Krylov subspace method for fuzzy eigenvalue problem

The eigenvalue problem arises in many application areas and in the fuzzy setting, focus has always been geared towards the finding of solution for the whole set of eigenvalues and corresponding eigenvectors. This paper introduces the computation of a few eigenpairs of a matrix with triangular fuzzy numbers as elements, where the modal matrix is assumed to be sparse and real symmetric. A two-step procedure is developed for the solution of this type of fuzzy eigenvalue problem. The first step solves the 1-cut of the problem, where the well-known Krylov subspace method, implicitly restarted Lanczos algorithm, is employed to approximate part of the spectrum with respect to either smallest or largest eigenvalues. The second step assigns unknown symmetric linear spreads to the approximate eigenpairs. Numerical experiments are provided to illustrate the efficiency of the proposed scheme for determining fuzzy symmetric eigenpairs of a fuzzy matrix.

[1]  D. Calvetti,et al.  AN IMPLICITLY RESTARTED LANCZOS METHOD FOR LARGE SYMMETRIC EIGENVALUE PROBLEMS , 1994 .

[2]  Zhongxiao Jia,et al.  Polynomial characterizations of the approximate eigenvectors by the refined Arnoldi method and an implicitly restarted refined Arnoldi algorithm , 1999 .

[3]  Zhongxiao Jia,et al.  The Convergence of Generalized Lanczos Methods for Large Unsymmetric Eigenproblems , 1995, SIAM J. Matrix Anal. Appl..

[4]  Y. Saad,et al.  Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems , 1984 .

[5]  Silvia Muzzioli,et al.  A note on fuzzy linear systems , 2003 .

[6]  Tofigh Allahviranloo,et al.  Homomorphic solution of fully fuzzy linear systems , 2008 .

[7]  Zengfeng Tian.  Fuzzy Eigenvectors of Real Matrix , 2010 .

[8]  Hans-Jürgen Zimmermann,et al.  Fuzzy Set Theory - and Its Applications , 1985 .

[9]  Yiannis A. Theodorou,et al.  The fuzzy eigenvalue problem of fuzzy correspondence analysis , 2006 .

[10]  Kesheng Wu,et al.  Dynamic Thick Restarting of the Davidson, and the Implicitly Restarted Arnoldi Methods , 1998, SIAM J. Sci. Comput..

[11]  Ashvin Gopaul,et al.  A-posteriori residual bounds for Arnoldi’s methods for nonsymmetric eigenvalue problems , 2011, Numerical Algorithms.

[12]  Tien-Yien Li,et al.  A homotopy algorithm for a symmetric generalized eigenproblem , 1993, Numerical Algorithms.

[13]  H. Carter.  Fuzzy Sets and Systems — Theory and Applications , 1982 .

[14]  Tofigh Allahviranloo,et al.  A comment on fuzzy linear systems , 2003, Fuzzy Sets Syst..

[15]  Anthony Tzes,et al.  A hybrid frequency—time domain adaptive fuzzy control scheme for flexible link manipulators , 1994, J. Intell. Robotic Syst..

[16]  Danny C. Sorensen,et al.  Implicit Application of Polynomial Filters in a k-Step Arnoldi Method , 1992, SIAM J. Matrix Anal. Appl..

[17]  Lotfi A. Zadeh,et al.  Fuzzy Logic , 2009, Encyclopedia of Complexity and Systems Science.

[18]  Tofigh Allahviranloo,et al.  Numerical methods for fuzzy system of linear equations , 2004, Appl. Math. Comput..

[19]  Neetu,et al.  A new computational method for solving fully fuzzy linear systems of triangular fuzzy numbers , 2012 .

[20]  Tofigh Allahviranloo,et al.  A new method for solving an arbitrary fully fuzzy linear system , 2013, Soft Comput..

[21]  Tofigh Allahviranloo,et al.  ON THE ALGEBRAIC SOLUTION OF FUZZY LINEAR SYSTEMS BASED ON INTERVAL THEORY , 2012 .

[22]  Tofigh Allahviranloo,et al.  FINDING AN INNER ESTIMATION OF THE SOLUTION SET OF A FUZZY LINEAR SYSTEM , 2013 .

