Full Diagonalisation

Routines are available to diagonalise real symmetric, Hermitian, tridiagonal and general matrices. In the first 2 cases this is usually a 2 step process in which the matrix is first tridiagonalised (transformed to tridiagonal form) and then passed to a routine for diagonalising a tridiagonal matrix. Routines are available which find only the eigenvalues or both eigenvalues and eigenvectors. The former are usually much faster than the latter. Usually the eigenvalues of a Hermitian matrix are returned sorted into ascending order, but this is not always the case (check the description of the routine). Also the eigenvectors are usually normalised to unity. For non-Hermitian matrices only the right-handed eigenvectors are returned and are not normalised. In fact it is not always clear what normalisation means in the general case. Some older FORTRAN and all C (not C++) and PASCAL routines for complex matrices store the real and imaginary parts as separate arrays. The eigenvalues and eigenvectors may also be returned in this form. This is due to 2 facts

• The original routines from Wilkinson and Reinsch (Wilkinson, 1964) were written in ALGOL, which had no complex type.
• Many FORTRAN compilers (even recent ones) handle complex numbers very inefficiently, in that the use a function even for complex addition rather than inline code. In C++ it is worthwhile checking the complex header file to see how this is implemented.