ex2.py: Standard symmetric eigenproblem for the 2-D Laplacian — slepc4py 3.24.3 documentation

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# ex2.py: Standard symmetric eigenproblem for the 2-D Laplacian

This example computes eigenvalues and eigenvectors of the discrete Laplacian on a two-dimensional domain with finite differences.

The full source code for this demo can be downloaded here.

Initialization is similar to previous examples.

    try: range = xrange
    except: pass

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

    from petsc4py import PETSc
    from slepc4py import SLEPc

    Print = PETSc.Sys.Print

In this example we have organized the code in several functions. This one builds the finite-difference Laplacian matrix by computing the indices of each entry. An alternative would be to use the functionality offered by `DMDA`.

    def construct_operator(m, n):
        # Create matrix for 2D Laplacian operator
        A = PETSc.Mat().create()
        A.setSizes([m*n, m*n])
        A.setFromOptions()
        # Fill matrix
        hx = 1.0/(m-1) # x grid spacing
        hy = 1.0/(n-1) # y grid spacing
        diagv = 2.0*hy/hx + 2.0*hx/hy
        offdx = -1.0*hy/hx
        offdy = -1.0*hx/hy
        Istart, Iend = A.getOwnershipRange()
        for I in range(Istart, Iend) :
            A[I,I] = diagv
            i = I//n    # map row number to
            j = I - i*n # grid coordinates
            if i> 0  : J = I-n; A[I,J] = offdx
            if i< m-1: J = I+n; A[I,J] = offdx
            if j> 0  : J = I-1; A[I,J] = offdy
            if j< n-1: J = I+1; A[I,J] = offdy
        A.assemble()
        return A

This function receives the matrix and the problem type, then solves the eigenvalue problem and prints information about the computed solution. Although we know that eigenvalues and eigenvectors are real in this example, the function is prepared to solve it as a non-symmetric problem, by passing `SLEPc.EPS.ProblemType.NHEP`, that is why the code handles possibly complex eigenvalues and eigenvectors.

    def solve_eigensystem(A, problem_type=SLEPc.EPS.ProblemType.HEP):
        # Create the result vectors
        xr, xi = A.createVecs()

        # Setup the eigensolver
        E = SLEPc.EPS().create()
        E.setOperators(A,None)
        E.setDimensions(3,PETSc.DECIDE)
        E.setProblemType(problem_type)
        E.setFromOptions()

        # Solve the eigensystem
        E.solve()

        Print("")
        its = E.getIterationNumber()
        Print("Number of iterations of the method: %i" % its)
        sol_type = E.getType()
        Print("Solution method: %s" % sol_type)
        nev, ncv, mpd = E.getDimensions()
        Print("Number of requested eigenvalues: %i" % nev)
        tol, maxit = E.getTolerances()
        Print("Stopping condition: tol=%.4g, maxit=%d" % (tol, maxit))
        nconv = E.getConverged()
        Print("Number of converged eigenpairs: %d" % nconv)
        if nconv > 0:
            Print("")
            Print("        k          ||Ax-kx||/||kx|| ")
            Print("----------------- ------------------")
            for i in range(nconv):
                k = E.getEigenpair(i, xr, xi)
                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("")

The main program simply processes three user-defined command-line options and calls the other two functions.

    def main():
        opts = PETSc.Options()
        N = opts.getInt('N', 32)
        m = opts.getInt('m', N)
        n = opts.getInt('n', m)
        Print("Symmetric Eigenproblem (sparse matrix), "
              "N=%d (%dx%d grid)" % (m*n, m, n))
        A = construct_operator(m,n)
        solve_eigensystem(A)

    if __name__ == '__main__':
        main()