Tutorial — SLEPc for Python 3.15.1 documentation

# Tutorial

This tutorial is intended for basic use of slepc4py. For more advanced use, the reader is referred to SLEPc tutorials as well as to slepc4py reference documentation.

## Commented source of a simple example

In this section, we include the source code of example `demo/ex1.py` available in the slepc4py distribution, with comments inserted inline.

The first thing to do is initialize the libraries. This is normally not required, as it is done automatically at import time. However, if you want to gain access to the facilities for accesing command-line options, the following lines must be executed by the main script prior to any petsc4py or slepc4py calls:

    import sys, slepc4py
    slepc4py.init(sys.argv)

Next, we have to import the relevant modules. Normally, both PETSc and SLEPc modules have to be imported in all slepc4py programs. It may be useful to import NumPy as well:

    from petsc4py import PETSc
    from slepc4py import SLEPc
    import numpy

At this point, we can use any petsc4py and slepc4py operations. For instance, the following lines allow the user to specify an integer command-line argument `n` with a default value of 30 (see the next section for example usage of command-line options):

    opts = PETSc.Options()
    n = opts.getInt('n', 30)

It is necessary to build a matrix to define an eigenproblem (or two in the case of generalized eigenproblems). The following fragment of code creates the matrix object and then fills the non-zero elements one by one. The matrix of this particular example is tridiagonal, with value 2 in the diagonal, and -1 in off-diagonal positions. See petsc4py documentation for details about matrix objects:

    A = PETSc.Mat().create()
    A.setSizes([n, n])
    A.setFromOptions()
    A.setUp()

    rstart, rend = A.getOwnershipRange()

    # first row
    if rstart == 0:
        A[0, :2] = [2, -1]
        rstart += 1
    # last row
    if rend == n:
        A[n-1, -2:] = [-1, 2]
        rend -= 1
    # other rows
    for i in range(rstart, rend):
        A[i, i-1:i+2] = [-1, 2, -1]

    A.assemble()

The solver object is created in a similar way as other objects in petsc4py:

    E = SLEPc.EPS(); E.create()

Once the object is created, the eigenvalue problem must be specified. At least one matrix must be provided. The problem type must be indicated as well, in this case it is HEP (Hermitian eigenvalue problem). Apart from these, other settings could be provided here (for instance, the tolerance for the computation). After all options have been set, the user should call the `setFromOptions()` operation, so that any options specified at run time in the command line are passed to the solver object:

    E.setOperators(A)
    E.setProblemType(SLEPc.EPS.ProblemType.HEP)
    E.setFromOptions()

After that, the `solve()` method will run the selected eigensolver, keeping the solution stored internally:

    E.solve()

Once the computation has finished, we are ready to print the results. First, some informative data can be retrieved from the solver object:

    Print = PETSc.Sys.Print

    Print()
    Print("******************************")
    Print("*** SLEPc Solution Results ***")
    Print("******************************")
    Print()

    its = E.getIterationNumber()
    Print("Number of iterations of the method: %d" % its)

    eps_type = E.getType()
    Print("Solution method: %s" % eps_type)

    nev, ncv, mpd = E.getDimensions()
    Print("Number of requested eigenvalues: %d" % nev)

    tol, maxit = E.getTolerances()
    Print("Stopping condition: tol=%.4g, maxit=%d" % (tol, maxit))

For retrieving the solution, it is necessary to find out how many eigenpairs have converged to the requested precision:

    nconv = E.getConverged()
    Print("Number of converged eigenpairs %d" % nconv)

For each of the `nconv` eigenpairs, we can retrieve the eigenvalue `k`, and the eigenvector, which is represented by means of two petsc4py vectors `vr` and `vi` (the real and imaginary part of the eigenvector, since for real matrices the eigenvalue and eigenvector may be complex). We also compute the corresponding relative errors in order to make sure that the computed solution is indeed correct:

    if nconv > 0:
        # Create the results vectors
        vr, wr = A.getVecs()
        vi, wi = A.getVecs()
        #
        Print()
        Print("        k          ||Ax-kx||/||kx|| ")
        Print("----------------- ------------------")
        for i in range(nconv):
            k = E.getEigenpair(i, vr, vi)
            error = E.computeError(i)
            if k.imag != 0.0:
                Print(" %9f%+9f j %12g" % (k.real, k.imag, error))
            else:
                Print(" %12f      %12g" % (k.real, error))
        Print()

## Example of command-line usage

Now we illustrate how to specify command-line options in order to extract the full potential of slepc4py.

A simple execution of the `demo/ex1.py` script will result in the following output:

    $ python demo/ex1.py

    ******************************
    *** SLEPc Solution Results ***
    ******************************

    Number of iterations of the method: 4
    Solution method: krylovschur
    Number of requested eigenvalues: 1
    Stopping condition: tol=1e-07, maxit=100
    Number of converged eigenpairs 4

        k          ||Ax-kx||/||kx||
    ----------------- ------------------
         3.989739        5.76012e-09
         3.959060        1.41957e-08
         3.908279        6.74118e-08
         3.837916        8.34269e-08

For specifying different setting for the solver parameters, we can use SLEPc command-line options with the `-eps` prefix. For instance, to change the number of requested eigenvalues and the tolerance:

    $ python demo/ex1.py -eps_nev 10 -eps_tol 1e-11

The method used by the solver object can also be set at run time:

    $ python demo/ex1.py -eps_type subspace

All the above settings can also be changed within the source code by making use of the appropriate slepc4py method. Since options can be set from within the code and the command-line, it is often useful to view the particular settings that are currently being used:

    $ python demo/ex1.py -eps_view

    EPS Object: 1 MPI processes
      type: krylovschur
        50% of basis vectors kept after restart
        using the locking variant
      problem type: symmetric eigenvalue problem
      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
    BV Object: 1 MPI processes
      type: svec
      17 columns of global length 30
      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: hep
      solving the problem with: Implicit QR method (_steqr)
    ST Object: 1 MPI processes
      type: shift
      shift: 0
      number of matrices: 1

Note that for computing eigenvalues of smallest magnitude we can use the option `-eps_smallest_magnitude`, but for interior eigenvalues things are not so straightforward. One possibility is to try with harmonic extraction, for instance to get the eigenvalues closest to 0.6:

    $ python demo/ex1.py -eps_harmonic -eps_target 0.6

Depending on the problem, harmonic extraction may fail to converge. In those cases, it is necessary to specify a spectral transformation other than the default. In the command-line, this is indicated with the `-st_` prefix. For example, shift-and-invert with a value of the shift equal to 0.6 would be:

    $ python demo/ex1.py -st_type sinvert -eps_target 0.6