Department of Mathematics
University of Houston
Numerical Analysis I
Dr. Ronald H.W. Hoppe
Numerical Mathematics I
(10. Homework Assignment)
Exercise 34
(
L
2
estimate for piecewise linear interpolation
)
The interpolation error estimates presented in class provide upper bounds for
the pointwise error in polynomial interpolation. Often, it is of interest to have
estimates of the error in specific norms as, for instance, the L
2
norm
k
f
k
L
2
([
a,b
])
:= (
b
Z
a

f
(
x
)

2
dx
)
1
/
2
.
Let Δ
h
:=
{
x
i
:=
ih
}
n
i
=0
, h
:=
b/n, n
∈
N
be a uniform partition of the intervale
[0
, b
](
b >
0). For
f
∈
C
2
([0
, b
]) we want to estimate the L
2
norm of the interpo
lation error
f

s
, where
s
represents the polygon which interpolates
f
at the
nodes
x
i
,
0
≤
i
≤
n
.
(i) Show that for
x
∈
[0
, h
] there holds

f
(
x
)

s
(
x
)
 ≤
√
3
x
√
h
(
h
Z
0

f
00
(
t
)

2
dt
)
1
/
2
.
[Hinweis: Use the integral representation of the remainder in the Taylor expansion,
the Cauchy Schwarz inequality, and the fact that for a continuously differentiable
function Φ there holds
Φ(
x
)

Φ(0) =
x
Z
0
Φ
0
(
t
)
dt .
]
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 Spring '12
 Staff
 Numerical Analysis, Department of Mathematics, Runge's phenomenon, INTERPOLATION ERROR, Dr. Ronald H.W. Hoppe

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