×

zbMATH — the first resource for mathematics

Gram-Schmidt orthogonalization: 100 years and more. (English) Zbl 1313.65086
Summary: In 1907, Erhard Schmidt published a paper in which he introduced an orthogonalization algorithm that has since become known as the classical Gram-Schmidt process. Schmidt claimed that his procedure was essentially the same as an earlier one published by J. P. Gram in 1883. The Schmidt version was the first to become popular and widely used. An algorithm related to a modified version of the process appeared in an 1820 treatise by P. S. Laplace. Although related algorithms have been around for almost 200 years, it is the Schmidt paper that led to the popularization of orthogonalization techniques. The year 2007 marked the 100th anniversary of that paper. In celebration of that anniversary, we present a comprehensive survey of the research on Gram-Schmidt orthogonalization and its related QR factorization. Its application for solving least squares problems and in Krylov subspace methods are also reviewed. Software and implementation aspects are also discussed.

MSC:
65F25 Orthogonalization in numerical linear algebra
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65-03 History of numerical analysis
65F10 Iterative numerical methods for linear systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abdelmalek, Roundoff error analysis for Gram-Schmidt method and solution of linear least squares problems, BIT 11 pp 345– (1971) · Zbl 0236.65031 · doi:10.1007/BF01939404
[2] Anderson, LAPACK Users’ Guide (1999) · Zbl 0934.65030 · doi:10.1137/1.9780898719604
[3] Anderson M Ballard G Demmel J Keutzer K Communication-avoiding QR decomposition for GPUs Technical Report UCB/EECS-2010-31 2008
[4] Arnoldi, The principle of minimized iteration in the solution of the matrix eigenvalue problem, Quarterly of Applied Mathematics. 9 pp 17– (1951) · Zbl 0042.12801 · doi:10.1090/qam/42792
[5] Bauer, Elimination with weighted row combinations for solving linear equations and least squares problems, Numerische Mathematik 7 pp 338– (1965) · Zbl 0142.11504 · doi:10.1007/BF01436528
[6] Bauer, Linear Algebra II, in: Handbook for Automatic Computation pp 1– (1971)
[7] Baumgärtel, Mathematics in Berlin, in: Erhard Schmidt, John von Neumann pp 97– (1998)
[8] Bernkopf, Dictionary of Scientific Biography XII, in: Erhard Schmidt pp 187– (1970-80)
[9] Bienaymé, Remarques sur les differences qui distinguent l’interpolation de M. Cauchy de la methode des moindre carres et qui assurent la superiorite de cette methode, Compte Rendu de l’Academie des Sciences 37 pp 5– (1853)
[10] Björck Å A method for solving systems of linear equations by orthogonalization including underdetermined and overdetermined systems Technical Report Feb. 1962 1962
[11] Björck, Iterative refinement of linear least squares solutions I, BIT 7 pp 257– (1967) · Zbl 0159.20404 · doi:10.1007/BF01939321
[12] Björck, Solving least squares problems by Gram-Schmidt orthogonalization, BIT 7 pp 1– (1967) · Zbl 0183.17802 · doi:10.1007/BF01934122
[13] Björck, Iterative refinement of linear least squares solutions II, BIT 8 pp 8– (1968) · Zbl 0177.43204 · doi:10.1007/BF01939974
[14] Björck, Component-wise perturbation analysis and error bounds for linear least squares solutions, BIT 31 pp 238– (1991) · Zbl 0732.65043
[15] Björck, Numerical Methods for Least Squares Problems (1996) · doi:10.1137/1.9781611971484
[16] Björck, Iterative refinement of linear least squares solution by Householder transformation, BIT 7 pp 322– (1967)
[17] Björck, Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm, SIAM Journal on Matrix Analysis and Applications 13 pp 176– (1992) · Zbl 0747.65026 · doi:10.1137/0613015
