Modeling Tools for Environmental Engineers and Scientists

Another method, known as the Newton-Raphson method, requires onlyone guess to start the iterative process. It is based on Taylor’s expansion ofthe form:f (x n h) f (x n1 ) f (x n ) hf(x n ) h f(x n ) + . . . (3.8)22If h 2 and higher terms are ignored, it can be seen that the step from x n to x n+1moves the function value closer to a root so that f (x n h) = 0. Then,x n1 x n – f ( xn) (3.9)f(x )Here, the computational process starts with a guess value for x n . Then, usingf (x n ) and f(x n ), a value for x n1 is calculated. If f (x n1 ) is sufficiently small,the root is taken as x n1 ; or, a new value for x n1 is calculated using the currentvalue of x n1 for x n in Equation (3.7), and the process is repeated. Amodification to this method is the Secant method, which is preferable whenit is difficult to get the derivative f(x n ) to be used in Equation (3.7). In suchcases, the Secant method uses the following approximation for f(x n ):nf (x n ) ≈ slope at x n f(x n)–f(xn–1) (3.10)x – xEven though these methods can be implemented in a spreadsheet with relativeease, most equation solver-based software packages feature built-infunctions that can return the solutions to such equations more efficiently andaccurately, without requiring any programming. Spreadsheet packages suchas Excel ® also include some built-in functions that are preprogrammed to performiterative calculations for solving simple equations.The use of the built-in Goal Seek feature of Excel ® in solving the problemin Worked Example 3.2 is illustrated in Figure 3.7. In this worksheet, theright-hand side of the equation to be solved is entered in cell C4. Then, theGoal Seek option is selected from the Tools menu. To start the Goal Seekprocess, cell C4 is specified to be 10 by changing the value of cell C6. Theprocess is instantly executed, and the result is returned as 9.21014.Alternatively, in equation solver-based software packages, such equationscan be solved readily by calling appropriate built-in routines. For example,Figure 3.8 shows how the above problem can be solved in Mathematica ® . Thevariables in the equation are defined first. The built-in routine Solve is calledwhere the first argument contains the equation to be solved. The secondargument is the variable for which the equation is to be solved. When executed,the solution is returned in line Out[2] as 9.21034. Note that theMathematica ® sheet is set up so that one can change the values of the variablesand readily solve the equation for t. In addition, the same setup can beused to solve for any one of the four variables in the equation, provided thenn–1© 2002 by CRC Press LLC