-^
f^
x ( 1
–^
x^
x^ o
x^ o
e x (
)^
x^
x^ o
(^
)^
x^
x 1
(^
d - 2
f d x
2 --------
x^
ξ =
x^ o^
ξ^
x 1
x^ o
x^1 , [^
]
d 2 f d x
2
x^
ξ
d 2 f d x
2
x^
x^ o
d 2
f d x
2
x^
x^1
d 2
f d x
2
x^
x^ m
x^ m
x^ o
x^1 +^2
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
• Thus • Notes on Error for linear interpolation
• The error expression has a polynomial and a derivative portion.• Maximum error occurs approximately at the midpoint between
and
• Error increases as the interval
increases
• Error increases as
increases. Again note that
can be approximated
with finite difference (F.D.) formulae if at least 3 surrounding functional values areavailable. (We will discuss F.D. formulae later.)
max
e x
(^
)^
x^0
x^
x^1
<^
<
h
d 2 f d x
2 --------
x^ m
x^ o^
x^1
h
f^
(^2) ( )
x (
)^
f^
(^2) ( )
x (
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
Example 2 • Compute an error estimate for the problem in Example 1.• Recall we found that• Error is estimated as:• Since
is the midpoint at the interval
, we have
⇒
g^
)^
e x (
)^
1 ---^2
x^
x^ o
(^
)^
x^
x^1
(^
d ) 2 f d x
2
x^
x^ m
x^
[^
]
e^
)^
(^
)^
(^
d ) 2
f d x
2
x^
=
e^
)^
d 2 f d x
2
x^
=
