>> % NEWTON
>> f = @(x) cos(x) - x
f =

@(x) cos (x) - x

>> x = -10:.01:10;  plot(x,f(x)),  grid on
>> format long
>> x = 1
x =  1
>> df = @(x) -sin(x) - 1
df =

@(x) -sin (x) - 1

>> x = x - f(x) / df(x)
x =  0.750363867840244
>> x = x - f(x) / df(x)
x =  0.739112890911362
>> x = x - f(x) / df(x)
x =  0.739085133385284
>> x = x - f(x) / df(x)
x =  0.739085133215161
>> x = x - f(x) / df(x)
x =  0.739085133215161
>> f(x)
ans = 0
>> 
>> % SECANT
>> xold = 1
xold =  1
>> x = 0.8
x =  0.800000000000000
>> xnew = x - f(x) / ( (f(x) - f(xold)) / (x - xold) ),  xold = x;  x = xnew;
xnew =  0.742035906602420
>> xnew = x - f(x) / ( (f(x) - f(xold)) / (x - xold) ),  xold = x;  x = xnew;
xnew =  0.739123508315782
>> xnew = x - f(x) / ( (f(x) - f(xold)) / (x - xold) ),  xold = x;  x = xnew;
xnew =  0.739085158179144
>> xnew = x - f(x) / ( (f(x) - f(xold)) / (x - xold) ),  xold = x;  x = xnew;
xnew =  0.739085133215372
>> xnew = x - f(x) / ( (f(x) - f(xold)) / (x - xold) ),  xold = x;  x = xnew;
xnew =  0.739085133215161
>> xnew = x - f(x) / ( (f(x) - f(xold)) / (x - xold) ),  xold = x;  x = xnew;
xnew =  0.739085133215161
>> 
>> % BISECTION
>> a = 0.5, b= 1
a =  0.500000000000000
b =  1
>> f(a)*f(b)
ans = -0.173573833045405
>> c = (a+b)/2,  f(a),  f(b),  f(c)
c =  0.750000000000000
ans =  0.377582561890373
ans = -0.459697694131860
ans = -0.0183111311261791
>> b = c
b =  0.750000000000000
>> c = (a+b)/2,  f(a),  f(b),  f(c)
c =  0.625000000000000
ans =  0.377582561890373
ans = -0.0183111311261791
ans =  0.185963119505218
>> a = c
a =  0.625000000000000
>> c = (a+b)/2,  f(a),  f(b),  f(c)
c =  0.687500000000000
ans =  0.185963119505218
ans = -0.0183111311261791
ans =  0.0853349461524715
>> a = c
a =  0.687500000000000
>> c = (a+b)/2,  f(a),  f(b),  f(c)
c =  0.718750000000000
ans =  0.0853349461524715
ans = -0.0183111311261791
ans =  0.0338793724180665
>> a = c
a =  0.718750000000000
>> c = (a+b)/2,  f(a),  f(b),  f(c)
c =  0.734375000000000
ans =  0.0338793724180665
ans = -0.0183111311261791
ans =  0.00787472545850132
>> a = c
a =  0.734375000000000
>> c = (a+b)/2,  f(a),  f(b),  f(c)
c =  0.742187500000000
ans =  0.00787472545850132
ans = -0.0183111311261791
ans = -0.00519571174375921
>> b = c
b =  0.742187500000000
>> c = (a+b)/2,  f(a),  f(b),  f(c)
c =  0.738281250000000
ans =  0.00787472545850132
ans = -0.00519571174375921
ans =  0.00134514975180511
>> a = c
a =  0.738281250000000
>> c = (a+b)/2,  f(a),  f(b),  f(c)
c =  0.740234375000000
ans =  0.00134514975180511
ans = -0.00519571174375921
ans = -0.00192387278089767
>> b = c
b =  0.740234375000000
>> c = (a+b)/2,  f(a),  f(b),  f(c)
c =  0.739257812500000
ans =  0.00134514975180511
ans = -0.00192387278089767
ans = -2.89009146790087e-04
>> c
c =  0.739257812500000
>> c   % bisection estimate after 9 steps
c =  0.739257812500000
>> % use Newton to recall solution
>> x=1; for k=1:10, x = x - f(x) / df(x); end,  x
x =  0.739085133215161
>> diary off