companion_matrix, a Fortran77 code which computes the companion matrix for a polynomial. The polynomial may be represented in the standard monomial basis, or as a sum of Chebyshev, Gegenbauer, Hermite, Laguerre, or Lagrange basis polynomials. All the roots of the polynomial can be determined as the eigenvalues of the corresponding companion matrix.
The Fortran77 test code must compute eigenvalues and polynomial roots. To do so, it relies on help from the eigs() function, which in turn calls the lapack() library, and the r8poly_roots() function, which calls the toms493() library.
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companion_matrix is available in a Fortran77 version and a MATLAB version and an Octave version and a Python version.
chebyshev_polynomial, a Fortran77 code which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.
eigs, a Fortran77 code which computes the eigenvalues and eigenvectors of a matrix.
gegenbauer_polynomial, a Fortran77 code which evaluates the Gegenbauer polynomial and associated functions.
hermite_polynomial, a Fortran77 code which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.
laguerre_polynomial, a Fortran77 code which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.
lapack_test, a Fortran77 code which calls lapack(), which is a standard linear algebra package for solving linear systems, computing matrix factorizations, and solving eigenvalue problems.
legendre_polynomial, a Fortran77 code which evaluates the Legendre polynomial and associated functions.
polynomial_conversion, a Fortran77 code which converts representations of a polynomial between monomial, Bernstein, Chebyshev, Hermite, Laguerre and Legendre forms.
test_matrix, a Fortran77 code which defines test matrices for which the condition number, determinant, eigenvalues, eigenvectors, inverse, null vectors, P*L*U factorization or linear system solution are known. Examples include the Fibonacci, Hilbert, Redheffer, Vandermonde, Wathen and Wilkinson matrices.
toms493, a Fortran77 code which seeks the roots of a real polynomial; this is commonly called rpoly(); this is a version of ACM TOMS algorithm 493, by Michael Jenkins.