False Position and Secant Methods

secant method
to circumvent the problem of the derivative evaluation in newton’s method, we introduce a slight variation. by definition
f (p_(n-1) )=lim?(x?p_(n-1) )??(f(x)-f(p_(n-1) ))/(x-p_(n-1) )?
if p_(n-2) is close to p_(n-1), then

f^ (p_(n-1) )?(f(p_(n-2) )-f(p_(n-1) ))/(p_(n-2)-p_(n-1) )=(f(p_(n-1) )-f(p_(n-2) ))/(p_(n-1)-p_(n-2) )

using this approximation for f(p_(n-1) ) in newton’s formula gives

p_n=p_(n-1)-f(p_(n-1) )(p_(n-1)-p_(n-2) )/(f(p_(n-1) )-f(p_(n-2) ) ) ?

this technique is called the secant method and is presented in algorithm 2. (see figure 2.) starting with the two initial approximations p_0 and p_1, the approximation p2 is the x-intercept of the line joining (p_0,f(p_0 )) and (p_1,f(p_1 )).


algorithm 2: secant method
input initial approximation p_0, p_1 epsilon ? maximum number of iterations n0.
output approximate solution p or message of failure.

step 1 set i = 2
q_0=f(p_0 )
q_1=f(p_1 )

step 2 while i ? n0 do steps 3–6.
step 3 set p=p_1-(q_1 (p_1-p_0 ))?((q_1-q_0 ) ) (compute pi.)

step 4 if |p-p_1 | < ? then
output (p) (procedure completed successfully.)
stop

step 5 set i = i + 1.

step 6 set p_0=p_1 (updating p_0,q_0,p_1,q_1)
q_0=q_1
p_1=p
q_1=f(p).

step 7 output (‘method failed after n0 iterations, n0 =’, n0)
(the procedure was unsuccessful.)
stop.