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用重正交化Lanczos法求解大型非正交归一基稀疏矩阵的特征值问题

焦宝宝

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用重正交化Lanczos法求解大型非正交归一基稀疏矩阵的特征值问题

焦宝宝

Eigenvalue problems solved by reorthogonalization Lanczos method for the large non-orthonormal sparse matrix

Jiao Bao-Bao
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  • 在116Sn原子核壳模型结构下,利用广义辛弱数截断多体空间得到的哈密顿矩阵是一个大型的实对称非正交归一基稀疏矩阵,因此求解大型矩阵的能量特征值和能量特征向量是原子核物理上的一个重要问题.为此,利用重正交化Lanczos法与Cholesky分解法和Elementary transformation法相结合的方法,实现了用内存较小的计算机求解大型实对称非正交归一基稀疏矩阵的特征值和特征向量.用这种方法计算小型矩阵得到的特征值和精确值符合得较好,且运用这个方法计算了116Sn壳模型截断后的大型非正交归一基稀疏矩阵的能量特征值,得到的原子核低态能量与实验测量能量相吻合,计算结果表明Lanczos法在Matlab编程和大型壳模型计算中的精确性和可行性.此方法也有助于求解一些中质核或者重核的低态能量,同时也有利于用内存稍大的计算机求解更大的非正交归一基矩阵的特征值问题.
    Using shell model to calculate the nuclear systems in a large model space is an important method in the field of nuclear physics.On the basis of the nuclear shell model,a large symmetric non-orthonormal sparse Hamiltonian matrix is generated when adopting the generalized seniority method to truncate the many-body space.Calculating the energy eigenvalues and energy eigenvectors of the large symmetric non-orthonormal sparse Hamiltonian matrix is of indispensable steps before energies of nucleus are further calculated.In the mean time,some low-lying energy eigenvalues are always the focus of attention on the occasion of large scale shell model calculation.In this paper,by combining reorthogonalization Lanczos method with Cholesky decomposition method and Elementary transformation method,converting the generalized eigenvalue problems into the standard eigenvalue problems,and transforming the large standard eigenvalue problems into the small standard eigenvalue problems,we successfully calculate the eigenvalues and eigenvectors of large non-orthonormal sparse matrices with the help of computers with limited memory.The values obtained by using this method to calculate the small matrix agree with the exact values,which demonstrates that this method is accurate and can be used to calculate the energy eigenvalues and energy eigenvectors of large symmetric nonorthonormal sparse matrix.We take 116Sn (s=8,the number of unpaired particles,namely the generalized seniority) as an example in which there are active valence neutrons but inert protons at the magic number,and calculate ten of its lowest energy eigenvalues.Through calculation,we find that among these low-lying energy eigenvalues,the lowest energy eigenvalue converges fastest.A comparison between the calculation values and the experiment values shows that the difference between the calculated high-lying energy eigenvalue and its corresponding experimental one arrives at hundreds of keV,while for the low-lying energy eigenvalue,its calculation value can reach an accuracy of a few tens of keV.The results demonstrate that the Lanczos method is feasible in Matlab programming and shell model calculations. The significance of this research lies in the fact that this method will not only greatly help to calculate and obtain the low-lying energy eigenvalues of some medium-mass and heavy nuclei,but also possess great importance in calculating partial eigenvalues involved in large matrices in other theoretical researches and engineering designs.
      Corresponding author: Jiao Bao-Bao, baobaojiao91@126.com
    [1]

    Ring P, Schuck P 1980 The Nuclear Many-body Problem (Berlin: Springer-Verlag) pp36-95

    [2]

    Shen J J, Zhao Y M 2009 Sci. China: Ser. G 52 1477

    [3]

    Shen J J, Arima A, Zhao Y M, Yoshinaga N 2008 Phys. Rev. C 78 044305

    [4]

    Zhang L H, Shen J J, Lei Y, Zhao Y M 2008 Int. J. Mod. Phys. E 17 342

    [5]

    Jia L Y 2013 Phy. Rev. C 88 044303

    [6]

    Pittel S, Sandulescu N 2006 Phys. Rev. C 73 014301

    [7]

    Thakur B, Pittel S, Sandulescu N 2008 Phys. Rev. C 78 041303

    [8]

    Papenbrock T, Dean D J 2005 J. Phys. G: Nucl. Part. Phys. 31 S1377

    [9]

    Kruse M K G, Jurgenson E D, Navratil P, Barrett B R, Ormand W E 2013 Phys. Rev. C 87 044301

    [10]

    Han H, Wu L Y, Song N N 2014 Acta Phys. Sin. 63 138901 (in Chinese) [韩华, 吴翎燕, 宋宁宁2014物理学报63 138901]

    [11]

    Li S, Wang B, Hu J Z 2003 Appl. Math. Mech. 24 92

    [12]

    Morris N F 1990 J. Struct. Eng. 116 2049

    [13]

    Jia L Y 2015 J. Phys. G: Nucl. Part. Phys. 42 115105

    [14]

    Qi C, Xu Z X 2012 Phys. Rev. C 86 044323

    [15]

    Simon H D 1984 Math. Comput. 42 115

    [16]

    Zhao X H, Chen F W, Wu J, Zhou Q L 2008 Acta Phys. Chim. Sin. 24 823 (in Chinese) [赵小红, 陈飞武, 吴健, 周巧龙2008物理化学学报24 823]

    [17]