[18] Björck, Solution of augmented linear systems using orthogonal factorizations, BIT 34 pp 1– (1994) · Zbl 0822.65021 · doi:10.1007/BF01935013
[19] Bischof, Computing rank-revealing factorizations for dense matrices, ACM Transactions on Mathematical Software 24 (2) pp 226– (1998) · Zbl 0932.65033 · doi:10.1145/290200.287637
[20] Businger, Linear least squares solutions by Householder transformations, Numerische Mathematik 7 pp 269– (1965) · Zbl 0142.11503 · doi:10.1007/BF01436084
[21] Cauchy, Memoire sur l’interpolation, Journal de Mathématiques Pures et Appliquées 2 pp 193– (1837)
[22] Cauchy, Mémoire sur la détermination des orbites des planètes et des cométes, Compte Rendu de l’Academie des Sciences 25 (1847)
[23] Chan, Rank revealing QR factorizations, Linear Algebra and its Applications 88/89 pp 67– (1987) · Zbl 0624.65025 · doi:10.1016/0024-3795(87)90103-0
[24] Chandrasekaran, On rank-revealing factorisations, SIAM Journal on Matrix Analysis and Applications 15 (2) pp 592– (1994) · Zbl 0796.65030 · doi:10.1137/S0895479891223781
[25] Cox, Pitman Research Notes in mathematics 380, in: Numerical Analysis 1997: Proceedings of the 17th Dundee Biennial Conference pp 57– (1998)
[26] Dahlquist, Numerical Methods in Scientific Computing (2008) · Zbl 1153.65001 · doi:10.1137/1.9780898717785
[27] Dahlquist, Comparison of the method of averages with the method of least squares, Mathematics of Computation 22 (104) pp 833– (1968) · Zbl 0176.46303
[28] Daniel, Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization, Mathematics of Computation 30 pp 772– (1976) · Zbl 0345.65021
[29] Davis, Survey of Numerical Analysis pp 558– (1962)
[30] Davis, A multiple purpose orthonormalizing code and its uses, Journal of the Association for Computing Machinery 1 (4) pp 183– (1954) · doi:10.1145/320783.320788
[31] Dax, A modified Gram-Schmidt algorithm with iterative orthogonalization and pivoting, Linear Algebra and its Applications 310 pp 42– (2000) · Zbl 0990.65047 · doi:10.1016/S0024-3795(00)00022-7
[32] Demmel J Grigori L Hoemmen M Langou J Communication-avoiding parellel and sequential QR factorizations Technical Report UCB/EECS-2008-74 2008 · Zbl 1241.65028
[33] Demmel, Extra-precise iterative refinement for overdetermined least squares problems, ACM Transactions on Mathematical Software 35 (4) pp 1– (2009) · Zbl 05517421 · doi:10.1145/1462173.1462177
[34] Dongarra, Linpack Users’ Guide (1979) · doi:10.1137/1.9781611971811
[35] Dongarra, An extended set of FORTRAN Basic Linear Algebra Subprograms, ACM Transactions on Mathematical Software 14 pp 1– (1988) · Zbl 0639.65016 · doi:10.1145/42288.42291
[36] Dongarra, Beautiful Code pp 229– (2007)
[37] Elmroth, Applying recursion to serial and parallel QR factorization, IBM Journal of Research & Development 44 (4) pp 605– (2000) · Zbl 05420288 · doi:10.1147/rd.444.0605
[38] Faddeev, Programmation en Mathématiques Numériques pp 161– (1968)
[39] Fierro, Low-rank revealing two-sided orthogonal decompositions, Numerical Algorithms 15 pp 37– (1997) · Zbl 0887.65043 · doi:10.1023/A:1019254318361
[40] Fierro, UTV Tools: MATLAB templates for rank revealing UTV decompositions, Numerical Algorithms 20 pp 165– (1999) · Zbl 0936.65054 · doi:10.1023/A:1019112103049
[41] Foster, Rank and null space calculations using matrix decomposition without column interchanges, Linear Algebra and its Applications 74 pp 47– (1986) · Zbl 0589.65031 · doi:10.1016/0024-3795(86)90115-1
[42] Frayssé, Algorithm 842: A set of GMRES routines for real and complex arithmetic on high performance computers, ACM Transactions on Mathematical Software 31 (2) pp 228– (2005) · Zbl 1070.65527 · doi:10.1145/1067967.1067970
