STQP2033/34 Kaedah Berangka

UKSD/270710

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STQP2033/34: Kaedah Berangka/Numerical Methods

TUTORIAL 3 Semester 1, 2010-11

1. Use iterative refinement techniques to improve

2, 3 and 81 2 3x x x= = − = , which are approximate solutions of

1 2 3

1 2 3

1 2 3

2 5 5

5 2 12

2 3

x x x

x x x

x x x

+ + = −

+ + =

+ + =

2. Given the data

x 1.6 2 2.5 3.2 4 4.5

f(x) 2 8 14 15 8 2

Calculate f(2.8) using Newton’s interpolating polynomials of order 1

through 3. Choose the sequence of the points for your estimates to attain

the best possible accuracy.

3. Given the data

x 1 2 3 5 7 8

f(x) 3 6 19 99 291 444

Calculate f(4) using Newton’s interpolating polynomials of order 1

through 4. Choose the base points to attain the good accuracy. What do

your results indicate regarding the order of the polynomial used to

generate the data in the table?

4. Repeat question 3 using Lagrange polynomials of order 1 through 3. (18.7)

5. The following data come from a table that was measured with high

precision. Use the best numerical method (for this type of problem) to determine y at x = 3.5. Note that a polynomial will yield an exact value.

Your solution should prove that your result is exact.

x 0 1.8 5 6 8.2 9.2 12

y 26 16.415 5.375 3.5 2.015 2.54 8

STQP2033/34 Kaedah Berangka

UKSD/270710

2

6. Use Newton’s interpolating polynomial to determine y at x = 3.5 to the best possible accuracy. Compute the finite divided differences and order your

points to attain optimal accuracy and convergence.

x 0 1 2.5 3 4.5 5 6

y 2 5.4375 7.3516 7.5625 8.4453 9.1875 12

Check your answer using Lagrange polynomials.

7. Develop quadratic splines for the first 5 data points in question 2 and

predict f(3.4) and f(2.2).

8. Evaluate the following integral: / 2

0

(8 4cos )x dx

π

+∫

(a) analytically; (b) single application of the trapezoidal rule;

(c) multiple-application trapezoidal rule, with n=2 and 4;

9. Evaluate the following integral: 4

3 5

2

(1 4 2 )x x x dx−

− − +∫

(a) analytically;

(b) single application of the trapezoidal rule;

(c) multiple-application trapezoidal rule, with n=2 and 4;

For each of the numerical estimates in (b) and (c), determine the percent

relative error based on (a).

10. Integrate the following function analytically and using the trapezoidal rule, with n=1, 2, 3, and 4:

2

2

1

( 1/ )x x dx+∫

Use the analytical solution to compute true percent relative errors to evaluate the accuracy of the trapezoidal approximations.

Solutions to be submitted:

Group 1 to 8: Questions 1, 3, 4, 5 and 9.

Submit to me before 12.00 noon, Monday – 30th

August 2010.