Abstract
We propose a “one-sided” or “implicit” variant of the Jacobi diagonalization algorithm for positive definite matrices. The variant is based on a previous Cholesky decomposition and currently uses essentially one square array which, on output, contains the matrix of eigenvectors thus reaching the storage economy of the symmetric QL algorithm. The current array is accessed only columnwise which makes the algorithm attractive for various parallelized and/or vectorized implementations. Even on a serial computer our algorithm shows improved efficiency, in particular if the Cholesky step is made with diagonal pivoting. On matrices of ordern=25–200 our algorithm is about 2–3.5 times slower than QL thus being almost on the halfway between the standard Jacobi and QL algorithms. The previous Cholesky decomposition can be performed with higher precision without extra time or storage costs thus offering considerable gains in accuracy with highly conditioned input matrices.
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