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Year 2021, , 133 - 141, 30.09.2021
https://doi.org/10.36753/mathenot.868902

Abstract

References

  • [1] Abbassi, M. T. K., Sarih, M.: On Natural Metrics on Tangent Bundles of Riemannian Manifolds. Arch. Math. (Brno) 41, 71-92 (2005).
  • [2] Cengiz, N., Salimov, A. A.: Diagonal lift in the tensor bundle and its applications. Appl. Math. Comput. 142(2-3), 309-319 (2003).
  • [3] Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. of Math. (2) 96, 413-443 (1972). https://doi.org/10.2307/1970819
  • [4] Djaa, M., Gancarzewicz, J.: The geometry of tangent bundles of order r. boletin Academia Galega de Ciencias 4, 147-165 (1985).
  • [5] Djaa, N. E. H., Boulal, A., Zagane, A.: Generalized Warped Product Manifolds And Biharmonic Maps. Acta Math. Univ. Comenian. (N.S.) 81 (2), 283-298 (2012).
  • [6] Dombrowski, P.: On the Geometry of the Tangent Bundle. J. Reine Angew. Math. 210, 73-88 (1962). https: //doi.org/10.1515/crll.1962.210.73
  • [7] Gezer, A.: On the Tangent Bundle With Deformed Sasaki Metric. Int. Electron. J. Geom. 6 (2), 19-31 (2013).
  • [8] Gudmundsson, S., Kappos, E.:On the Geometry of the Tangent Bundle with the Cheeger-Gromoll Metric. Tokyo J. Math. 25 (1), 75-83 (2002).
  • [9] Jian, W., Yong, W.: On the Geometry of Tangent Bundles with the Rescaled Metric. arXiv:1104.5584v1 [math.DG] 29 Apr 2011.
  • [10] Kada Ben Otmane, R., Zagane, A., Djaa, M.: On generalized Cheeger-Gromoll metric and harmonicity. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 69 (1), 629-645 (2020). https://doi.org/10.31801/cfsuasmas. 487296
  • [11] Kowalski, O., Sekizawa, M.: On Riemannian Geometry Of Tangent Sphere Bundles With Arbitrary Constant Radius. Arch. Math. (Brno) 44, 391-401 (2008).
  • [12] Latti, F., Djaa, M., Zagane, A.: Mus-Sasaki Metric and Harmonicity. Mathematical Sciences and Applications E-Notes 6 (1) 29-36 (2018). https://doi.org/10.36753/mathenot.421753
  • [13] Musso,E.,Tricerri,F.:RiemannianMetricsonTangentBundles.Ann.Mat.Pura.Appl.150,1-19(1988).https: //doi.org/10.1007/BF01761461
  • [14] Salimov, A. A., Gezer, A., Akbulut, K.: Geodesics of Sasakian metrics on tensor bundles. Mediterr. J. Math. 6 (2), 135-147 (2009). https://doi.org/10.1007/s00009-009-0001-z
  • [15] Salimov, A. A, Gezer, A.: On the geometry of the (1, 1)-tensor bundle with Sasaki type metric. Chin. Ann. Math. Ser. B 32 (3), 369-386 (2011). https://doi.org/10.1007/s11401-011-0646-3
  • [16] Salimov, A. A., Agca, F.: Some Properties of Sasakian Metrics in Cotangent Bundles. Mediterr. J. Math. 8 (2), 243-255 (2011). https://doi.org/10.1007/s00009-010-0080-x
  • [17] Salimov, A. A., Kazimova, S.: Geodesics of the Cheeger-Gromoll Metric. Turkish J. Math. 33, 99-105 (2009). doi:10.3906/mat-0804-24
  • [18] Sasaki,S.: OnthedifferentialgeometryoftangentbundlesofRiemannianmanifolds,II.TohokuMath.J.(2)14(2), 146-155 (1962). https://doi.org/10.2748/tmj/1178244169
  • [19] Sekizawa,M.: CurvaturesoftangentbundleswithCheeger-Gromollmetric.TokyoJ.Math.14(2),407-417(1991). DOI10.3836/tjm/1270130381
  • [20] Yano, K., Ishihara, S.: Tangent and tangent bundles. Marcel Dekker. INC. New York (1973).
  • [21] Zagane, A., Djaa, M.: On Geodesics of Warped Sasaki Metric. Mathematical Sciences and Applications E-Notes 5 (1), 85-92 (2017). https://doi.org/10.36753/mathenot.421709
  • [22] Zagane, A., Djaa, M.: Geometry of Mus-Sasaki metric. Commun. Math. 26 (2), 113-126 (2018). https://doi. org/10.2478/cm-2018-0008

