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3 Problems on Fixed Point Iteration Method - Assignment 2 | MATH 417, Assignments of Mathematical Methods for Numerical Analysis and Optimization

Texas A&M University (A&M)Mathematical Methods for Numerical Analysis and Optimization

Material Type: Assignment; Class: NUMERICAL METHODS; Subject: MATHEMATICS; University: Texas A&M University; Term: Unknown 1989;

Typology: Assignments

Pre 2010

Uploaded on 02/10/2009

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MATH 417: Numerical Analysis
Instructors: Prof. Wolfgang Bangerth, Prof. Guido Kanschat
[email protected],
[email protected]
Teaching Assistants: Seungil Kim, Yan Li
[email protected],
[email protected]
Homework assignment 2 – due 9/14/06 and 9/15/06
Problem 1 (Fixed-point iteration). Let abe a positive number and
g(x) = 2x−ax2.
(a) Show that the limit of this fixed-point iteration is either 0 or 1/a.
(b) This iteration gives us a means to compute the reciprocal value of ausing
only multiplication and subtraction. Find an interval about 1/a in which
this iteration converges to 1/a.(6 points)
Problem 2 (Bisection method). A ball is dropped from a tower. During
its fall it is subject to the forces of gravity and air resistance. Thus, the height
sin ft as a function of time tin sis given by the formula
h(t) = h0−
mg
kt+m2g
k21−e−
kt
m.
Here, h0= 300ft is the height of the tower, m= 1/2lb the mass of the ball,
g= 32.17ft/s2is the gravitation of earth and k= 1/4lb s/ft is the air resistance
coefficient.
Use the bisection method (on paper, with pen and calculator) to find out
when the ball hits the ground with an accuracy of 0.1 seconds. (6 points)
(please turn over)
1
pf2

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Download 3 Problems on Fixed Point Iteration Method - Assignment 2 | MATH 417 and more Assignments Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!

MATH 417: Numerical Analysis

Instructors: Prof. Wolfgang Bangerth, Prof. Guido Kanschat [email protected], [email protected] Teaching Assistants: Seungil Kim, Yan Li [email protected], [email protected]

Homework assignment 2 – due 9/14/06 and 9/15/

Problem 1 (Fixed-point iteration). Let a be a positive number and

g(x) = 2x − ax^2.

(a) Show that the limit of this fixed-point iteration is either 0 or 1/a.

(b) This iteration gives us a means to compute the reciprocal value of a using only multiplication and subtraction. Find an interval about 1/a in which this iteration converges to 1/a. (6 points)

Problem 2 (Bisection method). A ball is dropped from a tower. During its fall it is subject to the forces of gravity and air resistance. Thus, the height s in f t as a function of time t in s is given by the formula

h(t) = h 0 −

mg k t +

m^2 g k^2

1 − e−^

ktm^ ) .

Here, h 0 = 300f t is the height of the tower, m = 1/ 2 lb the mass of the ball, g = 32. 17 f t/s^2 is the gravitation of earth and k = 1/ 4 lb s/f t is the air resistance coefficient. Use the bisection method (on paper, with pen and calculator) to find out when the ball hits the ground with an accuracy of 0.1 seconds. (6 points)

(please turn over)

Problem 3 (Regula falsi). This scheme can be obtained from the bisection method by not using the midpoint of the interval [an, bn] to subdivide it into two parts, but instead using the point

anf (bn) − bnf (an) f (bn) − f (an)

and continuing the method as before, i.e. choosing either the left or the right part of the interval. Show that this point is where the the secant of the function between the points a and b crosses the x-axis. Show that it is indeed within the interval (an, bn). Program this method and compare it to the bisection method you worked on in the lab to compute the roots of

(a) x^2 − 2 in [0, 2]

(b) cos x in [0.3]

(c) x^10 − 0 .1 in [0, 1]

to an accuracy of 10−^6. Plot or draw the estimated errors in each step of both methods in a diagram. (8 points)