1. Trilinos 102: Advanced Concepts November 7, 2007 8:30-9:30 a.m. Mike Heroux Jim Willenbring

2. Overview  How to Create a Trilinos (Compatible) Package  Adding Files to the Build System and Tarball  Adding Configure Options  Using Makefile.export for Tests and Examples  2D Objects.  Parallel Data Redistribution.

3. Outline  Creating Objects.  2D Objects.  Teuchos tidbits.  Performance Optimizations.

4. How to Create a Trilinos (Compatible) Package  Two primary cases  Using Autotools with an existing package  Starting a new package using Autotools  Both cases are similar  In either case, the package might be  Stand alone  Used via Trilinos/packages/external  Added to Trilinos

5. How to Create a Trilinos (Compatible) Package  Look at the new_package package  Customize the following files for your package  configure.ac  Makefile.am  src/Makefile.am  test/Makefile.am  example/Makefile.am  Makefile.export..in  (Some of the necessary changes can be made using scripts supplied by new_package)  Additional instructions are supplied with new_package

6.  Library source files: CORE = \ $(srcdir)/Epetra_BLAS.cpp \... $(srcdir)/Epetra_Object.cpp CORE_H = \ $(srcdir)/Epetra_BLAS.h \ … $(srcdir)/Epetra_ConfigDefs.h  Conditionally compiled files listed with ‘EXTRA_’ prefix  Don’t forget to list header files! Adding Files to the Build System and Tarball

7.  Makefile.am/.in:  Add the directory the new files are in to ‘SUBDIRS’ in the Makefile.am one level up SUBDIRS = DIR1 DIR2  Add the Makefile that will be generated to ‘AC_CONFIG_FILES’ in configure.ac AC_CONFIG_FILES([Makefile … src/Makefile …])  Don’t forget to ‘cvs add’ both files ./bootstrap  Other types of files (scripts, plain text, etc):  Add the name of the file to EXTRA_DIST EXTRA_DIST = script1 README … ./bootstrap

8. Adding Configure Options  TAC_ARG_ENABLE_CAN_USE_PACKAGE(epetra, teuchos, …)  ‘#ifdef HAVE_EPETRA_TEUCHOS’ in source code  TAC_ARG_ENABLE_FEATURE_SUB( epetra, abc, …)  ‘#ifdef HAVE_EPETRA_ARRAY_BOUNDS_CHECK’ in source

9. Adding Configure Options  TAC_ARG_WITH_PACKAGE(zoltan, [Enable Zoltan interface support], ZOLTAN, no)  AM_CONDITIONAL(HAVE_ZOLTAN, [test "X$ac_cv_use_zoltan" != "Xno"]) ‘if HAVE_ZOLTAN’ in Makefile.am  AC_SEARCH_LIBS(pow,[m],,AC_MSG_ERROR(Cannot find math library))

10. Using Makefile.export for Tests / Examples  Makefile.am: include $(top_builddir)/Makefile.export.epetra EXEEXT =.exe noinst_PROGRAMS = CrsMatrix_test CrsMatrix_test_SOURCES = $(srcdir)/cxx_main.cpp CrsMatrix_test_DEPENDENCIES=$(top_builddir)/src/libepetra.a CrsMatrix_test_CXXFLAGS = $(EPETRA_INCLUDES) CrsMatrix_test_LDADD = $(EPETRA_LIBS)

11. LAL Foundation: Petra  Petra provides a “common language” for distributed linear algebra objects (operator, matrix, vector)  Petra provides distributed matrix and vector services.  Has 3 implementations under development.

12. Perform redistribution of distributed objects: Parallel permutations. “Ghosting” of values for local computations. Collection of partial results from remote processors. Petra Object Model Abstract Interface to Parallel Machine Shameless mimic of MPI interface. Keeps MPI dependence to a single class (through all of Trilinos!). Allow trivial serial implementation. Opens door to novel parallel libraries (shmem, UPC, etc…) Abstract Interface for Sparse All-to-All Communication Supports construction of pre-recorded “plan” for data-driven communications. Examples: Supports gathering/scatter of off-processor x/y values when computing y = Ax. Gathering overlap rows for Overlapping Schwarz. Redistribution of matrices, vectors, etc… Describes layout of distributed objects: Vectors: Number of vector entries on each processor and global ID Matrices/graphs: Rows/Columns managed by a processor. Called “Maps” in Epetra. Dense Distributed Vector and Matrices: Simple local data structure. BLAS-able, LAPACK-able. Ghostable, redistributable. RTOp-able. Base Class for All Distributed Objects: Performs all communication. Requires Check, Pack, Unpack methods from derived class. Graph class for structure-only computations: Reusable matrix structure. Pattern-based preconditioners. Pattern-based load balancing tools. Basic sparse matrix class: Flexible construction process. Arbitrary entry placement on parallel machine.

