Publikationsliste

A non-iterative domain decomposition time integrator for linear wave equations
Buchholz, T.; Hochbruck, M.
2025. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000183505  

Time-integration of Gaussian variational approximation for the magnetic Schrödinger equation
Scheifinger, M.; Busch, K.; Hochbruck, M.; Lasser, C.
2025. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000181084  

Local time-integration for Friedrich’s systems
Hochbruck, M.; Scheifinger, M.
2025. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000179811  

Error analysis of DGTD for linear Maxwell equations with inhomogeneous interface conditions
Dörich, B.; Dörner, J.; Hochbruck, M.
2024. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000173144  

Variational Gaussian approximation for the magnetic Schrödinger equation *
Burkhard, S.; Dörich, B.; Hochbruck, M.; Lasser, C.
2024. Journal of Physics A: Mathematical and Theoretical, 57 (29), 295202. doi:10.1088/1751-8121/ad591e  

Error analysis of an implicit-explicit time discretization scheme for semilinear wave equations with application to multiscale problems
Eckhardt, D.; Hochbruck, M.; Verfürth, B.
2024. Karlsruher Institut für Technologie (KIT). doi:10.48550/arXiv.2406.09889  

Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations
Carle, C.; Hochbruck, M.
2024. Mathematics of Computation, 93 (350), 2611 – 2641. doi:10.1090/mcom/3952 

Variational Gaussian approximation for the magnetic Schrödinger equation
Burkhard, S.; Dörich, B.; Hochbruck, M.; Lasser, C.
2023. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000154431/v2  

Dynamical low-rank integrators for second-order matrix differential equations
Hochbruck, M.; Neher, M.; Schrammer, S.
2023. BIT Numerical Mathematics, 63 (1), Art.-Nr.: 4. doi:10.1007/s10543-023-00941-7  

Rank-adaptive dynamical low-rank integrators for first-order and second-order matrix differential equations
Hochbruck, M.; Neher, M.; Schrammer, S.
2023. BIT Numerical Mathematics, 63 (1), Art.-Nr.: 9. doi:10.1007/s10543-023-00942-6  

Error analysis of second-order locally implicit and local time-stepping methods for discontinuous Galerkin discretizations of linear wave equations
Carle, C.; Hochbruck, M.
2023. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000154254  

Error Analysis of Multirate Leapfrog-Type Methods for Second-Order Semilinear Odes
Carle, C.; Hochbruck, M.
2022. SIAM Journal on Numerical Analysis, 60 (5), 2897–2924. doi:10.1137/21M1427255 

Error analysis for space discretizations of quasilinear wave-type equations
Hochbruck, M.; Maier, B.
2022. IMA Journal of Numerical Analysis, 42 (3), 1963–1990. doi:10.1093/imanum/drab073 

Exponential Integrators for Quasilinear Wave-Type Equations
Dörich, B.; Hochbruck, M.
2022. SIAM Journal on Numerical Analysis, 60 (3), 1472–1493. doi:10.1137/21M1410579 

Preconditioned implicit time integration schemes for Maxwell’s equations on locally refined grids
Hochbruck, M.; Köhler, J.; Kumbhar, P. M.
2022. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000148078  

Rank-adaptive dynamical low-rank integrators for first-order and second-order matrix differential equations
Hochbruck, M.; Neher, M.; Schrammer, S.
2022. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000143198  

Dynamical low-rank integrators for second-order matrix differential equations
Hochbruck, M.; Neher, M.; Schrammer, S.
2022. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000143003  

Error analysis of a fully discrete discontinuous Galerkin alternating direction implicit discretization of a class of linear wave-type problems
Hochbruck, M.; Köhler, J.
2022. Numerische Mathematik, 150 (3), 893–927. doi:10.1007/s00211-021-01262-z  

Error analysis of multirate leapfrog-type methods for second-order semilinear ODEs
Carle, C.; Hochbruck, M.
2021. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000133957  

On averaged exponential integrators for semilinear wave equations with solutions of low-regularity
Buchholz, S.; Dörich, B.; Hochbruck, M.
2021. SN Partial Differential Equations and Applications, 2 (2), Art.-Nr.: 23. doi:10.1007/s42985-020-00045-9  

An implicit–explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions
Hochbruck, M.; Leibold, J.
2021. Numerische Mathematik, 147 (4), 869–899. doi:10.1007/s00211-021-01184-w  

Correction to: An implicit–explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions
Hochbruck, M.; Leibold, J.
2021. Numerische Mathematik, 147 (4), Art. Nr.: 901. doi:10.1007/s00211-021-01190-y 

