Transcendental functions are functions that cannot be expressed by simple polynomials. Examples of transcendental functions include the exponential function, the trigonometric functions, and their inverses. Given a transcendental function f(x), how do you find r so that f(r) = 0? This book discusses methods for finding such rs.

The book is divided into eight sections. The first offers an introduction to the key issues in root finding. The second section looks at the Chebyshev-proxy (or the companion matrix) root finder. MATLAB and Maple codes are briefly discussed. The adaptive Chebyshev interpolation technique is considered for various scenarios including adaption without a priori knowledge. Adaptive Fourier interpolation is also studied. The Delves-Lyness algorithm is analyzed in the context of complex zeros.

Section 3 looks at the Newton iteration and its kin. There is some discussion of bifurcation theory and continuation in a parameter. The fourth section looks at polynomials and the consequences of Galois theory. The quadratic equation, the cubic polynomial, and the quartic polynomial are studied. Section 5 looks at methods for obtaining explicit solutions. Here, perturbation theory is examined along with singular perturbation methods.

The sixth section considers classical methods for solving one equation in one unknown. Algorithms for special functions, inverse functions, and oracles (techniques for finding out the existence/nonexistence of zeros and their count) are studied. Section 7 looks at techniques for solving two equations in two unknowns. The Section 8 takes a very abbreviated glimpse at the challenges in the field.

There are four appendices on companion matrices, Chebyshev interpolation and quadrature, matching triangles, and imbricate-Fourier series and the Poisson summation theorem. A glossary and a helpful index come with an extensive bibliography.

This well-written book by a highly experienced author will be useful for specialists in numerical analysis, and it will also interest other mathematicians. It can surely be used for teaching courses in numerical analysis. It will also serve as a handy reference book for practitioners and applied scientists for solving transcendental equations.