Secant Method Convergence  Roots of a Single-Variable  Convergence of Newton's Method

The Secant Method

One drawback associated with Newton's method is the requirement that there be some mechanism to compute the derivative of the function. One may circumvent this requirement and still adhere to the conceptual ideas that generated Newton's method. Recall that description of Newton's method involved replacing the function f(x) with a line which approximated the function in some sense. In particular, Newton's method approximated the function with a line that passed through the point  and had the slope  . Instead of defining the approximating line via the derivative  , the secant method defines the line by requiring that it pass through  , values which are readily available since they were used in the previous iteration.    Figure 4.2:   The secant method approximates the function f(x) by a line which passes through the points  and  . The next approximation,  , is taken to be the x-intercept of the approximating secant line.

    To formulate the mathematical definition of the secant method, begin with the secant line approximation passing through  and  :

    The x-intercept will define the next approximation  . To find the x-intercept, set y=0 and solve for x. Doing this, we find

  

    Note the similarity with the equation which defines Newton's method. The latter part of Equation 4.7 may be viewed as an approximation to  . That is, the secant method may be viewed as a modification of Newton's method, where the modification takes the form of the approximation

    Two initial approximations are needed for the secant method. These approximations do not need to bracket a root. In other words, the secant method as described below is not a bracketing method, although variations on the bracketing requirements leads to alternate bracketing methods such as the ``Regula-Falsi'' (false-position) method.

 

• Secant Method Convergence

  
 Secant Method Convergence  Roots of a Single-Variable  Convergence of Newton's Method
Wed Oct 28 11:42:13 EST 1998