Exponential integrators for quasilinear wave-type equations
Dörich, B.; Hochbruck, M.
2021. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000131210  

Error analysis for space discretizations of quasilinear wave-type equations
Hochbruck, M.; Maier, B.
2021. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000128952  

Error analysis of a fully discrete discontinuous Galerkin alternating direction implicit discretization of a class of linear wave-type problems
Hochbruck, M.; Köhler, J.
2020. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000127911  

An implicit-explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions
Hochbruck, M.; Leibold, J.
2020. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000122353  

On the convergence of Lawson methods for semilinear stiff problems
Hochbruck, M.; Leibold, J.; Ostermann, A.
2020. Numerische Mathematik, 145 (3), 553–580. doi:10.1007/s00211-020-01120-4  

On Leapfrog-Chebyshev Schemes
Carle, C.; Hochbruck, M.; Sturm, A.
2020. SIAM journal on numerical analysis, 58 (4), 2404–2433. doi:10.1137/18M1209453 

Finite element discretization of semilinear acoustic wave equations with kinetic boundary conditions
Hochbruck, M.; Leibold, J.
2020. Electronic transactions on numerical analysis, 53, 522–540. doi:10.1553/ETNA_VOL53S522 

On averaged exponential integrators for semilinear wave equations with solutions of low-regularity
Buchholz, S.; Dörich, B.; Hochbruck, M.
2020. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000117802  

Unified error analysis for nonconforming space discretizations of wave-type equations
Hipp, D.; Hochbruck, M.; Stohrer, C.
2019. IMA journal of numerical analysis, 39 (3), 1206–1245. doi:10.1093/imanum/dry036 

Heterogeneous Multiscale Method for Maxwell’s Equations
Hochbruck, M.; Maier, B.; Stohrer, C.
2019. Multiscale modeling & simulation, 17 (4), 1147–1171. doi:10.1137/18M1234072 

Finite element discretization of semilinear acoustic wave : equations with kinetic boundary conditions
Hochbruck, M.; Leibold, J.
2019. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000105549  

On the convergence of Lawson methods for semilinear stiff problems
Hochbruck, M.; Leibold, J.; Ostermann, A.
2019. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000099482  

On leapfrog-Chebyshev schemes
Carle, C.; Hochbruck, M.; Sturm, A.
2019. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000099118  

Error analysis of discontinuous Galerkin discretizations of a class of linear wave-type problems
Hochbruck, M.; Köhler, J.
2019. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000098066  

Analytical and numerical analysis of linear and nonlinear properties of an rf-SQUID based metasurface
Müller, M. M.; Maier, B.; Rockstuhl, C.; Hochbruck, M.
2019. Physical review / B, 99 (7), Art.-Nr.: 075401. doi:10.1103/PhysRevB.99.075401 

On the efficiency of the Peaceman–Rachford ADI-dG method for wave-type problems
Hochbruck, M.; Köhler, J.
2019. European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2017; Voss; Norway; 25 September 2017 through 29 September 2017. Ed.: F.A. Radu, 135–144, Springer. doi:10.1007/978-3-319-96415-7_10 

Upwind discontinuous Galerkin space discretization and locally implicit time integration for linear Maxwell’s equations
Hochbruck, M.; Sturm, A.
2018. Mathematics of computation, 88 (317), 1121–1153. doi:10.1090/MCOM/3365 

Error analysis of implicit Runge–Kutta methods for quasilinear hyperbolic evolution equations
Hochbruck, M.; Pažur, T.; Schnaubelt, R.
2018. Numerische Mathematik, 138 (3), 557–579. doi:10.1007/s00211-017-0914-6 

Heterogeneous multiscale method for Maxwell’s equations
Hochbruck, M.; Maier, B.; Stohrer, C.
2018. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000088934  

Analytical and numerical analysis of linear and nonlinear properties of an rf-SQUID based metasurface
Müller, M. M.; Maier, B.; Rockstuhl, C.; Hochbruck, M.
2018. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000088565  

On leap-frog-Chebyshev schemes
Hochbruck, M.; Sturm, A.
2018. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000085527  

Closing the gap between trigonometric integrators and splitting methods for highly oscillatory differential equations
Buchholz, S.; Gauckler, L.; Grimm, V.; Hochbruck, M.; Jahnke, T.
2018. IMA journal of numerical analysis, 38 (1), 57–74. doi:10.1093/imanum/drx007 

Error analysis of implicit Euler methods for quasilinear hyperbolic evolution equations
Hochbruck, M.; Pažur, T.
2017. Numerische Mathematik, 135 (2), 547–569. doi:10.1007/s00211-016-0810-5 

