Establishing secure connection… Loading editor… Preparing document…
Navigation

Fill and Sign the Pharmacy Purchase Order Format

Fill and Sign the Pharmacy Purchase Order Format

How it works

Open the document and fill out all its fields.
Apply your legally-binding eSignature.
Save and invite other recipients to sign it.

Rate template

4.5
42 votes
Generalized Inverses: Theory and Applications Bibliography for the 2nd Edition August 29, 2002 [2145] items Adi Ben-Israel Thomas N.E. Greville† RUTCOR–Rutgers Center for Operations Research, Rutgers University, 640 Bartholomew Rd, Piscataway, NJ 08854-8003, USA E-mail address: bisrael@rutcor.rutgers.edu Bibliography 15. 16. 1. K. Abdel-Malek and Harn-Jou Yeh, On the determination of starting points for parametric surface intersections, Computer-aided Design 29 (1997), no. 1, 21–35. 2. N. N. Abdelmalek, On the solutions of the linear least squares problems and pseudo–inverses, Computing 13 (1974), no. 3-4, 215–228. 3. V. M. Adukov, Generalized inversion of block Toeplitz matrices, Linear Algebra and its Applications 274 (1998), 85–124. 4. , Generalized inversion of finite rank Hankel and Toeplitz operators with rational matrix symbols, Linear Algebra and its Applications 290 (1999), no. 1-3, 119–134. 5. S. N. Afriat, On the latent vectors and characteristic values of products of pairs of symmetric idempotents, Quart. J. Math. Oxford Ser. (2) 7 (1956), 76–78. , Orthogonal and oblique projectors and the 6. characteristics of pairs of vector spaces, Proc. Cambridge Philos. Soc. 53 (1957), 800–816. 7. J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and their Applications, Academic Press, New York, 1967. 8. A. C. Aitken, On least squares and linear combinations of observations, Proceedings of the Royal Society of Edinburgh, Sec A 55 (1934), 42–47. 9. Y. Akatsuka and T. Matsuo, Optimal control of linear discrete systems using the generalized inverse of a matrix, Techn Rept. 13, Institute of Automatic Control, Nagoya Univ., Nagoya, Japan, 1965. 10. F. Akdeniz, A note concerning the Gauss-Markov theorem, J. Fac. Sci. Karadeniz Tech. Univ. 1 (1977), 129–133. 11. F. Akdeniz and M. Sweilem, An application of the generalized inverse to the two way classification for estimating variance components, J. Karadeniz Univ. Fac. Arts Sci. Ser. Math.-Phys. 9 (1986), 1– 10, (extension of [1732]). 12. I. S. Alalouf and G. P. H. Styan, Characterizations of estimability in the general linear model, Ann. Statist. 7 (1979), no. 1, 194–200. 13. , Estimability and testability in restricted linear models, Math. Operationsforsch. Statist. Ser. Statist. 10 (1979), no. 2, 189–201. 14. A. Albert, Conditions for positive and nonnegative definiteness in terms of pseudo–inverses, SIAM J. Appl. Math. 17 (1969), 434–440. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 3 , Regression and the Moore–Penrose Pseudoinverse, Academic Press, New York, 1972. , The Gauss–Markov theorem for regression models with possibly singular covariances, SIAM J. Appl. Math. 24 (1973), 182–187. , Statistical applications of the pseudo inverse, In Nashed [1402], pp. 525–548. A. Albert and R. W. Sittler, A method for computing least squares estimators that keep up with the data, SIAM J. Control 3 (1965), 384–417. V. Aleksić and V. Rakočević, Approximate properties of the Moore-Penrose inverse, VIII Conference on Applied Mathematics (Tivat, 1993), Univ. Montenegro, Podgorica, 1994, pp. 1–14. B. A. Aliev, The pseudo-inverse operator for the product of two operators, Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. (1979), no. 2, 11– 15, 70. M. F. Allam and M. A. Laughton, The use of pseudo and oblique pseudo-inverse matrices in power system state estimation algorithms, Internat. J. Control 24 (1976), no. 5, 661–671. E. L. Allgower, K. Böhmer, A. Hoy, and V. Janovský, Direct methods for solving singular nonlinear equations, ZAMM Z. Angew. Math. Mech. 79 (1999), 219–231. E. L. Allgower and K. Georg, Numerical path following, Handbook of Numerical Analysis, Vol. V, North-Holland, Amsterdam, 1997, pp. 3–207. D. Alpay, J. A. Ball, and V. Bolotnikov, On the bitangential interpolation problem for contractive valued functions in the polydisk, J. Operator Theory 44 (2000), no. 2, 277–301. D. Alpay, V. Bolotnikov, and Ph. Loubaton, One two-sided residue interpolation for matrix-valued H2 -functions with symmetries, J. Math. Anal. Appl. 200 (1996), no. 1, 76–105. D. Alpay, V. Bolotnikov, and L. Rodman, Onesided tangential interpolation for operator-valued Hardy functions on polydisks, Integral Equations Operator Theory 35 (1999), no. 3, 253–270. , Two-sided tangential interpolation for Hilbert-Schmidt operator functions on polydisks, Operator Theory: Advances and Applications 124 (2001), 21–62. M. Altman, A generalization of Newton’s method, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 3 (1955), 189–193. , On a generalization of Newton’s method, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 5 (1957), 789–795. 4 BIBLIOGRAPHY 30. J. K. Amburgey, T. O. Lewis, and T. L. Boullion, On computing generalized characteristic vectors and values for a rectangular matrix, In Boullion and Odell [255], pp. 267–275. 31. A. R. Amir-Moéz, Geometry of generalized inverses, Math. Mag. 43 (1970), 33–36. 32. , Quasi-singular values of linear transformations, Rend. Circ. Mat. Palermo (2) 22 (1973), no. 3, 314–316 (1974). 33. , Geometry of determinants and Grassmann algebras, Delta (Waukesha) 5 (1975), no. 2, 71–83. 34. , Extreme Properties of Linear Transformations, Polygonal Publ. House, Washington, NJ, 1990. 35. A. R. Amir-Moéz and R. E. Byerly, Pythagorean theorem in unitary spaces, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 7 (1996), 85–89. 36. A. R. Amir-Moéz and A. Horn, Singular values of a matrix, Amer. Math. Monthly 65 (1958), 742–748. 37. C. L. Anderson, A Geometric Theory of Pseudoinverses and some Applications in Statistics, Master’s thesis in statistics, Southern Methodist Univ., 1967. 38. R. D. Anderson, H. V. Henderson, F. Pukelsheim, and S. R. Searle, Best estimation of variance components from balanced data, with arbitrary kurtosis, Math. Operationsforsch. Statist. Ser. Statist. 15 (1984), no. 2, 163–176. 39. W. N. Anderson, Jr., Shorted operators, SIAM J. Appl. Math. 20 (1971), 520–525. 40. W. N. Anderson, Jr. and R. J. Duffin, Series and parallel addition of matrices, SIAM J. Appl. Math. 26 (1969), 576–594, (see [1085]). 41. W. N. Anderson, Jr. and M. Schreiber, The infimum of two projections, Acta Sci. Math. (Szeged) 33 (1972), 165–168. 42. W. N. Anderson, Jr. and G. E. Trapp, Inequalities for the parallel connection of resistive n-port networks, J. Franklin Inst. 209 (1975), no. 5, 305–313. 43. , Shorted operators. II, SIAM J. Appl. Math. 28 (1975), 60–71, (this concept first introduced by Krein [1088]). 44. , Analytic operator functions and electrical networks, In Campbell [320], pp. 12–26. 45. , Inverse problems for means of matrices, SIAM J. Algebraic Discrete Methods 7 (1986), no. 2, 188–192. 46. T. Ando, Generalized Schur complements, Linear Algebra and its Applications 27 (1979), 173–186. 47. Mihai Anitescu, Dan I. Coroian, M. Zuhair Nashed, and Florian A. Potra, Outer inverses and multibody system simulation, Numer. Funct. Anal. Optim. 17 (1996), no. 7-8, 661–678. 48. P. M. Anselone and P. J. Laurent, A general method for the construction of interpolating or smoothing spline-functions, Numer. Math. 12 (1968), 66–82. 49. H. Anton and C. S. Duris, On minimum norm and best approximate solutions of Av = b in normed 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. spaces, J. Approximation Theory 16 (1976), no. 3, 245–250. A. C. Antoulas, Approximation of linear operators in the 2-norm, Linear Algebra and its Applications 278 (1998), no. 1-3, 309–316. R. C. Archibald, A Semicentennial History of the American Mathematical Society 1888–1938 (2 volumes), American Mathematical Society, New York, 1938. E. Arghiriade, Sur les matrices qui sont permutables avec leur inverse généralisée, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. Ser. VIII 35 (1963), 244–251. , On the generalized inverse of a product of matrices, An. Univ. Timişoara Ser. Şti. Mat.