• [1] Eduard Batschelet, Über die numerische Auflösung von Ranswertproblemen bei elliptischen partiellen Differentialgleichungen, Z. Angew. Math. Phys. 3 (1952), 165–193 (German). MR 60912, https://doi.org/10.1007/bf02008824
• [2] Garrett Birkhoff and Richard S. Varga, Implicit alternating direction methods, Trans. Amer. Math. Soc. 92 (1959), 13–24. MR 105814, https://doi.org/10.1090/S0002-9947-1959-0105814-4
• [3] Garrett Birkhoff, Richard S. Varga, and David Young, Alternating direction implicit methods, Advances in Computers, Vol. 3, Academic Press, New York, 1962, pp. 189–273. MR 0168131
• [4] J. H. Bramble and B. E. Hubbard, On a finite difference analogue of an elliptic boundary problem which is neither diagonally dominant nor of non-negative type, J. Math. and Phys. 43 (1964), 117–132. MR 162367
• [5] James H. Bramble and Bert E. Hubbard, New monotone type approximations for elliptic problems, Math. Comp. 18 (1964), 349–367. MR 165702, https://doi.org/10.1090/S0025-5718-1964-0165702-X
• [6] L. Collatz, ``Bermerkungen zur Fehlerabschätzung für das Differenzenverfahren bei partiellen Differentialgleichungen,'' Z. Angew. Math. Mech., v. 13, 1933, pp. 56-57.
• [7] Lothar Collatz, The numerical treatment of differential equations. 3d ed, Translated from a supplemented version of the 2d German edition by P. G. Williams. Die Grundlehren der mathematischen Wissenschaften, Bd. 60, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0109436
• [8] David Feingold and Daniel Spohn, Un théorème simple sur les normes de matrices et ses conséquences, C. R. Acad. Sci. Paris 256 (1963), 2758–2760 (French). MR 150937
• [9] George E. Forsythe and Wolfgang R. Wasow, Finite-difference methods for partial differential equations, Applied Mathematics Series, John Wiley & Sons, Inc., New York-London, 1960. MR 0130124
• [10] F. P. Gantmacher & M. G. Krein, Oscillation Matrices and Small Vibrations of Mechanical Systems, GITTL, Moscow, 1950; English transl., Office of Technical Service, Dept. of Commerce, Washington, D. C. MR 14, 178.
• [11] F. P. Gantmacher, The Theory of Matrices, Vol. II, GITTL, Moscow, 1953; English transl., Chelsea, New York, 1959. MR 16, 438; MR 21 #6372c.
• [12] S. Gerschgorin, ``Fehlerabschätzung für das Differenzenverfahren zur Lösung partieller Differentialgleichungen,'' Z. Angew. Math. Mech., v. 10, 1930, pp. 373-382.
• [13] S. Gerschgorin, ``Über die Abrenzung der Eigenwerte einer Matrix,'' Izv. Akad. Nauk SSSR Ser. Mat., v. 7, 1931, pp. 749-754.
• [14] Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729
• [15] Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. MR 0175290
• [16] Eugene Isaacson, Error estimates for parabolic equations, Comm. Pure Appl. Math. 14 (1961), 381–389. MR 137311, https://doi.org/10.1002/cpa.3160140315
• [17] Alexander Ostrowski, Determinanten mit überwiegender Hauptdiagonale und die absolute Konvergenz von linearen Iterationsprozessen, Comment. Math. Helv. 30 (1956), 175–210 (German). MR 76433, https://doi.org/10.1007/BF02564340
• [18] D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. 3 (1955), 28–41. MR 71874
• [19] Carl Pearcy, On convergence of alternating direction procedures, Numer. Math. 4 (1962), 172–176. MR 145677, https://doi.org/10.1007/BF01386310
• [20] H. S. Price, Monotone and Oscillation Matrices Applied to Finite Difference Approximations, Ph.D. Thesis, Case Institute of Technology, 1965.
• [21] H. S. Price & R. S. Varga, Recent Numerical Experiments Comparing Successive Overrelaxation Iterative Methods with Implicit Alternating Direction Methods, Report No. 91, Gulf Research & Development Company, Reservoir Mechanics Division, 1962.
• [22] Harvey S. Price, Richard S. Varga, and Joseph E. Warren, Application of oscillation matrices to diffusion-convection equations, J. Math. and Phys. 45 (1966), 301–311. MR 207230
• [23] M. L. Rockoff, Comparison of Some Iterative Methods for Solving Large Systems of Linear Equations, National Bureau of Standards Report No. 8577, 1964.
• [24] W. H. Roudebush, Analysis of Discretization Error for Differential Equations with Discontinuous Coefficients, Ph.D. Thesis, Case Institute of Technology, 1963.
• [25] E. L. Wachspress, Optimum alternating-direction-implicit iteration parameters for a model problem, J. Soc. Indust. Appl. Math. 10 (1962), 339–350. MR 150935
• [26] Eugene L. Wachspress, Extended application of alternating direction implicit iteration model problem theory, J. Soc. Indust. Appl. Math. 11 (1963), 994–1016. MR 169373
• [27] E. L. Wachspress and G. J. Habetler, An alternating-direction-implicit iteration technique, J. Soc. Indust. Appl. Math. 8 (1960), 403–424. MR 114308
• [28] Olof B. Widlund, On the rate of convergence of an alternating direction implicit method in a noncommutative case, Math. Comp. 20 (1966), 500–515. MR 231551, https://doi.org/10.1090/S0025-5718-1966-0231551-9
• [29] Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502
• [30] D. M. Young & L. Ehrlich, ``Some numerical studies of iterative methods for solving elliptic difference equations'' in Boundary Problems in Differential Equations, Univ. of Wisconsin Press, Madison, Wis., 1960, pp. 143-162. MR 22 #5127.

