E X E R C I S E S E T 2.1
1.
2.
3.
4.
Use the Bisection method to find p3 for f (x) = f i - cosx on [0, 11.
Let f (x) = 3(x l)(x - ;)(x - 1). Use the Bisection method on the following intervals to findp,.
a. [-2, 1.51
b. [- 1.25,2.5]
Use the Bisection method to find solutions accurate to within lo-' for x4 - 2x3 - 4x2 4x 4 = 0
on each interval.
a. [-2, -11
b. [o, 21
C.
LA31
d. [-1,0]
Use the Bisection method to find solutions accurate to within
for x3 - 7x2 14x - 6 = 0 on
each interval.
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+
[O, 11
b. [1,3.2]
C.
[3.2,4]
Use the Bisection method to find solutions, accurate to within
for the following problems.
a. 3 x - e y = O f o r 1 5 x 5 2
b. 2 x f 3 c o s x - d = O
for0 j x s 1
c. x 2 - 4 x + 4 - I n x = O
f o r 1 5 x 2 and 2 5 x 5 4
d. x + 1 - 2 s i n i r x = O
f o r O s x 0 . 5 and 0 . 5 5 ~ 5 1
Use the Bisection method to find solutions accurate to within
for the following problems.
a. x - 2 - " = O
forOsx5 1
b. e r - x 2 + 3 x - 2 = 0
forO(xs1
c. 2 x c 0 ~ ( 2 w ) - ( x + l ) ~ = O f o r - 3 5 x 5 - 2
and - 1 5 x 5 0
d. x c o s x - 2 x 2 + 3 x - 1 = 0
f o r 0 . 2 I : x 5 0 . 3 and 1 . 2 5 x 5 1 . 3
a.
5.
6.
E X E R C I S E S E T 2.2
1.
Use algebraic manipulation to show that each of the following functions has a fixed point a t p precisely
when f (p) = 0, where f ( x ) = x4 2x2 - .u - 3.
+
2.2
Fixed-Point Iteration
65
a.
Perform four iterations, if possible, on each of the functions g defined in Exercise 1. Let po = 1
a n d p , ~ , ~= g(p,,), for rz = 0, 1,2,3.
b.
Which function do you think gives the best approximation to the solution?
The following four methods are proposed to compute 7'1'. Rank them in order, based on their apparent
speed of convergence, assuming po = 1.
The following four methods are proposed to compute 21'1" Rank them in order, based on their
apparent speed of convergence, assuming po = 1.
112
d.
(E)
p,, = 1711-1
Use a fixed-point iteration method to determine a solution accurate to within
on [I, 21. Usepo = 1.
for x3 - x - 1 = 0
Use a fixed-point iteration method to determine a solution accurate to within lo-' forx4 - 3x2 - 3 = 0
on [1,2]. Usepo = 1.
+
Use Theorem 2.3 to show that g(x) = IT 0.5 sin(s/2) has a unique fixed point on [O,2n]. Use
fixed-point iteration to find an approximation to the fixed point that is accurate to within lop2. Use
Corollary 2.5 to estimate the number of iterations required to achieve lo-' accuracy, and compare
this theoretical estimate to the number actually needed.
Use Theorem 2.3 to show that g(x) = 2-" has a unique fixed point on [f , 11. Use fixed-point iteration
to find an approximation to the fixed point accurate to within
Use Corollary 2.5 to estimate the
number of iterations required to achieve
accuracy, and compare this theoretical estimate to the
number actually needed.
Use a fixed-point iteration method to find an approximation to f i that is accurate to within
Compare your result and the number of iterations required with the answer obtained in Exercise 12
of Section 2.1.
Use a fixed-point iteration method to find an approximation to -,?%that is accurate to within
Compare your result and the number of iterations required with the answer obtained in Exercise 13
of Section 2.1.
For each of the following equations, determine an interval [o, b] on which fixed-point iteration will
converge. Estimate the number of iterations necessary to obtain approximations accurate to within
and perform the calculations.
2-er+x'
5
a. x =
b. x = - + 2
3
x2
c. x = (cY/3)'12
d. x = S - "
e. x = 6-"
f. s = O.S(sinx cosx)
For each of the following equations, use the given interval or determine an interval [a,O] on which
fixed-point iteration will converge. Estimate the number of iterations necessary to obtain approximations accurate to within lo-', and perform the calculations.
a. 2 sinx - x = 0 use [2,3]
b. s3- 2x - 5 = 0 use [2,3]
C.
3x2 - e): = 0
d. x - c o s x = O
Find all the zeros o f f (x) = x2 10 cosx by using the fixed-point iteration method for an appropriate
iteration function g. Find the zeros accurate to within
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