Fundamentals of Numerical Mathematics for Physicists and Engineers. Alvaro Meseguer

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 equation²

(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 equations³:

(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