Numerical linear algebra with applications vol:2 issue:2 pages:95-113
Various plane rotation patterns are presented, which provide stable algorithms for reducing a b-band matrix bordered by p rows and/or columns to (b + p)-band form. These schemes generalize previously presented O(N-2) reduction algorithms for matrices of order N, b = 1, and p = 1 to the reduction of more general b-band, p-bordered matrices where b greater than or equal to 1 and p greater than or equal to 1. Moreover, by splitting the matrix into two similarly structured submatrices and chasing nonzeros to the corners in two directions, the newly proposed patterns reduce the number of required rotations and hence the computational cost by one half compared to the other existing one-way chasing algorithms. Symmetric, as well as more general matrices, are considered. An example of the first type is the symmetric arrowhead matrix that arises in solving inverse eigenvalue problems. Examples of the second type are found in updating the singular value decomposition (SVD) and the partial SVD.