The ARKode infrastructure provides adaptive-step time integration modules for stiff, nonstiff and mixed stiff/nonstiff systems of ordinary differential equations (ODEs). ARKode itself is structured to support a wide range of one-step (but multi-stage) methods, allowing for rapid development of parallel implementations of state-of-the-art time integration methods. At present, ARKode is packaged with two time-stepping modules, ARKStep and ERKStep.

ARKStep supports ODE systems posed in split, linearly-implicit form,

(1)\[M \dot{y} = f_E(t,y) + f_I(t,y), \qquad y(t_0) = y_0,\]

where \(t\) is the independent variable, \(y\) is the set of dependent variables (in \(\mathbb{R}^N\)), \(M\) is a user-specified, nonsingular operator from \(\mathbb{R}^N\) to \(\mathbb{R}^N\), and the right-hand side function is partitioned into up to two components:

Either of these operators may be disabled, allowing for fully explicit, fully implicit, or combination implicit-explicit (ImEx) time integration.

The algorithms used in ARKStep are adaptive- and fixed-step additive Runge Kutta methods. Such methods are defined through combining two complementary Runge-Kutta methods: one explicit (ERK) and the other diagonally implicit (DIRK). Through appropriately partitioning the ODE right-hand side into explicit and implicit components (1), such methods have the potential to enable accurate and efficient time integration of stiff, nonstiff, and mixed stiff/nonstiff systems of ordinary differential equations. A key feature allowing for high efficiency of these methods is that only the components in \(f_I(t,y)\) must be solved implicitly, allowing for splittings tuned for use with optimal implicit solver algorithms.

This framework allows for significant freedom over the constitutive methods used for each component, and ARKode is packaged with a wide array of built-in methods for use. These built-in Butcher tables include adaptive explicit methods of orders 2-8, adaptive implicit methods of orders 2-5, and adaptive ImEx methods of orders 3-5.

ERKStep focuses specifically on problems posed in explicit form,

(2)\[\dot{y} = f(t,y), \qquad y(t_0) = y_0.\]

allowing for increased computational efficiency and memory savings. The algorithms used in ERKStep are adaptive- and fixed-step explicit Runge Kutta methods. As with ARKStep, the ERKStep module is packaged with adaptive explicit methods of orders 2-8.

For problems that include nonzero implicit term \(f_I(t,y)\), the resulting implicit system (assumed nonlinear, unless specified otherwise) is solved approximately at each integration step, using a modified Newton method, inexact Newton method, or an accelerated fixed-point solver. For the Newton-based methods and the serial or threaded NVECTOR modules in SUNDIALS, ARKode may use a variety of linear solvers provided with SUNDIALS, including both direct (dense, band, or sparse) and preconditioned Krylov iterative (GMRES [SS1986], BiCGStab [V1992], TFQMR [F1993], FGMRES [S1993], or PCG [HS1952]) linear solvers. When used with the MPI-based parallel, PETSc, hypre, CUDA, and Raja NVECTOR modules, or a user-provided vector data structure, only the Krylov solvers are available, although a user may supply their own linear solver for any data structures if desired. For the serial or threaded vector structures, we provide a banded preconditioner module called ARKBANDPRE that may be used with the Krylov solvers, while for the MPI-based parallel vector structure there is a preconditioner module called ARKBBDPRE which provides a band-block-diagonal preconditioner. Additionally, a user may supply more optimal, problem-specific preconditioner routines.

Changes from previous versions

Changes in v3.1.0

An additional NVECTOR implementation was added for the Tpetra vector from the Trilinos library to facilitate interoperability between SUNDIALS and Trilinos. This implementation is accompanied by additions to user documentation and SUNDIALS examples.

A bug was fixed where a nonlinear solver object could be freed twice in some use cases.

The EXAMPLES_ENABLE_RAJA CMake option has been removed. The option EXAMPLES_ENABLE_CUDA enables all examples that use CUDA including the RAJA examples with a CUDA back end (if the RAJA NVECTOR is enabled).

The implementation header file arkode_impl.h is no longer installed. This means users who are directly manipulating the ARKodeMem structure will need to update their code to use ARKode’s public API.

Python is no longer required to run make test and make test_install.