-Fiz. No. 5 (1967), 37–42. , Remarques sur l’inverse généralisée d’un produit de matrices, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. Ser. VIII 42 (1967), 621– 625. , Sur quelques équations fonctionnelles de matrices, Rev. Roumaine Math. Pures Appl. 12 (1967), 1127–1133. , Sur l’inverse généralisée d’un operateur lineaire dans les espaces de Hilbert, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. Ser. VIII 45 (1968), 471–477. E. Arghiriade and A. Dragomir, Une nouvelle définition de l’inverse généralisée d’une matrice, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 35 (1963), 158–165. , Remarques sur quelques théoremes relatives a l’inverse généralisée d’un operateur lineaire dans les espaces de hilbert, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. Ser. VIII 46 (1969), 333–338. I. K. Argyros, Local convergence theorems of Newton’s method for nonlinear equations using outer or generalized inverses, Czechoslovak Math. J. 50(125) (2000), no. 3, 603–614. S. Aronowitz and B. E. Eichinger, Petrie matrices and generalized inverses, J. Math. Phys. 16 (1975), 1278–1283. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. F. V. Atkinson, The normal solvability of linear equations in normed spaces (russian), Mat. Sbornik N.S. 28(70) (1951), 3–14. , On relatively regular operators, Acta Sci. Math. Szeged 15 (1953), 38–56. K. E. Atkinson, The solution of non-unique linear integral equations, Numer. Math. 10 (1967), 117– 124, (see also [1368]). L. Autonne, Sur les groupes linéaires, réels et orthogonaux, Bull. Soc. Math. France 30 (1902), 121– 134. , Sur les matrices hypohermitiennes et sur les matrices unitaires, Ann. Univ. Lyon, Nouvelle BIBLIOGRAPHY 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. Sér. I 38 (1915), 1–77, (see history of SVD in [1765]). K. E. Avrachenkov, M. Haviv, and P. G. Howlett, Inversion of analytic matrix functions that are singular at the origin, SIAM J. Matrix Anal. Appl. 22 (2001), no. 4, 1175–1189 (electronic). Gorô Azumaya, Strongly π-regular rings, J. Fac. Sci. Hokkaido Univ. Ser. I. 13 (1954), 34–39. G. Backus, Inference from inadequate and inaccurate data. I, II, Proc. Nat. Acad. Sci. U.S.A. 65 (1970), 1–7; ibid. 65 (1970), 281–287. G. Backus and F. Gilbert, Uniqueness in the inversion of inaccurate gross Earth data, Philos. Trans. Roy. Soc. London Ser. A 266 (1970), no. 1173, 123– 192. C. Badea and M. Mbekhta, Generalized inverses and the maximal radius of regularity of a Fredholm operator, Integral Equations Operator Theory 28 (1997), no. 2, 133–146. , Compressions of resolvents and maximal radius of regularity, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2949–2960. J. K. Baksalary, A relationship between the star and minus orderings, Linear Algebra and its Applications 82 (1986), 163–167. J. K. Baksalary and O. M. Baksalary, Commutativity of projectors, Linear Algebra and its Applications 341 (2002), 129–142. J. K. Baksalary and J. Hauke, Partial orderings of matrices referring to singular values or eigenvalues, Linear Algebra and its Applications 96 (1987), 17– 26. J. K. Baksalary and R. Kala, The matrix equation AX−Y B = C, Linear Algebra and its Applications 25 (1979), 41–43. , The matrix equation AXB + CY D = E, Linear Algebra and its Applications 30 (1980), 141–147. , Two properties of a nonnegative definite matrix, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), no. 5-6, 233–235 (1981). , On equalities between BLUEs, WLSEs, and SLSEs, Canad. J. Statist. 11 (1983), no. 2, 119–123, (extension of [1204]). , Range invariance of certain matrix products, Linear and Multilinear Algebra 14 (1983), no. 1, 89–96. , Corrigendum: “On equalities between BLUEs, WLSEs, and SLSEs”, Canad. J. Statist. 12 (1984), no. 3, 240. J. K. Baksalary and T. Mathew, Rank invariance criterion and its application to the unified theory of least squares, Linear Algebra and its Applications 127 (1990), 393–401. J. K. Baksalary and S. K. Mitra, Left-star and right-star partial orderings, Linear Algebra and its Applications 149 (1991), 73–89. J. K. Baksalary, P. R. Pordzik, and G. Trenkler, A note on generalized ridge estimators, Comm. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 5 Statist. Theory Methods 19 (1990), no. 8, 2871– 2877. J. K. Baksalary and F. Pukelsheim, On the Löwner, minus, and star partial orderings of nonnegative definite matrices and their squares, Linear Algebra and its Applications 151 (1991), 135–141. J. K. Baksalary, F. Pukelsheim, and G. P. H. Styan, Some properties of matrix partial orderings, Linear Algebra and its Applications 119 (1989), 57–85. J. K. Baksalary, S. Puntanen, and H. Yanai, Canonical correlations associated with symmetric reflexive generalized inverses of the dispersion matrix, Linear Algebra and its Applications 176 (1992), 61–74. J. K. Baksalary, P. Šemrl, and G. P. H. Styan, A note on rank–aditivity and range–aditivity, Linear Algebra and its Applications 237–238 (1996), 489– 498. A. V. Balakrishnan, An operator theoretic formulation of a class of control problems and a steepest descent method of solution, J. Soc. Indust. Appl. Math. Ser. A: Control 1 (1963), 109–127. K. F. Baldwin and A. E. Hoerl, Bounds of minimum mean squared error in ridge regression, Comm. Statist. A—Theory Methods 7 (1978), no. 13, 1209–1218. J. A. Ball, M. Rakowski, and B. F. Wyman, Coupling operators, Wedderburn-Forney spaces, and generalized inverses, Linear Algebra and its Applications 203/204 (1994), 111–138. K. S. Banerjee, Singularity in Hotelling’s weighing designs and generalized inverses, Ann. Math. Statist. 37 (1966), 1021–1032, (erratum, ibid 40(1969), 710). K. S. Banerjee and W. T. Federer, On the structure and analysis of singular fractional replicates, Ann. Math. Statist. 39 (1968), 657–663. R. B. Bapat, Generalized inverses with proportional minors, Linear Algebra and its Applications 211 (1994), 27–33. , Moore–Penrose inverse of the incidence matrix of a tree, Linear and Multilinear Algebra 42 (1997), no. 2, 159–167. , Structure of a nonnegative regular matrix and its generalized inverses, Linear Algebra and its Applications 268 (1998), 31–39. , Linear Algebra and Linear Models, second ed., Hindustan Book Agency, New Delhi, 1999. , Linear estimation in models based on a graph, Linear Algebra and its Applications 302/303 (1999), 223–230. , Resistance distance in graphs, Mathematics Student 68 (1999), 87–98. , Moore-Penrose inverse of set inclusion matrices, Linear Algebra and its Applications 318 (2000), no. 1-3, 35–44. , Outer inverses: Jacobi type identities and nullities of submatrices, To appear (2002). 6 BIBLIOGRAPHY 102. R. B. Bapat and A. Ben-Israel, Singular values and maximum rank minors of generalized inverses, Linear and Multilinear Algebra 40 (1995), no. 2, 153– 161. 103. R. B. Bapat and R. E. Hartwig, A master of the row space and the column space: the mathematical work of Sujit Kumar Mitra, In Bapat et al. [108], (special issue of Linear Algebra and its Applications 211 (1994)), pp. 5–14. 104. R. B. Bapat, S. K. Jain, and S. Pati, Weighted Moore-Penrose inverse of a Boolean matrix, Linear Algebra and its Applications 255 (1997), 267–279. 105. R. B. Bapat, S. K. Jain, and K. M. Prasad, Generalized power symmetric stochastic matrices, Proc. Amer. Math. Soc. 127 (1999), no. 7, 1987–1994. 106. R. B. Bapat, S. K. Jain, and L. E. Snyder, Nonnegative idempotent matrices and the minus partial order, Linear Algebra and its Applications 261 (1997), 143–154. 107. R. B. Bapat and D. M. Kulkarni, Minors of some matrices associated with a tree, Algebra and Its Applications. Contemporary Mathematics 259 (D. V. Huynh, S. K. Jain, S. R. Lopez-Permouth Ed.), American Math Society, Providence, RI, 2000, pp. 45–66. 108. R. B. Bapat, S. K. Mitra, and R. Hartwig (eds.), Generalized inverses. papers from the workshop on g-inverses held in calcutta, december 11–16, 1993, New York, North-Holland Publishing Co., 1994, (special issue of Linear Algebra and its Applications 211 (1994)). 109. R. B. Bapat and Sukanta Pati, Algebraic connectivity and the characteristic set of a graph, Linear and Multilinear Algebra 45 (1998), no. 2-3, 247–273. 110. R. B. Bapat and K. M. Prasad, Cochran’s theorem and related results on matrix rank over a commutative ring, Statistical Inference and Design of Experiments (U. J. Dixit and M. R. Satam Ed.), Narosa Publishing House, 1999, pp. 125–133. 