[23]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[24]  M. Bhuruth,et al.  An Implicit Partial Pivoting Gauss Elimination Algorithm for Linear System of Equations with Fuzzy Parameters , 2012 .

[25]  R. Goetschel,et al.  Elementary fuzzy calculus , 1986 .

[26]  Amit Kumar,et al.  Fully fuzzy linear systems of triangular fuzzy numbers (a, b, c) , 2013, Int. J. Intell. Comput. Cybern..

[27]  Abraham Kandel,et al.  Fuzzy linear systems , 1998, Fuzzy Sets Syst..

[28]  Tofigh Allahviranloo,et al.  Maximal- and minimal symmetric solutions of fully fuzzy linear systems , 2011, J. Comput. Appl. Math..

[29]  Ronald B. Morgan,et al.  On restarting the Arnoldi method for large nonsymmetric eigenvalue problems , 1996, Math. Comput..

[30]  C. Drossos,et al.  Correspondence analysis with fuzzy data: The fuzzy eigenvalue problem , 2007, Fuzzy Sets Syst..

[31]  S. Salahshour,et al.  Computing the eigenvalues and eigenvectors of a fuzzy matrix , 2012 .

[32]  W. Arnoldi.  The principle of minimized iterations in the solution of the matrix eigenvalue problem , 1951 .

[33]  Y. Saad.  Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices , 1980 .

[34]  Mehdi Dehghan,et al.  Computational methods for solving fully fuzzy linear systems , 2006, Appl. Math. Comput..

[35]  Tofigh Allahviranloo,et al.  Fuzzy linear matrix equation , 2009, Fuzzy Optim. Decis. Mak..

[36]  Tofigh Allahviranloo,et al.  Signed Decomposition of Fully Fuzzy Linear Systems , 2008 .

[37]  Silvia Muzzioli,et al.  Fuzzy linear systems of the form A1x+b1=A2x+b2 , 2006, Fuzzy Sets Syst..

[38]  Ravindra Boojhawon,et al.  A new method for accelerating Arnoldi algorithms for large scale Eigenproblems , 2009, Math. Comput. Simul..

[39]  D. S. Dinagar,et al.  SOME TYPES OF TYPE-2 TRIANGULAR FUZZY MATRICES , 2013 .

[40]  D. Sorensen.  Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[41]  Richard B. Lehoucq,et al.  Implicitly Restarted Arnoldi Methods and Subspace Iteration , 2001, SIAM J. Matrix Anal. Appl..

[42]  Kumar Dookhitram,et al.  PRECONDITIONING ALGORITHM FOR THE POSITIVE SOLUTION OF FULLY FUZZY LINEAR SYSTEM , 2012 .

[43]  J. Cullum,et al.  Lanczos algorithms for large symmetric eigenvalue computations , 1985 .

[44]  C. Paige.  Computational variants of the Lanczos method for the eigenproblem , 1972 .

[45]  Tofigh Allahviranloo,et al.  NON ZERO SOLUTIONS OF THE FULLY FUZZY LINEAR SYSTEMS , 2011 .

[46]  Mehdi Dehghan,et al.  Iterative solution of fuzzy linear systems , 2006, Appl. Math. Comput..

[47]  Tofigh Allahviranloo,et al.  A method to find fuzzy eigenvalues and fuzzy eigenvectors of fuzzy matrix , 2012, Neural Computing and Applications.

[48]  Z. Jia,et al.  Refined iterative algorithms based on Arnoldi's process for large unsymmetric eigenproblems , 1997 .

[49]  Tofigh Allahviranloo,et al.  Fuzzy symmetric solutions of fuzzy linear systems , 2011, J. Comput. Appl. Math..

[50]  Snehashish Chakraverty,et al.  A new method for solving real and complex fuzzy systems of linear equations , 2012 .

[51]  G. W. Stewart,et al.  A Krylov-Schur Algorithm for Large Eigenproblems , 2001, SIAM J. Matrix Anal. Appl..

[52]  Tofigh Allahviranloo,et al.  Solution of a fuzzy system of linear equation , 2006, Appl. Math. Comput..

[53]  Pavel V. Sevastjanov,et al.  A new method for solving interval and fuzzy equations: Linear case , 2009, Inf. Sci..

[54]  D. Sorensen,et al.  4. The Implicitly Restarted Arnoldi Method , 1998 .