Fixed a bug in ARKodeButcherTable_Write when printing a Butcher table without an embedding.

Changes in v3.0.2

Added information on how to contribute to SUNDIALS and a contributing agreement.

Changes in v3.0.1

A bug in ARKode where single precision builds would fail to compile has been fixed.

Changes in v3.0.0

The ARKode library has been entirely rewritten to support a modular approach to one-step methods, which should allow rapid research and development of novel integration methods without affecting existing solver functionality. To support this, the existing ARK-based methods have been encapsulated inside the new ARKStep time-stepping module. Two new time-stepping modules have been added:

  • The ERKStep module provides an optimized implementation for explicit Runge-Kutta methods with reduced storage and number of calls to the ODE right-hand side function.
  • The MRIStep module implements two-rate explicit-explicit multirate infinitesimal step methods utilizing different step sizes for slow and fast processes in an additive splitting.

This restructure has resulted in numerous small changes to the user interface, particularly the suite of “Set” routines for user-provided solver parameters and and “Get” routines to access solver statistics, that are now prefixed with the name of time-stepping module (e.g., ARKStep or ERKStep) instead of ARKode. Aside from affecting the names of these routines, user-level changes have been kept to a minimum. However, we recommend that users consult both this documentation and the ARKode example programs for further details on the updated infrastructure.

As part of the ARKode restructuring an ARKodeButcherTable structure has been added for storing Butcher tables. Functions for creating new Butcher tables and checking their analytic order are provided along with other utility routines. For more details see Butcher Table Data Structure.

Two changes were made in the initial step size algorithm:

  • Fixed an efficiency bug where an extra call to the right hand side function was made.
  • Changed the behavior of the algorithm if the max-iterations case is hit. Before the algorithm would exit with the step size calculated on the penultimate iteration. Now it will exit with the step size calculated on the final iteration.

ARKode’s dense output infrastructure has been improved to support higher-degree Hermite polynomial interpolants (up to degree 5) over the last successful time step.

ARKode’s previous direct and iterative linear solver interfaces, ARKDLS and ARKSPILS, have been merged into a single unified linear solver interface, ARKLS, to support any valid SUNLINSOL module. This includes DIRECT and ITERATIVE types as well as the new MATRIX_ITERATIVE type. Details regarding how ARKLS utilizes linear solvers of each type as well as discussion regarding intended use cases for user-supplied SUNLinSol implementations are included in the chapter Description of the SUNLinearSolver module. All ARKode examples programs and the standalone linear solver examples have been updated to use the unified linear solver interface.

The user interface for the new ARKLS module is very similar to the previous ARKDLS and ARKSPILS interfaces. Additionally, we note that Fortran users will need to enlarge their iout array of optional integer outputs, and update the indices that they query for certain linear-solver-related statistics.

The names of all constructor routines for SUNDIALS-provided SUNLinSol implementations have been updated to follow the naming convention SUNLinSol_* where * is the name of the linear solver. The new names are SUNLinSol_Band, SUNLinSol_Dense, SUNLinSol_KLU, SUNLinSol_LapackBand, SUNLinSol_LapackDense, SUNLinSol_PCG, SUNLinSol_SPBCGS, SUNLinSol_SPFGMR, SUNLinSol_SPGMR, SUNLinSol_SPTFQMR, and SUNLinSol_SuperLUMT. Solver-specific “set” routine names have been similarly standardized. To minimize challenges in user migration to the new names, the previous routine names may still be used; these will be deprecated in future releases, so we recommend that users migrate to the new names soon. All ARKode example programs and the standalone linear solver examples have been updated to use the new naming convention.

The SUNBandMatrix constructor has been simplified to remove the storage upper bandwidth argument.

SUNDIALS integrators have been updated to utilize generic nonlinear solver modules defined through the SUNNONLINSOL API. This API will ease the addition of new nonlinear solver options and allow for external or user-supplied nonlinear solvers. The SUNNONLINSOL API and SUNDIALS provided modules are described in Nonlinear Solver Data Structures and follow the same object oriented design and implementation used by the NVector, SUNMatrix, and SUNLinSol modules. Currently two SUNNONLINSOL implementations are provided, SUNNonlinSol_Newton and SUNNonlinSol_FixedPoint. These replicate the previous integrator specific implementations of a Newton iteration and an accelerated fixed-point iteration, respectively. Example programs using each of these nonlinear solver modules in a standalone manner have been added and all ARKode example programs have been updated to use generic SUNNonlinSol modules.

