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Файл:Derive_v5_05 / Derive / SECANT
.docSECANT.DOC - by Steven Schonefeld The Secant method for approximating solutions to F(x) = 0 is contained in the file SECANT.MTH. To use the secant method, you simply define (or re- define) the function F(x), then [A]uthor and appro[X]imate SECANT(a, �, n) where a is an initial guess for the desired root, � is a small (non-zero) number, and n is the number of iterations to be performed (usually 5-9). In the following example, we approximate the solution, near a = 1, to the equation x - COS(x) = 0. We define F(x) on line #8, then line #9 is appro[X]imated to give the matrix #10. The first column of #10 contains the approximations a = x0, x1, .., x5 to the desired root. The second column contains � = �0, �1, .., �5 such that x1 = x0 + �1, x2 = x1 + �2, .., x5 = x4 + �5. The third and forth columns contain values F(xn - �n) and F(xn). In this example, the secant method "converges" to the approximate solution x � 0.7390851 to F(x) = 0. When the secant method converges, the values xn in the first column begin to repeat and the other values toward the bottom of the matrix become small in magnitude. #8: F(x) := x - COS(x) #9: SECANT(1, 0.1, 5) #10: [[1 , 0.1 , 0.27839003 , 0.45969769], [0.74645434, -0.25354565 , 0.45969769 , 0.012353230], [0.73945279, -0.0070015570 , 0.012353230 , 6.1536687 10^(-4)], [0.73908572, -3.6706223 10^(-4) , 6.1536687 10^(-4) , 9.9716390 10^(-7)], [0.73908513, -5.9576701 10^(-7) , 9.9716390 10^(-7) , 7.9466089 10^(-11)], [0.73908513, -4.7481710 10^(-11), 7.9466089 10^(-11), -2.3431856 10^(-13)]] For more information on the implementation of this method see the book: NUMERICAL ANALYSIS via DERIVE, by Steven Schonefeld, MathWare, 604 E Mumford Dr., Urbana, IL 61801 1-800-255-2468.