HOMEWORK ASSIGNMENT No.2 Due Date: 23Rd March, 2007

HOMEWORK ASSIGNMENT No.2 Due Date: 23rd March, 2007 COURSE: AMAT314 -NUMERICAL METHODS

INSTRUCTIONS TO STUDENTS

A presentable report should be prepared with all questions answered. All necessary working must be shown. You are reminded of the necessity for proper English and orderly presentation of your answers. Calculators may be used.

DEFINITIONS:

a) The “true percent relative error” ,  t ,is defined as: true error  = x100% t true value where “true error =true value-approximation”.

b) The “approximate percent relative error” ,  a , is defined as: present.approximation  previous.approximation  = x100% a present.approximation

c) The “Scarborough Termination Criterion”:

In was shown by Scarborough(1966) that if the following criterion is met, we can be assured that the computational result is correct to at least n significant figures: 2n  a <  s where  s  (0.5x10 )% .

ITERATION METHODS:

f (xn ) a) Standard Newton-Raphson Method: xn1  xn  f (xn )

f (xn ) f (xn ) b) Modified Newton-Raphson Method: xn1  xn  2  f (xn )  f (xn ) f (xn )

f (xn )(xn1  xn ) c) Secant Method: xn1  xn  f (xn1 )  f (xn )

1 FITAMAT314.HOMEWORK ASSIGNMENT.No2.SPRING2007.doc HOMEWORK ASSIGNMENT No.2 COURSE: AMAT314 -NUMERICAL METHODS

PROBLEM STATEMENT:

The function f (x)  x3  5x 2  7x  3 has a double real root at x  1.0 and a single real root at x  3.0 in the interval [0,4].Can you verify that?

(a) Using the Standard Newton-Raphson Method and starting with the initial

guess of x0  0 , write down the first 6 terms of the sequence, keeping 8 digits after the decimal point.

(b) Using the Modified Newton-Raphson Method and starting with the initial

guesses of x0  0 , write down the first 6 terms of the sequence, keeping 8 digits after the decimal point.

(c) Using the Secant Method and starting with the initial guesses of x1  0 ,and

x0  1.5 write down the first 6 terms of the sequence, keeping 8 digits after the decimal point.

(d) At each step calculate the true percent relative error ,  t ,and the

approximate percent relative error ,  a , and deduce the root of f (x) to 9 significant figures.

(e) Record the data of the above steps (a)-(d) in tabular form(tables) with entries

xn ,  t ,  a for each iteration.

(f) Compare all three methods as to their effectiveness in locating the true root, their accuracy and the convergence speed.

(g) Could you have terminated the iteration process earlier in the above computational results if you had used the “Scarborough Criterion” instead of the number of iterations? Explain your answer!

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