Fundamentals of Numerical Mathematics for Physicists and Engineers. Alvaro Meseguer

Fundamentals of Numerical Mathematics for Physicists and Engineers - Alvaro Meseguer


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relation

      (1.21)equation

      or images for short, where the prime symbols acting on images and images must be understood as derivatives with respect to images and images, respectively. We can now estimate the location of the new root. Since images is small,

equation

      Since the new root is precisely located at images,

equation

      Finally, since images, we conclude from the last equation that images. Taking the absolute value and recalling that images,

Graphs depict (a) (x,f(x)) intercepting ordinates y equal to b and y equal to b plus delta-b at abscissas x equal to a and x equal to a plus delta-b, respectively. (b) Roots of the equation x cube-two point four x square plus one point eight x minus zero point four minus b equal to zero for b equal to zero indicated by solid curve, b equal to negative zero point zero one indicated by dashed gray, and b equal to zero point zero one indicated by solid gray line.
intercepting ordinates
and
at abscissas
and
, respectively. (b) Roots of the equation images for images (solid black curve), images (dashed gray), and images (solid gray).

      From the previous analysis, we can clearly conclude that the double root images is more sensitive (or ill‐conditioned) than the simple root images. This phenomenon could have been predicted in advance just by evaluating the denominator images appearing in (1.22) with images and images or images, since images, whereas images.

      In general, for a given numerical problem, it is common practice to quantify its conditioning by the simple relation