Error analysis of non-conforming FE methods for wave-type problems and its application to heterogeneous multiscale methods
Hipp, D.; Hochbruck, M.; Stohrer, C.
2017. ICTCA 2017 Vienna : 13th International Conference on Theoretical and Computational Acoustics : book of abstracts : 30. Juli-03. August 2017. Ed.: P. Borejko, 130, TU 

On the efficiency of the Peaceman-Rachford ADI-dG method forwave-type methods
Hochbruck, M.; Köhler, J.
2017. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000078144  

Finite Element Heterogeneous Multiscale Method for Time-Dependent Maxwell’s Equations
Hochbruck, M.; Stohrer, C.
2017. Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016 : Selected Papers from the ICOSAHOM conference, Rio de Janeiro, Brazil, 27th June - 1st July , 2016. Ed.: M. L. Bittencourt, 269–281, Springer. doi:10.1007/978-3-319-65870-4_18 

Unified error analysis for non-conforming space discretizations ofwave-type equations
Hipp, D.; Hochbruck, M.; Stohrer, C.
2017. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000076637  

Upwind discontinuous Galerkin space discretization and locally implicit time integration for linear Maxwell’s equations
Hochbruck, M.; Sturm, A.
2017. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000070153  

On the convergence of Lawson methods for semilinear stiff problems
Hochbruck, M.; Ostermann, A.
2017. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000068916  

Error analysis of implicit Runge-Kutta methods for quasilinear hyperbolic evolution equations
Hochbruck, M.; Pažur, T.; Schnaubelt, R.
2017. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000067352  

Upwind discontinuous Galerkin space discretization and locally implicit time integration for linear Maxwell’s equations
Hochbruck, M.; Sturm, A.
2017. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000067269  

A fast mollified impulse method for biomolecular atomistic simulations
Fath, L.; Hochbruck, M.; Singh, C. V.
2017. Journal of computational physics, 333, 180–198. doi:10.1016/j.jcp.2016.12.024 

Error Analysis of a Second-Order Locally Implicit Method for Linear Maxwell’s Equations
Hochbruck, M.; Sturm, A.
2016. SIAM journal on numerical analysis, 54 (5), 3167–3191. doi:10.1137/15M1038037 

Finite element heterogeneous multiscale method for time-dependent Maxwell’s equation
Hochbruck, M.; Stohrer, C.
2016. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000060075  

Convergence of viscoelastic constraints to nonholonomic idealization
Deppler, J.; Braun, B.; Fidlin, A. Y.; Hochbruck, M.
2016. European Journal of Mechanics / A: Solids, 58, 140–147. doi:10.1016/j.euromechsol.2016.01.003 

Error analysis of implicit Euler methods for quasilinear hyperbolic evolution systems
Hochbruck, M.; Pažur, T.
2016. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000053752  

Error analysis of a second order locally implicit method for linear Maxwell’s equations
Hochbruck, M.; Sturm, A.
2015. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000049414  

Efficient time integration for discontinuous Galerkin approximations of linear wave equations
Hochbruck, M.; Pazur, T.; Schulz, A.; Thawinan, E.; Wieners, C.
2015. ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik, 95 (3), 237–259. doi:10.1002/zamm.201300306 

Implicit Runge-Kutta methods and discontinuous Galerkin discretizations for linear Maxwell’s equations
Hochbruck, M.; Pazur, T.
2015. SIAM journal on numerical analysis, 53 (1), 485–507. doi:10.1137/130944114 

Efficient multiple time-stepping algorithms of higher order
Demirel, A.; Niegemann, J.; Busch, K.; Hochbruck, M.
2015. Journal of Computational Physics, 285, 133–148. doi:10.1016/j.jcp.2015.01.018 

A preconditioned Krylov method for an exponential integrator for non-autonomous parabolic systems
Hipp, D.; Hochbruck, M.
2014. Oberwolfach reports, 11, 822–824 

An exponential integrator for non-autonomous parabolic problems
Hipp, D.; Hochbruck, M.; Ostermann, A.
2014. Electronic transactions on numerical analysis, 41, 497–511 

Convergence of an ADI splitting for Maxwell’s equations
Hochbruck, M.; Jahnke, T.; Schnaubelt, R.
2014. Numerische Mathematik, 129 (3), 535–561. doi:10.1007/s00211-014-0642-0  

Residual, restarting and Richardson iteration for the matrix exponential
Botchev, M. A.; Grimm, V.; Hochbruck, M.
2013. SIAM Journal on Scientific Computing, 35 (3), A1376-A1397. doi:10.1137/110820191  

