Graphics Programs Reference
In-Depth Information
EXAMPLE 4.9
Find a solution of
y
2
sin
x
+
+
ln
z
−
7
=
0
2
y
z
3
3
x
+
−
+
1
=
0
x
+
y
+
z
−
5
=
0
using
newtonRaphson2
.Start with the point(1
,
1
,
1).
Solution
Letting
x
=
x
1
,
y
=
x
2
and
z
=
x
3
, the code defining the function array
f
(
x
) is
functiony=fex4
9(x)
% Function used in Example 4.9
y=[sin(x(1))+x(2)ˆ2+log(x(3))-7;...
3*x(1) + 2ˆx(2) - x(3)ˆ3 + 1;
_
...
x(1) + x(2) + x(3) - 5];
The solution can nowbeobtainedwith the single command
_
>> newtonRaphson2(@fex4
9,[1;1;1])
which results in
ans =
0.5991
2.3959
2.0050
Hence the solutionis
x
=
0
.
5991,
y
=
2
.
3959 and
z
=
2
.
0050.
PROBLEM SET 4.1
1.
Use the Newton-Raphson
me
thod and a four-function calculator (
+−×÷
oper-
√
75 with four significant figure accuracy.
2. Find the smallest positive(real) root of
x
3
ationsonly)tocompute
3
23
x
2
−
3
.
−
5
.
54
x
+
9
.
84
=
0 by the
method of bisection.
3.
The smallest positive, nonzero root of cosh
x
cos
x
−
1
=
0lies in the interval(4
,
5)
.
Compute this root by Brent's method.
4.SolveProb.3bythe Newton-Raphsonmethod.
5. A root of the equation tan
x
−
tanh
x
=
0lies in (7
.
0
,
7
.
4)
.
Find this root with three
decimal place accuracybythe method of bisection.
6. Determine the two roots of sin
x
+
−
=
−
,
3 cos
x
2
0thatlie in the interval(
2
2).
Use the Newton-Raphsonmethod.
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