111. R. B. Bapat and T. E. S. Raghavan, Nonnegative Matrices and Applications, Cambridge University Press, Cambridge, 1997. 112. R. B. Bapat, K. P. S. Bhaskara Rao, and K. M. Prasad, Generalized inverses over integral domains, Linear Algebra and its Applications 140 (1990), 181–196. 113. R. B. Bapat and D. W. Robinson, The MoorePenrose inverse over a commutative ring, Linear Algebra and its Applications 177 (1992), 89–103. 114. G. P. Barker and S. L. Campbell, Internal stability of two-dimensional systems, Linear and Multilinear Algebra 14 (1983), no. 4, 365–369. 115. G. A. Barnard, The logic of least squares, J. Roy. Statist. Soc. Ser. B 25 (1963), 124–127. 116. S. Barnett, Matrices in Control Theory, Van Nostrand Reinhold, London, 1971. , Matrices: Methods and Applications, 117. Clarendon Press, Oxford, 1990. 118. H. Bart, M. A. Kaashoek, and D. C. Lay, Relative inverses of meromorphic operator functions and associated holomorphic projection functions, Math. Ann. 218 (1975), no. 3, 199–210. 119. G. Basile, Alcune osservazioni sulla pseudoinversa di una matrice rettangolare., Atti Accad. Sci. Ist. Bologna Cl. Sci. Fis. Rend. (12) 6 (1968/1969), no. fasc., 1–2, 236–240. 120. T. S. Baskett and I. J. Katz, Theorems on products of EPr matrices, Linear Algebra and its Applications 2 (1969), 87–103. 121. H. Bateman, A formula for the solving function of a certain integral equation of the second kind, Transactions of the Cambridge Philosophical Society 20 (1908), 179–187. , On the application of integral equations 122. to the determination of upper and lower limits of a double integral, Transactions of the Cambridge Philosophical Society 21 (1908), 123–128. 123. , The reality of the roots of certain transcendental equations occurring in the theory of integral equations, Transactions of the Cambridge Philosophical Society 20 (1908), 371–381. 124. , On the numerical solution of linear integral equations, Proc. Roy. Soc. London Ser. A 100 (1922), 441–449. 125. D. Batigne, Integral generalized inverses of integral matrices, Linear Algebra and its Applications 22 (1978), 125–134. 126. D. R. Batigne, F. J. Hall, and I. J. Katz, Further results on integral generalized inverses of integral matrices, Linear and Multilinear Algebra 6 (1978/79), no. 3, 233–241. 127. F. L. Bauer, A further generalization of the Kantorovič inequality, Numer. Math. 3 (1961), 117– 119. 128. , Elimination with weighted row combinations for solving linear equations and least squares problems, Numer. Math. 7 (1965), 338–352, (republished, pp. 119–133 in [2058]). 129. , Theory of norms, Computer Science Dept. CS 75, Stanford University, Stanford, 1967. 130. F. L. Bauer, J. Stoer, and C. Witzgall, Absolute and monotonic norms, Numer. Math. 3 (1961), 257–264. 131. E. F. Beckenbach and R. Bellman, Inequalities, 3rd ed., Springer-Verlag, New York, 1971. 132. R. Bellman, Introduction to Matrix Analysis, 2nd ed., McGraw-Hill Book Co., New York, 1970. 133. E. Beltrami, Sulle funzioni bilineari, Giornale di Matematiche ad Uso degli Studenti Delle Universita 11 (1873), 98–106, (an English translation by D. Boley is available as University of Minnesota, Department of Computer Science, Technical Report 90–37, 1990. see history of SVD in [1765]). 134. E. J. Beltrami, A constructive proof of the Kuhn– Tucker multiplier rule, J. Math. Anal. Appl. 26 (1969), 297–306. BIBLIOGRAPHY 135. A. Ben-Israel, On direct sum decompositions of Hestenes algebras, Israel J. Math. 2 (1964), 50–54. , An iterative method for computing the gen136. eralized inverse of an arbitrary matrix, Math. Comput. 19 (1965), 452–455. 137. , A modified Newton-Raphson method for the solution of systems of equations, Israel J. Math. 3 (1965), 94–98. 138. , A Newton-Raphson method for the solution of systems of equations, J. Math. Anal. Appl. 15 (1966), 243–252. 139. , A note on an iterative method for generalized inversion of matrices, Math. Comput. 20 (1966), 439–440. 140. , A note on the Cayley transform, Notices Amer. Math. Soc. 13 (1966), 599. 141. , On error bounds for generalized inverses, SIAM J. Numer. Anal. 3 (1966), 585–592, (see also [1759]). 142. , On iterative methods for solving nonlinear least squares problems over convex sets, Israel J. Math. 5 (1967), 211–214. 143. , On the geometry of subspaces in Euclidean n–spaces, SIAM J. Appl. Math. 15 (1967), 1184– 1198. 144. , On applications of generalized inverses in nonlinear analysis, In Boullion and Odell [255], pp. 183–202. 145. , On decompositions of matrix spaces with applications to matrix equations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 45 (1968), 122–128. 146. , On optimal solutions of 2-person 0-sum games, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 44 (1968), 512–516. 147. , A note on partitioned matrices and equations, SIAM Rev. 11 (1969), 247–250. 148. , On matrices of index zero or one, SIAM J. Appl. Math. 17 (1969), 1118–1121, (see [1281], [1630]). 149. , On Newton’s method in nonlinear programming, In Kuhn [1098], pp. 339–352. 150. , Applications of generalized inverses to programming, games, and networks, In Nashed [1402], pp. 495–523. 151. , A Cramer rule for least-norm solutions of consistent linear equations, Linear Algebra and its Applications 43 (1982), 223–226, (see [1906], [373], [375], [374], [1781], [1927], [1932], [2034]). 152. , Generalized inverses of matrices: a perspective of the work of Penrose, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 3, 407–425. 153. , A volume associated with m × n matrices, Linear Algebra and its Applications 167 (1992), 87–111, (this concept was introduced by Good [687]). 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 7 , The change-of-variables formula using matrix volume, SIAM J. Matrix Anal. Appl. 21 (1999), no. 1, 300–312 (electronic). , A local inverse for nonlinear mappings, Numer. Algorithms 25 (2000), no. 1-4, 37–46, Mathematical journey through analysis, matrix theory and scientific computation (Kent, OH, 1999). , An application of the matrix volume in probability, Linear Algebra and its Applications 321 (2001), 9–25. , The Moore of the Moore–Penrose inverse, To appear (2002). A. Ben-Israel and A. Charnes, Contributions to the theory of generalized inverses, J. Soc. Indust. Appl. Math. 11 (1963), 667–699. , Generalized inverses and the Bott-Duffin network analysis, J. Math. Anal. Appl. 7 (1963), 428–435, (corrigendum in J. Math. Anal. Appl. 18(1967), 393). , An explicit solution of a special class of linear programming problems, Operations Res. 16 (1968), 1166–1175, (see [181], [209], [1597], [1645]). A. Ben-Israel, A. Charnes, and P. D. Robers, On generalized inverses and interval linear programming, In Boullion and Odell [255], pp. 53–70. A. Ben-Israel and D. Cohen, On iterative computation of generalized inverses and associated projections, SIAM J. Numer. Anal. 3 (1966), 410–419. A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, WileyInterscience [John Wiley & Sons], New York, 1974, (reprinted by Robert E. Krieger Publishing Co. Inc., Huntington, NY, 1980.). , Some topics in generalized inverses of matrices, In Nashed [1402], pp. 125–147. A. Ben-Israel and M. J. L. Kirby, A characterization of equilibrium points of bimatrix games, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 46 (1969), 402–407. A. Ben-Israel and S. J. Wersan, An elimination method for computing the generalized inverse of an arbitrary complex matrix, J. Assoc. Comput. Mach. 10 (1963), 532–537. A. Ben-Tal and M. Teboulle, A geometric property of the least squares solution of linear equations, Linear Algebra and its Applications 139 (1990), 165–170, (see [177], [487], [153], [168], [233], [612]). , Addenda: “A geometric property of the least squares solution of linear equations” [Linear Algebra Appl. 139 (1990), 165–170], Linear Algebra and its Applications 180 (1993), 5. J. M. Bennett and J. J. Edwards, A graph isomorphism algorithm using pseudoinverses, BIT 36 (1996), no. 1, 41–53. M. W. Benson and P. O. Frederickson, Fast parallel algorithms for the Moore-Penrose pseudo-inverse, 8 BIBLIOGRAPHY 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. Hypercube Multiprocessors 1987 (Knoxville, TN, 1986), SIAM, Philadelphia, PA, 1987, pp. 597–604. , Fast pseudo-inverse algorithms on hypercubes, Multigrid Methods (Copper Mountain, CO, 1987), Dekker, New York, 1988, pp. 23–33. M. Benzi and C. D. Meyer, Jr., A direct projection method for sparse linear systems, SIAM J. Sci. Comput. 