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1.6.3 Numerical Equation Solving

Numerical root finding and minimization.

  • NSolve gives you numerical approximations to all the roots of a polynomial equation.
  • In[1]:= NSolve[ x^5 + x + 1 == 0, x ]


  • You can also use NSolve to solve sets of simultaneous equations numerically.
  • In[2]:= NSolve[{x + y == 2, x - 3 y + z == 3, x - y + z == 0},
    {x, y, z}]


    If your equations involve only linear functions or polynomials, then you can use NSolve to get numerical approximations to all the solutions. However, when your equations involve more complicated functions, there is in general no systematic procedure for finding all solutions, even numerically. In such cases, you can use FindRoot to search for solutions. You have to give FindRoot a place to start its search.

  • This searches for a numerical solution, starting at

  • In[3]:= FindRoot[ 3 Cos[x] == Log[x], {x, 1} ]


  • The equation has several solutions. If you start at a different , FindRoot

    may return a different solution.
  • In[4]:= FindRoot[ 3 Cos[x] == Log[x], {x, 10} ]


  • You can search for solutions to sets of equations. Here the solution involves complex numbers.
  • In[5]:= FindRoot[{x==Log[y], y==Log[x]}, {x, I}, {y, 2}]