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Secant Iteration Procedure

Component

Utility

 

Definition

The Secant method is a root-finding algorithm which assumes a function to be approximately linear in the region of interest. It uses the zero-crossing of the line connecting the limits of the interval as the new reference point. The next iteration starts from evaluating the function at the new reference point and then forms another line. This process is repeated until the root is found.

 

 

Formula

The following algorithm finds a root of f(x) = 0 in the interval of (xo, x1) with which f(x0)f(x1) < 0:

Equation Template

 

Convergence Criteria

The secant method converges more quickly than the bisection method, however as the secant method retains only the most recent estimate, the root does not necessarily remain bracketed. This implies that the algorithm may not converge for functions that are not sufficiently smooth.

Reasonable upper (x1) and lower (x0) bounds are preset for each procedure. If the algorithm does not converge on a root within these bounds, the bounds are adjusted until they contain the root.

The number of iterations required depends considerably upon how close the root is to either of the bounds as well as the desired accuracy level (xk - xk-1). The desired accuracy level varies depending on the scale of x0. Base line accuracy is set at six decimal places for x0 = 1, and decreases as the order of magnitude of x0 increases. For example, if x0 = 10 then the accuracy is set to five decimal places; if x0 = 100 then the iteration procedure will converge to four decimal places; and so on.

If after 40 iterations the value is not within the desired accuracy level, then the entire process will start again with another estimate of x0. This will occur up to five times, and if the procedure still fails to converge then an error message will be displayed.

On average, the procedure will require ten or more iterations to converge on the root.

 

 

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