Structure Preserving Algorithm for the Logarithm of Symplectic Matrices

Year 2021, , 133 - 141, 30.09.2021
https://doi.org/10.36753/mathenot.868902

Abstract

The current algorithms use either the full form or the Schur decomposition of the matrix in the inverse scaling and squaring method to compute the matrix logarithm. The inverse scaling and squaring method consists of two main calculations: taking a square root and evaluating the Padé approximants. In this work, we suggest using the structure preserving iteration as an alternative to Denman-Beavers iteration for taking a square root. Numerical experiments show that while using the structure preserving square root iteration in the inverse scaling and squaring method preserves the Hamiltonian structure of matrix logarithm, Denman-Beavers iteration and Schur decomposition cause a structure loss.

References

  • [1] Abbassi, M. T. K., Sarih, M.: On Natural Metrics on Tangent Bundles of Riemannian Manifolds. Arch. Math. (Brno) 41, 71-92 (2005).
  • [2] Cengiz, N., Salimov, A. A.: Diagonal lift in the tensor bundle and its applications. Appl. Math. Comput. 142(2-3), 309-319 (2003).
  • [3] Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. of Math. (2) 96, 413-443 (1972). https://doi.org/10.2307/1970819
  • [4] Djaa, M., Gancarzewicz, J.: The geometry of tangent bundles of order r. boletin Academia Galega de Ciencias 4, 147-165 (1985).
  • [5] Djaa, N. E. H., Boulal, A., Zagane, A.: Generalized Warped Product Manifolds And Biharmonic Maps. Acta Math. Univ. Comenian. (N.S.) 81 (2), 283-298 (2012).
  • [6] Dombrowski, P.: On the Geometry of the Tangent Bundle. J. Reine Angew. Math. 210, 73-88 (1962). https: //doi.org/10.1515/crll.1962.210.73
  • [7] Gezer, A.: On the Tangent Bundle With Deformed Sasaki Metric. Int. Electron. J. Geom. 6 (2), 19-31 (2013).
  • [8] Gudmundsson, S., Kappos, E.:On the Geometry of the Tangent Bundle with the Cheeger-Gromoll Metric. Tokyo J. Math. 25 (1), 75-83 (2002).
  • [9] Jian, W., Yong, W.: On the Geometry of Tangent Bundles with the Rescaled Metric. arXiv:1104.5584v1 [math.DG] 29 Apr 2011.
  • [10] Kada Ben Otmane, R., Zagane, A., Djaa, M.: On generalized Cheeger-Gromoll metric and harmonicity. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 69 (1), 629-645 (2020). https://doi.org/10.31801/cfsuasmas. 487296
  • [11] Kowalski, O., Sekizawa, M.: On Riemannian Geometry Of Tangent Sphere Bundles With Arbitrary Constant Radius. Arch. Math. (Brno) 44, 391-401 (2008).
  • [12] Latti, F., Djaa, M., Zagane, A.: Mus-Sasaki Metric and Harmonicity. Mathematical Sciences and Applications E-Notes 6 (1) 29-36 (2018). https://doi.org/10.36753/mathenot.421753
  • [13] Musso,E.,Tricerri,F.:RiemannianMetricsonTangentBundles.Ann.Mat.Pura.Appl.150,1-19(1988).https: //doi.org/10.1007/BF01761461
  • [14] Salimov, A. A., Gezer, A., Akbulut, K.: Geodesics of Sasakian metrics on tensor bundles. Mediterr. J. Math. 6 (2), 135-147 (2009). https://doi.org/10.1007/s00009-009-0001-z
  • [15] Salimov, A. A, Gezer, A.: On the geometry of the (1, 1)-tensor bundle with Sasaki type metric. Chin. Ann. Math. Ser. B 32 (3), 369-386 (2011). https://doi.org/10.1007/s11401-011-0646-3
  • [16] Salimov, A. A., Agca, F.: Some Properties of Sasakian Metrics in Cotangent Bundles. Mediterr. J. Math. 8 (2), 243-255 (2011). https://doi.org/10.1007/s00009-010-0080-x
  • [17] Salimov, A. A., Kazimova, S.: Geodesics of the Cheeger-Gromoll Metric. Turkish J. Math. 33, 99-105 (2009). doi:10.3906/mat-0804-24
  • [18] Sasaki,S.: OnthedifferentialgeometryoftangentbundlesofRiemannianmanifolds,II.TohokuMath.J.(2)14(2), 146-155 (1962). https://doi.org/10.2748/tmj/1178244169
  • [19] Sekizawa,M.: CurvaturesoftangentbundleswithCheeger-Gromollmetric.TokyoJ.Math.14(2),407-417(1991). DOI10.3836/tjm/1270130381
  • [20] Yano, K., Ishihara, S.: Tangent and tangent bundles. Marcel Dekker. INC. New York (1973).
  • [21] Zagane, A., Djaa, M.: On Geodesics of Warped Sasaki Metric. Mathematical Sciences and Applications E-Notes 5 (1), 85-92 (2017). https://doi.org/10.36753/mathenot.421709
  • [22] Zagane, A., Djaa, M.: Geometry of Mus-Sasaki metric. Commun. Math. 26 (2), 113-126 (2018). https://doi. org/10.2478/cm-2018-0008
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Bahar Arslan 0000-0002-7750-4325

