Mechanical Engineering Assignment Help

Mechanical Engineering Assignment Hel

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Mechanical Engineering The following three problems aim to get you familiarize with MATLAB functions. You don’t need to worry about exceptional cases such as when n is not a non-negative integer in problem 4.3. It is assumed that input arguments are well-defined.

Problem 1 : Writing a function to perform multiple matrix operations Write a function that will calculate the sum, difference, element-by-element multiplication, and matrix multiplication of two arbitrary same size matrices at once. The results of function with 1 3

given matrices >> [s,d,

ep,mp

s = 2 5

5 8

0 1

-1 0

1 6

6 16

5 11

11 25

2

 

4

1 3   2

4

 and

are shown as below in the command window:

]=bop([1 2 ; 3 4],[1 3 ; 2 4])

d =

ep =

mp =

Function name (and m-file name) should be ‘bop_your_kerberos_name’, and upload it to 2.003 MIT Server site. You also submit print-out of your function. Calculate sum, difference, element-

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by-element multiplication, and matrix multiplication of

10 35  and 70 100  



5 9

 25 with your

3

function in this problem.

Problem 2 : Integrating two functions numerically over a given interval using function handlers Evaluate the following two integrals using the ‘quad’ function. You don’t need to turn in the code electronically. Just manually write down the MATLAB codes that you have used and the evaluation result that you get.  1   bx ax 2  c dx where a  1, b  2, and c  3 10 2 ii) y   a exp    2c  dx xb  where a1  , b  0, and c  1       2  Does the ‘quad’ function give reasonable results for both functions i) and ii)? For ii), analytically evaluate the Gaussian i)

y

function (or look it up in an integral table) and compare with your numerical result. Explain any differences.

Problem 3 : Calculating the factorial of a non-negative integer Write a function to calculate the factorial of a non-negative integer, n!. It is not allowed to use recursive algorithm you will learn next week in this problem. Function name (and m-file name) should be ‘fctrl_your_kerberos_name’ and upload it to 2.003 MIT Server site. You also submit print-out of your function. Calculate 125! with this function.

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Solutions Problem 1 : Writing a function to perform multiple matrix operations We have two matrices to be used in the operations (A,B), and four outputs, the results of sum(s), difference(d), element-by-element product(ep), and matrix product(mp) The function can be implemented as below. function [s,d,ep,mp]=bop(A,B) % % Problme 4.1: Writing a function to perform multiple matrix operations % % Input arguments: two same size matrices % Output arguments: sum(s), difference(d), element by element product(ep) %

and matrix product(mp) of two matrices

% % sum of two matrices

s=A+B;

% difference of two matrices

d=A-B;

% element-by-element product of two matrices % matrix product of two matrices

ep=A.*B;

mp=A*B;

The output of calculating sum, difference, element-by-element multiplication, and matrix multiplication of of

35  10  and 70 100  



5 9

 25 with this function is shown as below.

3

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>> [s,d,

ep,mp]

=bop([10

s = 15 79

38 125

5 61

32 75

35 ; 70 100],[5 3 ; 9 25])

d =

ep = 50 630

105 2500

365 1250

905 2710

mp =

Problem 2 : Integrating two functions numerically over the interval with function handler i) With ‘quad’ function, you can obtain numerical integration over [1,10]. First, you should define the function you will evaluate, and calculate numerical value for given integration interval. >> P=@(x)x.^2+2*x+3; 4.2 i) >> quad(P,1,10) ans = 459

% Define Polynomial function given at problem % Integrate function from 1 to 10

ii) The procedure to evaluate Gaussian function is same as i), but we have indefinite integration interval. ‘quad’ function can be used only for the integration over definite interval. However, we can assume that indefinite number is approximated to very large number such as 1010, since Gaussian function goes to 0 as x goes to infinity (or – infinity), and we get very close number to what we get with infinite evaluations. >>

G=@(x)1/sqrt(2*pi)*exp(-(x/sqrt(2)).^2);

%

Define

Gaussian

function

given at 4.2 ii) >> quad(G,-10^10,10^10)

% Integrate function from -

(almost -infinity) to 10^10 (almost +infinity) ans =

1.0000

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10^10

For i), numerical solution and analytical solution is identical. Due to some limitations of MATLAB number expression, the integration result for i) seems to be 1.0000 which is equal to analytical solution for this Gaussian function, but actually 1.000001123047690….(with more significant digits) The reason why they are different is that we just evaluate function up to certain finite number, and there always exists error between numerical value and true value. Therefore, evaluating tolerance is quite important in the numerical analysis. This evaluation is acceptable if we have tolerance of 10-5, but it’s not if we have 10-6.

Problem 3 : Calculating the factorial of non-negative integer The factorial of integer n, n!, is defined as below: n !  n  n  1 n  2  L  2 1 0!  1 Therefore, n! can be calculated with ‘for’ loop by multiplying loop counter to the output. However, special case of n  0 should be considered. Input argument (n) is non-negative integer, and output argument (o) is the factorial of input argument. function o=fctrl(n) % % Problem 4.3: Calculating the factorial of non-negative integer with % % Define 0!=1

o=1;

% Calculate factorial % if n<1, MATLAB skips 'for' loop (for n=0)

for i=1:n

o=o*i; % n!=(n-1)!*n end The output of 125! will be displayed as below. >> fctrl(125) % Calculate 125! ans = 1.8827e+209

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function

function [s,d,ep,mp]=bop(A,B) % % Problme 4.1: Writing a function to perform multiple matrix operations % % Input arguments: two same size matrices % Output arguments: sum(s), difference(d), element by element product(ep) % and matrix product(mp) of two matrices %

% sum of two matrices s=A+B; % difference of two matrices d=A-B; % element-by-element product of two matrices ep=A.*B; % matrix product of two matrices mp=A*B; function o=fctrl(n) % % Problem 4.3: Calculating the factorial of non-negative integer with function %

% Define 0!=1 o=1;

% Calculate factorial % if n<1, Matlab skips 'for' loop (for n=0)

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for i=1:n o=o*i; % n!=(n-1)!*n end function integral % % Problem 4.2 : Integrating two functions numerically over the interval with function handler %

% First method

P=@(x)x.^2+2*x+3; % Define Polynomial function given at 4.2 i) quad(P,1,10) % Integrate function from 1 to 10 G=@(x)1/sqrt(2*pi)*exp(-(x/sqrt(2)).^2); % Define Gaussian function given at 4.2 ii) quad(G,-10^10,10^10) % Integrate function from -10^10 (almost -infinity) to 10^10 (almost +infinity)

% Second method disp(sprintf('Intrgral from 1 ro 10 for x^2+2x+3 is %f',quad(@Polynomial,1,10))); disp(sprintf('Intrgral from -inf to +inf exp(-((x)/(sqrt(2)*c))^2)is %20.15f',quad(@Gaussian,-10^10,10^10)));

function y=Polynomial(x) a=1; b=2; c=3; y=a*x.^2+b*x+c;

function y=Gaussian(x) a=1/sqrt(2*pi); b=0; c=1; y=a*exp(-((x-b)/(sqrt(2)*c)).^2);

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