MTH 221 - Discrete Math for Information Technology

Nov 27th, 2013
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Exercise 5.2, problems 27(a & b); p 259
One version of Ackermann’s function A(m,n) is defined recursively for m, n ∈ N by:            
       
 
A(0, n)= n + 1, n ≥ 0;    
A(m, 0) = A(m − 1, 1), m > 0; and
A(m, n) = A(m − 1, A(m, n − 1)), m, n > 0.
       
a. Calculate A(1, 3) and A(2, 3).                      
b. Prove that A(1, n) =  n + 2 for all n ∈ N.                    
c. For all n ∈ N show that A(2, n) = 3 + 2n                    
d. Verify that A(3, n) =  2n+3 − 3 for all n ∈ N.                    
                           
Exercise 5.8, problem 6; p 301 (take a look at problem 5 first)
We first note how the polynomial in Exercise 5 can be written in the nested multiplication method:        
                           
8 + x(−10 + x(7 + x(−2 + x(3 + 12x)))).                    
                         
Using this representation, the following pseudocode procedure (implementing Horner’s method) can be used to evaluate the given polynomial.      
                           
procedure PolynomialEvaluation2                      
(n: nonnegative integer;                      
r,a0,a1,a2,. . .,an: real)                      
begin                          
value := an                        
for j := n - 1 down to 0 do                      
  value := aj + r * value                    
end                            
                           
a. How many additions take place in the evaluation of the given polynomial? (Do not include the n − 1 additions needed to increment the loop variable i.) How many multiplications?  
b. Answer the questions in part (a) for the general polynomial a0 + a1x + a2x2 + a3x3 + · · · + an−1xn−1 + anxn, where a0, a1, a2, a3, . . . , an−1, an are real numbers and n is a positive integer.  

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