16 (1995), no. 5, 1159–1176. H. Berens and M. Finzel, A continuous selection of the metric projection in matrix spaces, Numerical Methods of Approximation Theory, Vol. 8 (Oberwolfach, 1986), Birkhäuser, Basel, 1987, (see [174]), pp. 21–29. , Addendum: “A continuous selection of the metric projection in matrix spaces” [in numerical methods of approximation theory, vol. 8 (Oberwolfach, 1986), 21–29, Birkhäuser, Basel, 1987; MR 90i:41040], Numer. Math. 57 (1990), no. 6-7, 663– 667. I.S. Berezin and N.P. Zhidkov, Computing Methods, Pergamon Press, London, 1965. L. Berg, Über quasivertauschbare Matrixinversen, Rostock. Math. Kolloq. (1980), no. 15, 5–10. , Three results in connection with inverse matrices, Linear Algebra and its Applications 84 (1986), 63–77, (see also [167]). M. Berger and B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces, Graduate Texts in Mathematics No. 115, Springer-Verlag, New York, 1988, (translated by S. Levy). P. G. Bergman, R. Penfield, R. Schiller, and H. Zatkis, The Hamiltonian of the general theory of relativity with electromagnetic field, Physical Review 52 (1950), 1950. A. Berman, Nonnegative matrices which are equal to their generalized inverse, Linear Algebra and its Applications 9 (1974), 261–265. , Generalized interval programming, Bull. Calcutta Math. Soc. 71 (1979), no. 3, 169–176. A. Berman and S. K. Jain, Nonnegative generalized inverses of powers of nonnegative matrices, Linear Algebra and its Applications 107 (1988), 169–179. A. Berman and M. Neumann, Consistency and splittings, SIAM J. Numer. Anal. 13 (1976), no. 6, 877–888. , Proper splittings of rectangular matrices, SIAM J. Appl. Math. 31 (1976), no. 2, 307–312. A. Berman and R. J. Plemmons, Monotonicity and the generalized inverse, SIAM J. Appl. Math. 22 (1972), 155–161. , Cones and iterative methods for best least squares solutions of linear systems, SIAM J. Numer. Anal. 11 (1974), 145–154. , Inverses of nonnegative matrices, Linear and Multilinear Algebra 2 (1974), 161–172. , Nonnegative Matrices in the Mathematical Sciences, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994, (revised reprint of the 1979 original). 189. L. Bernard, A generalized inverse method for asymptotic linear programming, Mathematical Programming 43 (1989), 71–86. , An efficient basis update for asymptotic 190. linear programming, Linear Algebra and its Applications 184 (1993), 83–102. 191. Kenneth W. Berryman, Mario E. Inchiosa, Arthur M. Jaffe, and Steven A. Janowsky, Extending the pseudoinverse rule, Neural Networks and Spin Glasses (Pôrto Alegre, 1989), World Sci. Publishing, Teaneck, NJ, 1990, pp. 169–175. 192. M. Bertero, C. De Mol, and E. R. Pike, Linear inverse problems with discrete data. I. General formulation and singular system analysis, Inverse Problems 1 (1985), no. 4, 301–330. 193. , Linear inverse problems with discrete data. II. Stability and regularisation, Inverse Problems 4 (1988), no. 3, 573–594. 194. J. Bérubé, R. E. Hartwig, and G. P. H. Styan, On canonical correlations and the degrees of nonorthogonality in the three-way layout, Statistical Sciences and Data Analysis (Tokyo, 1991), VSP, Utrecht, 1993, pp. 247–252. 195. F. J. Beutler, The operator theory of the pseudoinverse. I. Bounded operators, J. Math. Anal. Appl. 10 (1965), 451–470. 196. , The operator theory of the pseudo-inverse. II. Unbounded operators with arbitrary range, J. Math. Anal. Appl. 10 (1965), 471–493. 197. F. J. Beutler and W. L. Root, The operator pseudoinverse in control and systems identification, In Nashed [1402], pp. 397–494. 198. J. H. Bevis, F. J. Hall, and R. E. Hartwig, Consimilarity and the matrix equation AX − XB = C, Current Trends in Matrix Theory (Auburn, Ala., 1986), North-Holland, New York, 1987, pp. 51–64. 199. , The Drazin inverse of a semilinear transformation and its matrix representation, Linear Algebra and its Applications 97 (1987), 229–242. 200. J. H. Bevis, F. J. Hall, and I. J. Katz, Integer generalized inverses of incidence matrices, Linear Algebra and its Applications 39 (1981), 247–258. 201. R. Bhatia, Perturbation Bounds for Matrix Eigenvalues, Longman Scientific & Technical, Harlow, 1987. 202. , Letter to the editor: “The n-dimensional Pythagorean theorem” [Linear and Multilinear Algebra 26(1990), no. 1-2, 9–13; MR 90k:51031] by S.-Y. T. Lin and Y. F. Lin, Linear and Multilinear Algebra 30 (1991), no. 1-2, 155, (see [1178]). 203. , Matrix Analysis, Springer-Verlag, New York, 1997. 204. P. Bhimasankaram, A characterization of subclasses of generalized inverses of specified rank, Sankhyā Ser. A 36 (1974), no. 2, 214–218. 205. , On generalized inverses of a block in a partitioned matrix, Linear Algebra and its Applications 109 (1988), 131–143. BIBLIOGRAPHY 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. , Rank factorization of a matrix and its applications, Math. Sci. 13 (1988), no. 1, 4–14, (see [1587]). P. Bhimasankaram and T. Mathew, On ordering properties of generalized inverses of nonnegative definite matrices, Linear Algebra and its Applications 183 (1993), 131–146. P. Bhimasankaram and R. SahaRay, On a partitioned linear model and some associated reduced models, Linear Algebra and its Applications 264 (1997), 329–339. M. Bilodeau, Sur une représentation explicite des solutions optimales d’un programme linéaire, Canad. Math. Bull. 29 (1986), no. 4, 419–425. G. D. Birkhoff, Review of “The New Haven Colloquium. by E. H. Moore, E. J. Wilczynski, M. Mason. Yale University Press, 1910, x + 222 p.”, Bull. Amer. Math. Soc. 17 (1911), 414–428. Z. W. Birnbaum, Introduction to Probability and Mathematical Statistics, Harper & Brothers, Publishers, New York, 1962. A. Bjerhammar, Application of calculus of matrices to method of least squares with special reference to geodetic calculations, Trans. Roy. Inst. Tech. Stockholm 1951 (1951), no. 49, 86 pp. (2 plates). , Rectangular reciprocal matrices, with special reference to geodetic calculations, Bull. Géodésique (1951), 188–220. , A generalized matrix algebra, Trans. Roy. Inst. Tech. Stockholm 1958 (1958), no. 124, 32 pp. , Studies with generalized matrix algebra, Bull. Géodésique (N.S.) No. 85 (1967), 193–210. , Theory of Errors and Generalized Matrix Inverses, Elsevier Scientific Publishing Co., Amsterdam, 1973. Å. Björck, Iterative refinement of linear least squares solutions I, BIT 7 (1967), 257–278. , Solving linear least squares problems by Gram–Schmidt orthogonalization, BIT 7 (1967), 1– 21. , Iterative refinement of linear least squares solutions II, BIT 8 (1968), 8–30. , A uniform numerical method for linear estimation from general Gauss-Markov models, Proceedings of the First Symposium on Comutational Statistics (COMPSTAT), (G. Bruckmann, F. Ferschl and L. Schmetterer, Editors), Physica Verlag, Vienna, 1974, pp. 131–140. , Numerical Methods for Least Squares Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. Å. Björck and C. Bowie, An iterative algorithm for computing the best estimate of an orthogonal matrix, SIAJN 8 (1971), no. 2, 358–364. Å. Björck and T. Elfving, Accelerated projection methods for computing pseudoinverse solutions of systems of linear equations, BIT 19 (1979), 145– 163. 9 224. Å. Björck and G. H. Golub, Iterative refinement of linear least squares solutions by householder transformation, BIT 7 (1967), 322–337. , Numerical methods for computing angles 225. between linear subspaces, Mathematics of Computation 27 (1973), 579–594. 226. B. Blaschke, A. Neubauer, and O. Scherzer, On convergence rates for the iteratively regularized Gauss-Newton method, IMA J. Numer. Anal. 17 (1997), no. 3, 421–436. 227. J. Blatter and E. W. Cheney, On the existence of extremal projections, J. Approximation Theory 6 (1972), 72–79. 228. J. W. Blattner, Bordered matrices, J. Soc. Indust. Appl. Math. 10 (1962), 528–536. 229. , On the convergence of a certain matrix iteration, Bul. Inst. Politehn. Iaşi (N.S.) 10 (14) (1964), no. 3-4, 43–46. 