产生于Euler方程数值解中的广义H-矩阵和矩阵谱估计研究
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摘要
本文的研究涉及三类特殊矩阵: 广义正定矩阵, 广义H-矩阵和非负不可约矩阵. 全文共分五章三部分,创新成果着重体现在第二和第三部分.
第一部分(第二章)给出了广义正定矩阵的一些新性质, 得到了两个有关广义正定矩阵的行列式不等式.
第二部分(第三章)主要讨论了广义H-矩阵. 利用广义M-矩阵, 广义H-矩阵与M-矩阵和正定矩阵的关系, 获得了若干广义H-矩阵新的等价条件,并指出文[R. Nabben, On a class of matrices which arise in the numericalsolution of Euler equations, Numer. Math. 1992, 63: 411-431]中的两个结论是错误的,并进行了修正.
第三部分(第四章和第五章)构造非负不可约矩阵Perron 根的新上界序列, 通过数值例子说明该序列在许多情况下优于一些著名结果. 最后, 利用非负矩阵的性质研究矩阵的最大奇异值和谱半径, 给出了若干不等式.
This thesis presents a systematic research on some types of special matricessuch as generalized positive definition matrices, generalized H-matrices andnonnegative irreducible matrices. The thesis consists three parts with fivechapters.
In part one (chapter two), we give some new properties of generalizedpositive definition matrices, and two determinant inequalities about generalizedpositive definition matrices are presented.
The second part (chapter three) contributes to generalized H-matrices.Based on the relationships between generalized M-matrices and generalizedH-matrices with M-matrices and positive definition matrices, we obtain severalequivalent conditions for generalized H-matrices, some remarks on two resultsin [R. Nabben, on a class of matrices which arise in the numerical solution ofEuler equations, Numer. Math. 1992, 63: 411-431] are given.
In the last part (chapter four and five), for the Perron root of nonnegativeirreducible matrices, a new sequence of upper bounds is presented, whichconvergence is discussed. The comparison of the new sequence with the knownones is supplemented with two numerical examples. Finally, by the properties ofnonnegative matrices, we give some inequalities about the largest singularvalues and spectral radius of matrices.
第一部分(第二章)给出了广义正定矩阵的一些新性质, 得到了两个有关广义正定矩阵的行列式不等式.
第二部分(第三章)主要讨论了广义H-矩阵. 利用广义M-矩阵, 广义H-矩阵与M-矩阵和正定矩阵的关系, 获得了若干广义H-矩阵新的等价条件,并指出文[R. Nabben, On a class of matrices which arise in the numericalsolution of Euler equations, Numer. Math. 1992, 63: 411-431]中的两个结论是错误的,并进行了修正.
第三部分(第四章和第五章)构造非负不可约矩阵Perron 根的新上界序列, 通过数值例子说明该序列在许多情况下优于一些著名结果. 最后, 利用非负矩阵的性质研究矩阵的最大奇异值和谱半径, 给出了若干不等式.
This thesis presents a systematic research on some types of special matricessuch as generalized positive definition matrices, generalized H-matrices andnonnegative irreducible matrices. The thesis consists three parts with fivechapters.
In part one (chapter two), we give some new properties of generalizedpositive definition matrices, and two determinant inequalities about generalizedpositive definition matrices are presented.
The second part (chapter three) contributes to generalized H-matrices.Based on the relationships between generalized M-matrices and generalizedH-matrices with M-matrices and positive definition matrices, we obtain severalequivalent conditions for generalized H-matrices, some remarks on two resultsin [R. Nabben, on a class of matrices which arise in the numerical solution ofEuler equations, Numer. Math. 1992, 63: 411-431] are given.
In the last part (chapter four and five), for the Perron root of nonnegativeirreducible matrices, a new sequence of upper bounds is presented, whichconvergence is discussed. The comparison of the new sequence with the knownones is supplemented with two numerical examples. Finally, by the properties ofnonnegative matrices, we give some inequalities about the largest singularvalues and spectral radius of matrices.
