# 7.1. Introduction

KINSOL is part of a software family called SUNDIALS: SUite of Nonlinear and DIfferential/ALgebraic equation Solvers [70]. This suite consists of CVODE, ARKODE, KINSOL, and IDA, and variants of these with sensitivity analysis capabilities.

KINSOL is a general-purpose nonlinear system solver based on Newton-Krylov solver technology. A fixed point iteration is also included with the release of KINSOL v.2.8.0 and higher.

## 7.1.1. Historical Background

The first nonlinear solver packages based on Newton-Krylov methods were written in Fortran. In particular, the NKSOL package, written at LLNL, was the first Newton-Krylov solver package written for solution of systems arising in the solution of partial differential equations [25]. This Fortran code made use of Newton’s method to solve the discrete nonlinear systems and applied a preconditioned Krylov linear solver for solution of the Jacobian system at each nonlinear iteration. The key to the Newton-Krylov method was that the matrix-vector multiplies required by the Krylov method could effectively be approximated by a finite difference of the nonlinear system-defining function, avoiding a requirement for the formation of the actual Jacobian matrix. Significantly less memory was required for the solver as a result.

In the late 1990s, there was a push at LLNL to rewrite the nonlinear solver in C and port it to distributed memory parallel machines. Both Newton and Krylov methods are easily implemented in parallel, and this effort gave rise to the KINSOL package. KINSOL is similar to NKSOL in functionality, except that it provides for more options in the choice of linear system methods and tolerances, and has a more modular design to provide flexibility for future enhancements.

At present, KINSOL may utilize a variety of Krylov methods provided in SUNDIALS. These methods include the GMRES (Generalized Minimal RESidual) [106], FGMRES (Flexible Generalized Minimum RESidual) [105], Bi-CGStab (Bi-Conjugate Gradient Stabilized) [131], TFQMR (Transpose-Free Quasi-Minimal Residual) [56], and PCG (Preconditioned Conjugate Gradient) [65] linear iterative methods. As Krylov methods, these require little matrix storage for solving the Newton equations as compared to direct methods. However, the algorithms allow for a user-supplied preconditioner, and, for most problems, preconditioning is essential for an efficient solution. For very large nonlinear algebraic systems, the Krylov methods are preferable over direct linear solver methods, and are often the only feasible choice. Among the Krylov methods in SUNDIALS, we recommend GMRES as the best overall choice. However, users are encouraged to compare all options, especially if encountering convergence failures with GMRES. Bi-CGStab and TFQMR have an advantage in storage requirements, in that the number of workspace vectors they require is fixed, while that number for GMRES depends on the desired Krylov subspace size. FGMRES has an advantage in that it is designed to support preconditioners that vary between iterations (e.g., iterative methods). PCG exhibits rapid convergence and minimal workspace vectors, but only works for symmetric linear systems.

For the sake of completeness in functionality, direct linear system solvers are included in KINSOL. These include methods for both dense and banded linear systems, with Jacobians that are either user-supplied or generated internally by difference quotients. KINSOL also includes interfaces to sparse direct solvers, including KLU [4, 40] and the threaded sparse direct solver, SuperLU_MT [9, 42, 89], among others (see Chapter §10 for further details).

In the process of translating NKSOL into C, the overall KINSOL organization has been changed considerably.
One key feature of the KINSOL organization is that a separate module devoted to vector operations was created.
This module facilitated extension to multiprosessor environments with minimal impact on the rest of the solver. The
vector module design is shared across the SUNDIALS suite. This `N_Vector`

module is written in terms of
abstract vector operations with the actual routines attached by a particular implementation (such as serial or parallel)
of `N_Vector`

. This abstraction allows writing the SUNDIALS solvers in a manner independent of the actual
`N_Vector`

implementation (which can be user-supplied), as well as allowing more than one `N_Vector`

module linked
into an executable file. SUNDIALS (and thus KINSOL) is supplied with serial, MPI-parallel, OpenMP
and Pthreads thread-parallel `N_Vector`

implementations, as well as multiple `N_Vector`

implementations designed to leverage GPU architectures (see Chapter §8 for
further details).

There are several motivations for choosing the C language for KINSOL. First, a general movement away from Fortran and toward C in scientific computing was apparent. Second, the pointer, structure, and dynamic memory allocation features in C are extremely useful in software of this complexity, with the great variety of method options offered. Finally, we prefer C over C++ for KINSOL because of the wider availability of C compilers, the potentially greater efficiency of C, and the greater ease of interfacing the solver to applications written in Fortran.

## 7.1.2. Changes to SUNDIALS in release 7.1.0

**Bug Fixes**

Fixed a bug in v7.1.0 with the SYCL N_Vector `N_VSpace`

function.

For changes in prior versions of SUNDIALS see §15.

## 7.1.3. Reading this User Guide

This user guide is a combination of general usage instructions and specific examples. 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.

There are different possible levels of usage of KINSOL. The most casual user, with a small nonlinear system, can
get by with reading all of Chapter §7.2, then Chapter :numref:KINSOL.Usage.CC through
§7.4 only, and looking at examples in [39]. In a different
direction, a more expert user with a nonlinear system may want to (a) use a package preconditioner
(§7.4.5), (b) supply his/her own Jacobian or preconditioner routines
(§7.4.4), (c) supply a new `N_Vector`

module
(Chapter §8), or even (d) supply a different linear solver module
(§7.4.3.2 and Chapter §10).

The structure of this document is as follows:

In Chapter §7.2, we provide short descriptions of the numerical methods implemented by KINSOL for the solution of nonlinear systems.

The following chapter describes the software organization of the KINSOL solver (§7.3).

Chapter :numref:KINSOL.Usage.CC is the main usage document for KINSOL for C applications. It includes a complete description of the user interface for the solution of nonlinear algebraic systems.

Chapter §8 gives a brief overview of the generic

`N_Vector`

module shared among the various components of SUNDIALS, and details on the four`N_Vector`

implementations provided with SUNDIALS.Chapter §9 gives a brief overview of the generic

`SUNMatrix`

module shared among the various components of SUNDIALS, and details on the`SUNMatrix`

implementations provided with SUNDIALS.Chapter §10 gives a brief overview of the generic

`SUNLinearSolver`

module shared among the various components of SUNDIALS. This chapter contains details on the`SUNLinearSolver`

implementations provided with SUNDIALS. The chapter also contains details on the`SUNLinearSolver`

implementations provided with SUNDIALS that interface with external linear solver libraries.Finally, in the appendices, we provide detailed instructions for the installation of KINSOL, within the structure of SUNDIALS (Appendix §1.1), as well as a list of all the constants used for input to and output from KINSOL functions (Appendix §7.5).

Finally, the reader should be aware of the following notational conventions in this user guide: program listings and
identifiers (such as `KINInit`

) within textual explanations appear in typewriter type style; fields in C
structures (such as *content*) appear in italics; and packages or modules are written in all capitals. Usage and

## 7.1.4. SUNDIALS License and Notices

All SUNDIALS packages are released open source, under the BSD 3-Clause license for more details see the LICENSE and NOTICE files provided with all SUNDIALS packages.

## 7.1.5. Acknowledgments

We wish to acknowledge the contributions to previous versions of the KINSOL code and user guide by Allan G. Taylor.