Computation of minimum eigenvalue through minimization of rayleigh's quotient for large sparse matrices using vector computer:

References

• Hestenes , M. R. and Karush , W. 1951 . A method of gradients for the calculation of the characteristic roots and vectors of a real symmetric matrix . J. Res. Nat. Bur. Stand. , 47 : 45 – 61 .
   Google Scholar
• Hestenes , M. R. and Karush , W. 1951 . Solution of Ax = λBx . J. Res. Nat. Bur. Stand. , 47 : 471 – 478 .
   Google Scholar
• Faddeev D. K. Faddeva V. N. Computational Methods of Linear Algebra 1963 Freeman, San Francisco
   Google Scholar
• Bradbury , W. W. and Fletcher , R. 1966 . New iterative methods for solution of the eigenproblem . Numer. Math. , 9 : 259 – 267 .
   Google Scholar
• Fried , I. 1971 . Optimal gradient minimization schema for finite element eigenproblems . J. Sound Vibration , 19 : 319 – 331 .
   Google Scholar
• Schwarz , H. R. 1974 . The eigenvalue problem (A − λB)x = 0 for symmetric matrices of high order . Comput. Methods Appl. Mech. Eng , 3 : 11 – 28 .
   Google Scholar
• Ruhe , A. “ Computation of eigenvalues and eigenvectors ” . In Sparse Matrix Techniques Edited by: Barker , V. A. 130 – 184 .
   Google Scholar
• Paige , C. C. 1972 . Computational variants of the Lanczos method for the eigenproblem . J. Inst. Math. Appl. , 10 : 373 – 373 .
   Google Scholar
• Golub , G. H. and Underwood , R. R. 1977 . “ The block Lanczos method for computing eigenvalues ” . In Mathematical Softward III , Edited by: Rice , J. R. 361 – 361 . New York : Academic Press .
   Google Scholar
• Ericson , T. 1980 . A. Ruhe, The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems . Math. Comput. , 35 : 1251 – 1251 .
   Google Scholar
• Longsine , D. E. and Mc Cormick , S. F. 1980 . Simultaneous Rayleigh quotient minimization for Ax = λBx . Lin. Alg. Appl. , 34 : 195 – 234 .
   Google Scholar
• Schwarz H. R. Simultaneous Rayleigh-quotient iteration methods for large sparse generalized eigenvalue problems Hinze J. Springer, Berlin 1982 384 398 Lecture Notes in Mathematics, 968,
   Google Scholar
• Gambolati , G. and Perdon , A. M. 1983 . Minimal eigenvalue of large sparse matrices by an efficient reverse power-conjugate gradient scheme . Comput. Methods Appl. Mech. Eng. , 41 : 1 – 10 .
   Google Scholar
• Perdon , A. M. and Gambolati , G. 1986 . Extreme eigenvalues of large sparse matrices by Rayleigh quotient and modified conjugate gradients . Comput.Methods Appl. Mech. Eng. , 56 : 251 – 264 .
   Web of Science ®Google Scholar
• Gambolati , G. , Pini , G. and Sartoretto , F. 1985 . Solution to large symmetric eigenproblems by an accelerated conjugate gradient technique . Environ. Software , 1 : 31 – 39 .
   Google Scholar
• Gambolati , G. , Pini , G. and Sartoretto , F. 1988 . An improved iterative optimization technique for the leftmost eigenpairs of large symmetric matrices . J. Comput. Phys. , 74 : 41 – 60 .
   Google Scholar
• Sartoretto , F. , Pini , G. and Gambolati , G. 1989 . Accelerated simultaneous iterations for large finite element eigenproblems . J. Comput. Phys. , 81 : 53 – 69 .
   Google Scholar
• Gambolati G. Sartoretto F. Florian P. An orthogonal accelerated deflation technique for large symmetric eigenproblems Comput. Methods Appl. Mech. Eng. to appear
   Google Scholar
• Barlow , R. H. , Evans , D. J. and Shanehchi , J. 1982 . An asynchronous parallel version of the power method . Int. J. Comput. Math. , 11 ( No. 2 ) : 143 – 154 .
   Web of Science ®Google Scholar
• Evans , D. J. and Hatzopoulos , M. 1978 . The parallel calculation of the eigenvalues of a real matrix A . Comput. Math. Appl. , 4 ( No. 2 ) : 211 – 218 .
   Google Scholar
• Modi , J. J. and Price , J. D. 1985 . Efficient implementation of Jacobi's diagonalization method on the DAP . Numer. Math. , 46 ( No. 2 ) : 443 – 454 .
   Google Scholar
• Modi , J. J. and Pryce , J. D. 1986 . Mobile Jacobi schemes for parallel computation . Comput. Math. Appl. Part B , 12 ( No. 2 ) : 1217 – 1224 .
   Google Scholar
• Natvig , J. , Nour-Omid , B. and Parlett , B. N. 1986 . Effect of the CYBER 205 on the choice of method for solving the eigenvalue problem (A−λM)x = 0 . J. Comput. Appl. Math. , 15 ( No. 2 ) : 137 – 159 .
   Web of Science ®Google Scholar
• Karp , A. H. and Greenstadt , J. 1987 . An improved parallel Jacobi for diagonalizing a symmetric matrix . Parallel Comput. , 5 ( No. 2 ) : 281 – 294 .
   Google Scholar
• Dongarra , J. J. and Sorensen , D. C. 1987 . A fully parallel algorithm for symmetric eigenvalue problem . SI AM J. Sci. Stat. Comput. , 8 ( No. 2 ) : 139 – 154 .
