Homework 6 Foundations of Computational Math 2 Spring
2011
Solutions will be posted Monday, 2/28/11
Problem 6.1
Consider a minimax approximation to a function
f
(
x
) on [
a, b
]. Assume that
f
(
x
) is contin
uous with continuous Frst and second order derivatives. Also, assume that
f
′′
(
x
)
<
0 on for
a
≤
x
≤
b
, i.e.,
f
is concave on the interval.
6.1.a
. Derive the equations you would solve to determine the linear minimax approx
imation,
p
1
(
x
) =
αx
+
β
, to
f
(
x
) on [
a, b
] and describe their use to solve the
problem.
6.1.b
. Apply your approach to determine
p
1
(
x
) =
αx
+
β
for
f
(
x
) =

x
2
on [

1
,
1].
6.1.c
. How does
p
1
(
x
) relate to the quadratic monic Chebyshev polynomial
t
2
(
x
)?
6.1.d
. Apply your approach to determine ˜
p
1
(
x
) = ˜
αx
+
˜
β
for
f
(
x
) =

x
2
on [0
,
1].
6.1.e
. How could the quadratic monic Chebyshev polynomial
t
2
(
y
) on

1
≤
y
≤
1 be
used to provide and alternative derivation of ˜
p
1
(
x
) on 0
≤
x
≤
1?
6.1.f
. Suppose you adapt your approach to derive a constant approximation,
p
0
(
x
).
What points will you use as the extrema of the error?
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 Spring '11
 gallivan
 Polynomials, Derivative, TI, Chebyshev polynomial t2

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