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KCEP2105 Numerical Methods & Statistics Dr Liew HL

Homework 2

Out 21 Mar

[GS] A. Gilat, V. Subramaniam. Numerical Methods An Introduction withApplications Using Matlab, 2nd edn, Wiley, 2011.

[CK] W. Cheney, D. Kincaid. Numerical Mathematics and Computing, 6th Edn,

Thomson Brooks/Cole, 2008.

[K] J. Kiusalaas. Numerical Methods in Engineering with Matlab, Cambridge, 2005.

A reminder for guidelines on problems involving Matlab:

(a) You must turn in the Matlab codes you write to solve problems.

(b) Graphs must be properly annotated.

(c) Include a brief write-up on the mathematical steps used in your m-files by way

of a sort of pseudo-code. In other words, tell me what is going on in your m-files.

Grading (#-pts): 1-30, 2-10, 3-20, 4-20, 5-20, 6-20. (Total: 120)

1. [CK-3.1.1] Find the intersection of = 3 and = . Tabulate your work.

a. Use bisection method with range 1,2. Do five recursive steps.

b. Use Newtons method. Use = 1. Do five recursive steps.

c. Use secant method. Use = 1 and = 2. Do five recursive steps.

2. [CK-4.1.1] Use the Lagrange interpolation process to obtain a polynomial

that assumes these values:

0

7

2

11

3

28

4

63

Problem 3, 4, 5 (computer problems):

The natural frequencies of a uniform cantilever beam are related to theroots of the frequency equation = cosh cos + 1 = 0, where

= 2

= thnaturalfrequencycps

= massofthebeamdensity=7850kg/m

= lengthofthebeam=0.9m

= modulusofelasticity=200GPa

= momentofcrosssectionalinertiacrosssectionis25mmx2.5mm

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KCEP2105 Numerical Methods & Statistics Dr Liew HL

Write a program to find the lowest natural frequency, use accurate to 6

decimal places. See [GS-Sec2.9] for more details on the physics of this

problem.

3. Use bisection method, and 0,4.Can you find a priorithe number ofsteps required? Provide a print-out for the result at each recursive step.

4. Use Newtons method. Let = 2. Provide a print-out for the result at each

recursive step.

5. Use secant method. Let = 2, and = 2.5. Provide a print-out for the

result at each recursive step.

Note: To obtain the print-out of solutions at each recursive step:

a. You may copy and paste from the on-screen display. See, e.g. Ex

2.1 on p. 29 of [GS].

b. You can also write the answers to an output data file, from which

you could print out the data.

6. (Computer problem) [K-3.1.20] Use Lagrange interpolation.

(a) Present clearly the pseudo-code for the loops in the construction of

the interpolation polynomial.

(b) Write an m-file to find the solution.