[43] Gander W Algorithms for the QR-decomposition Technical Report 80-02 1980
[44] Gander, Numerische Prozeduren aus Nachlass und Lehre von Heinz Rutishauser (1977)
[45] Gauss, Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections (1857) · Zbl 1234.01016
[46] Gauss, Werke, IV (1880)
[47] Gauss, Werke, IV (1880)
[48] Gauss, Theoria Motus Corporum Coelestium in Sectionibus Conices Solem Abientium (1909)
[49] Gauss, Theory of the Combination of Observations Least Subject to Errors. Part 1, Part 2, Supplement. Translation by G. W. Stewart (1995) · doi:10.1137/1.9781611971248
[50] Giraud, When modified Gram-Schmidt generates a well-conditioned set of vectors, IMA Journal of Numerical Analysis 22 (4) pp 521– (2002) · Zbl 1027.65050 · doi:10.1093/imanum/22.4.521
[51] Giraud, A robust criterion for the modified Gram-Schmidt algorithm with selective reorthogonalization, SIAM Journal on Scientific Computing 25 (2) pp 417– (2003) · Zbl 1042.65033 · doi:10.1137/S106482750340783X
[52] Giraud L Langou J Rozložnık M On the round-off error of the Gram-Schmidt algorithm with reorthogonalization Technical Report TR/PA/02/33 2002
[53] Giraud, Rounding error analysis of the classical Gram-Schmidt orthogonalization process, Numerische Mathematik 101 pp 87– (2005) · Zbl 1075.65060 · doi:10.1007/s00211-005-0615-4
[54] Goldstine, A History of Numerical Analysis from the 16th through the 19th Century (1977) · Zbl 0402.01005
[55] Golub, Numerical methods for solving least squares problems, Numerische Mathematik 7 pp 206– (1965) · Zbl 0142.11502 · doi:10.1007/BF01436075
[56] Golub, Note on the iterative refinement of least squares solutions, Numerische Mathematik 9 pp 139– (1966) · Zbl 0156.16106 · doi:10.1007/BF02166032
[57] Gram, Om Raekkenudviklinger bestemte ved Hjaelp af de mindste Kvadraters Methode (1879) · JFM 11.0166.01
[58] Gram, Ueber die Entwickelung reeller Functionen in Reihen mittelst der Methode der kleinsten Quadrate, Journal für die Reine und Angewandte Mathematik 94 pp 41– (1883)
[59] Grcar, Spectral condition numbers of orthogonal projections and full rank linear least squares residuals, SIAM Journal on Matrix Analysis and Applications 31 (5) pp 2934– (2010) · Zbl 1209.65047 · doi:10.1137/090777773
[60] Greenbaum, Numerical behaviour of the modified Gram-Schmidt GMRES implementation, BIT 37 (3) pp 706– (1997) · Zbl 0891.65031 · doi:10.1007/BF02510248
[61] Gu, Efficient algorithms for computing a strong rank-revealing QR factorization, SIAM Journal on Scientific Computing 12 (4) pp 948– (1996) · Zbl 0858.65044
[62] Hansen, Rank-deficient and Discrete Ill-posed Problems (1998) · Zbl 0890.65037 · doi:10.1137/1.9780898719697
[63] Hernandez, John von Neumann Institute for Computing Series 33, in: Parallel Computing: Current & Future Issues in High-End Computing pp 221– (2006)
[64] Hestenes, Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards 49 pp 409– (1952) · Zbl 0048.09901
[65] Higham, Accuracy and Stability of Numerical Algorithms (1996) · Zbl 0847.65010
[66] Higham, QR factorization with complete pivoting and accurate computation of the SVD, Linear Algebra and its Applications 309 pp 153– (2000) · Zbl 0953.65025 · doi:10.1016/S0024-3795(99)00230-X
[67] Hoffman, Iterative algorithms for Gram-Schmidt orthogonalization, Computing 41 pp 335– (1989) · Zbl 0667.65037 · doi:10.1007/BF02241222
[68] Hong, Rank-revealing QR factorizations and the singular value decomposition, Mathematics of Computation 58 pp 213– (1992) · Zbl 0743.65037
