Derive_v5_05 / Derive / SECANT

SECANT.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.