A new algorithm is developed which computes a specified number of eigenvalues in any part of the spectrum of a generalized symmetric matrix eigenvalue problem. It uses a linear system routine (factorization and solution) as a tool for applying the Lanczos algorithm to a shifted and inverted problem. The algorithm determines a sequence of shifts and checks that all eigenvalues get computed in the intervals between them. It is shown that for each shift several eigenvectors will converge after very few steps of the Lanczos algorithm, and the most effective combination of shifts and Lanczos runs is determined for different sizes and sparsity properties of the matrices. For large problems the operation counts are about five times smaller than for traditional subspace iteration methods. Tests on a numerical example, arising from a finite element computation of a nuclear power piping system, are reported, and it is shown how the performance predicted bears out in a practical situation.

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Abstract. The general, linear eigenvalue equations (H -XM)z = 0, where H and M are real symmetric matrices with M positive semidefimte, must be transformed if the Lanczos algorithm is to be used to compute eigenpairs (X,z). When the matrices are large and sparse (but not diagonal) some factorization must be performed as part of the transformation step. If we are interested in only a few eigenvalues a near a specified shift, then the spectral transformation of Ericsson and Ruhe [1] proved itself much superior to traditional methods of reduction.The purpose of this note is to show that a small variant of the spectral transformation is preferable in all respects. Perhaps the lack of symmetry in our formulation deterred previous investigators from choosing it. It arises in the use of inverse iteration.A second goal is to introduce a systematic modification of the computed Ritz vectors, which improves the accuracy when M is ill-conditioned or singular.We confine our attention to the simple Lanczos algorithm, although the first two sections apply directly to the block algorithms as well. Overview. This contribution is an addendum to the paper by Ericsson and Ruhe[1] and also [7]. The value of the spectral transformation is reiterated in a later section. Here we outline our implementation of this transformation.The equation to be solved, for an eigenvalue À and eigenvector z, is(1) (H -AM)z = 0, H and M are real symmetric n X n matrices, and M is positive semidefinite. A practical instance of (1) occurs in dynamic analysis of structures, where H and M are the stiffness and mass matrices, respectively. We assume that a linear combination of H and M is positive definite. It then follows that all eigenvalues X are real. In addition, one has a real scalar a, distinct from any eigenvalue, and we seek a few eigenvalues X close to a, together with their eigenvectors z. Ericsson and Ruhe replace (1) by a standard eigenvalue equation (2) [c(H-aM)"1CT-rl]y = 0, where C is the Choleski factor of M; M = CTC and y = Cz. If M is singular then so is C, but fortunately the eigenvector z can be recovered from y via z = (H -oM)"'CTy. Of course, there is no intention to invert (H -oM) explicitly. The

Various radionuclides have been proposed for systemic tumor therapy. However, in most dosimetric analysis of proposed radionuclides the charged particles are taken into consideration while the potential photons are ignored. The photons will cause undesirable irradiation of normal tissue, and increase the probability of toxicity in, e.g., the bone marrow. The aim of this study was to investigate the dosimetric properties according to particle range, photon emission, and subcellular radionuclide distribution, of a selection of radionuclides used or proposed for radionuclide therapy, and to investigate the possibility of dividing radionuclides into groups according to their dosimetric properties. The absorbed dose rate to the tumors divided by the absorbed dose rate to the normal tissue (TND) was estimated for different tumor sizes in a mathematical model of the human body. The body was simulated as a 70-kg ellipsoid and the tumors as spheres of different sizes (1 ng-100 g). The radionuclides were either assumed to be uniformly distributed throughout the entire tumor and normal tissue, or located in the nucleus or the cytoplasm of the tumor cells and on the cell membrane of the normal cells. Fifty-nine radionuclides were studied together with monoenergetic electrons, positrons, and alpha particles. The tumor and normal tissue were assumed to be of water density. The activity concentration ratio between the tumor and normal tissue was assumed to be 25. The radionuclides emitting low-energy electrons combined with a low photon contribution, and the alpha emitters showed high TND values for most tumor sizes. Electrons with higher energy gave reduced TND values for small tumors, while a higher photon contribution reduced the TND values for large tumors. Radionuclides with high photon contributions showed low TND value for all tumor sizes studied. The radionuclides studied could be divided into four main groups according to their TND values: beta emitters, Auger electron emitters, photon emitters, and alpha emitters. The TND values of the beta emitters were not affected by the subcellular distribution of the radionuclide. The TND values of the Auger electron emitters were affected by the subcellular radionuclide distribution. The photon emitters showed low TND values that were only slightly affected by the subcellular radionuclide distribution. The alpha emitters showed high TND values that were only slightly affected by the subcellular radionuclide distribution. This dosimetric characterization of radionuclides may be valuable in choosing the appropriate radionuclides for specific therapeutic applications.

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