As with previous versions, ARKode will use the Newton solver (now provided by SUNNonlinSol_Newton) by default. Use of the ARKStepSetLinear() routine (previously named ARKodeSetLinear) will indicate that the problem is linearly-implicit, using only a single Newton iteration per implicit stage. Users wishing to switch to the accelerated fixed-point solver are now required to create a SUNNonlinSol_FixedPoint object and attach that to ARKode, instead of calling the previous ARKodeSetFixedPoint routine. See the documentation sections A skeleton of the user’s main program, Nonlinear solver interface functions, and The SUNNonlinearSolver_FixedPoint implementation for further details, or the serial C example program ark_brusselator_fp.c for an example.

Three fused vector operations and seven vector array operations have been added to the NVECTOR API. These optional operations are disabled by default and may be activated by calling vector specific routines after creating an NVector (see Description of the NVECTOR Modules for more details). The new operations are intended to increase data reuse in vector operations, reduce parallel communication on distributed memory systems, and lower the number of kernel launches on systems with accelerators. The fused operations are N_VLinearCombination, N_VScaleAddMulti, and N_VDotProdMulti, and the vector array operations are N_VLinearCombinationVectorArray, N_VScaleVectorArray, N_VConstVectorArray, N_VWrmsNormVectorArray, N_VWrmsNormMaskVectorArray, N_VScaleAddMultiVectorArray, and N_VLinearCombinationVectorArray. If an NVector implementation defines any of these operations as NULL, then standard NVector operations will automatically be called as necessary to complete the computation.

Multiple changes to the CUDA NVECTOR were made:

  • Changed the N_VMake_Cuda function to take a host data pointer and a device data pointer instead of an N_VectorContent_Cuda object.
  • Changed N_VGetLength_Cuda to return the global vector length instead of the local vector length.
  • Added N_VGetLocalLength_Cuda to return the local vector length.
  • Added N_VGetMPIComm_Cuda to return the MPI communicator used.
  • Removed the accessor functions in the namespace suncudavec.
  • Added the ability to set the cudaStream_t used for execution of the CUDA NVECTOR kernels. See the function N_VSetCudaStreams_Cuda.
  • Added N_VNewManaged_Cuda, N_VMakeManaged_Cuda, and N_VIsManagedMemory_Cuda functions to accommodate using managed memory with the CUDA NVECTOR.

Multiple changes to the RAJA NVECTOR were made:

  • Changed N_VGetLength_Raja to return the global vector length instead of the local vector length.
  • Added N_VGetLocalLength_Raja to return the local vector length.
  • Added N_VGetMPIComm_Raja to return the MPI communicator used.
  • Removed the accessor functions in the namespace sunrajavec.

A new NVECTOR implementation for leveraging OpenMP 4.5+ device offloading has been added, NVECTOR_OpenMPDEV. See The NVECTOR_OPENMPDEV Module for more details.

Changes in v2.2.1

Fixed a bug in the CUDA NVECTOR where the N_VInvTest operation could write beyond the allocated vector data.

Fixed library installation path for multiarch systems. This fix changes the default library installation path to CMAKE_INSTALL_PREFIX/CMAKE_INSTALL_LIBDIR from CMAKE_INSTALL_PREFIX/lib. CMAKE_INSTALL_LIBDIR is automatically set, but is available as a CMAKE option that can modified.

Changes in v2.2.0

Fixed a problem with setting sunindextype which would occur with some compilers (e.g. armclang) that did not define __STDC_VERSION__.

Added hybrid MPI/CUDA and MPI/RAJA vectors to allow use of more than one MPI rank when using a GPU system. The vectors assume one GPU device per MPI rank.

Changed the name of the RAJA NVECTOR library to libsundials_nveccudaraja.lib from libsundials_nvecraja.lib to better reflect that we only support CUDA as a backend for RAJA currently.

