Steepest descent for singular linear operators with nonclosed range

References

• Arghiriade , E. 1968 . Sur Finverse généralisée d'un opérateur lineaire dans les espaces de Hilbert . Atti Accad. Naz. Lincei Rend. CI Sci. Fis. Mat. Natur , 8
   Google Scholar
• Ben-Israel , A. and Charnes , A. 1963 . Contributions to the theory of generalized inverses . J. Soc. Indust. Appl Math , 11 : 667 – 699 .
   Google Scholar
• Beutler , F. J. 1965 . The operator theory of the pseudoinverse I. Bounded operators, II. Unbounded operators with arbitrary range . J. Math. Anal. Appl , 10 : 451 – 470 .
   Web of Science ®Google Scholar
• Bialy , H. 1959 . Iterative Behandlung linearer Funcktionalgleichungen . Arch. Rational Mech. Anal , 4 : 166 – 176 .
   Google Scholar
• Desoer , C. A. and Whalen , B. H. 1963 . A note on pseudoinverses . J. Soc. Indust. Appl. Math , 11 : 442 – 447 .
   Google Scholar
• Douglas , J. and Pearcy , C. M. 1963 . On the convergence of alternating direction procedures in the presence of singular operators . Numer. Math , 5 : 175 – 184 .
   Google Scholar
• Friedrichs , K. O. 1934 . Spektral theorie halbeschrankter Operatoren . Math. Ann , 109 : 465 – 487 .
   Web of Science ®Google Scholar
• Hestenes , M. R. 1961 . Relative self-adjoint operators in Hilbert spaces . Pacific J. Math , 11 : 1315 – 1357 .
   Google Scholar
• Kammerer , W. J. and Nashed , M. Z. 1961 . On the convergence of the conjugate gradient method for singular linear operator equations . Siam J. Numer. Anal , 11 : 1315 – 1357 . To appear in
   Google Scholar
• Kantorovich , L. V. 1948 . Functional analysis and applied mathematics . Uspehi Mat. Nauk , 6 ( 3 ) : 89 – 185 .
   Google Scholar
• Kantorovich , L. V. and Akilov , G. P. 1964 . Functional Analysis in Normed Spaces , New York : Pergamon Press .
   Google Scholar
• Keller , H. B. 1965 . The solution of singular and semidefinite linear systems by iteration . SIAM J. Numer. Anal , 2 : 281 – 290 .
   Google Scholar
• Kellogg , R. B. and Spanier , J. 1965 . On optimal alternating direction parameters for singular matrices . Math. Comp , 19 : 448 – 452 .
   Google Scholar
• Martynjuk , A. E. 1959 . Certain new applications of the methods of Galerkin type . Mat. Sb , 49 : 85 – 108 .
   Google Scholar
• Martynjuk , A. E. 1964 . “ Some new criteria for convergence of the method of successive approximations ” . In Kazan. Gos. Univ. Veen. Zap , Vol. 6 , 183 – 188 . Mathematics Research Cente .
   Google Scholar
• Maurin , K. 1967 . Methods of Hilbert Spaces , Warsaw : Polish Scientific Publishers .
   Google Scholar
• Mikhlin , S. G. 1965 . The Problem of the Minimum of a Quadratic Functional , San Francisco : Holden-Day .
   Google Scholar
• Nashed , M. Z. 1970 . Steepest descent for singular linear operator equations . SIAM J. Numer. Anal , 1 : 358 – 362 .
   Google Scholar
• Petryshyn , W. V. 1962 . Direct and iterative methods for the solution of linear operator equations in Hilbert space . Trans. Amer.Math. Soc , 105 : 136 – 175 .
   Google Scholar
• Petryshyn , W. V. 1965 . On a class of K-p.d. and non-K-p.d. operators and operator equations . J. Math. Anal. Appl , 10 : l – 24 .
   Google Scholar
• Petryshyn , W. V. 1965 . On two variants of a method for the solution of linear equations with unbounded operators and their applications . J. Math. and Phys , XIIV : 296 – 312 .
   Google Scholar
• Petryshyn , W. V. 1967 . On generalized inverses and uniform convergence of (I-pK)n with applications to iterative methods . J. Math. Anal. Appl , 18 : 417 – 439 .
   Web of Science ®Google Scholar
• Rothe , E. R. 1956 . Remarks on the application of gradient mappings to the calculus of variations and the connected boundary value problems in partial differential equations . Comm. Pure and Appl. Math , 9 : 551 – 568 .
   Google Scholar
• Showalter , D. 1967 . Representation and computation of the pseudoinverse . Proc. Amer. Math. Soc , 18 : 584 – 586 .
   Web of Science ®Google Scholar
• Showalter , D. and Ben-Israel , A. 1969 . “ Series of Applied Mathematics ” . In Representation and Computation of the Generalized Inverse of a Bounded Linear Operator between Hilbert Spaces , Vol. 69 , Northwestern University .
   Google Scholar
• Tseng , Yu. Ya. 1949 . Centralized inverses of unbounded operators between two unitary spaces . Dokl. Akad. SSSR , 67 : 431 – 434 .
   Google Scholar
• Tseng , Yu. Ya. 1949 . Properties and classification of generalized inverses of closed operators . Dokl. Akad. Nauk. SSSR , 67 : 607 – 610 .
   Google Scholar
• Tseng , Yu. Ya. 1956 . Virtual solutions and general inversions . Uspehu Mat. Nauk , 11 : 213 – 215 .
   Google Scholar
• Votruba , G. F. 1963 . “ Generalized inverses and singular equations in functional analysis ” . In Ph.D. Dissertation , University of Michigan .
   Google Scholar
• Yosida , K. 1968 . Functional Analysis , New York : Springer-Verlag .
   Google Scholar