[69] Householder, Unitary triangularization of a nonsymmetric matrix, Journal of the Association for Computing Machinery 5 pp 339– (1958) · Zbl 0121.33802 · doi:10.1145/320941.320947
[70] Jalby, Stability analysis and improvement of the block Gram-Schmidt algorithm, SIAM Journal on Scientific and Statistical Computing 12 (5) pp 1058– (1991) · Zbl 0734.65034 · doi:10.1137/0912056
[71] James, Mathematics Dictionary (1968)
[72] Jennings, A direct error analysis for least squares, Numerische Mathematik 22 pp 325– (1974) · Zbl 0271.65029 · doi:10.1007/BF01406971
[73] Jordan, Experiments on error growth associated with some linear least-squares procedures, Mathematics of Computation 22 pp 579– (1968) · Zbl 0162.46801 · doi:10.1090/S0025-5718-1968-0229373-X
[74] Kahan, Numerical linear algebra, Canadian Mathematical Bulletin 9 pp 755– (1966) · Zbl 0236.65025 · doi:10.4153/CMB-1966-083-2
[75] Krylov, On the numerical solution of the equation by which in technical questions frequencies of small oscillations of material systems are determined (in Russian), Izvestija AN SSSR (News of Academy of Sciences of the USSR), Otdel. mat. i estest. nauk VII (4) pp 491– (1931)
[76] Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, Journal of Research of the National Bureau of Standards 45 pp 255– (1950) · Zbl 0045.39702
[77] Langou J Translation and modern interpretation of Laplace’s théorie analytique des probabilités, pages 505-512, 516-520 Technical Report CCM 280 2009
[78] Laplace, Mémoire sur les intégrales definiés et sur l’applications aux probabilités, Mémoire de l’Academie des Sciences de Paris 11 pp 755– (1810)
[79] Laplace, Théorie Analytique des Probabilités, Troisiéme Édition, Premier Supplément, Sur l’Application du Calcul des Probabilités à la Philosophie Naturelle (1820)
[80] Lawson, Solving Least Squares Problems (1974)
[81] Legendre, Nouvelles Méthodes pour la Détermination des Orbites des Comètes (1805)
[82] Lehoucq, The computations of elementary unitary matrices, ACM Transactions on Mathematical Software 22 (4) pp 393– (1996) · Zbl 0884.65039 · doi:10.1145/235815.235817
[83] Leon, Linear Algebra with Applications (2009)
[84] Lingen, Efficient Gram-Schmidt orthonormalization on parallel computers, Communications in Numerical. Methods in Engineering 16 (1) pp 57– (2000) · Zbl 0958.65048 · doi:10.1002/(SICI)1099-0887(200001)16:1<57::AID-CNM320>3.0.CO;2-I
[85] Longley, An appraisal of least squares programs for the electronic computer from the point of view of the user, Journal of the American Statistical Association 62 pp 819– (1967)
[86] Longley, Modified Gram-Schmidt process vs. classical Gram-Schmidt, Communications in Statistics Simulation, and Computation B10 (5) pp 517– (1981) · Zbl 0463.65026 · doi:10.1080/03610918108812227
[87] Malyshev, A unified theory of conditioning for linear least squares and Tikhonov regularization solutions, SIAM Journal on Matrix Analysis and Applications 24 (4) pp 1186– (2003) · Zbl 1036.65044 · doi:10.1137/S0895479801389564
[88] O’Connor JJ Robertson EF Erhard Schmidt 2001 http://www-history.mcs.st-and.ac.uk/Biographies/Schmidt.html
[89] O’Connor JJ Robertson EF Jørgen Pedersen Gram 2001 http://www-history.mcs.st-and.ac.uk/Biographies/Gram.html
[90] O’Leary, Parallel QR factorization by Householder and modified Gram-Schmidt algorithms, Parallel Computing 16 (1) pp 99– (1990) · Zbl 0715.65018 · doi:10.1016/0167-8191(90)90163-4
[91] Oliveira, Analysis of different partitioning schemes for parallel Gram-Schmidt algorithms, International Journal Parallel, Emergent and Distributed Systems 14 (4) pp 293– (2000) · Zbl 0955.68128
