Fundamentals of Numerical Mathematics for Physicists and Engineers. Alvaro Meseguer

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Название Fundamentals of Numerical Mathematics for Physicists and Engineers
Автор произведения Alvaro Meseguer
Жанр Математика
Серия
Издательство Математика
Год выпуска 0
isbn 9781119425755



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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 equation2

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:

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.

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

      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.

      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 images.

      However, computers work with finite precision and the condition images cannot be exactly satisfied, in general. Therefore, we need to reformulate our problem:

      Root‐finding (Approximate): For a given images, find images such that images for some images.

      This reformulation introduces a new component in the problem: the positive constant images, usually termed as tolerance, whose meaning is outlined in the plot on the right. Since the root condition images cannot be satisfied exactly, we must provide an interval containing the root images. In the figure on the right, the root images lies within the interval