Residual, Restarting and Richardson Iteration for the Matrix Exponential
Botchev, M. A.; Grimm, V.; Hochbruck, M.
2012. Karlsruher Institut für Technologie (KIT). doi:10.5445/IR/1000031191  

Exponential multistep methods of Adams-type
Hochbruck, M.; Ostermann, A.
2011. BIT Numerical Mathematics, 51 (4), 889–908. doi:10.1007/s10543-011-0332-6  

Three-dimensional relativistic particle-in-cell hybrid code based on an exponential integrator
Tückmantel, T.; Pukhov, A.; Liljo, J.; Hochbruck, M.
2010. IEEE Transactions on Plasma Science, 38 (9), 2383–2389. doi:10.1109/TPS.2010.2056706  

On the convergence of a regularizing Levenberg-Marquardt scheme for nonlinear ill-posed problems
Hochbruck, M.; Hoenig, M.
2010. Numerische Mathematik, 115 (1), 71–79. doi:10.1007/s00211-009-0268-9  

A multilevel Jacobi-Davidson method for polynomial PDE eigenvalue problems arising in plasma physics
Hochbruck, M.; Löchel, D.
2010. SIAM Journal on Scientific Computing, 32 (6), 3151–3169. doi:10.1137/090774604  

Exponential integrators
Hochbruck, M.; Ostermann, A.
2010. Acta numerica, 19 (May), 209–286. doi:10.1017/S0962492910000048  

Exponential Rosenbrock-type methods
Hochbruck, M.; Ostermann, A.; Schweitzer, J.
2009. SIAM Journal on Numerical Analysis, 47 (1), 786–803. doi:10.1137/080717717  

Regularization of nonlinear ill-posed problems by exponential integrators
Hochbruck, M.; Hönig, M.; Ostermann, A.
2009. ESAIM: Mathematical Modelling and Numerical Analysis, 43 (4), 709–720. doi:10.1051/m2an/2009021  

A convergence analysis of the exponential Euler iteration for nonlinear ill-posed problems
Hochbruck, M.; Hönig, M.; Ostermann, A.
2009. Inverse Problems, 25 (7), 075009. doi:10.1088/0266-5611/25/7/075009  

Effect of poloidal inhomogeneity in plasma parameters on edge anomalous transport
Löchel, D.; Tokar, M.; Hochbruck, M.; Reiser, D.
2009. Physics of Plasmas, 16 (4), 044508. doi:10.1063/1.3121222  

One dimensional electromagnetic relativistic PIC-hydrodynamic hybrid simulation code H-VLPL (Hybrid Virtual Laser Plasma Lab)
Liljo, J.; Karmakar, A.; Pukhov, A.; Hochbruck, M.
2008. Computer Physics Communication, 179 (6), 371–379. doi:10.1016/j.cpc.2008.03.008  

A parallel implementation of a two-dimensional fluid laser-plasma integrator for stratified plasma-vacuum systems
Karle, C.; Schweitzer, J.; Hochbruck, M.; Spatschek, K. H.
2008. Journal of Computational Physics, 227 (16), 7701–7719. doi:10.1016/j.jcp.2008.04.024  

Approximation of matrix operators applied to multiple vectors
Hochbruck, M.; Niehoff, J.
2008. Mathematics and Computers in Simulation, 79 (4), 1270–1283. doi:10.1016/j.matcom.2008.03.016  

Rational approximation to trigonometric operators
Grimm, V.; Hochbruck, M.
2008. BIT Numerical Mathematics, 48 (2), 215–229. doi:10.1007/s10543-008-0185-9  

Preconditioning Lanczos approximations to the matrix exponential
Eshof, J. van den; Hochbruck, M.
2006. SIAM Journal on Scientific Computing, 27 (4), 1438–1457. doi:10.1137/040605461  

Exponential integrators of Rosenbrock-type
Hochbruck, M.; Ostermann, A.; Schweitzer, J.
2006. Differential-Algebraic Equations. Hrsg.: S.L. Campbell, 1107–1110, Oberwolfach  

Numerical solution of nonlinear wave equations in stratified dispersive media
Karle, C.; Schweitzer, J.; Hochbruck, M.; Laedke, E. W.; Spatschek, K. H.
2006. Journal of Computational Physics, 216 (1), 138–152. doi:10.1016/j.jcp.2005.11.024  

Error analysis of exponential integrators for oscillatory second-order differential equations
Hochbruck, M.; Grimm, V.
2006. Journal of Physics A: Mathematical and General, 39, 5495–5507. doi:10.1088/0305-4470/39/19/S10  

