ISSAC 94 - Symbolic-Numeric Nonlinear Equation Solving Up
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Application of the Mean-Value Theorem Previous Next

From calculus we know that if then

for some .  If for all and , then for all where .  Thus, while searching for zeros of on , we can eliminate all of from consideration if we know and a finite bound .  If interval is sufficiently small, then will determine a finite bound on .  This leads to procedure Narrow(,) which trims the ends of interval by this technique before calling Zeros (if it must).  The recursive calls in Zeros are changed to be calls to procedure Narrow

We change lines 13 and 19-20 of procedure Zeros to be:

return Narrow Narrow;

return Narrow
Narrow;

and add procedure Narrow below:

proc Narrow
;
if then
for from to do
;
;
if then
return ;
fi;
od;
fi;
return Zeros;

proc Bound
;
return ;

is a "fudge factor" which we currently set as Next



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