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When do I use FindRoot instead of NSolve?

Like Solve, NSolve is designed primarily for solving polynomial equations.

In[1]:= NSolve[4 - r^2 == r, r]
Out[1]= {{r -> -2.56155}, {r -> 1.56155}}

The algebraic methods available in NSolve cannot handle general equations with transcendental functions—for example:

In[2]:= NSolve[4 Cos[r] == r, r]
  NSolve::nsmet: This system cannot be solved with the methods available to NSolve.
Out[2]= NSolve[4 Cos[r] == r, r]

FindRoot uses numerical methods for starting at an initial value for the independent variable and locating a solution:

In[3]:= fr = FindRoot[ 4 Cos[r] == r, {r, 1}]
Out[3]= {r -> 1.25235}

Verify that the returned value solves the equation:

In[4]:= (4 Cos[r] - r) /. fr // Chop
Out[4]= 0

FindRoot finds one solution per evaluation. To obtain further solutions with FindRoot, initial values need to be chosen sufficiently close to the other root locations:

In[5]:= Plot[ 4 Cos[r] - r, {r, -2 Pi, 2 Pi}]

4Cos[r] - r , 3 roots 

In[6]:= FindRoot[ 4 Cos[r] == r, {r, -4}]
Out[6]= {r -> -3.5953}

In[7]:= FindRoot[ 4 Cos[r] == r, {r, -2}]
Out[7]= {r -> -2.13333}

See also the tutorial on numerical equation solving.

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