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Matrix algorithms for solving (in)homogeneous bound state equations.

Blank M, Krassnigg A - Comput Phys Commun (2011)

Bottom Line: In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input.These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case).This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.

View Article: PubMed Central - PubMed

Affiliation: Institut für Physik, Universität Graz, Universitätsplatz 5, 8010 Graz, Austria.

ABSTRACT
In the functional approach to quantum chromodynamics, the properties of hadronic bound states are accessible via covariant integral equations, e.g. the Bethe-Salpeter equation for mesons. In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input. Analogously, inhomogeneous equations can be constructed to obtain off-shell information in addition to bound-state masses and other properties obtained from the covariant analogue to a wave function of the bound state. These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case). We demonstrate this by solving the homogeneous and inhomogeneous Bethe-Salpeter equations and find, e.g. that for the calculation of the mass spectrum it is as efficient or even advantageous to use the inhomogeneous equation as compared to the homogeneous. This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.

No MeSH data available.


Related in: MedlinePlus

The number of matrix–vector multiplications needed for convergence of the inhomogeneous BSE using BiCGstab (solid line) compared to the calculations of one eigenvalue (dash-dotted line) and three eigenvalues (dashed line) with the Arnoldi factorization. The vertical lines mark the ground and excited state in the  channel at  and .
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fg0110: The number of matrix–vector multiplications needed for convergence of the inhomogeneous BSE using BiCGstab (solid line) compared to the calculations of one eigenvalue (dash-dotted line) and three eigenvalues (dashed line) with the Arnoldi factorization. The vertical lines mark the ground and excited state in the channel at and .

Mentions: Still, it is interesting to investigate the convergence of the inhomogeneous equation when compared to the homogeneous, in analogy to Fig. 10. The results for the channel are presented in Fig. 11. For the excited state, the advantages of the inhomogeneous equation are even more pronounced than for the pseudoscalar channel, whereas for the ground state the two equations are almost equivalent in terms of efficiency.


Matrix algorithms for solving (in)homogeneous bound state equations.

Blank M, Krassnigg A - Comput Phys Commun (2011)

The number of matrix–vector multiplications needed for convergence of the inhomogeneous BSE using BiCGstab (solid line) compared to the calculations of one eigenvalue (dash-dotted line) and three eigenvalues (dashed line) with the Arnoldi factorization. The vertical lines mark the ground and excited state in the  channel at  and .
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3095096&req=5

fg0110: The number of matrix–vector multiplications needed for convergence of the inhomogeneous BSE using BiCGstab (solid line) compared to the calculations of one eigenvalue (dash-dotted line) and three eigenvalues (dashed line) with the Arnoldi factorization. The vertical lines mark the ground and excited state in the channel at and .
Mentions: Still, it is interesting to investigate the convergence of the inhomogeneous equation when compared to the homogeneous, in analogy to Fig. 10. The results for the channel are presented in Fig. 11. For the excited state, the advantages of the inhomogeneous equation are even more pronounced than for the pseudoscalar channel, whereas for the ground state the two equations are almost equivalent in terms of efficiency.

Bottom Line: In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input.These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case).This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.

View Article: PubMed Central - PubMed

Affiliation: Institut für Physik, Universität Graz, Universitätsplatz 5, 8010 Graz, Austria.

ABSTRACT
In the functional approach to quantum chromodynamics, the properties of hadronic bound states are accessible via covariant integral equations, e.g. the Bethe-Salpeter equation for mesons. In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input. Analogously, inhomogeneous equations can be constructed to obtain off-shell information in addition to bound-state masses and other properties obtained from the covariant analogue to a wave function of the bound state. These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case). We demonstrate this by solving the homogeneous and inhomogeneous Bethe-Salpeter equations and find, e.g. that for the calculation of the mass spectrum it is as efficient or even advantageous to use the inhomogeneous equation as compared to the homogeneous. This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.

No MeSH data available.


Related in: MedlinePlus