---
_id: '29240'
abstract:
- lang: eng
  text: "The principle of least action is one of the most fundamental physical principle.
    It says that among all possible motions connecting two points in a phase space,
    the system will exhibit those motions which extremise an action functional. Many
    qualitative features of dynamical systems, such as the presence of conservation
    laws and energy balance equations, are related to the existence of an action functional.
    Incorporating variational structure into learning algorithms for dynamical systems
    is, therefore, crucial in order to make sure that the learned model shares important
    features with the exact physical system. In this paper we show how to incorporate
    variational principles into trajectory predictions of learned dynamical systems.
    The novelty of this work is that (1) our technique relies only on discrete position
    data of observed trajectories. Velocities or conjugate momenta do not need to
    be observed or approximated and no prior knowledge about the form of the variational
    principle is assumed. Instead, they are recovered using backward error analysis.
    (2) Moreover, our technique compensates discretisation errors when trajectories
    are computed from the learned system. This is important when moderate to large
    step-sizes are used and high accuracy is required. For this,\r\nwe introduce and
    rigorously analyse the concept of inverse modified Lagrangians by developing an
    inverse version of variational backward error analysis. (3) Finally, we introduce
    a method to perform system identification from position observations only, based
    on variational backward error analysis."
article_type: original
author:
- first_name: Sina
  full_name: Ober-Blöbaum, Sina
  id: '16494'
  last_name: Ober-Blöbaum
- first_name: Christian
  full_name: Offen, Christian
  id: '85279'
  last_name: Offen
  orcid: 0000-0002-5940-8057
citation:
  ama: Ober-Blöbaum S, Offen C. Variational Learning of Euler–Lagrange Dynamics from
    Data. <i>Journal of Computational and Applied Mathematics</i>. 2023;421:114780.
    doi:<a href="https://doi.org/10.1016/j.cam.2022.114780">10.1016/j.cam.2022.114780</a>
  apa: Ober-Blöbaum, S., &#38; Offen, C. (2023). Variational Learning of Euler–Lagrange
    Dynamics from Data. <i>Journal of Computational and Applied Mathematics</i>, <i>421</i>,
    114780. <a href="https://doi.org/10.1016/j.cam.2022.114780">https://doi.org/10.1016/j.cam.2022.114780</a>
  bibtex: '@article{Ober-Blöbaum_Offen_2023, title={Variational Learning of Euler–Lagrange
    Dynamics from Data}, volume={421}, DOI={<a href="https://doi.org/10.1016/j.cam.2022.114780">10.1016/j.cam.2022.114780</a>},
    journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier},
    author={Ober-Blöbaum, Sina and Offen, Christian}, year={2023}, pages={114780}
    }'
  chicago: 'Ober-Blöbaum, Sina, and Christian Offen. “Variational Learning of Euler–Lagrange
    Dynamics from Data.” <i>Journal of Computational and Applied Mathematics</i> 421
    (2023): 114780. <a href="https://doi.org/10.1016/j.cam.2022.114780">https://doi.org/10.1016/j.cam.2022.114780</a>.'
  ieee: 'S. Ober-Blöbaum and C. Offen, “Variational Learning of Euler–Lagrange Dynamics
    from Data,” <i>Journal of Computational and Applied Mathematics</i>, vol. 421,
    p. 114780, 2023, doi: <a href="https://doi.org/10.1016/j.cam.2022.114780">10.1016/j.cam.2022.114780</a>.'
  mla: Ober-Blöbaum, Sina, and Christian Offen. “Variational Learning of Euler–Lagrange
    Dynamics from Data.” <i>Journal of Computational and Applied Mathematics</i>,
    vol. 421, Elsevier, 2023, p. 114780, doi:<a href="https://doi.org/10.1016/j.cam.2022.114780">10.1016/j.cam.2022.114780</a>.
  short: S. Ober-Blöbaum, C. Offen, Journal of Computational and Applied Mathematics
    421 (2023) 114780.
date_created: 2022-01-11T13:24:00Z
date_updated: 2023-08-10T08:42:39Z
ddc:
- '510'
department:
- _id: '636'
doi: 10.1016/j.cam.2022.114780
external_id:
  arxiv:
  - '2112.12619'
file:
- access_level: open_access
  content_type: application/pdf
  creator: coffen
  date_created: 2022-06-28T15:25:50Z
  date_updated: 2022-06-28T15:25:50Z
  description: |-
    The principle of least action is one of the most fundamental physical principle. It says that among all possible motions
    connecting two points in a phase space, the system will exhibit those motions which extremise an action functional.
    Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equa-
    tions, are related to the existence of an action functional. Incorporating variational structure into learning algorithms
    for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features
    with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predic-
    tions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position
    data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no
    prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward
    error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the
    learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this,
    we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of
    variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position
    observations only, based on variational backward error analysis.
  file_id: '32274'
  file_name: ShadowLagrangian_revision1_journal_style_arxiv.pdf
  file_size: 3640770
  relation: main_file
  title: Variational Learning of Euler–Lagrange Dynamics from Data
file_date_updated: 2022-06-28T15:25:50Z
has_accepted_license: '1'
intvolume: '       421'
keyword:
- Lagrangian learning
- variational backward error analysis
- modified Lagrangian
- variational integrators
- physics informed learning
language:
- iso: eng
oa: '1'
page: '114780'
publication: Journal of Computational and Applied Mathematics
publication_identifier:
  issn:
  - 0377-0427
publication_status: epub_ahead
publisher: Elsevier
quality_controlled: '1'
related_material:
  link:
  - relation: software
    url: https://github.com/Christian-Offen/LagrangianShadowIntegration
status: public
title: Variational Learning of Euler–Lagrange Dynamics from Data
type: journal_article
user_id: '85279'
volume: 421
year: '2023'
...
---
_id: '34633'
article_number: '114118'
author:
- first_name: Kerstin
  full_name: Hesse, Kerstin
  id: '42608'
  last_name: Hesse
  orcid: 0000-0003-4125-1941
- first_name: Quoc Thong
  full_name: Le Gia, Quoc Thong
  last_name: Le Gia
citation:
  ama: Hesse K, Le Gia QT. L_2 error estimates for polynomial discrete penalized least-squares
    approximation on the sphere from noisy data. <i>Journal of Computational and Applied
    Mathematics</i>. 2022;408. doi:<a href="https://doi.org/10.1016/j.cam.2022.114118">10.1016/j.cam.2022.114118</a>
  apa: Hesse, K., &#38; Le Gia, Q. T. (2022). L_2 error estimates for polynomial discrete
    penalized least-squares approximation on the sphere from noisy data. <i>Journal
    of Computational and Applied Mathematics</i>, <i>408</i>, Article 114118. <a href="https://doi.org/10.1016/j.cam.2022.114118">https://doi.org/10.1016/j.cam.2022.114118</a>
  bibtex: '@article{Hesse_Le Gia_2022, title={L_2 error estimates for polynomial discrete
    penalized least-squares approximation on the sphere from noisy data}, volume={408},
    DOI={<a href="https://doi.org/10.1016/j.cam.2022.114118">10.1016/j.cam.2022.114118</a>},
    number={114118}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier
    BV}, author={Hesse, Kerstin and Le Gia, Quoc Thong}, year={2022} }'
  chicago: Hesse, Kerstin, and Quoc Thong Le Gia. “L_2 Error Estimates for Polynomial
    Discrete Penalized Least-Squares Approximation on the Sphere from Noisy Data.”
    <i>Journal of Computational and Applied Mathematics</i> 408 (2022). <a href="https://doi.org/10.1016/j.cam.2022.114118">https://doi.org/10.1016/j.cam.2022.114118</a>.
  ieee: 'K. Hesse and Q. T. Le Gia, “L_2 error estimates for polynomial discrete penalized
    least-squares approximation on the sphere from noisy data,” <i>Journal of Computational
    and Applied Mathematics</i>, vol. 408, Art. no. 114118, 2022, doi: <a href="https://doi.org/10.1016/j.cam.2022.114118">10.1016/j.cam.2022.114118</a>.'
  mla: Hesse, Kerstin, and Quoc Thong Le Gia. “L_2 Error Estimates for Polynomial
    Discrete Penalized Least-Squares Approximation on the Sphere from Noisy Data.”