Bibliography

[1]

E. Batschelet, ``Uber die numerische Auflösung von Randwertproblem bei elliptischen partiellen Differentialgleichungen,'' Z. Angew. Math. Physik, v. 3, 1952, pp. 165-193. MR 15, 747. MR 0060912 (15:747b)

[2]

G. BiRKHOFF & R. S. Varga, ``Implicit alternating direction methods,'' Trans. Amer. Math. Soc., v. 92, 1959, pp. 13-24. MR 21 #4549. MR 0105814 (21:4549)

[3]

G. Birkhoff, R. S. Varga & D. M. Young, ``Alternating direction implicit methods'' in Advances in Computers, Vol. 3, Academic Press, New York, 1962, pp. 189-173. MR 29 #5395. MR 0168131 (29:5395)

[4]

J. H. Bramble & B. E. Hubbard, ``On a finite difference analogue of an elliptic boundary problem which is neither diagonally dominant nor of non-negative type,'' J. Math, and Phys., v. 43, 1964, pp. 117-132. MR 28 #5566. MR 0162367 (28:5566)

[5]

J. H. Bramble & B. E. Hubbard, ``New monotone type approximations for elliptic problems,'' Math. Comp., v. 18, 1964, pp. 349-367. MR 29 #2982. MR 0165702 (29:2982)

[6]

L. Collatz, ``Bermerkungen zur Fehlerabschätzung für das Differenzenverfahren bei partiellen Differentialgleichungen,'' Z. Angew. Math. Mech., v. 13, 1933, pp. 56-57.

[7]

L. Collatz, Numerical Treatment of Differential Equations, 3rd ed., Springer-Verlag, Berlin, 1960. MR 22 #322. MR 0109436 (22:322)

[8]

D. Feingold & D. Spohn, ``Un théorème simple sur les normes de matrices et ses consequences,'' C. R. Acad. Sci. Paris, v. 256, 1963, pp. 2758-2760. MR 27 #923. MR 0150937 (27:923)

[9]

G. E. Forsythe & W. R. Wasow, Finite-Difference Methods for Partial Differential Equations, Wiley, New York, 1960. MR 23 #B3156. MR 0130124 (23:B3156)

[10]

F. P. Gantmacher & M. G. Krein, Oscillation Matrices and Small Vibrations of Mechanical Systems, GITTL, Moscow, 1950; English transl., Office of Technical Service, Dept. of Commerce, Washington, D. C. MR 14, 178.