Several changes were made to the build system:

  • CMake 3.1.3 is now the minimum required CMake version.
  • Deprecate the behavior of the SUNDIALS_INDEX_TYPE CMake option and added the SUNDIALS_INDEX_SIZE CMake option to select the sunindextype integer size.
  • The native CMake FindMPI module is now used to locate an MPI installation.
  • If MPI is enabled and MPI compiler wrappers are not set, the build system will check if CMAKE_<language>_COMPILER can compile MPI programs before trying to locate and use an MPI installation.
  • The previous options for setting MPI compiler wrappers and the executable for running MPI programs have been have been depreated. The new options that align with those used in native CMake FindMPI module are MPI_C_COMPILER, MPI_CXX_COMPILER, MPI_Fortran_COMPILER, and MPIEXEC_EXECUTABLE.
  • When a Fortran name-mangling scheme is needed (e.g., LAPACK_ENABLE is ON) the build system will infer the scheme from the Fortran compiler. If a Fortran compiler is not available or the inferred or default scheme needs to be overridden, the advanced options SUNDIALS_F77_FUNC_CASE and SUNDIALS_F77_FUNC_UNDERSCORES can be used to manually set the name-mangling scheme and bypass trying to infer the scheme.
  • Parts of the main CMakeLists.txt file were moved to new files in the src and example directories to make the CMake configuration file structure more modular.

Changes in v2.1.2

Updated the minimum required version of CMake to 2.8.12 and enabled using rpath by default to locate shared libraries on OSX.

Fixed Windows specific problem where sunindextype was not correctly defined when using 64-bit integers for the SUNDIALS index type. On Windows sunindextype is now defined as the MSVC basic type __int64.

Added sparse SUNMatrix “Reallocate” routine to allow specification of the nonzero storage.

Updated the KLU SUNLinearSolver module to set constants for the two reinitialization types, and fixed a bug in the full reinitialization approach where the sparse SUNMatrix pointer would go out of scope on some architectures.

Updated the “ScaleAdd” and “ScaleAddI” implementations in the sparse SUNMatrix module to more optimally handle the case where the target matrix contained sufficient storage for the sum, but had the wrong sparsity pattern. The sum now occurs in-place, by performing the sum backwards in the existing storage. However, it is still more efficient if the user-supplied Jacobian routine allocates storage for the sum \(I+\gamma J\) or \(M+\gamma J\) manually (with zero entries if needed).

Changed LICENSE install path to instdir/include/sundials.

Changes in v2.1.1

Fixed a potential memory leak in the SPGMR and SPFGMR linear solvers: if “Initialize” was called multiple times then the solver memory was reallocated (without being freed).

Fixed a minor bug in the ARKReInit routine, where a flag was incorrectly set to indicate that the problem had been resized (instead of just re-initialized).

Fixed C++11 compiler errors/warnings about incompatible use of string literals.

Updated KLU SUNLinearSolver module to use a typedef for the precision-specific solve function to be used (to avoid compiler warnings).

Added missing typecasts for some (void*) pointers (again, to avoid compiler warnings).

Bugfix in sunmatrix_sparse.c where we had used int instead of sunindextype in one location.

Added missing #include <stdio.h> in NVECTOR and SUNMATRIX header files.

Added missing prototype for ARKSpilsGetNumMTSetups.

Fixed an indexing bug in the CUDA NVECTOR implementation of N_VWrmsNormMask and revised the RAJA NVECTOR implementation of N_VWrmsNormMask to work with mask arrays using values other than zero or one. Replaced double with realtype in the RAJA vector test functions.

Fixed compilation issue with GCC 7.3.0 and Fortran programs that do not require a SUNMatrix or SUNLinearSolver module (e.g. iterative linear solvers, explicit methods, fixed point solver, etc.).

Changes in v2.1.0

Added NVECTOR print functions that write vector data to a specified file (e.g. N_VPrintFile_Serial).

Added make test and make test_install options to the build system for testing SUNDIALS after building with make and installing with make install respectively.

Changes in v2.0.0

All interfaces to matrix structures and linear solvers have been reworked, and all example programs have been updated. The goal of the redesign of these interfaces was to provide more encapsulation and ease in interfacing custom linear solvers and interoperability with linear solver libraries.

