Motivation for solving equations numerically

Section 7.1 Motivation for solving equations numerically

We often want to solve an equation of the form

$f(x)=0\text{,}$ given a real-valued function

$f$ of one real variable

$x\text{.}$ Some such equations admit a symbolic solution, obtained by some algebraic manipulations. For example, the equation

$2e^x - 3 = 0$ has the solution

$x=\log(3/2)\text{.}$ But if we put together two or more unrelated functions, for example

$2e^x + x^3 - 1 = 0\text{,}$ algebra does not help.

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For this reason, Matlab has a built-in root-finding function fzero. Its first argument is the function which should be equated to zero, the second argument is our initial guess at where the solution may be (put 0 if you have no idea). For example,

fzero(@(x) 2*exp(x) + x^3 - 1, 0)

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finds the solution -0.5439. But even though we have fzero, we still have to understand the methods used for finding roots, because

fzero only gives one solution, there may be others;
the output of fzero may be wrong.

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An example of wrong answer:

fzero(@(x) exp(-x^2), 0)

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returns x=-28.9631. In reality,

$e^{-x^2} = 0$ has no solutions, and there is nothing special about the number produced by fzero: this is just where its algorithm decided to stop, thinking “close enough”.