BExercises for Egison as a Computer Algebra System

By writing some of the programs in the problems in this appendix, you will become proficient in using Egison as a computer algebra system. Many of these problems require mathematical knowledge more than programming difficulty, but please give them a try.

Solutions of Cubic and Quartic Equations

Extending the content of Section 9.4, write programs to find the solutions of cubic and quartic equations. The Cardano method and the Ferrari method are known algorithms for finding the solutions of cubic and quartic equations, respectively.

\(n\)th Roots of Unity

It is known that the \(n\)th roots of unity can be found algebraically. Try writing programs to compute the \(n\)th roots of unity for several values of \(n\). Sample code for the 5th, 7th, 9th, and 17th roots of unity is available in the Egison repository.

Computing Riemann Curvature

Referring to the program in Section 14.2, write programs to compute the Riemann curvature of a cylinder and the Riemann curvature of \(S^3\).

Hodge Laplacian in Spherical Coordinates

Referring to the program for computing the Hodge Laplacian in polar coordinates from Section 14.2, write a program to compute the Hodge Laplacian in spherical coordinates.