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=== =.= === Why is locating roots important?

=== =.= === Frequently, the solution to a scientific problem is a number about which we have little information other than that it satisfies smoe equation.

=== =.= === Since every equation can be written so that a function stands on one side and zero on the other, the desired number must be a zero of the function.

=== =.= === Thus, if we posses an arsenal of methods for locating zeros of functions, we shall be able to solve such problems.

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=== =.= === We illustrate this claim by use of a specific engineering problem whose solution is the root of an equation. In a certain electrical circuit, the voltage V and current I are related by two equation of the form.

I = a(e^(bV) - 1)
c = dI + V

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=== =.= === in which a,b , c and d are constants. For our purpose, thse four numbers are assumed to be known. When these equations are combined by elminating I between them, the result is a single equation.
c = ad(e^(bV) - 1) + V

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=== =.= === In a concrete cas, this might reduce to 12 = 14.3(e^(2V) - 1) + V
and its solution is required. (It turns out that V = approx 0.299 in this case.)

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In some problems in which a root of an equation is sought, we can perform the required calcuation with a hand calculator. But how can we locate zeros of complicated functions such as these?

f(x) = 3.24x^8 - 2.42x^7 + 10.34x^6 + 11.01x^2 + 47.98
g(x) = 2^x^2 - 19x + 1
h(x) = cosh( sqrt(x^2 + 1) - e^x)

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What is needed is a general numerical method that does not depend on sepcial properties of our functions. Of course, continuity and differenetiability are special properties, but they are common attributes of functions that are usually encountered. The sort of special property that we probably cannot easily explit in general-purpose codes is typified by the trigonometric identity mentioned previously.

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Hundreds of methods are available for locating zeros of functions, and three of the mose useful have been selected for study here: the bisection method, Newton's method, and the secant method.

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Let f be a function that has values of opposite sign at the two ends of an interval.
-Supposed also that f is continus on that interval.
- To fix the notation, let a < b and f(a)f(b) < 0.
---> It then follos that f has a root in the interval (a, b).
--- In other words, there must exist a number r that satisfies the two conditions a < r < b and f(r) = 0.
-How is this conclusion reached?
--->One must recall the Intermediate-value Theorem
-------> A formal statement is as follows: If the function f is continuous on the closed interval [a, b], and if f(a) <= y <= f(b) or f(b) <= y <= f(a), then there exists a point c such that a <= c <= b and f(c) = y.

If x traverses an interval [a, b], then the values of f(x) completely fill out the interval between f(a) and f(b) . No intermediate values can be skipped. Hence, a specific function f must take on the value zero somewhere in the interval (a, b) because f(a) and f(b) are opposite signs.