Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

• Relative error: |a − b| < ɛa. This may run into trouble close to x = 0.
E.1.2
• If the slope of f(x) close to zero is very small, there might be a finite
interval in which f(x) is indistinguishable from zero in machine representation.
Newton-Raphson method
The function is linearly approximated at each iteration in order to obtain a
better estimate of the zero. Let us assume that we know both f(x) and f ′ (x).
Then, close to x,
f(x + δ) ≃ f(x) + f ′ (x)δ
(E.1)
and thus, to first order,
δ = − f(x)
f ′ (x)
(E.2)
would yield f(x + δ) = 0. We proceed by iterating in this way. It is possible to
show that the rate of convergence is quadratic, i.e. the number of significant
figures approximately doubles at each iteration (while with bisection method it
grows linearly).
The problem of this method is that convergence is not guaranteed, notably
when f ′ (x) varies a lot close to the zero. Moreover, the method assumes that
f ′ (x) can be directly calculated for any given x. In cases in which this snot
true and the derivative must be calculated via finite differences, it is preferable
to use the secant method described below.
E.1.3
Secant method
This is based on a linear expansion of f(x) between two successive points, x n e
x n+1 , of an iteration:
f(x) = f(x n−1 ) + x − x n−1
x n − x n−1
[f(x n ) − f(x n−1 )]
(E.3)
that provides as an estimate for the zero
x n − x n−1
x n+1 = x n − f(x n )
f(x n ) − f(x n−1 )
(E.4)
The procedure consists in iterating the above step. It is not necessary that
the zero is contained inside the considered interval. This however may lead to
non-convergence in some pathological cases. In regular cases, the speed of convergence
is much better than for the bisection method, although slightly slower
that the Newton-Raphson method (which however requires the knowledge of
the derivative).
In most difficult cases, it is convenient to split the search into two step: first
bisection, with a sure and safe bracketing of the zero; then the secant method
that quickly stabilizes the searched value up to the required precision.
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