Appendix 1: Order of Convergence of the Secant Method

Let #tex2html_wrap_inline3392# be the root at which #tex2html_wrap_inline3394#. The error of #tex2html_wrap_inline3396# is:

#math451#

#tex2html_wrap_indisplay6946#	#tex2html_wrap_indisplay6947#	#tex2html_wrap_indisplay6948#
	#tex2html_wrap_indisplay6949#	#tex2html_wrap_indisplay6950#
	#tex2html_wrap_indisplay6951#	#tex2html_wrap_indisplay6952#	(91)

Consider the Taylor expansion of #tex2html_wrap_inline3398# around the root at which as #tex2html_wrap_inline3400#:

#math452#

#tex2html_wrap_indisplay6956#	#tex2html_wrap_indisplay6957#	#tex2html_wrap_indisplay6958#
	#tex2html_wrap_indisplay6959#	#tex2html_wrap_indisplay6960#	(92)

Similarly we also have

#math453# #tex2html_wrap_indisplay6962# (93)

Substituting these into the expression for #tex2html_wrap_inline3402# above we get

#math454#

#tex2html_wrap_indisplay6965#	#tex2html_wrap_indisplay6966#	#tex2html_wrap_indisplay6967#
	#tex2html_wrap_indisplay6968#	#tex2html_wrap_indisplay6969#
	#tex2html_wrap_indisplay6970#	#tex2html_wrap_indisplay6971#
	#tex2html_wrap_indisplay6972#	#tex2html_wrap_indisplay6973#	(94)

When #math455##tex2html_wrap_inline3404#, the lowest order terms in both the numerator and denominator become the dominant terms as all other higher order terms approach to zero, and we have

#math456# #tex2html_wrap_indisplay6976# (95)

where we have defined a constant #math457##tex2html_wrap_inline3406#. To find the order of convergence, we need to find #tex2html_wrap_inline3408# in

#math458# #tex2html_wrap_indisplay6980# (96)

Solving this equation for #tex2html_wrap_inline3410# we get

#math459# #tex2html_wrap_indisplay6983# (97)

On the other hand, when #math460##tex2html_wrap_inline3412# we also have

#math461# #tex2html_wrap_indisplay6986# (98)

Equating the right-hand sides of the two equations above we get

#math462# #tex2html_wrap_indisplay6988# (99)

which requires the following two equations to hold:

#math463# #tex2html_wrap_indisplay6990# (100)

These two equations can be solved separately to get

#math464# #tex2html_wrap_indisplay6992# (101)

i.e.,

#math465# #tex2html_wrap_indisplay6994# (102)

#math466# #tex2html_wrap_indisplay6996# (103)

We see that the secant method has an order of convergence #tex2html_wrap_inline3414# with rate of convergence #math467##tex2html_wrap_inline3416#.