[11]

F. P. Gantmacher, The Theory of Matrices, Vol. II, GITTL, Moscow, 1953; English transl., Chelsea, New York, 1959. MR 16, 438; MR 21 #6372c.

[12]

S. Gerschgorin, ``Fehlerabschätzung für das Differenzenverfahren zur Lösung partieller Differentialgleichungen,'' Z. Angew. Math. Mech., v. 10, 1930, pp. 373-382.

[13]

S. Gerschgorin, ``Über die Abrenzung der Eigenwerte einer Matrix,'' Izv. Akad. Nauk SSSR Ser. Mat., v. 7, 1931, pp. 749-754.

[14]

P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962. MR 24 #B1772. MR 0135729 (24:B1772)

[15]

A. S. Householder, The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964. MR 30 #5475. MR 0175290 (30:5475)

[16]

E. Isaacson, ``Error estimates for parabolic equations,'' Comm. Pure Appl. Math., v. 14, 1961, pp. 381-389. MR 25 #763. MR 0137311 (25:763)

[17]

A. M. Ostrowski, ``Determinanten mit überwiegender Hauptdiagonale und die absolute Konvergenz von linearen Iterationsprozessen,'' Comment. Math. Helv., v. 30, 1956, pp. 175-210. MR 17, 898. MR 0076433 (17:898e)

[18]

D. W. Peaceman & H. H. Rachford, ``The numerical solution of parabolic and elliptic differential equations,'' J. Soc. Indust. Appl. Math., v. 3, 1955, pp. 28-41. MR 17, 196. MR 0071874 (17:196d)

[19]

C. Pearcy, ``On convergence of alternating direction procedure,'' Numer. Math., v. 4, 1962, pp. 172-176. MR 26 #3206. MR 0145677 (26:3206)

[20]

H. S. Price, Monotone and Oscillation Matrices Applied to Finite Difference Approximations, Ph.D. Thesis, Case Institute of Technology, 1965.

[21]

H. S. Price & R. S. Varga, Recent Numerical Experiments Comparing Successive Overrelaxation Iterative Methods with Implicit Alternating Direction Methods, Report No. 91, Gulf Research & Development Company, Reservoir Mechanics Division, 1962.

[22]

H. S. Price, R. S. Varga & J. E. Warren, ``Application of oscillation matrices to diffusion-convection equations,'' J. Math, and Phys., v. 45, 1966, pp. 301-311. MR 34 #7046. MR 0207230 (34:7046)

[23]

M. L. Rockoff, Comparison of Some Iterative Methods for Solving Large Systems of Linear Equations, National Bureau of Standards Report No. 8577, 1964.

[24]

W. H. Roudebush, Analysis of Discretization Error for Differential Equations with Discontinuous Coefficients, Ph.D. Thesis, Case Institute of Technology, 1963.

[25]

E. L. Wachspress, ``Optimum alternating-direction-implicit iteration parameters for a model problem,'' J. Soc. Indust. Appl. Math., v. 10, 1962, pp. 339-350. MR 27 #921. MR 0150935 (27:921)

[26]

E. L. Wachspress, ``Extended application of alternating direction implicit iteration model problem theory,'' J. Soc. Indust. Appl. Math., v. 11, 1963, pp. 994-1016. MR 29 #6623. MR 0169373 (29:6623)

[27]

E. L. Wachspress & G. J. Habetler, ``An alternating-direction-implicit-iteration technique,'' J. Soc. Indust. Appl. Math., v. 8, 1960, pp. 403-424. MR 22 #5132. MR 0114308 (22:5132)

[28]

O. B. Widlund, ``On the rate of convergence of an alternating direction implicit method in a non-commutative case,'' Math. Comp., v. 20, 1966, pp. 500-515. MR 0231551 (37:7104)

[29]

R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1962 MR 28 #1725. MR 0158502 (28:1725)

[30]

D. M. Young & L. Ehrlich, ``Some numerical studies of iterative methods for solving elliptic difference equations'' in Boundary Problems in Differential Equations, Univ. of Wisconsin Press, Madison, Wis., 1960, pp. 143-162. MR 22 #5127.