Examples
E x a m p l e 1. Solve the cubic equation
S o l u t i o n . In this case
Let's check up, whether the given relative accuracy
Let's continue iterations:
Again we’ll check up, whether the given relative accuracy
Following iteration to within six decimal signs gives practically exact value of a root:
However, here again it is necessary to check up, whether the given relative accuracy
The found root of the equation is equal to 2.0000001. Thus, computing process has converged for 4 iterations, and we have received a required root with the given relative accuracy
E x a m p l e 2. Solve the cubic equation
S o l u t i o n . Let's copy the given equation in the form of (3):
where
The condition of convergence in this case looks like:
then
Let's accept again as zero approach
On this step the given relative accuracy
therefore process of a finding of a root of the equation can be considered completed. The found root of the equation is equal to 2.000050. Thus, here again the required root is found for 4 iterations with the given relative accuracy
E x a m p l e 3. Solve the cubic equation
S o l u t i o n . Here
On this step the given relative accuracy
and thus, process of a finding of a root can be considered completed. The found root of the equation is equal to 1.9986328. Here the required root is found for 12 iterations with the given relative accuracy
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4.
Solve the cubic equation
S o l u t i o n . Here
whence follows
The sequence of actions in the method of halving is reduced to the following: on each step the next segment is considered, we halve it and we calculate value of function F ( x ) in the middle of this segment. Then we choose that half of segment on which ends the function F ( x ) has different signs. Results of the lead consecutive actions from the method of halving are tabulated:
On 12-th step according to an estimation (11) the given accuracy of the solution is reached:
hence, for the solution of this equation with the given accuracy by the method halving into the segment [0, 3] it was required, predictably, 12 iterations. The found root of the equation is equal to 1.999878.
Let's notice, that if in the given example we have considered the segment [0, 4] as initial, that already on the first iteration we would receive the exact decision of the equation, as
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