Determining the eigenvalues of a quaternion matrix with a band structure Free

A Fortran program called QBAND has been developed which is capable of calculating the eigenvalues of relatively large quaternion matrices of dimension N≳10 000 with a band structure. For a quaternion band matrix with N=17 000 and bandwidth M=700, calculating eigenvalues with a general purpose program like the IMSL routine DEVLHF would require more than 2 Gbytes of random‐access memory. QBAND, however, would need only 200 Mbytes of random‐access memory. In addition, QBAND can be an order of magnitude faster than general purpose programs. Quaternion matrices are Hermitian but also have additional symmetries that were exploited to increase computational speed and to reduce computer memory requirements. QBAND reduces a band quaternion matrix of dimension N with M codiagonals to a Hermitian matrix with three codiagonals, and this three band matrix of dimension N forms the input to a routine called HBAND (previously discussed in this journal) which reduces a band Hermitian matrix to tridiagonal form. A standard routine can then be used to calculate the desired number of eigenvalues. Only the eigenvalues of the original matrix are calculated; eigenvectors are not determined. A simple application of QBAND, chosen to illustrate how a quaternion matrix arises and to demonstrate that QBAND calculates eigenvalues accurately, is presented by calculating the eigenvalues of a spin‐1/2 particle confined to a circular two‐dimensional box. Possible applications of QBAND to more complex problems are mentioned. © 1996 American Institute of Physics.