An improved parallel Jacobi method for diagonalizing a symmetric matrix

We compare five implementations of the Jacobi method for diagonalizing a symmetric matrix. Two of these, the classical Jacobi and sequential sweep Jacobi, have been used on sequential processors. The third method, the parallel sweep Jacobi, has been proposed as the method of choice for parallel processors. The fourth and fifth methods are believed to be new. They are similar to the parallel sweep method but use different schemes for selecting the rotations.

The classical Jacobi method is known to take O(n⁴) time to diagonalize a matrix of order n. We find that the parallel sweep Jacobi run on one processor is about as fast as the sequential sweep Jacobi. Both of these methods take O(n³ log₂n) time. One of our new methods also takes O(n³ log₂n) time, but the other one takes only O(n³) time. The choice among the methods for parallel processors depends on the degree of parallelism possible in the hardware. The time required to diagonalize a matrix on a variety of architectures is modeled.

Unfortunately for proponents of the Jacobi method, we find that the sequential QR method is always faster than the Jacobi method. The QR method is faster even for matrices that are nearly diagonal. If we perform the reduction to tridiagonal form in parallel, the QR method will be faster even on highly parallel systems.