Fundamentals of Numerical Mathematics for Physicists and Engineers. Alvaro Meseguer

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Название Fundamentals of Numerical Mathematics for Physicists and Engineers
Автор произведения Alvaro Meseguer
Жанр Математика
Серия
Издательство Математика
Год выпуска 0
isbn 9781119425755



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alt="images"/> may represent uncertainties in the parameters, numerical noise or, within the context of this book, numerical inaccuracies due to limited machine precision. The condition number images must be understood as a noise amplifier, which magnifies small uncertainties. A condition number of order 1 is an indication of well‐conditioning, whereas a problem with images is definitely ill‐conditioned.

      (1.24)equation

      as the condition number of the root images. As we will see in Section 1.6, the performance of Newton's method can be affected if the root we are looking for is ill‐conditioned.

      Newton's method converges properly only under certain conditions. One required condition is that the initial guess from which the iteration is initiated must be sufficiently close to the root, that is, a local initial guess. In that sense, it is said that Newton's method has only local convergence. Even if the sequence converges to the root, the order may not be always images, as in Figure 1.2a.

Graphs depicting the (a) convergence history of Newton's and secant methods with the Yk ordinates corresponding to the secant method shifted downwards three units to avoid overlap between two sets of data and (b) Newton's method iterating for the solution of logx-exp(sinx)=0.