Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

Appendix E
Useful algorithms
E.1 Search of the zeros of a function
Short notes on the problem of numerical determination of the zeros of a function,
i.e. the values of x that solve f(x) = 0 for a given function.
There are two different types of zeros:
• odd – when f(x) changes sign at x 0
• even – when f(x) does not change sign at x 0
The latter case is more difficult from a numerical point of view. Imagine to
add a small numerical noise to f(x): the zero disappears, or else it splits into
two odd zeros. It is usually convenient to resort to methods to localize extrema
(minima or maxima) rather than trying to pinpoint the zero. In the following
we will deal with odd zeros only.
E.1.1
Bisection method
We first have to find an interval [a, b] that includes a single zero, in such a way
that f(a)f(b) < 0. The bisection algorithm halves the interval at each iteration,
by refining the estimate of x 0 :
1. c = (a + b)/2
2. if f(a)f(c) < 0 we re-define b = c; else if f(b)f(c) < 0 we re-define a = c.
3. In this way we get a new interval [a, b] of half the width at the previous
step, and we repeat the procedure.
Convergence is guaranteed (it is impossible to ”miss” the zero), and the logarithm
of the uncertainty on the solution linearly decreases with the number of
iterations. Some difficulty may arise in the choice of the stopping criterion. For
instance:
• Absolute error: |a − b| < ɛ. This may run into trouble if x 0 is very large:
rounding errors |a − b| might be larger then ɛ.
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