230. G. A. Bliss, Eliakim Hastings Moore, Bull. Amer. Math. Soc. 39 (1933), 831–838. 231. , The scientific work of Eliakim Hastings Moore, Bull. Amer. Math. Soc. 40 (1934), 501–514. 232. L. Bober and P. Chrzan, Application of the generalized Moore-Penrose matrix inversion to the estimation of a classical econometric model under additional constraints, Przeglad Statyst. 25 (1978), ‘ no. 3, 315–324 (1979). 233. E. Y. Bobrovnikova and S. A. Vavasis, A norm bound for projections with complex weights, Linear Algebra and its Applications 307 (2000), no. 1-3, 69–75, (a complex version of the bounds in [1764], [1846]). 234. P. T. Boggs, The convergence of the Ben-Israel iteration for nonlinear least squares problems, Math. Comp. 30 (1976), no. 135, 512–522. 235. E. Bohl and P. Lancaster, Perturbation of spectral inverses applied to a boundary layer phenomenon arising in chemical networks, Linear Algebra and its Applications 180 (1993), 35–59. 236. F. Bohnenblust, A characterization of complex Hilbert spaces, Portugal. Math. 3 (1942), 103–109. 237. E. Boman and I. Koltracht, Computing preconditioners via subspace projection, Linear Algebra and its Applications 302/303 (1999), 347–353. 238. T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Springer, Berlin, 1934. 239. C. de Boor, The Method of Projections as applied to the Numerical Solution of Two Point Boundary Value Problems using Cubic Splines, Doctoral dissertation in mathematics, University of Michigan, Ann Arbor, MI, 1966. 240. , Convergence of abstract splines, J. Approx. Theory 31 (1981), no. 1, 80–89. 241. J. C. G. Boot, The computation of the generalized inverse of singular or rectangular matrices, Amer. Math. Monthly 70 (1963), 302–303. 242. E. Boroş, On the generalized inverse of an EPr matrix, An. Univ. Timişoara Ser. Şti. Mat.-Fiz. No. 2 (1964), 33–38. 10 243. 244. 245. 246. 247. 248. 249. 250. 251. 252. 253. 254. 255. 256. 257. 258. 259. 260. 261. 262. BIBLIOGRAPHY , On certain properties of EPr matrices, An. Univ. Timişoara Ser. Şti. Mat.-Fiz. No. 3 (1965), 77–84. , Das verallgemeinerte Inverse eines linearen Operators in Vektorräumen mit nicht ausgearteter Hermitescher Metrik über einem kommutativen Körper, J. Reine Angew. Math. 252 (1972), 68–78. , The generalized inverse of linear operators on spaces with indefinite metric, An. Univ. Timişoara Ser. Ştiinţ. Mat. 21 (1983), no. 1-2, 9– 44. E. Boroş and I. Sturz, On quasi-inverse matrices, An. Univ. Timişoara Ser. Şti. Mat.-Fiz. No. 1 (1963), 59–66. N. K. Bose and S. K. Mitra, Generalized inverse of polynomial matrices, IEEE Trans. Automatic Control 23 (1978), no. 3, 491–493, (see also [1730]). R. C. Bose, The fundamental theorem of linear estimation (abstract), Proc. 31st Indian Sci. Congress (1944), 2–3. R. Bott and R. J. Duffin, On the algebra of networks, Trans. Amer. Math. Soc. 74 (1953), 99–109. R. H. Bouldin, The product of operators with closed range, Tôhoku Math. J. (2) 25 (1973), 359–363. , The pseudo-inverse of a product, SIAM J. Appl. Math. 24 (1973), 489–495. , Selfadjoint approximants, Indiana Univ. Math. J. 27 (1978), no. 2, 299–307. , Generalized inverses and factorizations, In Campbell [320], pp. 233–249. T. L. Boullion, Contributions to the Theory of Pseudoinverses, Ph.D. thesis, University of Texas, Austin, 1966. T. L. Boullion and P. L. Odell (eds.), Proceedings of the Symposium on Theory and Applications Generalized Inverses of Matrices, Lubbock, Texas Tech. Press, 1968. , A note on the Scroggs-Odell pseudoinverse, SIAM J. Appl. Math. 17 (1969), 7–10, (correction of [1670, Theorem 6]). , Generalized Inverse Matrices, John Wiley & Sons, New York, 1971. T. L. Boullion and G. D. Poole, A characterization of the general solution of the matrix equation AX + XB = C, Indust. Math. 20 (1970), 91–95. Z. Boulmaarouf, M. Fernandez Miranda, and J-Ph. Labrousse, An algorithmic approach to orthogonal projections and Moore-Penrose inverses, Numer. Funct. Anal. Optim. 18 (1997), no. 1-2, 55–63. E. Bounitzky, Sur la fonction de Green des équations differentielles linéaires ordinaires, J. Math. Pures Appl. 5 (1909), no. 6, 65–125. N. Bourbaki, Eléments de Mathématiques. Livre V. Espaces Vectoriels Topologiques, Hermann & Cie, Paris, 1953. , Eléments de Mathématiques. Livre II. Algèbre, Hermann & Cie, Paris, 1958. 263. H. J. Bowdler, R. S. Martin, G. Peters, and J. H. Wilkinson, Solutions of real and complex systems of linear equations, Numer. Math. 8 (1966), 217–239, (republished, pp. 93–110 in [2058]). 264. V. J. Bowman and C.-A. Burdet, On the general solution to systems of mixed–integer linear equations, SIAM J. Appl. Math. 26 (1974), 120–125. 265. Yu. E. Boyarintsev, General solutions of boundary value problems for singular systems of ordinary differential equations, Čisl. Metody Meh. Splošn. Sredy 8 (1977), no. 7, 12–21. , A representation of the Drazin inverse 266. matrix, Numerical Methods of Optimization (Applied Mathematics) (Russian), Akad. Nauk SSSR Sibirsk. Otdel. Ènerget. Inst., Irkutsk, 1978, pp. 176–179. 267. , Regulyarnye i Singulyarnye Sistemy Lineinykh Obyknovennykh Differentsialnykh Uravnenii, “Nauka” Sibirsk. Otdel., Novosibirsk, 1980. 268. , Solving a pair of matrices, Approximate Methods for Solving Operator Equations and their Applications, Akad. Nauk SSSR Sibirsk. Otdel. Ènerget. Inst., Irkutsk, 1982, pp. 35–47. 269. , Representation of the solutions of a system of linear algebraic equations by means of generalized inverse matrices, Computational Methods in Linear Algebra (Russian) (Moscow, 1982), Akad. Nauk SSSR Otdel Vychisl. Mat., Moscow, 1983, pp. 33–45. 270. , The solving pair of matrices and its application, Current Problems in Numerical and Applied Mathematics (Novosibirsk, 1981), “Nauka” Sibirsk. Otdel., Novosibirsk, 1983, pp. 52–55. 271. , Degenerate systems and the index of a variable matrix, Differential Equations and Numerical Methods, “Nauka” Sibirsk. Otdel., Novosibirsk, 1986, pp. 105–114, 277. 272. , Methods of Solving Singular Systems of Ordinary Differential Equations, John Wiley & Sons Ltd., Chichester, 1992, (translation of the 1988 Russian original). 273. , A resolving transformation of unknowns in an implicit system of ordinary differential equations, Algebrodifferential Systems and Methods for their Solution (Russian), VO “Nauka”, Novosibirsk, 1993, pp. 4–19, 90. 274. Yu. E. Boyarintsev, V. A. Danilov, A. A. Loginov, and V. F. Chistyakov, Chislennye Metody Resheniya Singulyarnykh Sistem, “Nauka” Sibirsk. Otdel., Novosibirsk, 1989. 275. Yu. E. Boyarintsev and V. M. Korsukov, The structure of a general continuously differentiable solution of a boundary value problem for a singular system of ordinary differential equations, Questions in Applied Mathematics (Russian), Sibirsk. Ènerget. Inst., Akad. Nauk SSSR Sibirsk. Otdel., Irkutsk, 1977, pp. 73–93. 276. F. Brackx, R. Delanghe, and J. Van hamme, Generalized inverses of elliptic systems of differential BIBLIOGRAPHY 277. 278. 279. 280. 281. 282. 283. 284. 285. 286. 287. 288. 289. 290. 291. 