关于算法:MATH-2109数学计算

School of Sciences
MATH 2109 – Mathematical Computing Week 6 – Hand in Lab Class –
Due 5 : 00pm Friday 23rd April 2021.

1. Write a function in an m-file for y = cos5
   (x) sin3
   (x) + 2 that will accept a row vector of x
   values as input and output a row vector of y values. Now generate a row vector of x values in
   the interval [0, 7] (use 100 points) and input this into your function to find the corresponding
   y values. Plot the function using the plot() command. Also by looking up the xlabel()
   and ylabel() functions label the x and y axes.
2. A sequence is given by a5 = 2, a6 = −4, and a7 = 1, and the recurrence relation
   an = 5 an−1 − 3 an−2 + 6 an−3an−2
   when n ≥ 8.
   Write a function in an m-file that has n as its only input, where n > 5, and returns the value
   an. Put in a check that n ≥ 5 and exit with a printed error if this is not the case, assigning
   an the value NaN.
   Call your function in a main program, so that it prints, using fprintf, a table of values for
   n and f(n), for n = 5, 6, . . . 13. Your table should have headers.
3. The symbolic toolbox allows us to integrate symbolic expressions using the following commands
   (as an example):
   f = @(x) cos(x). ̂5.*sin(x). ̂3
   syms x ;
   int(f(x),x)
   ans =
   (cos(x) ̂5(5cos(x) ̂2 – 7))/35
   Write a function call IntFunct in an m-file that takes as input a function of one variable
   and values of a and b for x. As outputs it returns the value of the definite integral over the
   interval [a, b] and (only if asked for) additionally a symbolic expression for its derivative. Also
   include a check that not more than 2 outputs are requested and if they are, return a printed
   error message (of your making) and cease execution before Matlab complains. Test this by
   calculating the integral of y = cos5
   (x) sin3
   (x) over [0, 2]. You need to pass the function f
   and the values a = 0 and b = 2 as parameters when you call the function from the command
   window.
   [Hint: Use double(subs(Sym_expression, x, a)) to get a number after substituting. ]
4. The intersection of a circular cylinder of unit radius, oriented around the xn axis, and the
   sphere centred at the point e1 = (1, 0, . . . , 0) correspond to the points (x1, x2, . . . , xn) (in
   n-dimensions) that satisfy the equations:
   Write a function area_approx = Monte_Carlo_Test(n,N) that inputs the dimension n
   and the number of random samples N in the following Monte-Carlo estimation of the volume
   1
   of the intersection. Generate N uniformly distributed random samples in the following n
   dimensional box that contains the region defined in (1):
   and count the number k, of the N, events that land inside the region defined in (1). Include this function as a sub-functions (place it at
   the end of your work) and call it to print a table of the areas associated with the dimensions
   n = 3, 4, 5, 6 using N = 106.
   [Hint: Watch the week 5 Wednesday Lectorial Collaborate Ultra recording.]
   NOTE : Note that M-File functions MUST NOT contain any fprintf() or disp() commands,
   or echo print any values inside the function (other than for error messages). The Function MUST
   return a value or values, NOT just print them with a disp() command.
   You MUST include listings of your functions in your published output as follows. If your function
   is contained in the M-file with name myfunction.m, just include the line
   type myfunction.m
   at the point in your code where you want the listing of the function to appear.