    <i>Journal of Computational and Applied Mathematics</i>, vol. 408, 114118, Elsevier
    BV, 2022, doi:<a href="https://doi.org/10.1016/j.cam.2022.114118">10.1016/j.cam.2022.114118</a>.
  short: K. Hesse, Q.T. Le Gia, Journal of Computational and Applied Mathematics 408
    (2022).
date_created: 2022-12-20T17:37:16Z
date_updated: 2023-01-09T08:23:56Z
department:
- _id: '10'
doi: 10.1016/j.cam.2022.114118
intvolume: '       408'
keyword:
- Applied Mathematics
- Computational Mathematics
language:
- iso: eng
publication: Journal of Computational and Applied Mathematics
publication_identifier:
  issn:
  - 0377-0427
publication_status: published
publisher: Elsevier BV
status: public
title: L_2 error estimates for polynomial discrete penalized least-squares approximation
  on the sphere from noisy data
type: journal_article
user_id: '14931'
volume: 408
year: '2022'
...
---
_id: '34629'
article_number: '113061'
author:
- first_name: Kerstin
  full_name: Hesse, Kerstin
  id: '42608'
  last_name: Hesse
  orcid: 0000-0003-4125-1941
- first_name: Ian H.
  full_name: Sloan, Ian H.
  last_name: Sloan
- first_name: Robert S.
  full_name: Womersley, Robert S.
  last_name: Womersley
citation:
  ama: Hesse K, Sloan IH, Womersley RS. Local RBF-based penalized least-squares approximation
    on the sphere with noisy scattered data. <i>Journal of Computational and Applied
    Mathematics</i>. 2021;382. doi:<a href="https://doi.org/10.1016/j.cam.2020.113061">10.1016/j.cam.2020.113061</a>
  apa: Hesse, K., Sloan, I. H., &#38; Womersley, R. S. (2021). Local RBF-based penalized
    least-squares approximation on the sphere with noisy scattered data. <i>Journal
    of Computational and Applied Mathematics</i>, <i>382</i>, Article 113061. <a href="https://doi.org/10.1016/j.cam.2020.113061">https://doi.org/10.1016/j.cam.2020.113061</a>
  bibtex: '@article{Hesse_Sloan_Womersley_2021, title={Local RBF-based penalized least-squares
    approximation on the sphere with noisy scattered data}, volume={382}, DOI={<a
    href="https://doi.org/10.1016/j.cam.2020.113061">10.1016/j.cam.2020.113061</a>},
    number={113061}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier
    BV}, author={Hesse, Kerstin and Sloan, Ian H. and Womersley, Robert S.}, year={2021}
    }'
  chicago: Hesse, Kerstin, Ian H. Sloan, and Robert S. Womersley. “Local RBF-Based
    Penalized Least-Squares Approximation on the Sphere with Noisy Scattered Data.”
    <i>Journal of Computational and Applied Mathematics</i> 382 (2021). <a href="https://doi.org/10.1016/j.cam.2020.113061">https://doi.org/10.1016/j.cam.2020.113061</a>.
  ieee: 'K. Hesse, I. H. Sloan, and R. S. Womersley, “Local RBF-based penalized least-squares
    approximation on the sphere with noisy scattered data,” <i>Journal of Computational
    and Applied Mathematics</i>, vol. 382, Art. no. 113061, 2021, doi: <a href="https://doi.org/10.1016/j.cam.2020.113061">10.1016/j.cam.2020.113061</a>.'
