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Find the real roots of the equation correct to two significant figures by the method of tabulation

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ICS 2207: SCIENTIFIC COMPUTING DECEMBER 2011 ORDINARY EXAMINATION
Answer question ONE and ANY other TWO questions
Question one
a) Describe the numerical error and round-off error in approximation of x. Which one is more subtle?
(3 marks)
b) Find the real roots of the equation correct to two significant figures by the method of tabulation.
(6 marks)
c) Descibe the rounding of rule explaining the three cases (5 marks)
d) Let and be two exact numbers represented by rounded numbersand respectively, being correct up
to decimal places and up tp decimal places, . Carry out an error anlysis on approximating by
(8 marks)
e) Find the solution to the differential equation with the initial condition with step-length 0.05 by the
Euler’s method (8 marks)
Question two
Compute the value of the integral estimate of the errors by the following quadrature rulues.
Take 6 intervals
i. Composite Simpson’s rule (7 marks)
ii. Composite Weddle’s rule (6 marks)
iii. Composite trapezoidal rule (7 marks)
Question three
Solve the following system of linear equations correct to four places of decimals.
(20 marks)
Question four
a) Develop an algorithm to find the root of an equation by the bisection method (8 marks)
b) Find the root of equation correct to four significant figures. (12 marks)
Question five
Consider the following table for which is unknown and is to be approximated by polynomial
interpolation
x 3.141 3.142 3.143 3.144 3.145
y 0.4970679364 0.4972061807 0.4973443810 0.4974825374 0.4976206498
a) Construct an appropriate forward difference table
b) Using newton’s forward or backward interpolation formulae calculate
i.
ii.
ICS 2207: SCIENTIFIC COMPUTING DECEMBER 2011 ORDINARY EXAMINATION Answer question ONE and ANY other TWO questions Question one a) Describe the numerical error and round-off error in approximation of x. Which one is more subtle? (3 marks) b) Find the real roots of the equation correct to two significant figures by the method of tabulation. (6 marks) c) Descibe the rounding of rule explaining the three cases (5 marks) d) Let and be two exact numbers represented by rounded numbersand respectively, being correct up to decimal places and up tp decimal places, . Carry out an error anlysis on approximating by (8 marks) e) Find the solution to the differential equation with the initial condition with step-length 0.05 by the Euler’s method (8 marks) Question two Compute the value of the integral estimate of the errors by the following quadrature rulues. Take 6 intervals i. Composite Simpson’s rule (7 marks) ii. Composite Weddle’s rule (6 marks) iii. Composite trapezoidal rule (7 marks) Question three Solve the following system of linear equations correct to four places of decimals. (20 marks) Question four a) Develop an algorithm to find the root of an equation by the bisection method (8 marks) b) Find the root of equation correct to four significant figures. (12 marks) Question five Consider the following table for which is unknown and is to be approximated by polynomial interpolation x 3.141 3.142 3.143 3.144 3.145 y 0.4970679364 0.4972061807 0.4973443810 0.4974825374 0.4976206498 a) Construct an appropriate forward difference table b) Using newton’s forward or backward interpolation formulae calculate i. ii.

 
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