[92] Paige, A useful form of unitary matrix obtained from any sequence of unit 2-norm n-vectors, SIAM Journal on Matrix Analysis and Applications 31 (2) pp 565– (2009) · Zbl 1217.65072 · doi:10.1137/080725167
[93] Pan, Bounds on singular values revealed by QR factorizations, BIT 39 (4) pp 740– (1999) · Zbl 0944.65042 · doi:10.1023/A:1022395308695
[94] Parlett, The Symmetric Eigenvalue Problem (1980) · Zbl 0431.65017
[95] Peters, The least squares problem and pseudo-inverses, The Computer Journal 13 pp 309– (1970) · Zbl 0195.44804 · doi:10.1093/comjnl/13.3.309
[96] Placket, The discovery of the method of least squares, Biometrika 59 pp 239– (1972) · doi:10.2307/2334569
[97] Powell, Information Processing 68. Proceedings of the IFIP Congress 68 pp 122– (1969)
[98] Reichel, FORTRAN subroutines for updationg the QR decomposition, ACM Transactions on Mathematical Software 16 pp 369– (1990) · Zbl 0900.65063 · doi:10.1145/98267.98291
[99] Reid, A note on the least squares solution of a band system of linear equations by Householder reductions, The Computer Journal 10 pp 188– (1967) · Zbl 0168.13305
[100] Rice, Experiments on Gram-Schmidt orthogonalization, Mathematics of Computation 20 pp 325– (1966) · Zbl 0228.65034 · doi:10.1090/S0025-5718-1966-0192673-4
[101] Rohrbach, Erhard Schmidt: Ein Lebensbild, Jahresberichte der Deutschen Mathematiker-Vereinigung 69 pp 209– (1967/1968)
[102] Ruhe, Numerical aspects of Gram-Schmidt orthogonalization of vectors, Linear Algebra and its Applications 52/53 pp 591– (1983) · Zbl 0515.65036 · doi:10.1016/0024-3795(83)80037-8
[103] Rünger, Lecture Notes in Computer Science 3648, in: EURO-PAR Parallel Processing pp 826– (2005) · Zbl 05490596 · doi:10.1007/11549468_90
[104] Rutishauser, Description of Algol 60, Handbook for Automatic Computation 1A (1967) · doi:10.1007/978-3-642-86934-1
[105] Rutishauser, Handbook for Automatic Computation II pp 284– (1971)
[106] Saad, GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing 7 pp 856– (1986) · Zbl 0599.65018 · doi:10.1137/0907058
[107] Schmidt, Zur Theorie der linearen und nichtlinearen Integralgleichungen I. Teil: Entwicklung willkürlicher Funktionen nach Systemen vorgeschriebener, Mathematische Annalen 63 pp 433– (1907) · JFM 38.0377.02 · doi:10.1007/BF01449770
[108] Schreiber, A storage efficient WY representation for products of Householder transformations, SIAM Journal on Scientific and Statistical Computing 10 pp 53– (1989) · Zbl 0664.65025 · doi:10.1137/0910005
[109] Schröder, Erhard Schmidt, Mathematische Nachrichten 25 pp 1– (1963) · Zbl 0106.00316 · doi:10.1002/mana.19630250102
[110] Schwarz, Matrizen-Numerik (1968)
[111] Smoktunowicz, A note on the error analysis of classical Gram-Schmidt, Numerische Mathematik 105 (2) pp 299– (2006) · Zbl 1108.65021 · doi:10.1007/s00211-006-0042-1
[112] Sorensen, Numerical methods for large eigenvalue problems, Acta Numerica. 11 pp 519– (2002) · Zbl 1105.65325 · doi:10.1017/S0962492902000089
[113] Späth, Modified Gram-Schmidt for solving linear least squares problems is equivalent to Gaussian elimination for the normal equations, Applicationes Mathematicae 20 pp 587– (1990) · Zbl 0763.65029
[114] Stathopoulos, A block orthogonalization procedure with constant synchronization requirements, SIAM Journal on Scientific Computing 23 (6) pp 2165– (2002) · Zbl 1018.65050 · doi:10.1137/S1064827500370883
[115] Stewart, Mathematical Software III pp 1– (1977) · doi:10.1016/B978-0-12-587260-7.50005-4
[116] Stewart, Rank degenercy, SIAM Journal on Scientific and Statistical Computing 5 pp 403– (1984) · Zbl 0579.65034 · doi:10.1137/0905030
[117] Stewart, An updating algorithm for subspace tracking, IEEE Transactions on Signal Processing 40 pp 1535– (1992) · doi:10.1109/78.139256