  mla: Hesse, Kerstin, et al. “Local RBF-Based Penalized Least-Squares Approximation
    on the Sphere with Noisy Scattered Data.” <i>Journal of Computational and Applied
    Mathematics</i>, vol. 382, 113061, Elsevier BV, 2021, doi:<a href="https://doi.org/10.1016/j.cam.2020.113061">10.1016/j.cam.2020.113061</a>.
  short: K. Hesse, I.H. Sloan, R.S. Womersley, Journal of Computational and Applied
    Mathematics 382 (2021).
date_created: 2022-12-20T17:26:44Z
date_updated: 2024-04-03T11:04:23Z
department:
- _id: '10'
doi: 10.1016/j.cam.2020.113061
intvolume: '       382'
keyword:
- Applied Mathematics
- Computational Mathematics
language:
- iso: eng
publication: Journal of Computational and Applied Mathematics
publication_identifier:
  issn:
  - 0377-0427
publication_status: published
publisher: Elsevier BV
status: public
title: Local RBF-based penalized least-squares approximation on the sphere with noisy
  scattered data
type: journal_article
user_id: '49063'
volume: 382
year: '2021'
...
---
_id: '16624'
author:
- first_name: Stefan
  full_name: Klus, Stefan
  last_name: Klus
- first_name: Tuhin
  full_name: Sahai, Tuhin
  last_name: Sahai
- first_name: Cong
  full_name: Liu, Cong
  last_name: Liu
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
citation:
  ama: Klus S, Sahai T, Liu C, Dellnitz M. An efficient algorithm for the parallel
    solution of high-dimensional differential equations. <i>Journal of Computational
    and Applied Mathematics</i>. 2011:3053-3062. doi:<a href="https://doi.org/10.1016/j.cam.2010.12.026">10.1016/j.cam.2010.12.026</a>
  apa: Klus, S., Sahai, T., Liu, C., &#38; Dellnitz, M. (2011). An efficient algorithm
    for the parallel solution of high-dimensional differential equations. <i>Journal
    of Computational and Applied Mathematics</i>, 3053–3062. <a href="https://doi.org/10.1016/j.cam.2010.12.026">https://doi.org/10.1016/j.cam.2010.12.026</a>
  bibtex: '@article{Klus_Sahai_Liu_Dellnitz_2011, title={An efficient algorithm for
    the parallel solution of high-dimensional differential equations}, DOI={<a href="https://doi.org/10.1016/j.cam.2010.12.026">10.1016/j.cam.2010.12.026</a>},
    journal={Journal of Computational and Applied Mathematics}, author={Klus, Stefan
    and Sahai, Tuhin and Liu, Cong and Dellnitz, Michael}, year={2011}, pages={3053–3062}
    }'
  chicago: Klus, Stefan, Tuhin Sahai, Cong Liu, and Michael Dellnitz. “An Efficient
    Algorithm for the Parallel Solution of High-Dimensional Differential Equations.”
    <i>Journal of Computational and Applied Mathematics</i>, 2011, 3053–62. <a href="https://doi.org/10.1016/j.cam.2010.12.026">https://doi.org/10.1016/j.cam.2010.12.026</a>.
  ieee: S. Klus, T. Sahai, C. Liu, and M. Dellnitz, “An efficient algorithm for the
    parallel solution of high-dimensional differential equations,” <i>Journal of Computational
    and Applied Mathematics</i>, pp. 3053–3062, 2011.
  mla: Klus, Stefan, et al. “An Efficient Algorithm for the Parallel Solution of High-Dimensional
    Differential Equations.” <i>Journal of Computational and Applied Mathematics</i>,
    2011, pp. 3053–62, doi:<a href="https://doi.org/10.1016/j.cam.2010.12.026">10.1016/j.cam.2010.12.026</a>.
  short: S. Klus, T. Sahai, C. Liu, M. Dellnitz, Journal of Computational and Applied
    Mathematics (2011) 3053–3062.
date_created: 2020-04-16T08:18:33Z
date_updated: 2022-01-06T06:52:53Z
department:
- _id: '101'
doi: 10.1016/j.cam.2010.12.026
language:
- iso: eng
page: 3053-3062
publication: Journal of Computational and Applied Mathematics
publication_identifier:
  issn:
  - 0377-0427
publication_status: published
status: public
title: An efficient algorithm for the parallel solution of high-dimensional differential
  equations
type: journal_article
user_id: '15701'
year: '2011'
...