13. Petra Implementations  Three version under development:  Epetra (Essential Petra):  Current production version.  Restricted to real, double precision arithmetic.  Uses stable core subset of C++ (circa 2000).  Interfaces accessible to C and Fortran users.  Tpetra (Templated Petra):  Next generation C++ version.  Templated scalar and ordinal fields.  Uses namespaces, and STL: Improved usability/efficiency.  Jpetra (Java Petra):  Pure Java. Portable to any JVM.  Interfaces to Java versions of MPI, LAPACK and BLAS via interfaces.

14. Details about Epetra Maps  Note: Focus on Maps (not BlockMaps).  Getting beyond standard use case…

15. 1-to-1 Maps  1-to-1 map (defn): A map is 1-to-1 if each GID appears only once in the map (and is therefore associated with only a single processor).  Certain operations in parallel data repartitioning require 1- to-1 maps. Specifically:  The source map of an import must be 1-to-1.  The target map of an export must be 1-to-1.  The domain map of a 2D object must be 1-to-1.  The range map of a 2D object must be 1-to-1.

16. 2D Objects: Four Maps  Epetra 2D objects:  CrsMatrix, FECrsMatrix  CrsGraph  VbrMatrix, FEVbrMatrix  Have four maps:  RowMap: On each processor, the GIDs of the rows that processor will “manage”.  ColMap: On each processor, the GIDs of the columns that processor will “manage”.  DomainMap: The layout of domain objects (the x vector/multivector in y=Ax).  RangeMap: The layout of range objects (the y vector/multivector in y=Ax). Must be 1-to-1 maps!!! Typically a 1-to-1 map Typically NOT a 1-to-1 map

17. Sample Problem = yA x

18. Case 1: Standard Approach  RowMap = {0, 1}  ColMap = {0, 1, 2}  DomainMap = {0, 1}  RangeMap = {0, 1}  First 2 rows of A, elements of y and elements of x, kept on PE 0.  Last row of A, element of y and element of x, kept on PE 1. PE 0 ContentsPE 1 Contents  RowMap = {2}  ColMap = {1, 2}  DomainMap = {2}  RangeMap = {2} Notes:  Rows are wholly owned.  RowMap=DomainMap=RangeMap (all 1-to-1).  ColMap is NOT 1-to-1.  Call to FillComplete: A.FillComplete(); // Assumes = yAx Original Problem

19. Case 2: Twist 1  RowMap = {0, 1}  ColMap = {0, 1, 2}  DomainMap = {1, 2}  RangeMap = {0}  First 2 rows of A, first element of y and last 2 elements of x, kept on PE 0.  Last row of A, last 2 element of y and first element of x, kept on PE 1. PE 0 ContentsPE 1 Contents  RowMap = {2}  ColMap = {1, 2}  DomainMap = {0}  RangeMap = {1, 2} Notes:  Rows are wholly owned.  RowMap is NOT = DomainMap is NOT = RangeMap (all 1-to-1).  ColMap is NOT 1-to-1.  Call to FillComplete: A.FillComplete(DomainMap, RangeMap); = yAx Original Problem

20. Case 2: Twist 2  RowMap = {0, 1}  ColMap = {0, 1}  DomainMap = {1, 2}  RangeMap = {0}  First row of A, part of second row of A, first element of y and last 2 elements of x, kept on PE 0.  Last row, part of second row of A, last 2 element of y and first element of x, kept on PE 1. PE 0 ContentsPE 1 Contents  RowMap = {1, 2}  ColMap = {1, 2}  DomainMap = {0}  RangeMap = {1, 2} Notes:  Rows are NOT wholly owned.  RowMap is NOT = DomainMap is NOT = RangeMap (all 1-to-1).  RowMap and ColMap are NOT 1-to-1.  Call to FillComplete: A.FillComplete(DomainMap, RangeMap); = yAx Original Problem

21. What does FillComplete Do?  A bunch of stuff.  One task is to create (if needed) import/export objects to support distributed matrix-vector multiplication:  If ColMap ≠ DomainMap, create Import object.  If RowMap ≠ RangeMap, create Export object.  A few rules:  Rectangular matrices will always require: A.FillComplete(DomainMap,RangeMap);  DomainMap and RangeMap must be 1-to-1.