   Google Scholar
• Lo , S. , Philippe , B. and Sameh , A. 1987 . A multiprocessor algorithm for the symmetric tridiagonal eigenvalue problem . SI AMJ. Sci. Stat. Comput. , 8 ( No. 2 ) : 155 – 165 .
   Google Scholar
• Stewart , G. W. 1987 . A parallel implementation of the QR-algorithm . Parallel Comput. , 5 ( No. 2 ) : 187 – 196 .
   Web of Science ®Google Scholar
• Grimes , R. G. and Simon , H. D. 1988 . New software for large dense symmetric eigenvalue problems using secondary storage . J. Comput. Phys. , 77 ( No. 2 ) : 270 – 278 .
   Google Scholar
• Rousch , W. and Strauss , H. 1988 . A linear algebra package for a local memory multiprocessor problems, proposals and solutions . Parallel Comput. , 7 ( No. 2 ) : 413 – 418 .
   Google Scholar
• Gutheil , I. 1988 . SUPRENUM software for the symmetric eigenvalue problem . Parallel Comput. , 7 ( No. 2 ) : 419 – 424 .
   Google Scholar
• Hake , J. Fr. and Homberg , W. 1989 . Linear algebra software on a vector computer . Parallel Comput. , 10 ( No. 2 ) : 65 – 81 .
   Google Scholar
• Pini G. Perdon A. M. Calcolo dell' autovalore minimo di una matrice sparse su elaboratore vettoriale International Workshop on Supercomputing Tools for Science and Engineering Loughborough University of Technology Pisa 4-7 December 1989
   Google Scholar
• Pini G. Perdon A. M. Is a simple diagonal scaling the best preconditioner for conjugate gradients on supercomputers? Adv. Water Resources 1990
   Google Scholar
• Pini , G. and Zilli , G. 1990 . On vectorizing the preconditioned generalized conjugate residual methods . Int. J. ofComp. Math. , 33 ( No. 2 ) : 195 – 207 .
   Google Scholar
• Johnson , O. G. , Micchelli , C. A. and Paul , G. 1983 . Polynomial preconditioners for conjugate gradient calculations . SI AM J. Numer. Anal. , 20 ( No. 2 ) : 362 – 376 .
   Web of Science ®Google Scholar
• Saad , Y. 1985 . Practical use of polynomial preconditioning for the conjugate gradient method . SI AM J. Sci. Stat. Comput. , 6 ( No. 2 ) : 865 – 881 .
   Google Scholar
• Meijerink , J. A. and van der Vorst , H. A. 1977 . An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix . Math. Comp. , 31 ( No. 2 ) : 148 – 162 .
   Google Scholar
• Kershaw , D. S. 1978 . The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations . J. Comput. Phys. , 26 ( No. 2 ) : 43 – 65 .
   Google Scholar
• Axelsson , O. 1978 . Solution of linear systems of equations: Iterative methods . Sparse Matrix Techniques , 26 ( No. 2 ) : 1 – 51 .
   Google Scholar
• Gambolati , G. , Pini , G. and Zilli , G. 1988 . Numerical comparisons of preconditioning for large sparse finite element problems . Numerical Methods for PDE , 4 ( No. 2 ) : 139 – 157 .
   Google Scholar
• Kinkaid , D. R. , Oppe , T. C. , Respess , J. R. and Young , D. M. 1984 . “ Centre for Numerical Analysis ” . In ITPACKV 2C User's Guide , Austin : University of Texas . Report CNA-191
   Google Scholar
• Abdullah di Brozolo G. Vitaletti M. Sparse matrix vector product and storage represen-tations on the IBM 3090 with vector facility Loughborough University of Technology Rome 1986 Technical Report G 513-4098, IBM ECSEC
   Google Scholar
• Peters , A. , Romunde , B. and Sartoretto , F. 1988 . Proceedings of the International Conference on Computational Methods in Flow Analysis . Vectorized implementation of some MCG codes for F. E. solution of large groundwater flow problems . 1988 , Okayama, Japan.
   Google Scholar
• 1986 CRAY-CAL, Assembler Version 2, Reference Manual SR-2003
   Google Scholar
• Gambolati , G. , Gatto , P. and Ricceri , G. 1984976 . Land subsidence due to gas/oil removal in layered anisotropic soils by a fnite element model . 1984976 . pp. 29 – 41 . Venice : Third Symposium on Land Subsidence . IAHS-AISH Publ. no. 151
   Google Scholar
• Gambolati , G. and Perdon , A. M. 1984 . “ The conjugate gradients in subsurface flow and land subsidence modelling ” . In Fundamentals of Transport Phenomena in Porous Media, NATO-ASI Series, Applied Science 82 , Edited by: Bear , J. and Corapcioglu , M. J. 253 – 984 . The Hague : Martinus Nijhoff .
   Google Scholar
• Gambolati , G. , Sartoretto , F. and Uliana , F. 1986 . A conjugate gradient finite element model of flow for large multiaquifer systems . Water Resour. Res. , 22 : 1003 – 1015 .
   Web of Science ®Google Scholar
• Gambolati , G. , Pini , G. and Tucciarelli , T. 1986 . A 3-D finite element conjugate gradient model of subsurface flow with automatic mesh generation . Adv. Water Resources , 9 : 34 – 41 .
   Google Scholar
• CRAY-FORTRAN (CFT), Reference Manual SR-0009 Revision L-01 1986
   Google Scholar
• CRAY-Library Reference Manual SR-0014 Revision I 1984
   Google Scholar