Teuchos: Utilities for Developers & Users November 2nd, 3:30-4:30pm Roscoe Bartlett Mike Heroux Kris Kampshoff Kevin Long Paul Sexton Heidi Thornquist Questions from audience Can you use another matrix-vector multiply? Are multivectors supported? Are Petsc and Hypre available via TSF? How does Trilinos differ from Petsc? Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000.

What is Teuchos (TEF-hos)? Lightweight, portable utility package What’s new? First release in Trilinos 4.0 Templated BLAS/LAPACK interface Templated serial dense matrix / vector Timing classes Templated parameter list Numeric traits mechanisms ( ordinal / scalar ) Memory management: Reference-counted pointers Pre-allocated workspace Exception handling Command-line parsing “Extended” utilities: XML parsing MPI communicators / traits Users Developers

Templated Parameter List Design based on NOX parameter list Adopted by Amesos, ML, Ifpack, AztecOO, … Parameter lists can be queried for attributes. No mechanism for “smart” queries of tags. // Has a solver been chosen? bool solver_defined = My_List.isParameter(“Solver”); // Has a preconditioner been chosen? bool prec_defined = My_List.isSublist(“Preconditioner”); // Has a double-precision drop tolerance been chosen? (Helper function) bool drop_tol = Teuchos::isParameterType<double>(Prec_List, “Drop Tolerance”); // What if we use the wrong parameter name? bool solve_tol = My_List.isParameter(“tolerance”); ReturnType foo_solver( Teuchos::ParameterList& solverPL,… ) { // Getting parameters int max_iters = solverPL.template<int> get("Max Iters"); std::string solver = getParameter(solverPL, “Solver”) double tol = solverPL.get("Tolerance", 1e-8); // Get sublist ( same as making sublist ) Teuchos::ParameterList precPL = solverPL.sublist("Preconditioner"); … return Ok; } Parameters can be input and retrieved with templated “set” and “get” methods, as well as helper functions. Parameters can be numeric datatypes, pointers to vectors or functions, and other parameter lists. // Empty parameter list Teuchos::ParameterList My_List; // Setting parameters My_List.set("Max Iters", 1550); My_List.set("Tolerance", 1e-10); My_List.set("Solver", "GMRES"); // Create sublist Teuchos::ParameterList& Prec_List = My_List.sublist("Preconditioner"); // Setting parameters in sublist Prec_List.set("Type", "ILU"); Prec_List.set("Drop Tolerance", 1e-3); Parameters can be input and retrieved with templated “set” and “get” methods, as well as helper functions.

Numeric Traits Mechanisms Generic programming technique that uses templated interfaces to define the standard behavior of datatypes. Assumed: addition, subtraction, multiplication, and division OrdinalTraits zero & one int & long int ScalarTraits zero, one, magnitude type, absolute value, conjugate, square root, random number generator, … std::numeric_limits float, double, complex<float>, and complex<double> Required: none, a compile-time check is performed. What about arbitrary precision arithmetic?

Arbitrary Precision Libraries ARPREC & GNU MP ARPREC (Bailey, et. al.) uses arrays of 64-bit floating-point numbers to represent high-precision floating-point numbers. provides support for datatypes with differing precision levels. --enable-teuchos-arprec GNU MP uses fullwords, arbitrary precision mantissa, and limited precision exponent to represent high-precision floating-point numbers. platform specific kernels written in assembly language perform common inner-loop functions. --enable-teuchos-gmp Caveat: Arbitrary precision arithmetic is expensive, but can be used selectively to overcome numerical inaccuracy.

Templated BLAS Interface Basic Linear Algebra Subroutines (BLAS) Robust, high-performance dense vector-vector, matrix-vector, and matrix-matrix routines. Written in Fortran Fast and efficient Fortran is highly portable; however, the interface between it and C++ is platform-dependent Templated C++ wrappers for the BLAS Fortran routines are created through autoconf. Two main benefits of using a templated interface Template Specialization: template specialization can be used to support the four native datatypes of existing Fortran BLAS code. General Implementation: a generic C++ implementation can be created to handle many arbitrary datatypes.

Templated BLAS Interface Using Native Datatypes Teuchos::BLAS<int,double> blas; blas.AXPY(…) … blas.GEMV(…) … blas.GEMM(…) Using ARPREC mp::mp_init(200); Teuchos::BLAS<int,mp_real> blas; blas.AXPY(…) … blas.GEMV(…) … blas.GEMM(…)

Templated LAPACK Interface Linear Algebra PACKage (LAPACK) Robust, high-performance dense routines for solving systems of simultaneous linear equations, least-squares solutions of linear systems of equations, eigenvalue problems, and singular value problems. Templated C++ wrappers for the LAPACK Fortran routines are created for the four native datatypes. no general implementations at this time Templated serial dense matrix / vector class designed to be compatible with BLAS/LAPACK interfaces. Fortran-style storage Similar to Epetra_SerialDenseMatrix

Hilbert Matrices A Hilbert matrix HN is a square N-by-N matrix such that: Notoriously ill-conditioned κ(H3)  524 κ(H5)  476610 κ(H10)  1.6025 x 1013 κ(H20)  7.8413 x 1017 κ(H100)  1.7232 x 1020 Theoretically, Hilbert matrices are SPD

Cholesky Factorization and Hilbert Matrices In practice, floating-point rounding error causes higher-order Hilbert matrices to no longer be positive definite Cholesky factorization is often done to determine whether or not a given matrix is positive definite As such, a Cholesky factorization will often fail on higher-order Hilbert matrices, even though such matrices “should” be positive definite Precision Largest N for which Cholesky Factorization is successful Single Precision 8 Double Precision 13 Arbitrary Precision (20) 29 Arbitrary Precision (40) 40 Arbitrary Precision (200) 145 Arbitrary Precision (400) 233*

Cholesky Factorization and Hilbert Matrices In addition to providing a higher limit on N, using arbitrary precision reduces error. By solving the linear system: where the vector b is constructed such that the true solution x is equal to a 1-vector we can look at the forward error.

Timing Classes Teuchos::Time Teuchos::TimeMonitor Basic wall-clock timing class Teuchos::TimeMonitor Uses Teuchos::Time internally Constructor starts timer, destructor stops it Can also maintain an array of timers Teuchos::Time timer(“My Timer”); timer.start(); … timer.stop(); double comp_time = timer.totalElapsedTime()

Teuchos Users Overview Parameter list is easy to use and quickly being adopted by most Trilinos packages. Numeric traits mechanism allows the use of arbitrary datatypes in templated code. Such arithmetic may be expensive, but the increased precision may be required for ill-conditioned problems Arbitrary-precision arithmetic can be used in only the places where it is needed most in order to retain speed Templated tools libraries are essential for the development of templated solver libraries. See Teuchos website for more info: http://software.sandia.gov/Trilinos/packages/teuchos