Publication Date September 30, 2021
Submission Date January 26, 2021
Acceptance Date April 20, 2021
Published in Issue Year 2021

Cite

APA Arslan, B. (2021). Structure Preserving Algorithm for the Logarithm of Symplectic Matrices. Mathematical Sciences and Applications E-Notes, 9(3), 133-141. https://doi.org/10.36753/mathenot.868902
AMA Arslan B. Structure Preserving Algorithm for the Logarithm of Symplectic Matrices. Math. Sci. Appl. E-Notes. September 2021;9(3):133-141. doi:10.36753/mathenot.868902
Chicago Arslan, Bahar. “Structure Preserving Algorithm for the Logarithm of Symplectic Matrices”. Mathematical Sciences and Applications E-Notes 9, no. 3 (September 2021): 133-41. https://doi.org/10.36753/mathenot.868902.
EndNote Arslan B (September 1, 2021) Structure Preserving Algorithm for the Logarithm of Symplectic Matrices. Mathematical Sciences and Applications E-Notes 9 3 133–141.
IEEE B. Arslan, “Structure Preserving Algorithm for the Logarithm of Symplectic Matrices”, Math. Sci. Appl. E-Notes, vol. 9, no. 3, pp. 133–141, 2021, doi: 10.36753/mathenot.868902.
ISNAD Arslan, Bahar. “Structure Preserving Algorithm for the Logarithm of Symplectic Matrices”. Mathematical Sciences and Applications E-Notes 9/3 (September 2021), 133-141. https://doi.org/10.36753/mathenot.868902.
JAMA Arslan B. Structure Preserving Algorithm for the Logarithm of Symplectic Matrices. Math. Sci. Appl. E-Notes. 2021;9:133–141.
MLA Arslan, Bahar. “Structure Preserving Algorithm for the Logarithm of Symplectic Matrices”. Mathematical Sciences and Applications E-Notes, vol. 9, no. 3, 2021, pp. 133-41, doi:10.36753/mathenot.868902.
Vancouver Arslan B. Structure Preserving Algorithm for the Logarithm of Symplectic Matrices. Math. Sci. Appl. E-Notes. 2021;9(3):133-41.

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