### Eigensolver vs other solvers

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1

What types of solvers can be used for eigenvalue problems? Are eigenvalue solvers synonymous with linear or kylov solvers?

The following demo uses the SLEPcEigenSolver.

https://fenicsproject.org/olddocs/dolfin/1.3.0/python/demo/documented/eigenvalue/python/documentation.html

eigensolver.parameters.items() shows that the 'solver' is 'krylov-schur' (the method?). Why does this not show up in the following commands? Where could I find a list of methods for eigenvalue problems?

list_krylov_solver_methods()

list_linear_solver_methods()

The following demo uses the SLEPcEigenSolver.

https://fenicsproject.org/olddocs/dolfin/1.3.0/python/demo/documented/eigenvalue/python/documentation.html

eigensolver.parameters.items() shows that the 'solver' is 'krylov-schur' (the method?). Why does this not show up in the following commands? Where could I find a list of methods for eigenvalue problems?

list_krylov_solver_methods()

list_linear_solver_methods()

Community: FEniCS Project

### 1 Answer

2

Eigensolvers solve for the Eigenvalues \(\lambda\) and Eigenvectors \(v\) of a matrix \(A\) such that

\[

Av =\lambda v.

\]

The linear system solvers using the LU or Krylov methods listed by your function calls solve the system of equations

\[

Ax = b

\]

for the vector \(x\) using a known matrix \(A\) and vector \(b\).

Here is an example :

Av =\lambda v.

\]

The linear system solvers using the LU or Krylov methods listed by your function calls solve the system of equations

\[

Ax = b

\]

for the vector \(x\) using a known matrix \(A\) and vector \(b\).

Here is an example :

```
from pylab import *
from fenics import *
import sympy as sp
n = 8 # number of vertices in mesh
l = 1 # length of the domain
mesh = IntervalMesh(n, 0, l) # 1D mesh
Q = FunctionSpace(mesh, 'CG', 1) # linear-Lagrange F.S.
phi = TestFunction(Q)
u = TrialFunction(Q)
tol = 1e-14
def left(x, on_boundary): return on_boundary and abs(x[0]) < tol
def right(x, on_boundary): return on_boundary and abs(x[0] - l) < tol
gamma_l = DirichletBC(Q, 0.0, left) # left boundary condition
gamma_r = DirichletBC(Q, 0.0, right) # right boundary condition
k = inner(grad(phi), grad(u)) * dx # weak 2nd derivative matrix
m = phi * u * dx # mass matrix
K = PETScMatrix() # container for stiffness matrix
M = PETScMatrix() # container for mass matrix
K = assemble(k, tensor=K) # global assembly of stiffness mat.
M = assemble(m, tensor=M) # global assembly of mass mat.
gamma_r.apply(K) # apply right b.c. to K
gamma_l.apply(K) # apply left b.c. to K
gamma_r.apply(M) # apply right b.c. to M
gamma_l.apply(M) # apply left b.c. to M
eigensolver = SLEPcEigenSolver(K,M) # create solver for Kx = lambda Mx
eigensolver.parameters['solver'] = 'lapack'
eigensolver.solve() # solve with LAPACK
# generate the results and calculate the Fourier coef's with SymPy
x = sp.symbols('x')
xm = mesh.coordinates()[:,0][::-1]
xf = linspace(0,l,1000)
col = ['k', 'r', '#880cbc']
lam_m = []
lam_f = []
Am = []
Af = []
Av = []
cn = []
for i in range(n-1):
ep = eigensolver.get_eigenpair(n-2-i)
Afi = zeros(n)
lam_fi = ((i+1) * pi)**2
Afi = np.sin(np.sqrt(lam_fi) * xf)
cni = 2/l * sp.integrate(sp.sin((i+1)*pi*x/l), (x,0,l))
lam_m.append(ep[0])
lam_f.append(lam_fi)
e_v = ep[2].array()
if i == 1:
e_v = -e_v
Am.append(e_v)
Af.append(Afi)
cn.append(cni)
# plotting :
#===============================================================================
mpl.rcParams['font.family'] = 'serif'
mpl.rcParams['text.usetex'] = True
mpl.rcParams['text.latex.preamble'] = ['\usepackage{fouriernc}']
fig = figure(figsize=(5,3.5))
ax = fig.add_subplot(111)
for i, (am,af,lf,lm,c) in enumerate(zip(Am, Af, lam_f, lam_m, col)):
lbl = r'$n=%i,\ \lambda=%.2f$' % (i+1, lf)
ax.plot(xm, am, c, ls='-', lw=2.0, label=lbl)
lbl = r'$n=%i,\ \widetilde{\lambda}=%.2f$' % (i+1, lm)
ax.plot(xf, af, c, ls='--',lw=2.0, label=lbl)
leg = ax.legend(loc='lower left', handlelength=3, fontsize='x-small')
leg.get_frame().set_alpha(0.0)
ax.set_ylim([-1.1,1.1])
ax.set_xlabel(r'$x$')
ax.set_ylabel(r'$u$')
ax.grid()
tight_layout()
savefig('eigenvectors.pdf')
cn = array(cn, dtype='d')
af = zeros(1000)
am = zeros(n+1)
for i in range(n-1):
af += cn[i]*Af[i]
am += cn[i]*Am[i]
fig = figure(figsize=(5,3.5))
ax = fig.add_subplot(111)
ax.plot(xf, af, 'k--', lw=2.0, label=r'$u(x)$ - Fourier')
ax.plot(xm, am, 'k-', lw=2.0, label=r'$\widetilde{u}(x)$ - FEM')
leg = ax.legend(loc='center')
leg.get_frame().set_alpha(0.0)
ax.set_xlabel(r'$x$')
ax.set_ylabel(r'$u$')
ax.grid()
tight_layout()
savefig('eigen_solution.pdf')
```

Here is SLEPc documentation :

http://slepc.upv.es/documentation/

eigensolver.parameters.items() shows that the 'solver' is 'krylov-schur'. How can I find what other methods (if any) are available?

written
9 months ago by
Alden Pack

You can find other methods with the slepc user manual link I posted.
There is also this:
https://fenicsproject.org/qa/12526/how-to-set-the-parameters-of-slepceigensolver
And this:
http://slepc.upv.es/documentation/current/docs/manualpages/EPS/EPSSetType.html

written
9 months ago by
pf4d

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