A survey of matrix inverse eigenvalue problems
Open accessAbstract
The authors present a survey of some recent results regarding direct methods for solving certain symmetric inverse eigenvalue problems. The problems they discuss are those of generating a symmetric matrix either Jacobi, banded, or some variation thereof, given only some information on the eigenvalues of the matrix itself and some of its principal submatrices. Much of the motivation for the problems discussed came about from an interest in the inverse Sturm-Liouville problem.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Given an eigenvalue spectrum satisfying Eq. (5), the corresponding coupling constants can be uniquely determined.
“Given an eigenvalue spectrum { E k } satisfying Eq. (5), the corresponding coupling constants { ω j , λ j } can be uniquely determined.”
From Quantum speed limit for perfect state transfer in one dimension (2006) · cites this paper as [25]
The eigenvalues of the Hamiltonian must be non-degenerate.
“The eigenvalues, which must be non-degenerate [25], and the corresponding eigenvectors are denoted by E k and | e k 〉 respectively.”
From Quantum speed limit for perfect state transfer in one dimension (2006) · cites this paper as [25]
The eigenvector |e_k> has exactly k-1 sign changes in the position basis.
“On the other hand, it is known [25] that the eigenvector | e k 〉 has exactly k -1 sign changes in the position basis {| j 〉} .”
From Quantum speed limit for perfect state transfer in one dimension (2006) · cites this paper as [25]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
Introduction to matrix computations
Stewart, G. W. (1973) · International Linear Collider · Cited by 2,394 · see citing works
Calculation of Gauss quadrature rules
Golub, G., & Welsch, J. H. (1967) · Milestones in Matrix Computation · doi:10.1090/s0025-5718-69-99647-1 · Cited by 1,464 · see citing works
Self-citationFundamentals of Computer Algorithms
Horowitz, E., & Sahni, S. K. (1983) · CERN Document Server (European Organization for Nuclear Research) · Cited by 1,391 · see citing works
Elementary Numerical Analysis: An Algorithmic Approach
W., J. W., & Conte, S. D. (1967) · Mathematics of Computation · doi:10.2307/2003277 · Cited by 1,335 · see citing works
Some Modified Matrix Eigenvalue Problems
Golub, G. H. (1973) · SIAM Review · doi:10.1137/1015032 · Cited by 730 · see citing works
Self-citationKlema, V. C., Dongarra, J. J., Bunch, J. R., et al. (1980) · Mathematics of Computation · doi:10.2307/2006212 · Cited by 646 · see citing works
Numerical methods for establishing solutions to the inverse problem of electromagnetic induction
Parker, R. L., & Whaler, K. A. (1981) · Journal of Geophysical Research Atmospheres · doi:10.1029/jb086ib10p09574 · Cited by 203 · see citing works
The numerically stable reconstruction of a Jacobi matrix from spectral data
de Boor, C., & Golub, G. H. (1978) · Linear Algebra and its Applications · doi:10.1016/0024-3795(78)90086-1 · Cited by 185 · see citing works
Self-citationThe Formulation and Analysis of Numerical Methods for Inverse Eigenvalue Problems
Friedland, S., Nocedal, J., & Overton, M. L. (1987) · SIAM Journal on Numerical Analysis · doi:10.1137/0724043 · Cited by 164 · see citing works
Inverse eigenvalue problems for Jacobi matrices
Hald, O. H. (1976) · Linear Algebra and its Applications · doi:10.1016/0024-3795(76)90064-1 · Cited by 154 · see citing works
On the construction of a Jacobi matrix from spectral data
Hochstadt, H. (1974) · Linear Algebra and its Applications · doi:10.1016/0024-3795(74)90077-9 · Cited by 150 · see citing works
The formulation and analysis of numerical methods for inverse eigenvalue problems
