Graphics Programs Reference
In-Depth Information
across aresistor. The energy
E
dissipatedbythe resistoris
∞
R
[
i
(
t
)
]
2
dt
E
=
0
Find
E
using the data
i
0
=
100 A,
R
=
0
.
5
and
t
0
=
0
.
01 s.
6.4
Gaussian Integration
Gaussian Integration Formulas
Wefound thatNewton-Cotes formulasfor approximating
a
f
(
x
)
dx
work best if
f
(
x
)
is a smooth function,such as apolynomial. This is also truefor Gaussian quadrature.
However,Gaussian formulas are also goodat estimating integrals of the form
b
w
(
x
)
f
(
x
)
dx
(6.15)
a
where
w
(
x
), called the
weighting function
,can contain singularities, aslong as they
are integrable.Anexample of such an integral is
0
(1
x
2
)ln
x dx
+
.
Sometimes infinite
limits, as in
0
e
−
x
sin
x dx
,can also be accommodated.
Gaussian integration formulashave the sameform asNewton-Cotes rules:
n
I
=
A
i
f
(
x
i
)
(6.16)
i
=
1
where, as before,
I
represents the approximation to the integral in Eq. (6.15). The
difference lies in the way that the weights
A
i
and nodal abscissas
x
i
are determined. In
Newton-Cotes integration the nodes wereevenly spacedin(
a
b
), i.e., their locations
were predetermined. In Gaussian quadrature the nodes and weights are chosen so
that Eq. (6.16) yields the exact integral if
f
(
x
) is apolynomialofdegree 2
n
,
−
1 or less;
that is,
b
n
w
(
x
)
P
m
(
x
)
dx
=
A
i
P
m
(
x
i
),
m
≤
2
n
−
1
(6.17)
a
i
=
1
One way of determining the weights and abscissas istosubstitute
P
1
(
x
)
=
1
,
P
2
(
x
)
=
x
2
n
−
1
in Eq. (6.17) and solve the resulting 2
n
equations
b
x
,...,
P
2
n
−
1
(
x
)
=
n
A
i
x
i
w
(
x
)
x
j
dx
=
,
j
=
0
,
1
,...,
2
n
−
1
a
i
=
1
for the unknowns
A
i
and
x
i
,
i
=
1
,
2
,...,
n
.
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