引文
1. C.R. Johnson, Positive Definite Matrices, Amer. Math.. 1970, 77(3):499-504
2. 佟文廷, 广义正定矩阵, 数学学报, 1984, 27(6):801-810
3. 夏长富, 矩阵正定性的进一步推广, 数学研究与评论, 1988, 8(4):499-504
4. 沈光星, 广义正定矩阵及其性质, 高等学校计算数学学报, 2002, (2):186-192
5. 屠伯埙, 亚正定阵理论( Ι), 数学学报, 1990, 33(4):462-471
6. 屠伯埙, 亚正定阵理论( ΙΙ), 数学学报, 1992, 34(1):91-102
7. 顾安娜, 关于广义正定矩阵, 数学的实践与认识, 1994(4):48-50
8. R. S. Varga, Matrix iterative analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962.
9. L. Elsner, V. Mehrmann, Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations. Numer. Math. 1991, 59: 541-559
10. R. Nabben, On a class of matrices which arise in the numerical solution of Euler equations, Numer. Math. 1992, 63: 411-431
11. R.S. Varga, On recurring theorems on diagonal dominance, Linear Algebra Appl. 1976, 13: 1-9
12. A.M. Ostrowski, U ber die Determination mit u berwiegender Hauptdiagonale. && && Comment. Math. Helv. 1937, 10:69-96.
13. 黄廷祝, 蒲和平, (块)H-矩阵与亚正定矩阵, 电子科技大学学报, 1998, 27(2):216 -218
14. Y. Robert, Regular incomplete factorizations of real positive definite matrices, Linear Algebra Appl. 1982, 48: 105-117
15. E. Dick, J. Linden, A multigrid flux-difference splitting method for steady incompressible Navier Stokes equations, Proceedings of the GAMM Conference on Numerical Methods in Fluid Mechanics, Delft, 1989
16. B. Polman, Incomplete blockwise factorization of (block) H-matrices, Linear Algebra Appl. 1987, 90: 119-132
17. 游兆永, 王川龙, 块H-矩阵的进一步推广, 纯粹数学与应用数学, 1995, 11(2): 41-44
18. A. Berman, R.J. Plemmons, Nonnegative matrices in the mathematical sciences, New York, Academic Press, 1979
19. R.A. Horn, C.A. Johnson , Matrix Analysis. Cambridge University Press, 1985
20. 黄廷祝, 钟守铭, 李正良, 矩阵理论, 高等教育出版社, 2003
21. P.W. Hemker, S.P. Spekreijse, Multiple Grid and Osher’s Scheme for the efficient solution of the steady Euler equations, Appl. Numer. Math. 1986, 2: 475-493
22. L. Dieci, J. Lorenz, Block M-matrices and computation of invariant tori, SIAM J. Sci. Statist. Comput. 1992, 13:885-903
23. G. Frobenious, U ber Matrizen aus nichtnegative Elemeten, Sitzungsber, K o n.uss. Akad. Wiss. Berlin, 1912
24. E. Deutsch, Bounds for the Perron root of a nonnegative irreducible partitioned matrix, Pacific J. Math. 1981, 92:49-56
25. H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Zeitschr. 1950, 52:642-648
26. T. Yamamoto, On the extreme values of the roots of matrices, J. Math. Soc. Japan, 1967, 19:173-178
27. D.B. Szyld, A sequence of lower bounds for the spectral radius of nonnegative matrices, Linear Algebra Appl. 1992, 174:239-242
28. L.Y. Kolotilina, Lower bounds for the Perron root of a nonnegative matrix, Linear Algebra Appl. 1993, 180:133-151
29. D. Tasci, S. Kirkland, A sequence of upper bounds for the Perron root of a nonnegative matrix, Linear Algebra Appl. 1998, 273:23-28
30. O. Rojo, R. Jimenez, A decreasing sequence of upper bounds for the Perron root, Computers Math. Applic, 1994, 28:9-15
31. O. Rojo, R. L. Soto, H. Rojo, A decreasing sequence of eigenvalue localization regions, Linear Algebra Appl., 1994, 196:71-84
32. 卢琳璋, 马飞, 非负矩阵Perron 根的上下界, 计算数学, 2003, 25:193-198
33. S. Kirkland, An upper bound on the Perron value of an almost regular tournament matrix, Linear Algebra Appl., 2003, 361:7-22