Friedland, S., & Overton, M. L. (1987) · Deep Sea Research Part B Oceanographic Literature Review · doi:10.1016/0198-0254(87)96151-6 · Cited by 144 · see citing works
The numerically stable reconstruction of Jacobi matrices from spectral data
Gragg, W. B., & Harrod, W. J. (1984) · Numerische Mathematik · doi:10.1007/bf01405565 · Cited by 138 · see citing works
The inverse problem for the vibrating beam
Gladwell, G. M. L. (1984) · Proceedings of the Royal Society of London A Mathematical and Physical Sciences · doi:10.1098/rspa.1984.0058 · Cited by 66 · see citing works
A Numerical Method for the Inverse Sturm–Liouville Problem
Paine, J. W. (1984) · SIAM Journal on Scientific and Statistical Computing · doi:10.1137/0905011 · Cited by 57 · see citing works
Some modified eigenvalue problems
Golub, G. H. (1971) · Lecture notes in mathematics · doi:10.1007/bfb0069447 · Cited by 37 · see citing works
Self-citationTHE INVERSE MODE PROBLEM FOR LUMPED-MASS SYSTEMS
Gladwell, G. M. L. (1986) · The Quarterly Journal of Mechanics and Applied Mathematics · doi:10.1093/qjmam/39.2.297 · Cited by 31 · see citing works
Construction of band matrices from spectral data
Biegler‐König, F. (1981) · Linear Algebra and its Applications · doi:10.1016/0024-3795(81)90141-5 · Cited by 27 · see citing works
A modified method for reconstructing periodic Jacobi matrices
Boley, D. L., & Golub, G. H. (1984) · Mathematics of Computation · doi:10.1090/s0025-5718-1984-0725989-1 · Cited by 26 · see citing works
Self-citationON THE INVERSE PROBLEM OF THE VIBRATING STRING OR ROD
Gladwell, G. M. L., & Gbadeyan, J. A. (1985) · The Quarterly Journal of Mechanics and Applied Mathematics · doi:10.1093/qjmam/38.1.169 · Cited by 25 · see citing works
A numerical design procedure for general canonic LC one-ports
Sussman-Fort, S. E. (1982) · IEEE Transactions on Circuits and Systems · doi:10.1109/tcs.1982.1085202 · Cited by 2 · see citing works
Flagged first, then by citations
Research works printed but not linked: 4
- Lin. Alg. Appl. 21 245-60 York: McGraw Hill) Comput. 35 79-84 eigenvalue problems SIAM J. Num. Anal. 24 634-67 Appl. Math. 38 169-74
- Golub G H and Van Loan C 1983 Matrix Computations (Baltimore: Johns Hopkins University Press)
- Rutishauser H 1963 On Jacobi rotation patterns; Experimental Arithmetic, High Speed Computing and Mathematics, Proc. Symp. Appl. Math. 15 (American Mathematics Society) pp 2 19-39
- Smith B T, Boyle J M, Ikebe Y, Klema V C and Moler C B 1970 Matrix Eigenvalue Routines, EISPACK Guide 2nd edn (New York: Springer)
Cited by
Sorted by citations
Quantum speed limit for perfect state transfer in one dimension
Yung, M.-H. (2006) · Physical Review A · doi:10.1103/physreva.74.030303 · Cited by 128 · see citing works
- Authors
- Boley, D., & Golub, G. H.
- Affiliations
- University of Minnesota (US)
- ORCID
- Boley, D
- Venue
- Inverse Problems
- Publisher
- IOP Publishing
- Published
- 1987-11-01
- Volume
- 3
- Issue
- 4
- Pages
- 595-622
- ISSN
- 0266-5611 (ISSN-L)
- 1361-6420
- OpenAlex
- W1970376871
- Cited by
- 223
- References
- 21
- Citations by year
- 2026: 3 · 2025: 2 · 2024: 4 · 2023: 6 · 2022: 5 · 2021: 5 · 2020: 5 · 2019: 7 · 2018: 4 · 2017: 3 · 2016: 5 · 2015: 9 · 2014: 10 · 2013: 12 · 2012: 4
- FWCI
- 1.39
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Inverse Problems (publishedVersion)
- Funders
- National Science Foundation
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
- Physical Sciences › Mathematics › Mathematical Physics › Numerical methods in inverse problems
- Keywords
- symmetric matrices, direct method, Jacobi matrices, banded matrices, eigenvalues, principal submatrices, inverse Sturm-Liouville problem
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