---
_id: '17023'
author:
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
- first_name: Oliver
  full_name: Schütze, Oliver
  last_name: Schütze
- first_name: Qinghua
  full_name: Zheng, Qinghua
  last_name: Zheng
citation:
  ama: Dellnitz M, Schütze O, Zheng Q. Locating all the zeros of an analytic function
    in one complex variable. <i>Journal of Computational and Applied Mathematics</i>.
    2002:325-333. doi:<a href="https://doi.org/10.1016/s0377-0427(01)00371-5">10.1016/s0377-0427(01)00371-5</a>
  apa: Dellnitz, M., Schütze, O., &#38; Zheng, Q. (2002). Locating all the zeros of
    an analytic function in one complex variable. <i>Journal of Computational and
    Applied Mathematics</i>, 325–333. <a href="https://doi.org/10.1016/s0377-0427(01)00371-5">https://doi.org/10.1016/s0377-0427(01)00371-5</a>
  bibtex: '@article{Dellnitz_Schütze_Zheng_2002, title={Locating all the zeros of
    an analytic function in one complex variable}, DOI={<a href="https://doi.org/10.1016/s0377-0427(01)00371-5">10.1016/s0377-0427(01)00371-5</a>},
    journal={Journal of Computational and Applied Mathematics}, author={Dellnitz,
    Michael and Schütze, Oliver and Zheng, Qinghua}, year={2002}, pages={325–333}
    }'
  chicago: Dellnitz, Michael, Oliver Schütze, and Qinghua Zheng. “Locating All the
    Zeros of an Analytic Function in One Complex Variable.” <i>Journal of Computational
    and Applied Mathematics</i>, 2002, 325–33. <a href="https://doi.org/10.1016/s0377-0427(01)00371-5">https://doi.org/10.1016/s0377-0427(01)00371-5</a>.
  ieee: M. Dellnitz, O. Schütze, and Q. Zheng, “Locating all the zeros of an analytic
    function in one complex variable,” <i>Journal of Computational and Applied Mathematics</i>,
    pp. 325–333, 2002.
  mla: Dellnitz, Michael, et al. “Locating All the Zeros of an Analytic Function in
    One Complex Variable.” <i>Journal of Computational and Applied Mathematics</i>,
    2002, pp. 325–33, doi:<a href="https://doi.org/10.1016/s0377-0427(01)00371-5">10.1016/s0377-0427(01)00371-5</a>.
  short: M. Dellnitz, O. Schütze, Q. Zheng, Journal of Computational and Applied Mathematics
    (2002) 325–333.
date_created: 2020-05-19T20:05:37Z
date_updated: 2022-01-06T06:53:02Z
doi: 10.1016/s0377-0427(01)00371-5
language:
- iso: eng
page: 325-333
publication: Journal of Computational and Applied Mathematics
publication_identifier:
  issn:
  - 0377-0427
publication_status: published
status: public
title: Locating all the zeros of an analytic function in one complex variable
type: journal_article
user_id: '32829'
year: '2002'
...
---
_id: '40197'
author:
- first_name: Margit
  full_name: Rösler, Margit
  id: '37390'
  last_name: Rösler
- first_name: Michael
  full_name: Voit, Michael
  last_name: Voit
citation:
  ama: Rösler M, Voit M. Biorthogonal polynomials associated with reflection groups
    and a formula of Macdonald. <i>Journal of Computational and Applied Mathematics</i>.
