Experimental proof of the quantizing of direction in the magnetic field. The Magnetic Moment of the Silver Atom : W. Gerlach and O. Stern. ( Zeit. f. Physik, Vol. 9, 1922.)

Abstract

A modified multilevel algorithm for solving the excessive storage requirements and ill-conditioning encountered in the boundary-type discretization method is proposed. The modified multilevel algorithm is an extension of the modified dual-level algorithm from dual levels to multiple levels. The method is a kernel-independent method. The core idea is the layer-by-layer calculation and then layer-by-layer correction. Making use of a multilevel structure, the original sparse matrix of the modified dual-level algorithm breaks down into a series of smaller sparse matrices corresponding to different fine meshes. The final matrix to be solved is hereby transformed to a series of smaller sparse matrices instead of a fully-populated matrix. The preconditioning effect originating from the recursive computations among the coarse mesh and fine meshes constitutes its core competitive attribute. The method evaluates far-field contributions only by the coarse mesh and uses a gradual approach to evaluate the near-field contributions. The storage requirements and computing complexity are hereby further reduced significantly.