EGM 4313 S08
Homework Solution #3
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1. Problem Set 19.3, p.808, #3 (Quadratic interpolation)
Calculate the Lagrange polynomial
(
)
2
p
x
for the 4D-values of the Gamma function [(24),
App. 3.1]
(
)
(
)
(
)
1.00
1.0000,
1.02
0.9888,
1.04
0.9784
Γ
=
Γ
=
Γ
=
, and from it approximations of
(
)
x
Γ
for
x
= 1.005, 1.010, 1.015, 1.025, 1.030, 1.035.
Solution
:
Second-degree Lagrange polynomial
(
)
2
p
x
is given by
(
)
(
)
(
)
(
)
2
0
0
1
1
2
2
p
x
L
x
f
L
x
f
L
x
f
=
+
+
Where
(
)
(
)(
)
(
)(
)
1
2
0
0
1
0
2
x
x
x
x
L
x
x
x
x
x
−
−
=
−
−
,
(
)
(
)(
)
(
)(
)
0
2
1
1
0
1
2
x
x
x
x
L
x
x
x
x
x
−
−
=
−
−
,
(
)
(
)(
)
(
)(
)
0
1
2
2
0
2
1
x
x
x
x
L
x
x
x
x
x
−
−
=
−
−
with
0
1
2
0
1
2
1.00,
1.02,
1.04,
1.0000,
0.9888,
0.9784
x
x
x
f
f
f
=
=
=
=
=
=
.
Using the above values, we obtain
(
)
2
2
1.0000
2.5800
2.5800
p
x
x
x
=
−
+
Therefore,
(
)
x
Γ
with 4D values for each x is
(
)
1.005
0.9971
Γ
=
,
(
)
1.010
0.9943
Γ
=
,
(
)
1.015
0.9915
Γ
=
(
)
1.025
0.9861
Γ
=
,
(
)
1.030
0.9835
Γ
=
,
(
)
1.035
0.9809
Γ
=