A modified method for reconstructing periodic Jacobi matrices

Authors:
Daniel Boley and Gene H. Golub

Journal:
Math. Comp. **42** (1984), 143-150

MSC:
Primary 65F15; Secondary 15A18

DOI:
https://doi.org/10.1090/S0025-5718-1984-0725989-1

MathSciNet review:
725989

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Abstract: In this note, we discuss the reconstruction of periodic Jacobi matrices from spectral data. The method combines ideas and techniques from the algorithms given by Boley and Golub [1], [2], and Ferguson [3], resulting in a numerically stable algorithm applicable to a larger class of problems. The number of initial data items needed for this method equals the number of items in the resulting matrix, namely 2*n*.

**[1]**D. Boley and G. H. Golub,*Inverse eigenvalue problems for band matrices*, Numerical analysis (Proc. 7th Biennial Conf., Univ. Dundee, Dundee, 1977), Springer, Berlin, 1978, pp. 23–31. Lecture Notes in Math., Vol. 630. MR**0474741****[2]**D. Boley & G. H. Golub,*The Matrix Inverse Eigenvalue Problem for Periodic Matrices*, invited paper at Fourth Conference on Basic Problems of Numerical Anaysis (LIBLICE IV), Pilsin, Czechoslovakia, Sept. 1978.**[3]**Warren E. Ferguson Jr.,*The construction of Jacobi and periodic Jacobi matrices with prescribed spectra*, Math. Comp.**35**(1980), no. 152, 1203–1220. MR**583498**, https://doi.org/10.1090/S0025-5718-1980-0583498-3**[4]**Friedrich W. Biegler-König,*Construction of band matrices from spectral data*, Linear Algebra Appl.**40**(1981), 79–87. MR**629608**, https://doi.org/10.1016/0024-3795(81)90141-5**[5]**John Thompson, personal communication.**[6]**G. W. Stewart,*Introduction to matrix computations*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Computer Science and Applied Mathematics. MR**0458818****[7]**Cornelius Lanczos,*An iteration method for the solution of the eigenvalue problem of linear differential and integral operators*, J. Research Nat. Bur. Standards**45**(1950), 255–282. MR**0042791****[8]**J. H. Wilkinson,*The algebraic eigenvalue problem*, Clarendon Press, Oxford, 1965. MR**0184422****[9]**H. Flaschka,*The Toda lattice. I. Existence of integrals*, Phys. Rev. B (3)**9**(1974), 1924–1925. MR**0408647****[10]**H. Flaschka and D. W. McLaughlin,*Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions*, Progr. Theoret. Phys.**55**(1976), no. 2, 438–456. MR**0403368**, https://doi.org/10.1143/PTP.55.438**[11]**H. Simon,*The Lanczos Algorithm for Solving Symmetric Linear Systems*, Ph.D Thesis, Center for Pure and Applied Mathematics, Univ. of Calif., Berkeley report no. PAM-74, April 1982.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1984-0725989-1

Article copyright:
© Copyright 1984
American Mathematical Society