Specific changes include:

  • Added generic SUNMATRIX module with three provided implementations: dense, banded and sparse. These replicate previous SUNDIALS Dls and Sls matrix structures in a single object-oriented API.
  • Added example problems demonstrating use of generic SUNMATRIX modules.
  • Added generic SUNLINEARSOLVER module with eleven provided implementations: dense, banded, LAPACK dense, LAPACK band, KLU, SuperLU_MT, SPGMR, SPBCGS, SPTFQMR, SPFGMR, PCG. These replicate previous SUNDIALS generic linear solvers in a single object-oriented API.
  • Added example problems demonstrating use of generic SUNLINEARSOLVER modules.
  • Expanded package-provided direct linear solver (Dls) interfaces and scaled, preconditioned, iterative linear solver (Spils) interfaces to utilize generic SUNMATRIX and SUNLINEARSOLVER objects.
  • Removed package-specific, linear solver-specific, solver modules (e.g. CVDENSE, KINBAND, IDAKLU, ARKSPGMR) since their functionality is entirely replicated by the generic Dls/Spils interfaces and SUNLINEARSOLVER/SUNMATRIX modules. The exception is CVDIAG, a diagonal approximate Jacobian solver available to CVODE and CVODES.
  • Converted all SUNDIALS example problems to utilize new generic SUNMATRIX and SUNLINEARSOLVER objects, along with updated Dls and Spils linear solver interfaces.
  • Added Spils interface routines to ARKode, CVODE, CVODES, IDA and IDAS to allow specification of a user-provided “JTSetup” routine. This change supports users who wish to set up data structures for the user-provided Jacobian-times-vector (“JTimes”) routine, and where the cost of one JTSetup setup per Newton iteration can be amortized between multiple JTimes calls.

Two additional NVECTOR implementations were added – one for CUDA and one for RAJA vectors. These vectors are supplied to provide very basic support for running on GPU architectures. Users are advised that these vectors both move all data to the GPU device upon construction, and speedup will only be realized if the user also conducts the right-hand-side function evaluation on the device. In addition, these vectors assume the problem fits on one GPU. Further information about RAJA, users are referred to the web site, These additions are accompanied by additions to various interface functions and to user documentation.

All indices for data structures were updated to a new sunindextype that can be configured to be a 32- or 64-bit integer data index type. sunindextype is defined to be int32_t or int64_t when portable types are supported, otherwise it is defined as int or long int. The Fortran interfaces continue to use long int for indices, except for their sparse matrix interface that now uses the new sunindextype. This new flexible capability for index types includes interfaces to PETSc, hypre, SuperLU_MT, and KLU with either 32-bit or 64-bit capabilities depending how the user configures SUNDIALS.

To avoid potential namespace conflicts, the macros defining booleantype values TRUE and FALSE have been changed to SUNTRUE and SUNFALSE respectively.

Temporary vectors were removed from preconditioner setup and solve routines for all packages. It is assumed that all necessary data for user-provided preconditioner operations will be allocated and stored in user-provided data structures.

The file include/sundials_fconfig.h was added. This file contains SUNDIALS type information for use in Fortran programs.

Added functions SUNDIALSGetVersion and SUNDIALSGetVersionNumber to get SUNDIALS release version information at runtime.

The build system was expanded to support many of the xSDK-compliant keys. The xSDK is a movement in scientific software to provide a foundation for the rapid and efficient production of high-quality, sustainable extreme-scale scientific applications. More information can be found at,

In addition, numerous changes were made to the build system. These include the addition of separate BLAS_ENABLE and BLAS_LIBRARIES CMake variables, additional error checking during CMake configuration, minor bug fixes, and renaming CMake options to enable/disable examples for greater clarity and an added option to enable/disable Fortran 77 examples. These changes included changing ENABLE_EXAMPLES to ENABLE_EXAMPLES_C, changing CXX_ENABLE to EXAMPLES_ENABLE_CXX, changing F90_ENABLE to EXAMPLES_ENABLE_F90, and adding an EXAMPLES_ENABLE_F77 option.

Corrections and additions were made to the examples, to installation-related files, and to the user documentation.

Changes in v1.1.0

We have included numerous bugfixes and enhancements since the v1.0.2 release.