    1998;99(1-2):337-351. doi:<a href="https://doi.org/10.1016/s0377-0427(98)00168-x">10.1016/s0377-0427(98)00168-x</a>
  apa: Rösler, M., &#38; Voit, M. (1998). Biorthogonal polynomials associated with
    reflection groups and a formula of Macdonald. <i>Journal of Computational and
    Applied Mathematics</i>, <i>99</i>(1–2), 337–351. <a href="https://doi.org/10.1016/s0377-0427(98)00168-x">https://doi.org/10.1016/s0377-0427(98)00168-x</a>
  bibtex: '@article{Rösler_Voit_1998, title={Biorthogonal polynomials associated with
    reflection groups and a formula of Macdonald}, volume={99}, DOI={<a href="https://doi.org/10.1016/s0377-0427(98)00168-x">10.1016/s0377-0427(98)00168-x</a>},
    number={1–2}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier
    BV}, author={Rösler, Margit and Voit, Michael}, year={1998}, pages={337–351} }'
  chicago: 'Rösler, Margit, and Michael Voit. “Biorthogonal Polynomials Associated
    with Reflection Groups and a Formula of Macdonald.” <i>Journal of Computational
    and Applied Mathematics</i> 99, no. 1–2 (1998): 337–51. <a href="https://doi.org/10.1016/s0377-0427(98)00168-x">https://doi.org/10.1016/s0377-0427(98)00168-x</a>.'
  ieee: 'M. Rösler and M. Voit, “Biorthogonal polynomials associated with reflection
    groups and a formula of Macdonald,” <i>Journal of Computational and Applied Mathematics</i>,
    vol. 99, no. 1–2, pp. 337–351, 1998, doi: <a href="https://doi.org/10.1016/s0377-0427(98)00168-x">10.1016/s0377-0427(98)00168-x</a>.'
  mla: Rösler, Margit, and Michael Voit. “Biorthogonal Polynomials Associated with
    Reflection Groups and a Formula of Macdonald.” <i>Journal of Computational and
    Applied Mathematics</i>, vol. 99, no. 1–2, Elsevier BV, 1998, pp. 337–51, doi:<a
    href="https://doi.org/10.1016/s0377-0427(98)00168-x">10.1016/s0377-0427(98)00168-x</a>.
  short: M. Rösler, M. Voit, Journal of Computational and Applied Mathematics 99 (1998)
    337–351.
date_created: 2023-01-26T08:31:16Z
date_updated: 2023-01-26T17:41:01Z
department:
- _id: '555'
doi: 10.1016/s0377-0427(98)00168-x
extern: '1'
intvolume: '        99'
issue: 1-2
keyword:
- Applied Mathematics
- Computational Mathematics
language:
- iso: eng
page: 337-351
publication: Journal of Computational and Applied Mathematics
publication_identifier:
  issn:
  - 0377-0427
publication_status: published
publisher: Elsevier BV
status: public
title: Biorthogonal polynomials associated with reflection groups and a formula of
  Macdonald
type: journal_article
user_id: '93826'
volume: 99
year: '1998'
...
---
_id: '40207'
author:
- first_name: Margit
  full_name: Rösler, Margit
  id: '37390'
  last_name: Rösler
citation:
  ama: Rösler M. Trigonometric convolution structures on Z derived from Jacobi polynomials.
    <i>Journal of Computational and Applied Mathematics</i>. 1995;65(1-3):357-368.
    doi:<a href="https://doi.org/10.1016/0377-0427(95)00122-0">10.1016/0377-0427(95)00122-0</a>
  apa: Rösler, M. (1995). Trigonometric convolution structures on Z derived from Jacobi
    polynomials. <i>Journal of Computational and Applied Mathematics</i>, <i>65</i>(1–3),
    357–368. <a href="https://doi.org/10.1016/0377-0427(95)00122-0">https://doi.org/10.1016/0377-0427(95)00122-0</a>
  bibtex: '@article{Rösler_1995, title={Trigonometric convolution structures on Z
    derived from Jacobi polynomials}, volume={65}, DOI={<a href="https://doi.org/10.1016/0377-0427(95)00122-0">10.1016/0377-0427(95)00122-0</a>},
    number={1–3}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier
    BV}, author={Rösler, Margit}, year={1995}, pages={357–368} }'
  chicago: 'Rösler, Margit. “Trigonometric Convolution Structures on Z Derived from
    Jacobi Polynomials.” <i>Journal of Computational and Applied Mathematics</i> 65,
    no. 1–3 (1995): 357–68. <a href="https://doi.org/10.1016/0377-0427(95)00122-0">https://doi.org/10.1016/0377-0427(95)00122-0</a>.'
