Brent Method Implementation in Octave, Julia, and Mathematica

John Bryan

• Equations.
  
  Inverse Quadrature Interpolation \begin{equation} s = \frac{af(b)f(c)}{(f(a)-f(b))(f(a)-f(c))}+ \frac{bf(a)f(c)}{(f(b)-f(a))(f(b)-f(c))}+ \frac{cf(a)f(b)}{(f(c)-f(a))(f(c)-f(b))} \tag{eq. 1} \end{equation}
  
  
  Secant (linear interpolation) \begin{equation} s = b - \frac{f(b)(b-a)}{f(b)-f(a)} \tag{eq. 2} \end{equation}
  
  
  Midpoint (bisection) \begin{equation} s = \frac{b+a}{2} \tag{eq. 3} \end{equation}




• Octave implementation.



• Julia implementation.



• Mathematica implementation.



• Results.
  $$ \begin{array}{c|lcr} (x+4)(x+1),\quad [-5,-3] & -4 \\ \qquad \sin(x),\quad [3\pi/4,5\pi/4] & \pi \end{array} $$