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are called
transcendental or
nonlinear equations, i.e. equations involving combinations of rational, trigonometric, hyperbolic, or even special functions. An example of a nonlinear equation arising in the field of classical
celestial mechanics is Kepler's equation
2
(1.4)
where
and
are known constants. Another popular example can be found in
quantum physics when solving Schrödinger's equation for a particle of mass
in a
square well potential of finite depth
and width
. In this problem, the admissible energy levels
corresponding to the bounded states are the solutions of any of the two transcendental equations
3:
(1.5)
where
and
is the reduced Planck constant. In this chapter, we will study different methods to obtain
approximate solutions of algebraic and transcendental or nonlinear equations such as (
1.2), (
1.4), or (
1.5). That is, while Cardano's formula (
1.3) provides the
exact value
of one of the roots of (
1.2), the methods we are going to study here will provide just a numerical
approximation of that root. If you have a rigorous mathematical mind you may feel a bit disappointed since it seems always preferable to have an exact analytical expression rather than an approximation. However, we should first clarify the actual meaning of
exact solution within the context of physics.
It is obvious that if the coefficients appearing in the quadratic equation
are known to infinite precision, then the solutions appearing in (
1.1) are exact. The same can be said for Cardano's solution (
1.3) of cubic equation (
1.2). However, equations arising in physics or engineering such as (
1.4) or (
1.5) frequently involve universal constants (such as Planck constant
, the gravitational constant
, or the elementary electric charge
). All universal constants are known with limited precision. For example, the currently accepted value of the
Newtonian constant of gravitation is, according to NIST,
4
As of 2019, the most accurately measured universal constant is
known as Rydberg constant. In other words, the most accurate physical constant is known with 12 digits of precision. Other equations may also contain empirical parameters (such as the thermal conductivity
or the magnetic permeability
of a certain material), which are also known with limited (and usually much less) precision. Therefore, the solutions obtained from equations arising in empirical sciences or technology (even if they have been obtained by analytical methods) are, intrinsically, inaccurate.
Current standard double precision floating point operations are nearly 10 000 times more accurate than the most precise universal constant known in nature. In this book, we will study how to take advantage of this precision in order to implement computational methods capable of satisfying the required accuracy constraints, even in the most demanding situations.
1.2 Approximate Roots: Tolerance
Suppose we want to locate the zeros or roots of a given function
that is continuous within the interval
. Mathematically, the main goal of
exact root‐finding of
in
is as follows:
Root‐finding (Exact): Find
such that
.
However, computers work with finite precision and the condition cannot be exactly satisfied, in general. Therefore, we need to reformulate our problem:
Root‐finding (Approximate): For a given , find such that for some .
This reformulation introduces a new component in the problem: the positive constant , usually termed as tolerance, whose meaning is outlined in the plot on the right. Since the root condition cannot be satisfied exactly, we must provide an interval containing the root . In the figure on the right, the root lies within the interval
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