22. Parallel Data Redistribution  Epetra vectors, multivectors, graphs and matrices are distributed via one of the map objects.  A map is basically a partitioning of a list of global IDs:  IDs are simply labels, no need to use contiguous values (Directory class handles details for general ID lists).  No a priori restriction on replicated IDs.  If we are given:  A source map and  A set of vectors, multivectors, graphs and matrices (or other distributable objects) based on source map.  Redistribution is performed by: 1.Specifying a target map with a new distribution of the global IDs. 2.Creating Import or Export object using the source and target maps. 3.Creating vectors, multivectors, graphs and matrices that are redistributed (to target map layout) using the Import/Export object.

23. Example: epetra/ex9.cpp int main(int argc, char *argv[]) { MPI_Init(&argc, &argv); Epetra_MpiComm Comm(MPI_COMM_WORLD); int NumGlobalElements = 4; // global dimension of the problem int NumMyElements; // local nodes Epetra_IntSerialDenseVector MyGlobalElements; if( Comm.MyPID() == 0 ) { NumMyElements = 3; MyGlobalElements.Size(NumMyElements); MyGlobalElements[0] = 0; MyGlobalElements[1] = 1; MyGlobalElements[2] = 2; } else { NumMyElements = 3; MyGlobalElements.Size(NumMyElements); MyGlobalElements[0] = 1; MyGlobalElements[1] = 2; MyGlobalElements[2] = 3; } // create a map Epetra_Map Map(-1,MyGlobalElements.Length(), MyGlobalElements.Values(),0, Comm); // create a vector based on map Epetra_Vector xxx(Map); for( int i=0 ; i<NumMyElements ; ++i ) xxx[i] = 10*( Comm.MyPID()+1 ); if( Comm.MyPID() == 0 ){ double val = 12; int pos = 3; xxx.SumIntoGlobalValues(1,0,&val,&pos); } cout << xxx; // create a target map, in which all elements are on proc 0 int NumMyElements_target; if( Comm.MyPID() == 0 ) NumMyElements_target = NumGlobalElements; else NumMyElements_target = 0; Epetra_Map TargetMap(-1,NumMyElements_target,0,Comm); Epetra_Export Exporter(Map,TargetMap); // work on vectors Epetra_Vector yyy(TargetMap); yyy.Export(xxx,Exporter,Add); cout << yyy; MPI_Finalize(); return( EXIT_SUCCESS ); }

24. Output: epetra/ex9.cpp > mpirun -np 2./ex9.exe Epetra::Vector MyPID GID Value 0 0 10 0 1 10 0 2 10 Epetra::Vector 1 1 20 1 2 20 1 3 20 Epetra::Vector MyPID GID Value 0 0 10 0 1 30 0 2 30 0 3 20 Epetra::Vector PE 0 xxx(0)=10 xxx(1)=10 xxx(2)=10 PE 1 xxx(1)=20 xxx(2)=20 xxx(3)=20 PE 0 yyy(0)=10 yyy(1)=30 yyy(2)=30 yyy(3)=20 PE 1 Export/Add Before ExportAfter Export

25. Import vs. Export  Import (Export) means calling processor knows what it wants to receive (send).  Distinction between Import/Export is important to user, almost identical in implementation.  Import (Export) objects can be used to do an Export (Import) as a reverse operation.  When mapping is bijective (1-to-1 and onto), either Import or Export is appropriate.

26. Example: 1D Matrix Assembly a b -u xx = f u(a) =  0 u(b) =  1 x1x1 x2x2 x3x3 PE 0PE 1 3 Equations: Find u at x 1, x 2 and x 3 Equation for u at x 2 gets a contribution from PE 0 and PE 1. Would like to compute partial contributions independently. Then combine partial results.

27. Two Maps  We need two maps:  Assembly map: PE 0: { 1, 2 }. PE 1: { 2, 3 }.  Solver map: PE 0: { 1, 2 } (we arbitrate ownership of 2). PE 1: { 3 }.