[118] Stewart, Updating a rank revealing ULV decomposition, SIAM Journal on Matrix Analysis and Applications 14 pp 494– (1993) · Zbl 0771.65021 · doi:10.1137/0614034
[119] Stewart, Gauss, statistics, and Gaussian elimination, Journal of Computational and Graphical Statistics 4 (1) pp 1– (1995)
[120] Stewart, Matrix Algorithms Volume I: Basic Decompositions (1998) · Zbl 0910.65012
[121] Stewart, Block Gram-Schmidt orthogonalization, SIAM Journal on Scientific Computing 31 (1) pp 761– (2008) · Zbl 1185.65069 · doi:10.1137/070682563
[122] Stigler, Gauss and the invention of least squares, The Annals of Statistics 9 pp 465– (1981) · Zbl 0477.62001 · doi:10.1214/aos/1176345451
[123] Stigler, The History of Statistics. The Measurement of Uncertainty before 1900 (1986) · Zbl 0656.62005
[124] Trefethen, Numerical Linear Algebra (1997) · doi:10.1137/1.9780898719574
[125] Sluis, Condition numbers and equilibration of matrices, Numerische Mathematik 14 pp 14– (1969) · Zbl 0182.48906 · doi:10.1007/BF02165096
[126] Sluis, Stability of the solutions of linear least squares problems, Numerische Mathematik 23 pp 241– (1975) · Zbl 0308.65026 · doi:10.1007/BF01400307
[127] Vanderstraeten, An accurate parallel block Gram-Schmidt algorithm without reorthogonalization, Numerical Linear Algebra with Applications 7 (4) pp 219– (2000) · Zbl 1051.65046 · doi:10.1002/1099-1506(200005)7:4<219::AID-NLA196>3.0.CO;2-L
[128] Walker, Implementation of the GMRES method using Householder transformations, SIAM Journal on Scientific Statistical Computing 9 (1) pp 152– (1988) · Zbl 0698.65021 · doi:10.1137/0909010
[129] Walker, Implementations of the GMRES method, Computer Physics Communications 53 pp 311– (1989) · Zbl 0798.65041 · doi:10.1016/0010-4655(89)90168-9
[130] Walsh, Algorithm 127, ORTHO, Communications of the Association for Computing Machinery 5 pp 511– (1962) · doi:10.1145/368959.368979
[131] Wampler, An evaluation of linear least squares computer programs, Journal of Research of the National Bureau of Standards 73B pp 59– (1969) · Zbl 0176.47402
[132] Wampler, A report on the accuracy of some widely used least squares computer programs, Journal of the American Statistical Association 65 pp 549– (1970) · Zbl 0196.22405
[133] Wampler, Algorithm 544: L2A and L2B, weighted least squares solutions by modified Gram-Schmidt with iterative refinement, ACM Transactions on Mathematical Software 5 pp 494– (1979) · Zbl 0434.65023 · doi:10.1145/355853.355866
[134] Wampler, Solutions to weighted least squares problems by modified Gram-Schmidt with iterative refinement, ACM Transactions on Mathematical Software 5 pp 457– (1979) · Zbl 0429.65033 · doi:10.1145/355853.355862
[135] Wampler RH Problems used in testing the efficency and accuracy of the modified Gram-Schmidt least squares algorithm Technical Note 1126 1980
[136] Wei, Roundoff error estimates of the modified Gram-Schmidt algorithm with column pivoting, BIT 43 (3) pp 627– (2003) · Zbl 1050.65039 · doi:10.1023/B:BITN.0000007051.49808.04
[137] Whittaker, The Calculus of Observation (1944)
[138] Wilkinson, Error in Digital Computation pp 77– (1965)
[139] Wilkinson, The Algebraic Eigenvalue Problem (1965) · Zbl 0258.65037
[140] Wilkinson, Modern error analysis, SIAM Review 13 (4) pp 548– (1971) · Zbl 0243.65018 · doi:10.1137/1013095
[141] Wilkinson, Handbook for Automatic Computation II (1971)
[142] Wong, An application of orthogonalization process to the theory of least squares, Annals of Mathematical Statistics 6 pp 53– (1935) · Zbl 0012.11601 · doi:10.1214/aoms/1177732609
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.