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FindRoot

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FindRoot[f, {x, $x_0$}]

searches for a numerical root of f, starting from x=$x_0$.

FindRoot[lhs == rhs, {x, $x_0$}]

tries to solve the equation lhs == rhs.

FindRoot by default uses Newton's method, so the function of interest should have a first derivative.

• • FindRoot[Cos[x], {x, 1}]

• • FindRoot[Sin[x] + Exp[x],{x, 0}]

• • FindRoot[Sin[x] + Exp[x] == Pi,{x, 0}]

FindRoot has attribute HoldAll and effectively uses Block to localize x.
However, in the result x will eventually still be replaced by its value.

• • x = "I am the result!";

• • FindRoot[Tan[x] + Sin[x] == Pi, {x, 1}]

• • Clear[x]

FindRoot stops after 100 iterations:

• • FindRoot[x^2 + x + 1, {x, 1}]

Find complex roots:

• • FindRoot[x ^ 2 + x + 1, {x, -I}]

The function has to return numerical values:

• • FindRoot[f[x] == 0, {x, 0}]

The derivative must not be 0:

• • FindRoot[Sin[x] == x, {x, 0}]

• • FindRoot[x^2 - 2, {x, 1,3}, Method->"Secant"]