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- W2500374157 abstract "We start our discussion of secant methods with the nonlinear equations problem, because secant approximations to the Jacobian are simpler than secant approximations to the Hessian, which we discuss in Chapter 9. Recall that in Chapter 2 we saw that we could approximate ƒ′(x+) at no additional cost in function evaluations by a+=(ƒ(x+)−ƒ(xc))/(x+−xc) , and that the price we paid was a reduction in the local q-convergence rate from 2 to (1+5)/2 . The idea in multiple dimensions is similar: we approximate J(x+) using only function values that we have already calculated. In fact, multivariable generalizations of the secant method have been proposed which, although they require some extra storage for the derivative, do have r-order equal to the largest root of rn+1−rn−1=0 ; but none of them seem robust enough for general use. Instead, in this chapter we will see the basic idea for a class of approximations that require no additional function evaluations or storage and that are very successful in practice. We will single out one that has a q-superlinear local convergence rate and r-order 21/2n ." @default.
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- W2500374157 date "1996-01-01" @default.
- W2500374157 modified "2023-10-16" @default.
- W2500374157 title "8. Secant Methods for Systems of Nonlinear Equations" @default.
- W2500374157 doi "https://doi.org/10.1137/1.9781611971200.ch8" @default.
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