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# Exercise 8: Quadratic Eigenvalue Problem

Now we are going to focus on the solution of quadratic eigenvalue problems with `PEP` solvers. In this case, the problem to be solved is formulated as \((\lambda^{2}M+ \lambda C+K)x=0\). In our simple example, \(M\) is a diagonal matrix, \(C\) is tridiagonal, and \(K\) is the 2-D Laplacian.

## Compiling

Copy the file ex16.c to your directory and add these lines to the makefile

    ex16: ex16.o
    	-${CLINKER} -o ex16 ex16.o ${SLEPC_PEP_LIB}
    	${RM} ex16.o

Build the executable with the command

    $ make ex16

## Running the Program

Run the program without arguments and see the output:

    Quadratic Eigenproblem, N=100 (10x10 grid)

     Number of requested eigenvalues: 1

               k          ||P(k)x||/||kx||
       ----------------- ------------------
     -1.164037+1.653625i    2.02021e-12
     -1.164037-1.653625i    2.02021e-12

## Source Code Details

The `PEP` object is used very much like `EPS` or `SVD`, as can be seen in the source code. Here is a summary of the main function calls:

- `PEPCreate(MPI_Comm comm,PEP *pep);`

- `PEPSetOperators`(`PEP` pep,PetscInt `nmat`,Mat `A[]`);

- `PEPSetProblemType(PEP pep,PEPProblemType type);`

- `PEPSetFromOptions(PEP pep);`

- `PEPSolve(PEP pep);`

- `PEPGetConverged`(`PEP` `pep`, PetscInt `*nconv`);

- `PEPGetEigenpair`(`PEP` `pep`,PetscInt `i`,PetscScalar `*kr`,PetscScalar `*ki`,Vec `xr`,Vec `xi`);

- `PEPDestroy(PEP pep)`;

First, the solver context (`PEP`) is created and the three problem matrices are specified. Then various options are set for customized solution. After that, the program solves the problem, retrieves the solution, and finally destroys the context.

## PEP Options

Most of the options available in the `EPS` object have their equivalent in `PEP`. A full list of command-line options can be obtained by running the example with the option `-help`.

To show information about the `PEP` solver, add the `-pep_view` option:

    PEP Object: 1 MPI processes
      type: toar
        50% of basis vectors kept after restart
        using the locking variant
      problem type: symmetric polynomial eigenvalue problem
      polynomial represented in MONOMIAL basis
      selected portion of the spectrum: largest eigenvalues in magnitude
      number of eigenvalues (nev): 1
      number of column vectors (ncv): 16
      maximum dimension of projected problem (mpd): 16
      maximum number of iterations: 100
      tolerance: 1e-08
      convergence test: relative to the eigenvalue
      extraction type: NORM
    BV Object: 1 MPI processes
      type: svec
      18 columns of global length 100
      vector orthogonalization method: classical Gram-Schmidt
      orthogonalization refinement: if needed (eta: 0.7071)
      block orthogonalization method: GS
      doing matmult as a single matrix-matrix product
    DS Object: 1 MPI processes
      type: nhep
    ST Object: 1 MPI processes
      type: shift
      shift: 0.
      number of matrices: 3
      all matrices have different nonzero pattern
      KSP Object: (st_) 1 MPI processes
        type: preonly
        maximum iterations=10000, initial guess is zero
        tolerances:  relative=1e-08, absolute=1e-50, divergence=10000.
        left preconditioning
        not checking for convergence
      PC Object: (st_) 1 MPI processes
        type: lu
          out-of-place factorization
          tolerance for zero pivot 2.22045e-14
          matrix ordering: nd
          factor fill ratio given 5., needed 1.
            Factored matrix:
              Mat Object: 1 MPI processes
                type: seqaij
                rows=100, cols=100
                package used to perform factorization: petsc
                total: nonzeros=100, allocated nonzeros=100
                total number of mallocs used during MatSetValues calls =0
                  not using I-node routines
        linear system matrix, which is also used to construct the preconditioner:
        Mat Object: 1 MPI processes
          type: seqaij
          rows=100, cols=100
          total: nonzeros=100, allocated nonzeros=500
          total number of mallocs used during MatSetValues calls =0
            not using I-node routines

Note

All the command-line options related to the `PEP` object have the `-pep_` prefix.

Try changing some of the values, for example:

    $ ./ex16 -pep_nev 4 -pep_ncv 24 -pep_smallest_magnitude -pep_tol 1e-5

## Choosing the Solver Method

Several polynomial eigensolvers are available, which can be selected in the source code with the function `PEPSetType()`, or at run time:

    $ ./ex16 -pep_type qarnoldi

Check the table of Polynomial eigenvalue solvers available in the PEP module.

Note

The default solver is `toar`.

The `linear` solver performs an explicit linearization of the quadratic eigenproblem, resulting in a generalized eigenproblem. This linearization can be customized, see section Linearization in the Users Manual for details. For instance:

    $ ./ex16 -pep_type linear -pep_linear_linearization 1,0 -pep_linear_explicitmatrix

Since in this problem all matrices are symmetric, the problem type is set to `PEP_HERMITIAN` in the source code with `PEPSetProblemType()`, and this obliges us to set the explicit matrix flag, see `PEPLinearSetExplicitMatrix()`. It is also possible to use a non-symmetric linearization by choosing the corresponding problem type:

    $ ./ex16 -pep_type linear -pep_general

In the `linear` solver it is also possible to tune any of the `EPS` options, including those corresponding to `ST` and the linear solvers. For instance:

    $ ./ex16 -pep_type linear -pep_general -pep_linear_st_ksp_type bcgs -pep_linear_st_pc_type jacobi