Fortran Program For Secant Method Equation
Although the Newton-Raphson method is very powerfull to solve non-linear equations, evaluating of the function derivative is the major difficulty of this method. To overcome this deficiency, the secant method starts the iteration by employing two starting points and approximates the function derivative by evaluating of the slope of the line passing through these points. The secant method has been shown in Fig. 1. As it is illustrated in Fig. 1, the new guess of the root of the function f(x) can be found as follows:
Fig. 1.The secant method
Solve the FS equation by RK4 method using the new initial value guess (obtained from secant method). Repeat the process until the required boundary value is obtained. A Fortran program for solving the Falkner-Skan equation implementing the above algorithm is provided below.
In the first glance, the secant method may be seemed similar to linear interpolation method, but there is a major difference between these two methods. In the secant method, it is not necessary that two starting points to be in opposite sign. Therefore, the secant method is not a kind of bracketing method but an open method. This major difference causes the secant method to be possibly divergent in some cases, but when this method is convergent, the convergent speed of this method is better than linear interpolation method in most of the problems.
The algorithm of the secant method can be written as follows:
Step 1: Choose two starting points x0 and x1.
Step 2: Let
Step 3: if x2-x1 <ethen let root = x2, else x0 = x1 ; x1 = x2 ; go to step 2.
Step 4: End.
e: Acceptable approximated error.
oBisection Method
oLinear Interpolation Method
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oModified Methods
oNewton-Raphson Method
oOne point Interpolation Method