28. End of Assembly Phase  At the end of assembly phase we have AssemblyMatrix: On PE 0: On PE 1:  Want to assign all of Equation 2 to PE 0 for use with solver.  NOTE: For a class of Neumann-Neumann preconditioners, the above layout is exactly what we want. Equation 1: Equation 2: Equation 2: Equation 3: Row 2 is shared

29. Export Assembly Matrix to Solver Matrix Epetra_Export Exporter(AssemblyMap, SolverMap); Epetra_CrsMatrix SolverMatrix (Copy, SolverMap, 0); SolverMatrix.Export(AssemblyMatrix, Exporter, Add); SolverMatrix.FillComplete();

30. Matrix Export Equation 1: Equation 2: Equation 3: Equation 1: Equation 2: Equation 2: Equation 3: PE 0 PE 1 Before Export After Export Export/Add

31. Example: epetraext/ex2.cpp int main(int argc, char *argv[]) { MPI_Init(&argc,&argv); Epetra_MpiComm Comm (MPI_COMM_WORLD); int MyPID = Comm.MyPID(); int n=4; // Generate Laplacian2d gallery matrix Trilinos_Util::CrsMatrixGallery G("laplace_2d", Comm); G.Set("problem_size", n*n); G.Set("map_type", "linear"); // Linear map initially // Get the LinearProblem. Epetra_LinearProblem *Prob = G.GetLinearProblem(); // Get the exact solution. Epetra_MultiVector *sol = G.GetExactSolution(); // Get the rhs (b) and lhs (x) Epetra_MultiVector *b = Prob->GetRHS(); Epetra_MultiVector *x = Prob->GetLHS(); // Repartition graph using Zoltan EpetraExt::Zoltan_CrsGraph * ZoltanTrans = new EpetraExt::Zoltan_CrsGraph(); EpetraExt::LinearProblem_GraphTrans * ZoltanLPTrans = new EpetraExt::LinearProblem_GraphTrans( *(dynamic_cast *>(ZoltanTrans)) ); cout << "Creating Load Balanced Linear Problem\n"; Epetra_LinearProblem &BalancedProb = (*ZoltanLPTrans)(*Prob); // Get the rhs (b) and lhs (x) Epetra_MultiVector *Balancedb = Prob->GetRHS(); Epetra_MultiVector *Balancedx = Prob->GetLHS(); cout << "Balanced b: " << *Balancedb << endl; cout << "Balanced x: " << *Balancedx << endl; MPI_Finalize() ; return 0 ; }

32. Need for Import/Export  Solvers for complex engineering applications need expressive, easy-to-use parallel data redistribution:  Allows better scaling for non-uniform overlapping Schwarz.  Necessary for robust solution of multiphysics problems.  We have found import and export facilities to be a very natural and powerful technique to address these issues.

33. Extending Capabilities: Preconditioners, Operators, Matrices Illustrated using AztecOO as example

34. Epetra User Class Categories  Sparse Matrices: RowMatrix, (CrsMatrix, VbrMatrix, FECrsMatrix, FEVbrMatrix)  Linear Operator:Operator: (AztecOO, ML, Ifpack)  Dense Matrices:DenseMatrix, DenseVector, BLAS, LAPACK, SerialDenseSolver  Vectors: Vector, MultiVector  Graphs:CrsGraph  Data Layout: Map, BlockMap, LocalMap  Redistribution:Import, Export, LbGraph, LbMatrix  Aggregates: LinearProblem  Parallel Machine: Comm, (SerialComm, MpiComm, MpiSmpComm)  Utilities: Time, Flops

35. LinearProblem Class  A linear problem is defined by:  Matrix A : An Epetra_RowMatrix or Epetra_Operator object. (often a CrsMatrix or VbrMatrix object.)  Vectors x, b : Vector objects.  To call AztecOO, first define a LinearProblem:  Constructed from A, x and b.  Once defined, can: Scale the problem (explicit preconditioning). Precondition it (implicitly). Change x and b.

36. AztecOO  Aztec is the previous workhorse solver at Sandia:  Extracted from the MPSalsa reacting flow code.  Installed in dozens of Sandia apps.  AztecOO leverages the investment in Aztec:  Uses Aztec iterative methods and preconditioners.  AztecOO improves on Aztec by:  Using Epetra objects for defining matrix and RHS.  Providing more preconditioners/scalings.  Using C++ class design to enable more sophisticated use.  AztecOO interfaces allows:  Continued use of Aztec for functionality.  Introduction of new solver capabilities outside of Aztec.  Belos is coming along as alternative.  AztecOO will not go away.  Will encourage new efforts and refactorings to use Belos.

