ABSTRACT
By the term secant-type methods we mean all methods of the form
z+ := F ( z , f(z)
) , (6.1)
which, like the generic secant method (0.3), require only one evaluation of the operator f per iteration (no derivatives). Such are, for example, the methods (3.4) and (4.1):
F
([ x A
] , f
([ x A
])) =
[ x−Af(x)
2A−A[x−Af(x), x | f ]A ]
for Ulm’s method, and
F
([ x A
] , f
([ x A
])) =
x−Af(x)
A− Af(x+)〈 A∗Af(x) , f(x+)− f(x)
〉〈A∗Af(x) , ·〉
for Broyden’s. Various members of this class differ from one another by the mapping F used to generate the next approximation z+ from the current iteration
( z , f(z)
) .
