Calculating elements of matrix functions using divided differences. (English) Zbl 1525.15004
Summary: We introduce a method for calculating individual elements of matrix functions. Our technique makes use of a novel series expansion for the action of matrix functions on basis vectors that is memory efficient even for very large matrices. We showcase our approach by calculating the matrix elements of the exponential of a transverse-field Ising model and evaluating quantum transition amplitudes for large many-body Hamiltonians of sizes up to \(2^{64} \times 2^{64}\) on a single workstation. We also discuss the application of the method to matrix inverses. We relate and compare our method to the state-of-the-art and demonstrate its advantages. We also discuss practical applications of our method.
MSC:
| 15A16 | Matrix exponential and similar functions of matrices |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 81V70 | Many-body theory; quantum Hall effect |
Keywords:
matrix functions; divided differences; off-diagonal series expansion; matrix exponential; transverse-field Ising model; quantum transition amplitudesReferences:
| [1] | Higham, J. N., Functions of Matrices: Theory and Computation (2008), Society for Industrial and Applied Mathematics · Zbl 1167.15001 |
| [2] | Güttel, S.; Kressner, D.; Lund, K., GAMM-Mitt., 43, Article e202000019 pp. (2020) · Zbl 1541.65028 |
| [3] | Higham, N. J.; Hopkins, E., A Catalogue of Software for Matrix Functions. Version 3.0 (2020), Manchester Institute for Mathematical Sciences, The University of Manchester: Manchester Institute for Mathematical Sciences, The University of Manchester UK, MIMS EPrint 2020.7 |
| [4] | Güttel, S., GAMM-Mitt., 36, 8 (2013) · Zbl 1292.65043 |
| [5] | Albash, T.; Wagenbreth, G.; Hen, I., Phys. Rev. E, 96, Article 063309 pp. (2017) |
| [6] | Hen, I., J. Stat. Mech. Theory Exp., 2018, Article 053102 pp. (2018) · Zbl 1459.82068 |
| [7] | Gupta, L.; Albash, T.; Hen, I., J. Stat. Mech. Theory Exp., 2020, Article 073105 pp. (2020) · Zbl 1459.82358 |
| [8] | Cvetković, D. M.; Rowlinson, P.; Simic, S., Eigenspaces of Graphs, vol. 66 (1997), Cambridge University Press · Zbl 0878.05057 |
| [9] | Estrada, E.; Higham, D. J., SIAM Rev., 52, 696 (2010) · Zbl 1214.05077 |
| [10] | Giscard, P.-L.; Thwaite, S. J.; Jaksch, D., SIAM J. Matrix Anal. Appl., 34, 445 (2013) · Zbl 1273.15007 |
| [11] | Joyner, D., Adventures in Group Theory. Rubik’s Cube, Merlin’s Machine, and Other Mathematical Toys (2008), Johns Hopkins University Press: Johns Hopkins University Press Baltimore, MD · Zbl 1221.00013 |
| [12] | Hen, I., Phys. Rev. E, 99, Article 033306 pp. (2019) |
| [13] | Whittaker, E. T.; Robinson, G., The Calculus of Observations: A Treatise on Numerical Mathematics (1940), Blackie and Sons Limited: Blackie and Sons Limited London · JFM 57.1507.02 |
| [14] | de Boor, C., Surv. Approx. Theory, 1, 46 (2005) · Zbl 1071.65027 |
| [15] | Gupta, L.; Barash, L.; Hen, I., Comput. Phys. Commun., 254, Article 107385 pp. (2020) · Zbl 1535.65038 |
| [16] | Pfeuty, P.; Elliott, R. J., J. Phys. C, Solid State Phys., 4, 2370 (1971) |
| [17] | Stinchcombe, R. B., J. Phys. C, Solid State Phys., 6, 2459 (1973) |
| [18] | Matrix functions program codes in c++ |
| [19] | Zivcovich, F., Dolomites Res. Notes Approx., 12, 28 (2019) · Zbl 1540.65059 |
| [20] | Farwig, R.; Zwick, D., J. Math. Anal. Appl., 108, 430 (1985) · Zbl 0579.41011 |
| [21] | Druskin, V. L.; Knizhnerman, L. A., USSR Comput. Math. Math. Phys., 29, 112 (1989) · Zbl 0719.65035 |
| [22] | Driscoll, T. A.; Hale, N.; Trefethen, L. N., Chebfun Guide (2014) |
| [23] | Kahan, W., Commun. ACM, 8, 40 (1965) |
| [24] | Higham, N. J., SIAM J. Sci. Comput., 14, 783 (1993) · Zbl 0788.65053 |
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