Fundamentals of Numerical Mathematics for Physicists and Engineers

Numerical methods are more than procedures for getting an answer: knowing why an algorithm works—and when its result may be unreliable—is essential in scientific computing. Alvaro Meseguer’s Fundamentals of Numerical Mathematics for Physicists and Engineers builds that understanding across the numerical tools used in physics and engineering, pairing mathematical analysis with computational examples.

From equations to approximation

The book begins with scalar nonlinear equations, including bisection, Newton’s method, and secant methods. It then develops polynomial interpolation, numerical differentiation, and numerical integration. Along the way, concepts such as tolerance, conditioning, convergence, and interpolation error help explain how method choices affect a computed result.

Linear algebra and nonlinear systems

The second part moves into numerical linear algebra, covering direct solvers, LU and QR factorization, least squares, matrix conditioning, and matrix-free Krylov methods. It also treats Newton’s method for systems of nonlinear equations and numerical continuation, connecting the algorithms to the challenges of solving more complex models.

Fourier tools and differential equations

Further chapters address the discrete Fourier transform, aliasing, and Fourier differentiation, followed by ordinary differential equations. Topics include boundary-value problems, Runge–Kutta and linear multistep formulas, convergence of time-stepping methods, and stiffness.

Mathematical ideas, put to work

MATLAB practicals and problem sets accompany the theoretical material, giving readers opportunities to implement methods as well as study their properties. Examples draw on areas including mechanics, thermodynamics, electrical networks, and quantum physics. Solutions to many problems and exercises appear in the book.

Who may find it useful?

Designed around a two-semester course, the text is suited to mathematically prepared undergraduate students in physics and engineering, and may also interest graduate students and applied mathematicians. The first part assumes a foundation in single-variable calculus and elementary linear algebra; the second calls for broader mathematical preparation.