292. operators with constant coefficients and related REDUCE programs for explicit calculations, Rend. Circ. Mat. Palermo (2) Suppl. (1987), no. 16, 21– 28. H. W. Braden, R-matrices and generalized inverses, J. Phys. A 30 (1997), no. 15, L485–L493. , The equations AT X ± X T A = B, SIAM J. Matrix Anal. Appl. 20 (1999), no. 2, 295–302 (electronic). J. S. Bradley, Adjoint quasi–differential operators of Euler type, Pacific J. Math. 16 (1966), 213–237. , Generalized Green’s matrices for compatible differential systems, Michigan Math. J. 13 (1966), 97–108. L. Brand, The solution of linear algebraic equations, Math. Gaz. 46 (1962), 203–237. C. Brezinski, Projection methods for linear systems, J. Comput. Appl. Math. 77 (1997), no. 1-2, 35–51, (ROLLS Symposium (Leipzig, 1996)). C. Brezinski, M. Morandi Cecchi, and M. Redivo Zaglia, The reverse bordering method, SIAM J. Matrix Anal. Appl. 15 (1994), no. 3, 922–937. C. Brezinski and M. Redivo Zaglia, Extrapolation Methods. Theory and practice, with 1 IBM-PC floppy disk (5.25 inch), North-Holland Publishing Co., Amsterdam, 1991. P. Broadbridge and H. G. Petersen, Use of generalized inverses in the construction of Hamiltonians for constrained dynamical systems, Confronting the Infinite (Adelaide, 1994), World Sci. Publishing, River Edge, NJ, 1995, pp. 307–318. C. G. den Broeder Jr. and A. Charnes, Contributions to the theory of generalized inverses for matrices, Dept. of math., Purdue University, Lafayette, IN, 1957, (Reprinted as ONR Res. Memo. No. 39, Northwestern University, Evanston, IL, 1962). D. S. Broomhead, R. Jones, G. P. King, and E. R. Pike, Singular system analysis with application to dynamical systems, Chaos, Noise and Fractals (Como, 1986), Hilger, Bristol, 1987, pp. 15–27. K. G. Brown, On ridge estimation in rank deficient models, Comm. Statist. A—Theory Methods 7 (1978), no. 2, 187–192. R. C. Brown, Generalized Green’s functions and generalized inverses for linear differential systems with Stieltjes boundary conditions, J. Differential Equations 16 (1974), 335–351. R. Bru and N. Thome, Group inverse and group involutory matrices, Linear and Multilinear Algebra 45 (1998), no. 2-3, 207–218. R. A. Brualdi, From the Editor-in-Chief. Comment on: “The explicit solutions and solvability of matrix equations” [Linear Algebra Appl. 311 (2000), no. 1-3, 195–199; MR 2000m:15019] by L. Huang, Linear Algebra and its Applications 320 (2000), no. 1-3, 216. R. A. Brualdi and H. Schneider, Determinantal identities : Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir and Cayley, 293. 294. 295. 296. 297. 298. 299. 300. 301. 302. 303. 304. 305. 306. 307. 11 Linear Algebra and its Applications 52 (1983), 769–791. J. T. Bruening, A new formula for the MoorePenrose inverse, Current Trends in Matrix Theory (Auburn, Ala., 1986), North-Holland, New York, 1987, pp. 65–74. R. S. Bucy, Comments on a paper by F. E. Udwadia and R. E. Kalaba: “A new perspective on constrained motion” [Proc. Roy. Soc. London Ser. A 439 (1992), no. 1906, 407–410; MR 94b:70027], Proc. Roy. Soc. London Ser. A 444 (1994), no. 1920, 253–255, (see [990]). Hamza Bulut and S. Aysun Bulut, Spectral decompositions and generalized inverses in a circularization network flow problem, J. Math. Anal. Appl. 174 (1993), 390–402. W. Burmeister, Inversionfreie Verfahren zur Lösung nichtlinearer Operatorgleichungen, Zeit. angew. Math. Mech. 52 (1972), 101–110. F. Burns, D. Carlson, E. V. Haynsworth, and T. Markham, Generalized inverse formulas using the Schur complement, SIAM J. Appl. Math. 26 (1974), 254–259. P. A. Businger and G. H. Golub, Linear least squares by Householder transformations, Numer. Math. 7 (1965), 269–276, (republished, pp. 111– 118 in [2058]). , Algorithm 358: Singular value decomposition of a complex matrix, Comm. ACM 12 (1969), 564–565. C. A. Butler and T. D. Morley, A note on the shorted operator, SIAM J. Matrix Anal. Appl. 9 (1988), no. 2, 147–155. , Six generalized Schur complements, Linear Algebra and its Applications 106 (1988), 259–269. Kim Ki-Hang Butler, A Moore-Penrose inverse for Boolean relation matrices, Combinatorial Mathematics (Proc. Second Australian Conf., Univ. Melbourne, Melbourne, 1973), Springer, Berlin, 1974, pp. 18–28. Lecture Notes in Math., Vol. 403. J. A. Cadzow, A finite algorithm for the minimum l∞ solution to a system of consistent linear equations, SIAM J. Numer. Anal. 10 (1973), 607–617. G. D. Callon and C. W. Groetsch, The method of weighting and approximation of restricted pseudosolutions, J. Approx. Theory 51 (1987), no. 1, 11–18. S. L. Campbell, Differentiation of the Drazin inverse, SIAM J. Appl. Math. 30 (1976), no. 4, 703– 707. , The Drazin inverse of an infinite matrix, SIAM J. Appl. Math. 31 (1976), no. 3, 492–503, (see [318]). , Optimal control of autonomous linear processes with singular matrices in the quadratic cost functional, SIAM J. Control Optimization 14 (1976), no. 6, 1092–1106. 12 308. 309. 310. 311. 312. 313. 314. 315. 316. 317. 318. 319. 320. 321. 322. 323. 324. 325. 326. 327. 328. BIBLIOGRAPHY , Linear systems of differential equations with singular coefficients, SIAM J. Math. Anal. 8 (1977), no. 6, 1057–1066. , On continuity of the Moore-Penrose and Drazin generalized inverses, Linear Algebra and Appl. 18 (1977), no. 1, 53–57. , Optimal control of discrete linear processes with quadratic cost, Internat. J. Systems Sci. 9 (1978), no. 8, 841–847. , Singular perturbation of autonomous linear systems. II, J. Differential Equations 29 (1978), no. 3, 362–373. , Limit behavior of solutions of singular difference equations, Linear Algebra and its Applications 23 (1979), 167–178. , Nonregular singular dynamic Leontief systems, Econometrica 47 (1979), no. 6, 1565–1568. , On a singularly perturbed autonomous linear control problem, IEEE Trans. Automat. Control 24 (1979), no. 1, 115–117. , Continuity of the Drazin inverse, Linear and Multilinear Algebra 8 (1979/80), no. 3, 265– 268. , Singular Systems of Differential Equations, Pitman (Advanced Publishing Program), Boston, Mass., 1980. , On an assumption guaranteeing boundary layer convergence of singularly perturbed systems, Automatica—J. IFAC 17 (1981), no. 4, 645–646. , The Drazin inverse of an operator, [320], pp. 250–260. , On positive controllers and linear quadratic optimal control problems, Internat. J. Control 36 (1982), no. 5, 885–888. S. L. Campbell (ed.), Recent Applications of Generalized Inverses, Boston, Mass., Pitman (Advanced Publishing Program), 1982. , Singular Systems of Differential Equations. II, Pitman (Advanced Publishing Program), Boston, Mass., 1982. , Index two linear time-varying singular systems of differential equations, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 237–243. , Control problem structure and the numerical solution of linear singular systems, Math. Control Signals Systems 1 (1988), no. 1, 73–87. , Uniqueness of completions for linear time varying differential algebraic equations, Linear Algebra and its Applications 161 (1992), 55–67. , Least squares completions for nonlinear differential algebraic equations, Numer. Math. 65 (1993), no. 1, 77–94. S. L. Campbell and C. D. Meyer, Jr., Continuity properties of the Drazin pseudoinverse, Linear Algebra and its Applications 10 (1975), 77–83. , EP operators and generalized inverses, Canad. Math. Bull 18 (1975), no. 3, 327–333. , Weak Drazin inverses, Linear Algebra and its Applications 20 (1978), no. 2, 167–178. 329. 330. 331. 332. 333. 334. 335. 336. 337. 338. 339. 340. 341. 342. 343. 344. 345. 346. , Generalized Inverses of Linear Transformations, Pitman (Advanced Publishing Program), Boston, Mass., 1979, (reprinted by Dover, 1991). S. L. Campbell, C. D. Meyer, Jr., and N. J. Rose, Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients, SIAM J. Appl. Math. 31 (1976), no. 3, 411–425, (see [1459], [400], [763], [2057]). S. L. Campbell and G. D. Poole, Computing nonnegative rank factorizations, Linear Algebra and its Applications 35 (1981), 175–182. , Convergent regular splittings for nonnegative matrices, Linear and Multilinear Algebra 10 (1981), no. 1, 63–73. S. L. Campbell and M. Rakowski, Explicit formulae for completions of linear time varying singular systems of differential equations, Circuits Systems Signal Process. 13 (1994), no. 2-3, 185–199. S. L. Campbell and N. J. Rose, Singular perturbation of autonomous linear systems. III, Houston J. Math. 4 (1978), no. 4, 527–539. , Singular perturbation of autonomous linear systems, SIAM J. Math. Anal. 10 (1979), no. 3, 542–551. , A second order singular linear system arising in electric power systems analysis, Internat. J. Systems Sci. 13 (1982), no. 1, 101–108. S. L. Campbell and K. D. Yeomans, Behavior of the nonunique terms in general DAE integrators, Appl. Numer. Math. 28 (1998), no. 2-4, 209–226. Wei Ping Cao and Ji Pu Ma, Perturbation of A+ 0 h, Numer. Math. J. Chinese Univ. (English Ser.) 3 (1994), no. 1, 96–103. , The pointwise continuity of the M-P generalized inverses A+ x , Acta Math. Sinica 40 (1997), no. 2, 287–295. , Moore-Penrose generalized inverses of closed operators, Nanjing Daxue Xuebao Shuxue Bannian Kan 16 (1999), no. 1, 75–81. S. R. Caradus, An equational approach to products of relatively regular operators, Aequationes Math. 15 (1977), no. 1, 55–62. , Generalized Inverses and Operator Theory, Queen’s University, Kingston, Ont., 1978. D. Carlson, Matrix decompositions involving the Schur complement, SIAM J. Appl. Math. 28 (1975), 577–587. , What are Schur complements, anyway?, Linear Algebra and its Applications 74 (1986), 257–275. , Generalized inverse invariance, partial orders, and rank-minimization problems for matrices, Current Trends in Matrix Theory (Auburn, Ala., 1986), North-Holland, New York, 1987, pp. 81–87. D. Carlson, E. V. Haynsworth, and T. Markham, A generalization of the Schur complement by means of the Moore–Penrose inverse, SIAM J. Appl. Math. 26 (1974), 169–175. BIBLIOGRAPHY 347. R. Caron, H. J. Greenberg, and A. Holder, Analytic centers and repelling inequalities, Tech. Report CCM 142, Center for Computational Mathematics, University of Colorado at Denver, 1999, (to appear in European Journal of Operations Research). 