IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i17p2018-d620346.html
   My bibliography  Save this article

An Improved Taylor Algorithm for Computing the Matrix Logarithm

Author

Listed:
  • Javier Ibáñez

    (Instituto de Instrumentación para Imagen Molecular, Universitat Politècnica de València, Av. dels Tarongers, 14, 46011 Valencia, Spain)

  • Jorge Sastre

    (Instituto de Telecomunicaciones y Aplicaciones Multimedia, Universitat Politècnica de València, Ed. 8G, Camino de Vera s/n, 46022 Valencia, Spain)

  • Pedro Ruiz

    (Instituto de Instrumentación para Imagen Molecular, Universitat Politècnica de València, Av. dels Tarongers, 14, 46011 Valencia, Spain)

  • José M. Alonso

    (Instituto de Instrumentación para Imagen Molecular, Universitat Politècnica de València, Av. dels Tarongers, 14, 46011 Valencia, Spain)

  • Emilio Defez

    (Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Ed. 8G, Camino de Vera s/n, 46022 Valencia, Spain)

Abstract

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the free-transformation approach of the inverse scaling and squaring technique, that uses recent matrix polynomial formulas for evaluating the Taylor approximation of the matrix logarithm more efficiently than the Paterson–Stockmeyer method. Two MATLAB implementations of this algorithm, related to relative forward or backward error analysis, were developed and compared with different state-of-the art MATLAB functions. Numerical tests showed that the new implementations are generally more accurate than the previously available codes, with an intermediate execution time among all the codes in comparison.

Suggested Citation

  • Javier Ibáñez & Jorge Sastre & Pedro Ruiz & José M. Alonso & Emilio Defez, 2021. "An Improved Taylor Algorithm for Computing the Matrix Logarithm," Mathematics, MDPI, vol. 9(17), pages 1-19, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:17:p:2018-:d:620346
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/17/2018/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/17/2018/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Robert B. Israel & Jeffrey S. Rosenthal & Jason Z. Wei, 2001. "Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings," Mathematical Finance, Wiley Blackwell, vol. 11(2), pages 245-265, April.
    2. Sastre, J. & Ibáñez, J. & Defez, E., 2019. "Boosting the computation of the matrix exponential," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 206-220.
    3. Yu, Philip L.H. & Wang, Xiaohang & Zhu, Yuanyuan, 2017. "High dimensional covariance matrix estimation by penalizing the matrix-logarithm transformed likelihood," Computational Statistics & Data Analysis, Elsevier, vol. 114(C), pages 12-25.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Georges Dionne & Geneviève Gauthier & Khemais Hammami & Mathieu Maurice & Jean‐Guy Simonato, 2010. "Default Risk in Corporate Yield Spreads," Financial Management, Financial Management Association International, vol. 39(2), pages 707-731, June.
    2. Areski Cousin & Mohamed Reda Kheliouen, 2016. "A comparative study on the estimation of factor migration models," Working Papers halshs-01351926, HAL.
    3. Fuertes, Ana-Maria & Kalotychou, Elena, 2007. "On sovereign credit migration: A study of alternative estimators and rating dynamics," Computational Statistics & Data Analysis, Elsevier, vol. 51(7), pages 3448-3469, April.
    4. Juergen Jung, 2022. "Estimating transition probabilities between health states using US longitudinal survey data," Empirical Economics, Springer, vol. 63(2), pages 901-943, August.
    5. Alexandre Ounnas, 2020. "Worker Flows and Occupations in the CPS 1976-2010: A Framework for Adjusting the Data," LIDAM Discussion Papers IRES 2020008, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
    6. Whitt, Ward, 2012. "Fitting birth-and-death queueing models to data," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 998-1004.
    7. Jorge Sastre & Javier Ibáñez, 2021. "Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method," Mathematics, MDPI, vol. 9(14), pages 1-23, July.
    8. Guogui Huang & Fei Guo & Zhiming Cheng & Massimiliano Tani & Gong Chen, 2023. "Projections of Future Demand and Costs of Aged Care Services in China," Population Research and Policy Review, Springer;Southern Demographic Association (SDA), vol. 42(4), pages 1-30, August.
    9. Camilla Ferretti & Giampaolo Gabbi & Piero Ganugi & Federica Sist & Pietro Vozzella, 2019. "Credit Risk Migration and Economic Cycles," Risks, MDPI, vol. 7(4), pages 1-18, October.
    10. Jia, Chen, 2016. "A solution to the reversible embedding problem for finite Markov chains," Statistics & Probability Letters, Elsevier, vol. 116(C), pages 122-130.
    11. Guglielmo D’Amico & Jacques Janssen & Raimondo Manca, 2006. "Homogeneous semi-Markov reliability models for credit risk management," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 28(2), pages 79-93, February.
    12. Albanese, Claudio & Chen, Oliver X., 2006. "Implied migration rates from credit barrier models," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 607-626, February.
    13. Simone Varotto, 2011. "Liquidity risk, credit risk, market risk and bank capital," International Journal of Managerial Finance, Emerald Group Publishing Limited, vol. 7(2), pages 134-152, April.
    14. Dionne, Georges & Gauthier, Geneviève & Hammami, Khemais & Maurice, Mathieu & Simonato, Jean-Guy, 2011. "A reduced form model of default spreads with Markov-switching macroeconomic factors," Journal of Banking & Finance, Elsevier, vol. 35(8), pages 1984-2000, August.
    15. Stefan Truck, 2008. "Forecasting credit migration matrices with business cycle effects—a model comparison," The European Journal of Finance, Taylor & Francis Journals, vol. 14(5), pages 359-379.
    16. Maximilian Hughes & Ralf Werner, 2016. "Choosing Markovian Credit Migration Matrices by Nonlinear Optimization," Risks, MDPI, vol. 4(3), pages 1-18, August.
    17. Greig Smith & Goncalo dos Reis, 2017. "Robust and Consistent Estimation of Generators in Credit Risk," Papers 1702.08867, arXiv.org, revised Oct 2017.
    18. Parnes, Dror, 2007. "Time series patterns in credit ratings," Finance Research Letters, Elsevier, vol. 4(4), pages 217-226, December.
    19. Simone Varotto, 2007. "Tests on the Accuracy of Basel II," ICMA Centre Discussion Papers in Finance icma-dp2007-09, Henley Business School, University of Reading.
    20. Pietro Veronesi, "undated". "Belief-dependent Utilities, Aversion to State-Uncertainty and Asset Prices,”," CRSP working papers 529, Center for Research in Security Prices, Graduate School of Business, University of Chicago.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:17:p:2018-:d:620346. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.