to determine a root of f(x)=0 that is accurate within a specified tolerance value

hhtt21

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"To determine a root of f(x)=0 that is accurate within a specified tolerance value, given values X1 and X2 such that f(X1)*f(X2)<0, ..."

Would you please explain the part "accurate within". That is where I do not understand in the sentence.

Source:Applied Numerical Analysis by Gerald/Wheatley.

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hhtt21

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Note: We usually say "accurate TO within a specified tolerance value." You have a constraint in this problem, i.e. f(x1)*f(x2)<0. Suppose that x' is the root that you compute using numerical analysis techniques, but f(x') does not equal exactly zero as the true solution would. However, X1 and X2 are your tolerance limits. Therefore we can say that the true value of x such that f(x) =0 lies within the interval x' - x1 < x < x' + x2.

For additional clarification, please send me a private message. (It is not clear from the information given why the constraint exists.)

This was not exactly what I mean to ask. How can a root of a function be accuraterate? Accurate v. non-accurate? How can we mention accuracy of a root? What is accuracy for a root. The root of a function is always constant and in its place in its axis.

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bubbha

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Surely the subject here is using numerical methods to estimate roots.

For example, if the root is 3, then:

Numbers between 2.9 and 3.1 are accurate within a tolerance of 0.1
Numbers between 2.99 and 3.01 are accurate within a tolerance of 0.01
Numbers between 2.999 and 3.001 are accurate within a tolerance of 0.001
...

They're not exactly accurate, but they are close enough given the tolerance level.

In the case of an irrational root, like pi, you will never come up with an exactly accurate numerical representation, so you have to satisfy yourself with approximations of varying accuracy. In the case of pi, 3.14159 is more accurate than 3.14.
 
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