Jacobs University, Bremen
School of Engineering and Science
Prof. Dr. Lars Linsen, Orif Ibrogimov
Spring Term 2010
Homework 9
120202: ESM4A  Numerical Methods
Homework Problems
9.1.
Least squares method
Let
f
(
x
) =
a
(
x
+ 1)
2
+
b
sin
πx
+
c
cos
πx
3
with
f
(0) = 0
, f
(
1
2
) = 1 and
f
(1) = 1.
(a) Derive the normal equations for the best approximative solution to
a
,
b
, and
c
in the
leastsquares sense.
(b) Find the leastsquares solution for
a
,
b
, and
c
.
(
? points
)
9.2.
Differentation
&
Richardson extrapolation
(a) Compute estimates for the derivative of function
f
(
x
) = (
x

1)(
x

2)(
x

3)(
x

4)(
x

5) at
x
= 0 using forward and central differencing as well as using the estimate you obtain when
applying two iterations of Richardson extrapolation to the central differencing estimate.
Choose
h
= 0
.
1. Compare the absolute errors.
(b) Derive an estimate for computing the secondorder derivative of a continuous function
f
with error term being
O
(
h
4
).
(c) Derive the error term for the approximation:
f
(
x
)
≈
f
(
x
)

2
f
(
x
+
h
)+
f
(
x
+2
h
)
h
2
(d) Derive a numerical differentation formula of order
O
(
h
4
) by applying Richardson’s ex
trapolation to
f
(
x
) =
f
(
x
+
h
)

f
(
x

h
)
2
h

h
2
6
f
(3)
(
x
)

h
4
4!
f
(5)
(
x
)

. . .
Give the error term of
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 Spring '10
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 Least Squares, Regression Analysis, error term, central differencing estimate

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