Question

The Golden Ratio is an irrational number whose existence has been known and whose properties have been studied for many centuries. It is symbolized by © and has the value o = 1.618033988749894848, exact to 18 digits. Many methods have been used to calculate 0. One method is the recursive formula = 1+ 1+ 1+ 1+V1+V... Start by guessing 0 = 1 and develop a recursive algorithm to estimate the true value of ø after N approximations (N = 0 is the guess 0 = 1). Write a MATLAB script HWYP3.m to implement your algorithm and reproduce Table 1 and print it to screen. Use disp() for the header and fprintf() for the data; use %4d for N and %20.16f for the other three columns. (HINT: use nested for-loops.) 01188 U.S.A. .

Answer 1

SOURCE CODE IN MATLAB:

%setting initial values
exact=1.618033988749894848;
N=0;
phi=1.0;
fprintf("____________________________________________\n"); %output header
fprintf(" N phi exact\n"); %output header
fprintf("____________________________________________\n"); %output header
fprintf("%4d%20.16f%20.16f\n", N, phi, exact); %output
for N=1:25
phi=sqrt(1+phi); %estimating new phi
fprintf("%4d%20.16f%20.16f\n", N, phi, exact); %output
end

OUTPUT:

octave: 8> source("HW7P3.m") N phi exact 0 1.0000000000000000 1.6180339887498949 1 1.4142135623730951 1.6180339887498949 2 1.5537739740300374 1.6180339887498949 3 1.5980531824786175 1.6180339887498949 4 1.6118477541252516 1.6180339887498949 5 1.6161212065081170 1.6180339887498949 6 1.6174427985273905 1.6180339887498949 7 1.6178512906096749 1.6180339887498949 8 1.6179775309347393 1.6180339887498949 9 1.6180165422314876 1.6180339887498949 10 1.6180285974702324 1.6180339887498949 11 1.6180323227519999 1.6180339887498949 12 1.6180334739281508 1.6180339887498949 13 1.6180338296612189 1.6180339887498949 14 1.6180339395887897 1.6180339887498949 15 1.6180339735582778 1.6180339887498949 16 1.6180339840554270 1.6180339887498949 17 1.6180339872992244 1.6180339887498949 18 1.6180339883016130 1.6180339887498949 19 1.6180339886113682 1.6180339887498949 20 1.6180339887070878 1.6180339887498949 21 1.6180339887366668 1.6180339887498949 22 1.6180339887458071 1.6180339887498949 23 1.6180339887486317 1.6180339887498949 24 1.6180339887495045 1.6180339887498949 25 1.6180339887497743 1.6180339887498949

Regards!