  ieee: 'M. Rösler, “Trigonometric convolution structures on Z derived from Jacobi
    polynomials,” <i>Journal of Computational and Applied Mathematics</i>, vol. 65,
    no. 1–3, pp. 357–368, 1995, doi: <a href="https://doi.org/10.1016/0377-0427(95)00122-0">10.1016/0377-0427(95)00122-0</a>.'
  mla: Rösler, Margit. “Trigonometric Convolution Structures on Z Derived from Jacobi
    Polynomials.” <i>Journal of Computational and Applied Mathematics</i>, vol. 65,
    no. 1–3, Elsevier BV, 1995, pp. 357–68, doi:<a href="https://doi.org/10.1016/0377-0427(95)00122-0">10.1016/0377-0427(95)00122-0</a>.
  short: M. Rösler, Journal of Computational and Applied Mathematics 65 (1995) 357–368.
date_created: 2023-01-26T08:42:19Z
date_updated: 2023-01-26T17:43:10Z
department:
- _id: '555'
doi: 10.1016/0377-0427(95)00122-0
extern: '1'
intvolume: '        65'
issue: 1-3
keyword:
- Applied Mathematics
- Computational Mathematics
language:
- iso: eng
page: 357-368
publication: Journal of Computational and Applied Mathematics
publication_identifier:
  issn:
  - 0377-0427
publication_status: published
publisher: Elsevier BV
status: public
title: Trigonometric convolution structures on Z derived from Jacobi polynomials
type: journal_article
user_id: '93826'
volume: 65
year: '1995'
...
---
_id: '16541'
author:
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
- first_name: Ian
  full_name: Melbourne, Ian
  last_name: Melbourne
citation:
  ama: Dellnitz M, Melbourne I. Generic movement of eigenvalues for equivariant self-adjoint
    matrices. <i>Journal of Computational and Applied Mathematics</i>. 1994:249-259.
    doi:<a href="https://doi.org/10.1016/0377-0427(94)90032-9">10.1016/0377-0427(94)90032-9</a>
  apa: Dellnitz, M., &#38; Melbourne, I. (1994). Generic movement of eigenvalues for
    equivariant self-adjoint matrices. <i>Journal of Computational and Applied Mathematics</i>,
    249–259. <a href="https://doi.org/10.1016/0377-0427(94)90032-9">https://doi.org/10.1016/0377-0427(94)90032-9</a>
  bibtex: '@article{Dellnitz_Melbourne_1994, title={Generic movement of eigenvalues
    for equivariant self-adjoint matrices}, DOI={<a href="https://doi.org/10.1016/0377-0427(94)90032-9">10.1016/0377-0427(94)90032-9</a>},
    journal={Journal of Computational and Applied Mathematics}, author={Dellnitz,
    Michael and Melbourne, Ian}, year={1994}, pages={249–259} }'
  chicago: Dellnitz, Michael, and Ian Melbourne. “Generic Movement of Eigenvalues
    for Equivariant Self-Adjoint Matrices.” <i>Journal of Computational and Applied
    Mathematics</i>, 1994, 249–59. <a href="https://doi.org/10.1016/0377-0427(94)90032-9">https://doi.org/10.1016/0377-0427(94)90032-9</a>.
  ieee: M. Dellnitz and I. Melbourne, “Generic movement of eigenvalues for equivariant
    self-adjoint matrices,” <i>Journal of Computational and Applied Mathematics</i>,
    pp. 249–259, 1994.
  mla: Dellnitz, Michael, and Ian Melbourne. “Generic Movement of Eigenvalues for
    Equivariant Self-Adjoint Matrices.” <i>Journal of Computational and Applied Mathematics</i>,
    1994, pp. 249–59, doi:<a href="https://doi.org/10.1016/0377-0427(94)90032-9">10.1016/0377-0427(94)90032-9</a>.