The bugfixes include:

  • For each linear solver, the various solver performance counters are now initialized to 0 in both the solver specification function and in the solver’s linit function. This ensures that these solver counters are initialized upon linear solver instantiation as well as at the beginning of the problem solution.
  • The choice of the method vs embedding the Billington and TRBDF2 explicit Runge-Kutta methods were swapped, since in those the lower-order coefficients result in an A-stable method, while the higher-order coefficients do not. This change results in significantly improved robustness when using those methods.
  • A bug was fixed for the situation where a user supplies a vector of absolute tolerances, and also uses the vector Resize() functionality.
  • A bug was fixed wherein a user-supplied Butcher table without an embedding is supplied, and the user is running with either fixed time steps (or they do adaptivity manually); previously this had resulted in an error since the embedding order was below 1.
  • Numerous aspects of the documentation were fixed and/or clarified.

The feature changes/enhancements include:

  • Two additional NVECTOR implementations were added – one for Hypre (parallel) ParVector vectors, and one for PETSc vectors. These additions are accompanied by additions to various interface functions and to user documentation.
  • Each NVECTOR module now includes a function, N_VGetVectorID, that returns the NVECTOR module name.
  • A memory leak was fixed in the banded preconditioner and banded-block-diagonal preconditioner interfaces. In addition, updates were done to return integers from linear solver and preconditioner ‘free’ routines.
  • The Krylov linear solver Bi-CGstab was enhanced by removing a redundant dot product. Various additions and corrections were made to the interfaces to the sparse solvers KLU and SuperLU_MT, including support for CSR format when using KLU.
  • The ARKode implicit predictor algorithms were updated: methods 2 and 3 were improved slightly, a new predictor approach was added, and the default choice was modified.
  • The underlying sparse matrix structure was enhanced to allow both CSR and CSC matrices, with CSR supported by the KLU linear solver interface. ARKode interfaces to the KLU solver from both C and Fortran were updated to enable selection of sparse matrix type, and a Fortran-90 CSR example program was added.
  • The missing ARKSpilsGetNumMtimesEvals() function was added – this had been included in the previous documentation but had not been implemented.
  • The handling of integer codes for specifying built-in ARKode Butcher tables was enhanced. While a global numbering system is still used, methods now have #defined names to simplify the user interface and to streamline incorporation of new Butcher tables into ARKode.
  • The maximum number of Butcher table stages was increased from 8 to 15 to accommodate very high order methods, and an 8th-order adaptive ERK method was added.
  • Support was added for the explicit and implicit methods in an additive Runge-Kutta method to utilize different stage times, solution and embedding coefficients, to support new SSP-ARK methods.
  • The FARKODE interface was extended to include a routine to set scalar/array-valued residual tolerances, to support Fortran applications with non-identity mass-matrices.

Reading this User Guide

This user guide is a combination of general usage instructions and specific example programs. We expect that some readers will want to concentrate on the general instructions, while others will refer mostly to the examples, and the organization is intended to accommodate both styles.

The structure of this document is as follows:

SUNDIALS Release License

All SUNDIALS packages are released open source, under the BSD 3-Clause license. The only requirements of the license are preservation of copyright and a standard disclaimer of liability. The full text of the license and an additional notice are provided below and may also be found in the LICENSE and NOTICE files provided with all SUNDIALS packages.

PLEASE NOTE If you are using SUNDIALS with any third party libraries linked in (e.g., LAPACK, KLU, SuperLU_MT, PETSc, or hypre), be sure to review the respective license of the package as that license may have more restrictive terms than the SUNDIALS license. For example, if someone builds SUNDIALS with a statically linked KLU, the build is subject to terms of the more-restrictive LGPL license (which is what KLU is released with) and not the SUNDIALS BSD license anymore.

BSD 3-Clause License

Copyright (c) 2002-2019, Lawrence Livermore National Security and Southern Methodist University.

All rights reserved.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

  • Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
  • Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
  • Neither the name of the copyright holder nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.


Additional Notice

This work was produced under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

This work was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.

Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or Lawrence Livermore National Security, LLC.

The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.

SUNDIALS Release Numbers




UCRL-CODE-155952 (IDA)