37. A Simple Epetra/AztecOO Program // Header files omitted… int main(int argc, char *argv[]) { MPI_Init(&argc,&argv); // Initialize MPI, MpiComm Epetra_MpiComm Comm( MPI_COMM_WORLD ); // ***** Create x and b vectors ***** Epetra_Vector x(Map); Epetra_Vector b(Map); b.Random(); // Fill RHS with random #s // ***** Create an Epetra_Matrix tridiag(-1,2,-1) ***** Epetra_CrsMatrix A(Copy, Map, 3); double negOne = -1.0; double posTwo = 2.0; for (int i=0; i<NumMyElements; i++) { int GlobalRow = A.GRID(i); int RowLess1 = GlobalRow - 1; int RowPlus1 = GlobalRow + 1; if (RowLess1!=-1) A.InsertGlobalValues(GlobalRow, 1, &negOne, &RowLess1); if (RowPlus1!=NumGlobalElements) A.InsertGlobalValues(GlobalRow, 1, &negOne, &RowPlus1); A.InsertGlobalValues(GlobalRow, 1, &posTwo, &GlobalRow); } A.FillComplete(); // Transform from GIDs to LIDs // ***** Map puts same number of equations on each pe ***** int NumMyElements = 1000 ; Epetra_Map Map(-1, NumMyElements, 0, Comm); int NumGlobalElements = Map.NumGlobalElements(); // ***** Report results, finish *********************** cout << "Solver performed " << solver.NumIters() << " iterations." << endl << "Norm of true residual = " << solver.TrueResidual() << endl; MPI_Finalize() ; return 0; } // ***** Create/define AztecOO instance, solve ***** AztecOO solver(problem); solver.SetAztecOption(AZ_precond, AZ_Jacobi); solver.Iterate(1000, 1.0E-8); // ***** Create Linear Problem ***** Epetra_LinearProblem problem(&A, &x, &b);

38. AztecOO Extensibility  AztecOO is designed to accept externally defined:  Operators (both A and M): The linear operator A is accessed as an Epetra_Operator. Users can register a preconstructed preconditioner as an Epetra_Operator.  RowMatrix: If A is registered as a RowMatrix, Aztec’s preconditioners are accessible. Alternatively M can be registered separately as an Epetra_RowMatrix, and Aztec’s preconditioners are accessible.  StatusTests: Aztec’s standard stopping criteria are accessible. Can override these mechanisms by registering a StatusTest Object.

39. AztecOO understands Epetra_Operator Epetra_Operator Methods Documentation  AztecOO is designed to accept externally defined:  Operators (both A and M).  RowMatrix (Facilitates use of AztecOO preconditioners with external A).  StatusTests (externally- defined stopping criteria).

40. AztecOO Understands Epetra_RowMatrix Epetra_RowMatrix Methods

41. AztecOO UserOp/UserMat Recursive Call Example Trilinos/packages/aztecoo/example/AztecOO_RecursiveCall 1. Poisson2dOperator A(nx, ny, comm); // Generate nx by ny Poisson operator 2. Epetra_CrsMatrix * precMatrix = A.GeneratePrecMatrix(); // Build tridiagonal approximate Poisson 3. Epetra_Vector xx(A.OperatorDomainMap()); // Generate vectors (xx will be used to generate RHS b) 4. Epetra_Vector x(A.OperatorDomainMap()); 5. Epetra_Vector b(A.OperatorRangeMap()); 6. xx.Random(); // Generate exact x and then rhs b 7. A.Apply(xx, b); 8. // Build AztecOO solver that will be used as a preconditioner 9. Epetra_LinearProblem precProblem; 10. precProblem.SetOperator(precMatrix); 11. AztecOO precSolver(precProblem); 12. precSolver.SetAztecOption(AZ_precond, AZ_ls); 13. precSolver.SetAztecOption(AZ_output, AZ_none); 14. precSolver.SetAztecOption(AZ_solver, AZ_cg); 15. AztecOO_Operator precOperator(&precSolver, 20); 16. Epetra_LinearProblem problem(&A, &x, &b); // Construct linear problem 17. AztecOO solver(problem); // Construct solver 18. solver.SetPrecOperator(&precOperator); // Register Preconditioner operator 19. solver.SetAztecOption(AZ_solver, AZ_cg); 20. solver.Iterate(Niters, 1.0E-12);

