Introduction to Scientific Computating-Homework 2 Solved

Problem 1 : If a regular fixed payment P is made n times a year to repay a loan of amount A over a period of k years, where the nominal annual interest rate is r, P is given by 

𝑟𝐴(1+𝑟/𝑛)𝑛𝑘

                                                                                  P =  𝑟 𝑛𝑘                   

                                                                                            𝑛[(1+𝑛)        −1]

You need to generate a table of repayments for a loan of $1000 over 15, 20, and 25 years, at interest rates that vary from 10% to 20% per annum in steps of 1%. (Tip: The table has 11 rows and 3 columns, with rows representing different interest rates and columns representing different years, n = 12). You need to use two different methods, the first is to use nested for loops, and the second is to vectorize the outer loop. You might use repmat(). The generated table is as follows:  

   10.7461    9.6502    9.0870    11.3660   10.3219    9.8011    12.0017   11.0109   10.5322    12.6524   11.7158   11.2784    13.3174   12.4352   12.0376    13.9959   13.1679   12.8083    14.6870   13.9126   13.5889    15.3900   14.6680   14.3780    16.1042   15.4331   15.1743    16.8288   16.2068   15.9768 

   17.5630   16.9882   16.7845 


 

Problem 2 [20]: Narcissistic number is a three-digit number, the sum of the third power of the digits in each bit of which is equal to itself (for example, 153 = 13 + 53 + 33). You may find more information by searching " Narcissistic number ". You need to write two functions, one for finding all the narcissistic numbers  and one for determining whether a three-digit number is a narcissistic number. The second function should be nested within the first function. 

 

Problem 3 [20]: Consider the following 3 x 3 matrix: A = 

1     3     2 

8     4     6 

7     9     5 

Use array subscripts or MATLAB functions to create the following arrays from A – no decisions, loops, etc. Each array should be created using only a single statement. 

i) B = 

     1     3 

     7     9 ii) C = 

     1     8 

     7     3 

iii)  D = 

     1     3     2 

     8     4     6 

iv)  Find the maximum value of each row 

E = 

     3 

     8      9 v) F = 

     1     0     8     0     7 

     0     0     0     0     0 

     3     0     4     0     9 

     0     0     0     0     0 

     2     0     6     0     5 

 
 

Problem 4 : Around 300 B.C. Euclid developed a wonderfully simple algorithm for determining the greatest common divisor of two positive integers. Please search online for “Euclidean Algorithm” to find out the details. Your job is to implement the following two functions: 

1) function out=my_gcd(a,b) 

which computes the greatest common divisor of two positive integers a and b. You may find the MATLAB function rem() useful. 2) function out=my_lcm(a,b) 

which computes the least common multiple of two positive integers a and b. This can be computed in part using my_gcd. 

Make sure you account for the case where a and/or b is zero in your my_gcd function. You can check this behavior by entering gcd(0,2), gcd(2,0), and gcd(0,0) into the MATLAB command line. Note that the least common multiple of a=0 or b=0 is not defined (only least common multiples of a and b larger than 0 should be returned). 

You can compare your implementations to MATLAB’s built-in gcd and lcm functions to determine if everything is working correctly. 

 

Problem 5 : Given a string of ‘1’ and ‘0’, find the maximum number of consecutive times of ‘1’. Write a function to do this. Your function declaration should be as follows. 

function y = lengthOnes(x) 

The sample is: 

x  = ‘110100111’ 

y  = 3