Question

Consider a function x^{2} − 3 = 0 . Then with the
starting point x_{0} = 1 , if we perform three iterations
of Newton-Rhapson Method, we have the following:

(Note that your answer format should be x.xxxx. For example, 2 ->, 2.0000 or 1.34 -> 1.3400, or 1.23474 - > 1.2374, or 1.23746->1.2375)

**x _{1} =**

**x _{2} =**

**x _{3} =**

Answer #1

Consider the function
g (x) = 12x + 4 - cos x. Given
g (x) = 0 has a unique solution
x = b in the interval (−1/2, 0), and you can use this
without justification.
(a) Show that Newton's method of starting point
x0
= 0 gives a number sequence with
b <··· <xn+1
<xn
<··· <x1
<x0
= 0
(The word "curvature" should be included in the argument!)
(b) Calculate
x1
and x2.
Use theorem 2 in section...

Consider the function g (x) = 12x + 4 - cos x.
Given g (x) = 0 has a unique solution x =
b in the interval (−1/2, 0), and you can use this without
justification.
(a) Show that Newton's method of starting point
x0 = 0 gives a number sequence with
b <··· <xn+1 <xn <···
<x1 <x0 = 0
(The word "curvature" should be included in the argument!)
(b) Calculate x1 and x2. Use
theorem 2 in section...

Consider the function g (x) = 12x + 4 - cos x.
Given g (x) = 0 has a unique solution x =
b in the interval (−1/2, 0), and you can use this without
justification.
(a) Show that Newton's method of starting point
x0 = 0 gives a number sequence with
b <··· <xn+1 <xn <···
<x1 <x0 = 0
(The word "curvature" should be included in the argument!)
(b) Calculate x1 and x2. Use
theorem 2 in section...

For the following function, determine the highest real root
of
f(x) = 2x3 – 11.7x2 + 17.7x - 5
by using (a) graphical methods, (b) fixed point iteration (three
iterations, x0 = 3) (Hint: Be certain that you develop a solution
that converges on the root), and (c) Newton-Raphson method (three
iterations, x0 = 3).
Perform an error check on each of your final root approximations
(e.g. for the last of the three iterations).

Part A.
Consider the nonlinear equation
x5-x=15
Attempt to find a root of this equation with Newton's method
(also known as Newton iteration).
Use a starting value of x0=4 and apply Newton's
method once to find x1
Enter your answer in the box below correct to four
decimal places.
Part B.
Using the value for x1 obtained in Part A, apply
Newton's method again to find x2
Note you should not round x1 when computing
x2

Consider three positive integers, x1, x2, x3, which satisfy the
inequality below:
x1 + x2 + x3 = 17.
Let’s assume each element in the sample space (consisting of
solution vectors (x1, x2, x3) satisfying the above conditions) is
equally likely to occur. For example, we have equal chances to have
(x1, x2, x3) = (1, 1, 15) or (x1, x2, x3) = (1, 2, 14). What is the
probability the events x1 + x2 ≤ 8 occurs, i.e., P(x1...

Consider the function, f(x) = - x4 - 2x3 -
8x2 - 5x
Use parabolic interpolation (x0 = -2, x1 =
-1, x2= 1, iterations = 4). Select new points
sequentially as in the secant method.

Newton's method: For a function ?(?)=ln?+?2−3f(x)=lnx+x2−3
a. Find the root of function ?(?)f(x) starting with
?0=1.0x0=1.0.
b. Compute the ratio |??−?|/|??−1−?|2|xn−r|/|xn−1−r|2, for
iterations 2, 3, 4 given ?=1.592142937058094r=1.592142937058094.
Show that this ratio's value approaches
|?″(?)/2?′(?)||f″(x)/2f′(x)| (i.e., the iteration converges
quadratically). In error computation, keep as many digits as you
can.

5. Write down the first few iterates of the secant
method for solving x2 − 3 = 0, starting with x0 = 0 and x1 = 1.

Let the LFSR be xn+5 = xn +
xn+3, where the initial values are x0=0,
x1=1, x2=0, x3=0,
x4=0
(a) Compute first 24 bits of the following LFSR.
(b) What is the period?

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