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# ------------------------------------------------------------------------
# Solve 1-D PDE
# -u'' = lambda*u
# on [0,1] subject to
# u(0)=0, u'(1)=u(1)*lambda*kappa/(kappa-lambda)
# ------------------------------------------------------------------------
import sys, slepc4py
slepc4py.init(sys.argv)
from petsc4py import PETSc
from slepc4py import SLEPc
from numpy import sqrt, sin
Print = PETSc.Sys.Print
class MyPDE(object):
def __init__(self, kappa, h):
self.kappa = kappa
self.h = h
def formFunction(self, nep, mu, F, B):
n, m = F.getSize()
Istart, Iend = F.getOwnershipRange()
i1 = Istart
if Istart==0: i1 = i1 + 1
i2 = Iend
if Iend==n: i2 = i2 - 1
h = self.h
c = self.kappa/(mu-self.kappa)
d = n
# Interior grid points
for i in range(i1,i2):
val = -d-mu*h/6.0
F[i,i-1] = val
F[i,i] = 2.0*(d-mu*h/3.0)
F[i,i+1] = val
# Boundary points
if Istart==0:
F[0,0] = 2.0*(d-mu*h/3.0)
F[0,1] = -d-mu*h/6.0
if Iend==n:
F[n-1,n-2] = -d-mu*h/6.0
F[n-1,n-1] = d-mu*h/3.0+mu*c
F.assemble()
if B != F: B.assemble()
return PETSc.Mat.Structure.SAME_NONZERO_PATTERN
def formJacobian(self, nep, mu, F):
n, m = J.getSize()
Istart, Iend = J.getOwnershipRange()
i1 = Istart
if Istart==0: i1 = i1 + 1
i2 = Iend
if Iend==n: i2 = i2 - 1
h = self.h
c = self.kappa/(mu-self.kappa)
# Interior grid points
for i in range(i1,i2):
J[i,i-1] = -h/6.0
J[i,i] = -2.0*h/3.0
J[i,i+1] = -h/6.0
# Boundary points
if Istart==0:
J[0,0] = -2.0*h/3.0
J[0,1] = -h/6.0
if Iend==n:
J[n-1,n-2] = -h/6.0
J[n-1,n-1] = -h/3.0-c*c
J.assemble()
return PETSc.Mat.Structure.SAME_NONZERO_PATTERN
def checkSolution(self, mu, y):
nu = sqrt(mu)
u = y.duplicate()
n = u.getSize()
Istart, Iend = J.getOwnershipRange()
h = self.h
for i in range(Istart,Iend):
x = (i+1)*h
u[i] = sin(nu*x);
u.assemble()
u.normalize()
u.axpy(-1.0,y)
return u.norm()
def FixSign(x):
# Force the eigenfunction to be real and positive, since
# some eigensolvers may return the eigenvector multiplied
# by a complex number of modulus one.
comm = x.getComm()
rank = comm.getRank()
n = 1 if rank == 0 else 0
aux = PETSc.Vec().createMPI((n, PETSc.DECIDE), comm=comm)
if rank == 0: aux[0] = x[0]
aux.assemble()
x0 = aux.sum()
sign = x0/abs(x0)
x.scale(1.0/sign)
opts = PETSc.Options()
n = opts.getInt('n', 128)
kappa = opts.getReal('kappa', 1.0)
pde = MyPDE(kappa, 1.0/n)
# Setup the solver
nep = SLEPc.NEP().create()
F = PETSc.Mat().create()
F.setSizes([n, n])
F.setType('aij')
F.setPreallocationNNZ(3)
nep.setFunction(pde.formFunction, F)
J = PETSc.Mat().create()
J.setSizes([n, n])
J.setType('aij')
J.setPreallocationNNZ(3)
nep.setJacobian(pde.formJacobian, J)
nep.setTolerances(tol=1e-9)
nep.setDimensions(1)
nep.setFromOptions()
# Solve the problem
x = F.createVecs('right')
x.set(1.0)
nep.setInitialSpace(x)
nep.solve()
its = nep.getIterationNumber()
Print("Number of iterations of the method: %i" % its)
sol_type = nep.getType()
Print("Solution method: %s" % sol_type)
nev, ncv, mpd = nep.getDimensions()
Print("")
Print("Subspace dimension: %i" % ncv)
tol, maxit = nep.getTolerances()
Print("Stopping condition: tol=%.4g" % tol)
Print("")
nconv = nep.getConverged()
Print( "Number of converged eigenpairs %d" % nconv )
if nconv > 0:
Print()
Print(" k ||T(k)x|| error ")
Print("----------------- ------------------ ------------------")
for i in range(nconv):
k = nep.getEigenpair(i, x)
FixSign(x)
res = nep.computeError(i)
error = pde.checkSolution(k.real, x)
if k.imag != 0.0:
Print( " %9f%+9f j %12g %12g" % (k.real, k.imag, res, error) )
else:
Print( " %12f %12g %12g" % (k.real, res, error) )
Print()
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