34. L.Z. Lu, Perron complement and perron root, Linear Algebra Appl., 2002, 341:239-248
35. L. Z. Lu, Michael K. Ng, Localization of Perron roots, Linear Algebra Appl., 2004, 392:103-117
36. J. K. Merikoski, A. Virtanen, The best possible lower bound for the Perron root using
2. 佟文廷, 广义正定矩阵, 数学学报, 1984, 27(6):801-810
3. 夏长富, 矩阵正定性的进一步推广, 数学研究与评论, 1988, 8(4):499-504
4. 沈光星, 广义正定矩阵及其性质, 高等学校计算数学学报, 2002, (2):186-192
5. 屠伯埙, 亚正定阵理论( Ι), 数学学报, 1990, 33(4):462-471
6. 屠伯埙, 亚正定阵理论( ΙΙ), 数学学报, 1992, 34(1):91-102
7. 顾安娜, 关于广义正定矩阵, 数学的实践与认识, 1994(4):48-50
8. R. S. Varga, Matrix iterative analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962.
9. L. Elsner, V. Mehrmann, Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations. Numer. Math. 1991, 59: 541-559
10. R. Nabben, On a class of matrices which arise in the numerical solution of Euler equations, Numer. Math. 1992, 63: 411-431
11. R.S. Varga, On recurring theorems on diagonal dominance, Linear Algebra Appl. 1976, 13: 1-9
12. A.M. Ostrowski, U ber die Determination mit u berwiegender Hauptdiagonale. && && Comment. Math. Helv. 1937, 10:69-96.
13. 黄廷祝, 蒲和平, (块)H-矩阵与亚正定矩阵, 电子科技大学学报, 1998, 27(2):216 -218
14. Y. Robert, Regular incomplete factorizations of real positive definite matrices, Linear Algebra Appl. 1982, 48: 105-117
15. E. Dick, J. Linden, A multigrid flux-difference splitting method for steady incompressible Navier Stokes equations, Proceedings of the GAMM Conference on Numerical Methods in Fluid Mechanics, Delft, 1989
16. B. Polman, Incomplete blockwise factorization of (block) H-matrices, Linear Algebra Appl. 1987, 90: 119-132
17. 游兆永, 王川龙, 块H-矩阵的进一步推广, 纯粹数学与应用数学, 1995, 11(2): 41-44
18. A. Berman, R.J. Plemmons, Nonnegative matrices in the mathematical sciences, New York, Academic Press, 1979
19. R.A. Horn, C.A. Johnson , Matrix Analysis. Cambridge University Press, 1985
20. 黄廷祝, 钟守铭, 李正良, 矩阵理论, 高等教育出版社, 2003
21. P.W. Hemker, S.P. Spekreijse, Multiple Grid and Osher’s Scheme for the efficient solution of the steady Euler equations, Appl. Numer. Math. 1986, 2: 475-493
22. L. Dieci, J. Lorenz, Block M-matrices and computation of invariant tori, SIAM J. Sci. Statist. Comput. 1992, 13:885-903
23. G. Frobenious, U ber Matrizen aus nichtnegative Elemeten, Sitzungsber, K o n.uss. Akad. Wiss. Berlin, 1912
24. E. Deutsch, Bounds for the Perron root of a nonnegative irreducible partitioned matrix, Pacific J. Math. 1981, 92:49-56
25. H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Zeitschr. 1950, 52:642-648
26. T. Yamamoto, On the extreme values of the roots of matrices, J. Math. Soc. Japan, 1967, 19:173-178
27. D.B. Szyld, A sequence of lower bounds for the spectral radius of nonnegative matrices, Linear Algebra Appl. 1992, 174:239-242
28. L.Y. Kolotilina, Lower bounds for the Perron root of a nonnegative matrix, Linear Algebra Appl. 1993, 180:133-151
29. D. Tasci, S. Kirkland, A sequence of upper bounds for the Perron root of a nonnegative matrix, Linear Algebra Appl. 1998, 273:23-28
30. O. Rojo, R. Jimenez, A decreasing sequence of upper bounds for the Perron root, Computers Math. Applic, 1994, 28:9-15
31. O. Rojo, R. L. Soto, H. Rojo, A decreasing sequence of eigenvalue localization regions, Linear Algebra Appl., 1994, 196:71-84
32. 卢琳璋, 马飞, 非负矩阵Perron 根的上下界, 计算数学, 2003, 25:193-198
33. S. Kirkland, An upper bound on the Perron value of an almost regular tournament matrix, Linear Algebra Appl., 2003, 361:7-22
34. L.Z. Lu, Perron complement and perron root, Linear Algebra Appl., 2002, 341:239-248
35. L. Z. Lu, Michael K. Ng, Localization of Perron roots, Linear Algebra Appl., 2004, 392:103-117
36. J. K. Merikoski, A. Virtanen, The best possible lower bound for the Perron root using