Exponential integrators for highly oscillatory differential equations
Grimm, V.; Hochbruck, M.
2006. Oberwolfach Reports, 3 (1), 868–869 

Exponential Runge-Kutta methods for parabolic problems
Hochbruck, M.; Ostermann, A.
2005. Applied Numerical Mathematics, 53 (2-4), 323–339. doi:10.1016/j.apnum.2004.08.005  

Explicit exponential Runge-Kutta methods for semilinear parabolic problems
Hochbruck, M.; Ostermann, A.
2005. SIAM Journal on Numerical Analysis, 43 (3), 1069–1090. doi:10.1137/040611434  

On Magnus integrators for time-dependent Schrödinger equations
Hochbruck, M.; Lubich, C.
2003. SIAM Journal on Numerical Analysis, 41 (3), 945–963. doi:10.1137/S0036142902403875  

Mathematik fürs Leben am Beispiel der Computertomographie
Hochbruck, M.; Sautter, J.-M.
2002. Mathematische Semesterberichte, 49 (1), 95–113. doi:10.1007/s005910200042  

A bunch of time integrators for quantum/classical molecular dynamics
Hochbruck, M.; Lubich, C.
1999. Algorithms for Macromolecular Modelling : challenges, methods, ideas; proceedings of the 2nd International Symposium on Algorithms for Macromolecular Modelling, Berlin, May 21 - 24, 1997. Ed.: P. Deuflhard, 421–432, Springer-Verlag. doi:10.1007/978-3-642-58360-5_24  

Exponential integrators for quantum-classical molecular dynamics
Hochbruck, M.; Lubich, C.
1999. BIT Numerical Mathematics, 39 (4), 620–645. doi:10.1023/A:1022335122807  

A Gautschi-type method for oscillatory second-order differential equations
Hochbruck, M.; Lubich, C.
1999. Numerische Mathematik, 83 (3), 403–426. doi:10.1007/s002110050456  

A numerical comparison of look-ahead Levinson and Schur algorithms for non-Hermitian Toeplitz systems
Hochbruck, M.
1998. High performance algorithms for structured matrix problems. Ed.: P. Arbenz, 127–148, Nova Science  

Exponential integrators for large systems of differential equations
Hochbruck, M.; Lubich, C.; Selhofer, H.
1998. SIAM Journal on Scientific Computing, 19 (5), 1552–1574. doi:10.1137/S1064827595295337  

Error analysis of Krylov methods in a nutshell
Hochbruck, M.; Lubich, C.
1998. SIAM Journal on Scientific Computing, 19 (2), 695–701. doi:10.1137/S1064827595290450 

Further optimized look-ahead recurrences for adjacent rows in the Padé table and Toeplitz matrix factorizations
Hochbruck, M.
1997. Journal of Computational and Applied Mathematics, 86 (1), 219–236. doi:10.1016/S0377-0427(97)00157-X  

On Krylov subspace approximations to the matrix exponential operator
Hochbruck, M.; Lubich, C.
1997. SIAM Journal on Numerical Analysis, 34 (5), 1911–1925. doi:10.1137/S0036142995280572  

Optimized look-ahead recurrences for adjacent rows in the Padé table
Gutknecht, M. H.; Hochbruck, M.
1996. BIT Numerical Mathematics, 36 (2), 264–286. doi:10.1007/BF01731983 

A note on conjugate-gradient type methods for indefinite and/or inconsistent linear systems
Fischer, B.; Hanke, M.; Hochbruck, M.
1996. Numer. algorithms 11 (1996) S. 181-187 

Preconditioned Krylov subspace methods for Lyapunov matrix equations
Starke, G.; Hochbruck, M.
1995. SIAM j. on matrix anal. and appl. 16 (1995) S. 156-171 

A Chebyshev-like semiiteration for inconsistent linear systems
Hanke, M.; Hochbruck, M.
1993. Electron. trans. on numer. anal. 1 (1993) S. 89-103 

Experiments with Krylov subspace methods on a massively parallel computer
Hanke, M.; Hochbruck, M.; Niethammer, W.
1992. Zuerich 1992. (IPS research report. No. 92-16.) 

A biconjugate gradient type algorithm on massively parallel architectures
Freund, R. W.; Hochbruck, M.
1991. In: Proceedings of the 13th IMACS World Congress on Computation and Applied Mathematics, Dublin 1991 

On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling
Freund, R. W.; Hochbruck, M.
1991. Palo Alto, Cal. 1991. (RIACS Technical report. 91.25.)