348. N. Castro González, On the convergence of semiiterative methods to the Drazin inverse solution of linear equations in Banach spaces, Collect. Math. 46 (1995), no. 3, 303–314. 349. N. Castro González and J. J. Koliha, Semi-iterative methods for the Drazin inverse solution of linear equations in Banach spaces, Numer. Funct. Anal. Optim. 20 (1999), no. 5-6, 405–418. 350. , Perturbation of the Drazin inverse for closed linear operators, Integral Equations Operator Theory 36 (2000), no. 1, 92–106. 351. N. Castro González, J. J. Koliha, and I. Straškraba, Perturbation of the Drazin inverse, Soochow J. Math. 27 (2001), no. 2, 201–211. 352. N. Castro González, J. J. Koliha, and Yimin Wei, Perturbation of the Drazin inverse for matrices with equal eigenprojections at zero, Linear Algebra and its Applications 312 (2000), no. 1-3, 181–189. 353. D. E. Catlin, Estimation, Control, and the Discrete Kalman Filter, Springer–Verlag, New York, 1989, (see, in particular, pp. 100–113). 354. Jian Miao Cen, Fuzzy matrix partial orderings and generalized inverses, Fuzzy Sets and Systems 105 (1999), no. 3, 453–458. 355. B. L. Chalmers, F. T. Metcalf, and B. Shekhtman, On the computation of minimal projections: millennium report, Applied Mathematics Reviews, Vol. 1, World Sci. Publishing, River Edge, NJ, 2000, pp. 119–156. 356. N. N. Chan, On a downdating formula for regression, J. Statist. Plann. Inference 46 (1995), no. 3, 347–350, (see [531]). 357. E. Chang, The generalized inverse and interpolation theory, In Campbell [320], pp. 196–219. 358. A. Charnes and W. W. Cooper, Structural sensitivity analysis in linear programming and an exact product form left inverse, Naval Res. Logist. Quart. 15 (1968), 517–522. 359. A. Charnes, W. W. Cooper, and G. L. Thompson, Constrained generalized medians and hypermedians as deterministic equivalents for two–stage linear programs under uncertainty, Management Sci. 12 (1965), 83–112. 360. A. Charnes and F. Granot, Existence and representation of Diophantine and mixed Diophantine solutions to linear equations and inequalities, Center for cybernetic studies, The University of Texas, Austin, TX, 1973. 361. A. Charnes and M. J. L. Kirby, Modular design, generalized inverses and convex programming, Operations Res. 13 (1965), 836–847. 362. Guoliang Chen, Musheng Wei, and Yifeng Xue, Perturbation analysis of the least squares solution 363. 364. 365. 366. 367. 368. 369. 370. 371. 372. 373. 374. 375. 376. 377. 378. 13 in Hilbert spaces, Linear Algebra and its Applications 244 (1996), 69–80. Guoliang Chen and Yimin Wei, Perturbation analysis for the projection of a point onto an affine set in a Hilbert space, Chinese Ann. Math. Ser. A 19 (1998), no. 4, 405–410, (translation in Chinese J. Contemp. Math. 19(1998), 245–252). Guoliang Chen and Yifeng Xue, Perturbation analysis for the operator equation T x = b in Banach spaces, J. Math. Anal. Appl. 212 (1997), no. 1, 107–125. , The expression of the generalized inverse of the perturbed operator under Type I perturbation in Hilbert spaces, Linear Algebra and its Applications 285 (1998), no. 1-3, 1–6. Han Fu Chen, Two kinds of linear estimators in Hilbert spaces, and their connection, Acta Math. Sinica 25 (1982), no. 6, 671–679. X. Chen, M. Z. Nashed, and L. Qi, Convergence of Newton’s method for singular smooth and nonsmooth equations using adaptive outer inverses, SIAM J. Optim. 7 (1997), 445–462. Xuzhou Chen and R. E. Hartwig, The group inverse of a triangular matrix, Linear Algebra and its Applications 237/238 (1996), 97–108. , The hyperpower iteration revisited, Linear Algebra Appl. 233 (1996), 207–229. , On the convergence of power scaled Cesàro sums, Linear Algebra and its Applications 267 (1997), 335–358. Yong-Lin Chen, On the weighted projector and weighted generalized inverse matrices, Acta Math. Appl. Sinica 6 (1983), no. 3, 282–291. , The generalized Bott-Duffin inverse and its applications, Linear Algebra and its Applications 134 (1990), 71–91. , A Cramer rule for solution of the general restricted linear equation, Linear and Multilinear Algebra 34 (1993), no. 2, 177–186. , An explicit representation of the general solution to a system of constrained linear equations and Cramer’s rule, Gaoxiao Yingyong Shuxue Xuebao 8 (1993), no. 1, Ser. A, 61–70. , Representations and Cramer rules for the solution of a restricted matrix equation, Linear and Multilinear Algebra 35 (1993), no. 3-4, 339–354. , Finite algorithms for the (2)-generalized (2) inverse AT,S , Linear and Multilinear Algebra 40 (1995), no. 1, 61–68. , Iterative methods for solving restricted linear equations, Appl. Math. Comput. 86 (1997), no. 2-3, 171–184. , Defining equations and explicit expressions (2) for the generalized inverse AT,S , J. Nanjing Norm. Univ. Nat. Sci. Ed. 23 (2000), no. 2, 5–8. 14 BIBLIOGRAPHY 379. Yong-Lin Chen and Xin Chen, Representation and (2) approximation of the outer inverse AT,S of a matrix A, Linear Algebra and its Applications 308 (2000), no. 1-3, 85–107. 380. Yonghong Chen, S. J. Kirkland, and M. Neumann, Group generalized inverses of M -matrices associated with periodic and nonperiodic Jacobi matrices, Linear and Multilinear Algebra 39 (1995), no. 4, 325–340. 381. , Nonnegative alternating circulants leading to M -matrix group inverses, Linear Algebra and its Applications 233 (1996), 81–97. 382. Yonghong Chen and M. Neumann, M -matrix generalized inverses of M -matrices, Linear Algebra and its Applications 256 (1997), 263–285. 383. Zi Kuan Chen, Finding the synthetic discriminant function for any sample set by means of the twostep pseudoinverse method, J. Numer. Methods Comput. Appl. 17 (1996), no. 1, 8–13. 384. E. W. Cheney, Introduction to Approximation Theory, McGraw–Hill Book Co., New York, 1966. 385. H. Chernoff, Locally optimal designs for estimating parameters, Ann. Math. Statist. 24 (1953), 586– 602. 386. V. A. Cheverda and V. I. Kostin, r-pseudoinverses for compact operators in Hilbert spaces: existence and stability, J. Inverse Ill-Posed Probl. 3 (1995), no. 2, 131–148. 387. J. S. Chipman, On least squares with insufficient observations, J. Amer. Statist. Assoc. 54 (1964), 1078–1111, (see [1978]). 388. , Specification problems in regression analysis, In Boullion and Odell [255], pp. 114–176. 389. , “Proofs” and proofs of the Eckart-Young theorem, Stochastic Processes and Functional Analysis (Riverside, CA, 1994), Dekker, New York, 1997, pp. 71–83. 390. , Linear restrictions, rank reduction, and biased estimation in linear regression, Linear Algebra and its Applications 289 (1999), no. 1-3, 55–74. 391. J. S. Chipman and M. M. Rao, On the treatment of linear restrictions in regression analysis, Econometrica 32 (1964), 198–209. 392. , Projections, generalized inverses and quadratic forms, J. Math. Anal. Appl. 9 (1964), 1–11. 393. H. Chitwood, Generalized Green’s matrices for linear differential systems, SIAM J. Math. Anal. 4 (1973), 104–110. 394. Han Hyuk Cho, Regular fuzzy matrices and fuzzy equations, Fuzzy Sets and Systems 105 (1999), no. 3, 445–451. 395. K. K. Choong and J. Y. Kim, A numerical strategy for computing the stability boundaries for multiloading systems by using generalized inverse and continuation method, Engineering Structures 23 (2001), 715–724. 396. Shui-Nee Chow and Yun Qiu Shen, Bifurcations via singular value decompositions, Appl. Math. Comput. 28 (1988), no. 3, part I, 231–245. 397. Ole Christensen, Frames and pseudo-inverses, J. Math. Anal. Appl. 195 (1995), no. 2, 401–414. 