Figure shows a plot of such a matrix after several passes of a Givens rotation scheme. Here I have used just the constant values of 1 and -4 from Table 1. In the migration case these coefficients will be complex, but the real values used are more suitable for a demonstration.
In Figure , one pass of Givens rotations (upper right) has turned the outer diagonals into a series of diagonals, all with amplitudes much smaller than the original. After three passes (lower right), these artifacts have effectively been suppressed. The matrix is now tridiagonal, and I can use traditional methods for solving the system.
Note that the rotation matrices that have been applied need also to be applied to the right-hand side of the system of equations being solved. Because the matrix remains sparse throughout the procedure, this method is rather inexpensive.
Figure illustrates what is happening during one pass of the Givens rotation technique. The snapshots here are all from the first pass. It can be seen that artifacts are introduced as the diagonal elements are zeroed, and that the zeroed elements are filled in again (with smaller values) as the algorithm proceeds.