ex9.py: Generalized symmetric-definite eigenproblem
===================================================

This example computes eigenvalues and eigenvectors of a generalized
symmetric-definite eigenvalue problem, where the first matrix is the
discrete Laplacian in two dimensions and the second matrix is quasi
diagonal.

The full source code for this demo can be `downloaded here
<../_static/ex9.py>`__.

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

This function builds the discretized Laplacian operator in 2 dimensions.

::

  def Laplacian2D(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 builds a quasi-diagonal matrix. It is two times the identity
matrix except for the 2x2 leading submatrix ``[6 -1; -1 1]``.

::

  def QuasiDiagonal(N):
      # Create matrix
      B = PETSc.Mat().create()
      B.setSizes([N, N])
      B.setFromOptions()
      # Fill matrix
      Istart, Iend = B.getOwnershipRange()
      for I in range(Istart, Iend):
          B[I,I] = 2.0
      if Istart==0:
          B[0,0] = 6.0
          B[0,1] = -1.0
          B[1,0] = -1.0
          B[1,1] = 1.0
      B.assemble()
      return B

The following function receives the two matrices and solves the
eigenproblem. In this example we illustrate how to pass objects
that have been created beforehand, instead of extracting the internal
objects. We are using a spectral transformation of type `ST.Type.PRECOND`
and a Block Jacobi preconditioner. We want to compute the leftmost
eigenvalues. The selected eigensolver is LOBPCG, which is appropriate
for this use case. After the solve, we print the computed solution.

::

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

      pc = PETSc.PC().create()
      # pc.setType(pc.Type.HYPRE)
      pc.setType(pc.Type.BJACOBI)

      ksp = PETSc.KSP().create()
      ksp.setType(ksp.Type.PREONLY)
      ksp.setPC( pc )

      F = SLEPc.ST().create()
      F.setType(F.Type.PRECOND)
      F.setKSP( ksp )
      F.setShift(0)

      # Setup the eigensolver
      E = SLEPc.EPS().create()
      E.setST(F)
      E.setOperators(A,B)
      E.setType(E.Type.LOBPCG)
      E.setDimensions(10,PETSc.DECIDE)
      E.setWhichEigenpairs(E.Which.SMALLEST_REAL)
      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 functions.

::

  def main():
      opts = PETSc.Options()
      N = opts.getInt('N', 10)
      m = opts.getInt('m', N)
      n = opts.getInt('n', m)
      Print("Symmetric-definite Eigenproblem, N=%d (%dx%d grid)" % (m*n, m, n))
      A = Laplacian2D(m,n)
      B = QuasiDiagonal(m*n)
      solve_eigensystem(A,B)

  if __name__ == '__main__':
      main()