  short: M. Dellnitz, I. Melbourne, Journal of Computational and Applied Mathematics
    (1994) 249–259.
date_created: 2020-04-15T08:44:12Z
date_updated: 2022-01-06T06:52:52Z
department:
- _id: '101'
doi: 10.1016/0377-0427(94)90032-9
language:
- iso: eng
page: 249-259
publication: Journal of Computational and Applied Mathematics
publication_identifier:
  issn:
  - 0377-0427
publication_status: published
status: public
title: Generic movement of eigenvalues for equivariant self-adjoint matrices
type: journal_article
user_id: '15701'
year: '1994'
...
---
_id: '16682'
author:
- first_name: Michael
  full_name: Dellnitz, Michael
  last_name: Dellnitz
- first_name: Bodo
  full_name: Werner, Bodo
  last_name: Werner
citation:
  ama: Dellnitz M, Werner B. Computational methods for bifurcation problems with symmetries—with
    special attention to steady state and Hopf bifurcation points. <i>Journal of Computational
    and Applied Mathematics</i>. 1989:97-123. doi:<a href="https://doi.org/10.1016/0377-0427(89)90150-7">10.1016/0377-0427(89)90150-7</a>
  apa: Dellnitz, M., &#38; Werner, B. (1989). Computational methods for bifurcation
    problems with symmetries—with special attention to steady state and Hopf bifurcation
    points. <i>Journal of Computational and Applied Mathematics</i>, 97–123. <a href="https://doi.org/10.1016/0377-0427(89)90150-7">https://doi.org/10.1016/0377-0427(89)90150-7</a>
  bibtex: '@article{Dellnitz_Werner_1989, title={Computational methods for bifurcation
    problems with symmetries—with special attention to steady state and Hopf bifurcation
    points}, DOI={<a href="https://doi.org/10.1016/0377-0427(89)90150-7">10.1016/0377-0427(89)90150-7</a>},
    journal={Journal of Computational and Applied Mathematics}, author={Dellnitz,
    Michael and Werner, Bodo}, year={1989}, pages={97–123} }'
  chicago: Dellnitz, Michael, and Bodo Werner. “Computational Methods for Bifurcation
    Problems with Symmetries—with Special Attention to Steady State and Hopf Bifurcation
    Points.” <i>Journal of Computational and Applied Mathematics</i>, 1989, 97–123.
    <a href="https://doi.org/10.1016/0377-0427(89)90150-7">https://doi.org/10.1016/0377-0427(89)90150-7</a>.
  ieee: M. Dellnitz and B. Werner, “Computational methods for bifurcation problems
    with symmetries—with special attention to steady state and Hopf bifurcation points,”
    <i>Journal of Computational and Applied Mathematics</i>, pp. 97–123, 1989.
  mla: Dellnitz, Michael, and Bodo Werner. “Computational Methods for Bifurcation
    Problems with Symmetries—with Special Attention to Steady State and Hopf Bifurcation
    Points.” <i>Journal of Computational and Applied Mathematics</i>, 1989, pp. 97–123,
    doi:<a href="https://doi.org/10.1016/0377-0427(89)90150-7">10.1016/0377-0427(89)90150-7</a>.
  short: M. Dellnitz, B. Werner, Journal of Computational and Applied Mathematics
    (1989) 97–123.
date_created: 2020-04-16T10:28:09Z
date_updated: 2022-01-06T06:52:54Z
department:
- _id: '101'
doi: 10.1016/0377-0427(89)90150-7
language:
- iso: eng
page: 97-123
publication: Journal of Computational and Applied Mathematics
publication_identifier:
  issn:
  - 0377-0427
publication_status: published
status: public
title: Computational methods for bifurcation problems with symmetries—with special
  attention to steady state and Hopf bifurcation points
type: journal_article
user_id: '15701'
year: '1989'
...