Module rgsl::polynomials

Polynomials

This chapter describes functions for evaluating and solving polynomials. There are routines for finding real and complex roots of quadratic and cubic equations using analytic methods. An iterative polynomial solver is also available for finding the roots of general polynomials with real coefficients (of any order).

References and Further Reading

The balanced-QR method and its error analysis are described in the following papers,

R.S. Martin, G. Peters and J.H. Wilkinson, “The QR Algorithm for Real Hessenberg Matrices”, Numerische Mathematik, 14 (1970), 219–231. B.N. Parlett and C. Reinsch, “Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors”, Numerische Mathematik, 13 (1969), 293–304. A. Edelman and H. Murakami, “Polynomial roots from companion matrix eigenvalues”, Mathematics of Computation, Vol. 64, No. 210 (1995), 763–776. The formulas for divided differences are given in the following texts,

Abramowitz and Stegun, Handbook of Mathematical Functions, Sections 25.1.4 and 25.2.26. R. L. Burden and J. D. Faires, Numerical Analysis, 9th edition, ISBN 0-538-73351-9, 2011.

Modules

cubic_equations

divided_difference_representation

The functions described here manipulate polynomials stored in Newton’s divided-difference representation. The use of divided-differences is described in Abramowitz & Stegun sections 25.1.4 and 25.2.26, and Burden and Faires, chapter 3, and discussed briefly below.

evaluation

The functions described here evaluate the polynomial P(x) = c[0] + c[1] x + c[2] x^2 + \dots + c[len-1] x^{len-1} using Horner’s method for stability.

quadratic_equations