398. , Operators with closed range, pseudoinverses, and perturbation of frames for a subspace, Canad. Math. Bull. 42 (1999), no. 1, 37–45. 399. , Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bull. Amer. Math. Soc. 38 (2001), no. 3, 273–291. 400. M. A. Christodoulou and P. N. Paraskevopoulos, Solvability, controllability, and observability of singular systems, J. Optim. Theory Appl. 45 (1985), no. 1, 53–72. 401. Moody T. Chu, On a numerical treatment for the curve-tracing of the homotopy method, Numer. Math. 42 (1983), no. 3, 323–329. 402. Moody T. Chu, R. E. Funderlic, and G. H. Golub, On a variational formulation of the generalized singular value decomposition, SIAM J. Matrix Anal. Appl. 18 (1997), no. 4, 1082–1092. 403. Kai Lai Chung, Elementary Probability Theory with Stochastic Processes, Springer–Verlag, New York, 1974. 404. G. Ciecierska, A note on another method of computing the Moore-Penrose inverse of a matrix, Demonstratio Math. 31 (1998), no. 4, 879–886. 405. G. Cimmino, Inversione delle corrispondenze funzionali lineari ed equazioni differenziali, Rivista Mat. Univ. Parma 1 (1950), 105–116. 406. , Cramer’s rule without the notion of determinant, Atti Accad. Sci. Istit. Bologna Cl. Sci. Fis. Rend. (14) 3 (1985/86), 115–138 (1987). 407. , An unusual way of solving linear systems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 80 (1986), no. 1-2, 6–7 (1987). 408. , On some identities involving spherical means, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 83 (1989), 69–72 (1990). 409. J. F. Claerbout, Geophysical Estimation by Example, Stanford Exploration Project, Stanford University, 2001, (on–line book). 410. J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414. 411. J.-J. Climent, M. Neumann, and A. Sidi, A semiiterative method for real spectrum singular linear systems with an arbitrary index, J. Comput. Appl. Math. 87 (1997), no. 1, 21–38. 412. J.-J. Climent, N. Thome, and Yimin Wei, A geometrical approach on generalized inverses by Neumann-type series, Linear Algebra and its Applications 332/334 (2001), 533–540. 413. R. E. Cline, Representations for the generalized inverse of a partitioned matrix, J. Soc. Indust. Appl. Math. 12 (1964), 588–600. 414. , Representations for the generalized inverse of sums of matrices, J. Soc. Indust. Appl. Math. Ser. B. Numer. Anal. 2 (1965), 99–114. 415. , Inverses of rank invariant powers of a matrix, SIAM J. Appl. Math. 5 (1968), 182–197. BIBLIOGRAPHY 416. 417. 418. 419. 420. 421. 422. 423. 424. 425. 426. 427. 428. 429. 430. 431. 432. , Elements of the Theory of Generalized Inverses for Matrices. (umap modules and monographs in undergraduate mathematics and its applications project). the umap expository monograph series), EDC/UMAP, Newton, Mass., 1979. , Note on an extension of the MoorePenrose inverse, Linear Algebra and its Applications 40 (1981), 19–23. R. E. Cline and R. E. Funderlic, The rank of a difference of matrices and associated generalized inverses, Linear Algebra and its Applications 24 (1979), 185–215. R. E. Cline and T. N. E. Greville, An extension of the generalized inverse of a matrix, SIAM J. Appl. Math. 19 (1970), 682–688. , A Drazin inverse for rectangular matrices, Linear Algebra and its Applications 29 (1980), 53– 62. R. E. Cline and R. J. Plemmons, l2 -solutions to underdetermined linear systems, SIAM Rev. 18 (1976), no. 1, 92–106. R. E. Cline and L. D. Pyle, The generalized inverse in linear programming. an intersecton projection method and the solution of a class of structured linear programming problems, SIAM J. Appl. Math. 24 (1973), 338–351. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw–Hill Book Co., 1955. L. Collatz, Aufgaben monotoner Art, Arch. Math. 3 (1952), 366–376. D. Constales, A closed formula for the MoorePenrose generalized inverse of a complex matrix of given rank, Acta Math. Hungar. 80 (1998), no. 1-2, 83–88. G. Corach, A. Maestripieri, and D. Stojanoff, Generalized orthogonal projections and shorted operators, Margarita mathematica, Univ. La Rioja, Logroño, 2001, pp. 607–625. , Oblique projections and Schur complements, Acta Sci. Math. (Szeged) 67 (2001), no. 1-2, 337–356. , Generalized Schur complements and oblique projections, Linear Algebra and its Applications 341 (2002), 259–272. G. Corach, H. Porta, and L. Recht, Differential geometry of spaces of relatively regular operators, Integral Equations Operator Theory 13 (1990), no. 6, 771–794. C. Corradi, A note on the solution of separable nonlinear least-squares problems with separable nonlinear equality constraints, SIAM J. Numer. Anal. 18 (1981), no. 6, 1134–1138. , Computing methods for restricted estimation in linear models, Statistica (Bologna) 42 (1982), no. 1, 55–68, (see [645]). R. W. Cottle, Manifestations of the Schur complement, Linear Algebra and Appl. 8 (1974), 189–211. 15 433. R. Courant, Differential and Integral Calculus, Interscience Publishers, New York, 1936, (translated by E.J. McShane). 434. R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I, Interscience Publishers, New York, 1953, (First published in German 1924). 435. D. E. Crabtree and E. V. Haynsworth, An identity for the Schur complement of a matrix, Proc. Amer. Math. Soc. 22 (1969), 364–366. 436. D. F. Cudia, Rotundity, Convexity, Proc. Sympos. Pure Math. Vol. VII (V. Klee, Editor), Amer. Math. Soc., Providence, R.I., 1963, pp. 73–97. 437. C. G. Cullen and K. J. Gale, A functional definition of the determinant, Amer. Math. Monthly 72 (1965), 403–406. 438. B. Cvetkov, A new method of computation in the theory of least squares, Austral. J. Appl. Sci. 6 (1955), 274–280. 439. Hua Dai, An algorithm for symmetric generalized inverse eigenvalue problems, Linear Algebra and its Applications 296 (1999), no. 1-3, 79–98. 440. J. F. Dalphin and V. Lovass-Nagy, Best least squares solutions to finite difference equations using the generalized inverse and tensor product methods, Journal of the ACM 20 (1973), no. 2, 279–289. 441. J. Dauxois and G. M. Nkiet, Canonical analysis of two Euclidean subspaces and its applications, Linear Algebra and its Applications 264 (1997), 355– 388. 442. D. F. Davidenko, On a new method of nu

Useful advice for finishing your 'Pharmacy Purchase Order Format' digitally

Are you fed up with the inconvenience of managing paperwork? Look no further than airSlate SignNow, the leading eSignature solution for individuals and businesses. Bid farewell to the labor-intensive process of printing and scanning documents. With airSlate SignNow, you can effortlessly complete and sign documents online. Utilize the powerful features included in this user-friendly and affordable platform to transform your method of document handling. Whether you need to approve forms or collect eSignatures, airSlate SignNow simplifies it all with just a few clicks.

Adhere to this detailed guide:

  1. Sign in to your account or initiate a free trial with our service.
  2. Select +Create to upload a file from your device, cloud storage, or our template collection.
  3. Edit your 'Pharmacy Purchase Order Format' in the workspace.
  4. Click Me (Fill Out Now) to set up the form on your end.
  5. Insert and allocate fillable fields for others (if required).
  6. Proceed with the Send Invite options to solicit eSignatures from others.
  7. Save, print your version, or turn it into a reusable template.

Don’t fret if you need to collaborate with your colleagues on your Pharmacy Purchase Order Format or send it for notarization—our platform provides everything you need to complete these tasks. Register with airSlate SignNow today and elevate your document management to a new level!

Here is a list of the most common customer questions. If you can’t find an answer to your question, please don’t hesitate to reach out to us.

Need help? Contact Support
Pharmacy purchase order format word
Pharmacy purchase order format pdf
Pharmacy purchase order format excel
Pharmacy purchase order format pdf download
Pharmacy purchase order format doc
Pharmacy purchase order format free download
Purchase order template Word
Free pharmacy purchase order format
Sign up and try Pharmacy purchase order format
  • Close deals faster
  • Improve productivity
  • Delight customers
  • Increase revenue
  • Save time & money
  • Reduce payment cycles