42. Ifpack/AztecOO Example Trilinos/packages/aztecoo/example/IfpackAztecOO 1. // Assume A, x, b are define, LevelFill and Overlap are specified 2. Ifpack_IlukGraph IlukGraph(A.Graph(), LevelFill, Overlap); 3. IlukGraph.ConstructFilledGraph(); 4. Ifpack_CrsRiluk ILUK (IlukGraph); 5. ILUK.InitValues(A); 6. assert(ILUK->Factor()==0); // Note: All Epetra/Ifpack/AztecOO method return int err codes 7. double Condest; 8. ILUK.Condest(false, Condest); // Get condition estimate 9. if (Condest > tooBig) { 10. ILUK.SetAbsoluteThreshold(Athresh); 11. ILUK.SetRelativeThreshold(Rthresh); 12. Go back to line 4 and try again 13. } 14. Epetra_LinearProblem problem(&A, &x, &b); // Construct linear problem 15. AztecOO solver(problem); // Construct solver 16. solver.SetPrecOperator(&ILUK); // Register Preconditioner operator 17. solver.SetAztecOption(AZ_solver, AZ_cg); 18. solver.Iterate(Niters, 1.0E-12); 19.// Once this linear solutions complete and the next nonlinear step is advanced, 20.// we will return to the solver, but only need to execute steps 5 on down…

43. Multiple Stopping Criteria  Possible scenario for stopping an iterative solver:  Test 1: Make sure residual is decreased by 6 orders of magnitude. And  Test 2: Make sure that the inf-norm of true residual is no more 1.0E-8. But  Test 3: do no more than 200 iterations.  Note: Test 1 is cheap. Do it before Test 2.

44. AztecOO StatusTest classes  AztecOO_StatusTest:  Abstract base class for defining stopping criteria.  Combo class: OR, AND, SEQ AztecOO_StatusTest Methods

45. AztecOO/StatusTest Example Trilinos/packages/aztecoo/example/AztecOO 1. // Assume A, x, b are define 2.Epetra_LinearProblem problem(&A, &x, &b); // Construct linear problem 3. AztecOO solver(problem); // Construct solver 4.AztecOO_StatusTestResNorm restest1(A, x, bb, 1.0E-6); 5.restest1.DefineResForm(AztecOO_StatusTestResNorm::Implicit, AztecOO_StatusTestResNorm::TwoNorm); 6.restest1.DefineScaleForm(AztecOO_StatusTestResNorm::NormOfInitRes, AztecOO_StatusTestResNorm::TwoNorm); 7.AztecOO_StatusTestResNorm restest2(A, x, bb, 1.0E-8); 8.restest2.DefineResForm(AztecOO_StatusTestResNorm::Explicit, AztecOO_StatusTestResNorm::InfNorm); 9.restest2.DefineScaleForm(AztecOO_StatusTestResNorm::NormOfRHS, AztecOO_StatusTestResNorm::InfNorm); 10.AztecOO_StatusTestCombo comboTest1(AztecOO_StatusTestCombo::SEQ, restest1, restest2); 11.AztecOO_StatusTestMaxIters maxItersTest(200); 12.AztecOO_StatusTestCombo comboTest2(AztecOO_StatusTestCombo::OR, maxItersTest1, comboTest1); 13.solver.SetStatusTest(&comboTest2); 14. solver.SetAztecOption(AZ_solver, AZ_cg); 15. solver.Iterate(Niters, 1.0E-12);

46. Summary: Extending Capabilities  Trilinos packages are designed to interoperate.  All packages (ML, IFPACK, AztecOO, …) that can provide linear operators:  Implement the Epetra_Operator interface.  Are available to any package that can use an linear operator.  All packages (ML, AztecOO, NOX, Belos, Anasazi, …) that can use linear operators:  Accept linear operator via Epetra_Operator interface.  Support easy user extensions.  All packages (ML, IFPACK, AztecOO, …) that need matrix coefficient data:  Can access that data from Epetra_RowMatrix interface.  Can use any concrete Epetra matrix class, or any user-provided adapter.

47. Summary: Extending Capabilities AztecOO is one example:  Flexibility comes from abstract base classes: Epetra_Operator: –All Epetra matrix classes implement. –Best way to define A and M when coefficient info not needed. Epetra_RowMatrix: –All Epetra matrix classes implement. –Best way to define A and M when coefficient info is needed. AztecOO_StatusTest: –A suite of parametrized status tests. –An abstract interface for users to define their own. –Ability to combine tests for sophisticated control of stopping.