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Find enclosing intervals for all zeros of f on the interval [1.05,1.5].

Here's the Maple Worksheet solving the task with intpakX. Doing 10 iteration steps, you can prove the existence of two zeros and find two intervals which must contain every further zero of the function on the interval. Further iteration steps and a smaller parameter for the accuracy would yield more detailed results.

> restart;

> libname:="/usr/maple/intpakX/lib", libname;

> with(intpakX);
 

> f:=x->sin(1/(x-1));

> X:=[1.05,1.5];

> compute_all_zeros_with_plot(f,X,0.001);

Enter maximal number of iteration steps: 10;

Digits = 10

Iteration step  1

xold= [1.05, 1.5]

xnew1= [1.276187073, 1.5]

xnew2= [1.05, 1.273812927]

Iteration step  2

xold= [1.276187073, 1.5]

xnew1= [1.276187073, 1.347259211]

Iteration step  3

xold= [1.276187073, 1.347259211]

xnew1= [1.316783008, 1.320737304]

Iteration step  4

xold= [1.316783008, 1.320737304]

xnew1= [1.318303507, 1.318314830]

Iteration step  5

xold= [1.05, 1.273812927]

xnew1= [1.162172903, 1.273812927]

xnew2= [1.05, 1.161640024]

Iteration step  6

xold= [1.05, 1.161640024]

xnew1= [1.105883079, 1.161640024]

xnew2= [1.05, 1.105756945]

Iteration step  7

xold= [1.105883079, 1.161640024]

xnew1= [1.144181348, 1.161640024]

xnew2= [1.105883079, 1.123341755]

Iteration step  8

xold= [1.144181348, 1.161640024]

xnew1= [1.158186203, 1.161255603]

Iteration step  9

xold= [1.158186203, 1.161255603]

xnew1= [1.159140063, 1.159163842]

Iteration step  10

xold= [1.105883079, 1.123341755]

xnew1= [1.105883079, 1.107392352]

Number of enclosures of zeros:  4

Number of iteration steps:  10

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