    Cao Z H 1980 Eigenvalue Problems of Matrices (Shanghai: Shanghai Scientific and Technical Publishers) pp212-220(in Chinese) [曹志浩1980矩阵特征值问题(上海:上海科学技术出版社)第212–220页]

    [18]

    Harbrecht H, Peters M, Schneider R 2012 Appl. Numer. Math. 62 428

    [19]

    D'azevedo E, Dongarra J 2000 Pract. Exper. 12 1481

    [20]

    Schweizer S, Kussmann J, Doser B, Ochsenfeld C 2008 J. Comput. Chem. 29 1004

    [21]

    Liu D, Gabrielli L H, Lipson M, Johnson S G 2013 Opt. Exp. 21 12

    [22]

    Giraud L, Langou J 2005 Comput. Math. Appl. 50 1069

    [23]

    Hoffmanm W, Amsterdam 1989 Computing 41 335

    [24]

    Xu S F 1995 Theory and Method of Matrix Calculation (Beijing: Peking University Publishers) pp307-319(in Chinese) [徐树方1995矩阵计算的理论与方法(北京:北京大学出版社)第307–319页]

    [25]

    Qiu Z, Wang X 2005 J. Sound Vib. 282 381

  • [1]

    Ring P, Schuck P 1980 The Nuclear Many-body Problem (Berlin: Springer-Verlag) pp36-95

    [2]

    Shen J J, Zhao Y M 2009 Sci. China: Ser. G 52 1477

    [3]

    Shen J J, Arima A, Zhao Y M, Yoshinaga N 2008 Phys. Rev. C 78 044305

    [4]

    Zhang L H, Shen J J, Lei Y, Zhao Y M 2008 Int. J. Mod. Phys. E 17 342

    [5]

    Jia L Y 2013 Phy. Rev. C 88 044303

    [6]

    Pittel S, Sandulescu N 2006 Phys. Rev. C 73 014301

    [7]

    Thakur B, Pittel S, Sandulescu N 2008 Phys. Rev. C 78 041303

    [8]

    Papenbrock T, Dean D J 2005 J. Phys. G: Nucl. Part. Phys. 31 S1377

    [9]

    Kruse M K G, Jurgenson E D, Navratil P, Barrett B R, Ormand W E 2013 Phys. Rev. C 87 044301

    [10]

    Han H, Wu L Y, Song N N 2014 Acta Phys. Sin. 63 138901 (in Chinese) [韩华, 吴翎燕, 宋宁宁2014物理学报63 138901]

    [11]

    Li S, Wang B, Hu J Z 2003 Appl. Math. Mech. 24 92

    [12]

    Morris N F 1990 J. Struct. Eng. 116 2049

    [13]

    Jia L Y 2015 J. Phys. G: Nucl. Part. Phys. 42 115105

    [14]

    Qi C, Xu Z X 2012 Phys. Rev. C 86 044323

    [15]

    Simon H D 1984 Math. Comput. 42 115

    [16]

    Zhao X H, Chen F W, Wu J, Zhou Q L 2008 Acta Phys. Chim. Sin. 24 823 (in Chinese) [赵小红, 陈飞武, 吴健, 周巧龙2008物理化学学报24 823]

    [17]

    Cao Z H 1980 Eigenvalue Problems of Matrices (Shanghai: Shanghai Scientific and Technical Publishers) pp212-220(in Chinese) [曹志浩1980矩阵特征值问题(上海:上海科学技术出版社)第212–220页]

    [18]

    Harbrecht H, Peters M, Schneider R 2012 Appl. Numer. Math. 62 428

    [19]

    D'azevedo E, Dongarra J 2000 Pract. Exper. 12 1481

    [20]

    Schweizer S, Kussmann J, Doser B, Ochsenfeld C 2008 J. Comput. Chem. 29 1004

    [21]

    Liu D, Gabrielli L H, Lipson M, Johnson S G 2013 Opt. Exp. 21 12

    [22]

    Giraud L, Langou J 2005 Comput. Math. Appl. 50 1069

    [23]

    Hoffmanm W, Amsterdam 1989 Computing 41 335

    [24]

    Xu S F 1995 Theory and Method of Matrix Calculation (Beijing: Peking University Publishers) pp307-319(in Chinese) [徐树方1995矩阵计算的理论与方法(北京:北京大学出版社)第307–319页]

    [25]

    Qiu Z, Wang X 2005 J. Sound Vib. 282 381

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出版历程
  • 收稿日期:  2016-04-27
  • 修回日期:  2016-07-05
  • 刊出日期:  2016-10-05

用重正交化Lanczos法求解大型非正交归一基稀疏矩阵的特征值问题

摘要: 在116Sn原子核壳模型结构下,利用广义辛弱数截断多体空间得到的哈密顿矩阵是一个大型的实对称非正交归一基稀疏矩阵,因此求解大型矩阵的能量特征值和能量特征向量是原子核物理上的一个重要问题.为此,利用重正交化Lanczos法与Cholesky分解法和Elementary transformation法相结合的方法,实现了用内存较小的计算机求解大型实对称非正交归一基稀疏矩阵的特征值和特征向量.用这种方法计算小型矩阵得到的特征值和精确值符合得较好,且运用这个方法计算了116Sn壳模型截断后的大型非正交归一基稀疏矩阵的能量特征值,得到的原子核低态能量与实验测量能量相吻合,计算结果表明Lanczos法在Matlab编程和大型壳模型计算中的精确性和可行性.此方法也有助于求解一些中质核或者重核的低态能量,同时也有利于用内存稍大的计算机求解更大的非正交归一基矩阵的特征值问题.

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