tips for Multiple roots of non-algebric equations

>>>>> "Luigi" == Luigi Marino <luigi_marino2 at alice.it> writes:

    Luigi> I try  to solve  non-algebric equations with multiple roots. :
    Luigi> 1. first example.
    Luigi> f:log(x^2+1)-sin(x)-4;

    Luigi> for k:-12 thru -3 step 1 do print("x="(newton(f,x,k,0.0000000001)))$

    Luigi> x=(-11.473978752837) 
    Luigi> x=(-11.47397875283699)

Are you bothered that you get the same roots again?

    Luigi> 3. third example
    Luigi>  h:log(x+1)-1/2*x^2+0.6;(%o13) 
    Luigi> log(x+1)-x^2/2+0.6(%i14) 
    Luigi> for k:0 thru 1 step 1 do print("x="(newton(h,x,k,0.0000000001)))$
    Luigi> x=(-0.40442128319509) 
    Luigi> x=(1.806649648414053) 

Are you bothered that you get roots outside of the interval [0,1]?

I think that's the nature of Newton's method.  Your initial guess has
to be close enough to the root to converge to the root; different
starting points can converge to the same root.  And nothing constrains
Newton's method to stay in any given interval.

Ray