Consider the causal nonlinear discretetime system characteri

Consider the causal non-linear discrete-time system characterized by the following difference equation: 2y[n] = y[n-1] + x[n] / y[n-1] If we use as input x[n] to this system (algorithm) a step function of amplitude A (i.e. x[n] = A u[n]), then y[n] will converge after several iterations to the square root of A

. • Write a MATLAB program that implements the above recursion to compute the square root of 16, 4, 5, and 3.

• How many iterations does it take to converge to the true value starting at y[-1] =0.5?

Solution

function [yn,iter] = squareroot2(A)
yminus1 = 0.5;
yn=0.5*((yminus1)+A/yminus1);%running the first iteration outside the loop
yminus1 = yn;%taking the initial value for next iteration
iter=1;
while yn~=sqrt(A)%loop stops when square root found
yn=0.5*((yminus1)+A/yminus1);
yminus1 = yn;
iter=iter+1;
end
end

Output:

>> [sqrt2,iterations]=squareroot2(16)

sqrt2 =

4

iterations =

8

>> [sqrt2,iterations]=squareroot2(4)

sqrt2 =

2

iterations =

7

>> [sqrt2,iterations]=squareroot2(5)

sqrt2 =

2.2361

iterations =

7

>> [sqrt2,iterations]=squareroot2(3)

sqrt2 =

1.7321

iterations =

7