getoperatordiagonalinverse(chebyshev)

Compute or retrieve the inverse of the symbol matrix diagonal for a Chebyshev preconditioner

Returns:

• symbol matrix diagonal inverse for the operator

Example:

# setup
mesh = Mesh1D(1.0);
diffusion = GalleryOperator("diffusion", 3, 3, mesh);

# preconditioner
chebyshev = Chebyshev(diffusion);

# verify operator diagonal inverse
@assert chebyshev.operatordiagonalinverse ≈ [6/7 0; 0 3/4]

# output

seteigenvalueestimatescaling(chebyshev, eigenvaluescaling)

Set the scaling of the eigenvalue estimates for a Chebyshev preconditioner

Arguments:

• chebyshev::Chebyshev: preconditioner to set eigenvalue estimate scaling
• eigenvaluescaling::Array{Float64,1}: array of 4 scaling factors to use when setting $\lambda_{\text{min}}$ and $\lambda_{\text{max}}$ based on eigenvalue estimates

$\lambda_{\text{min}}$ = a * estimated min + b * estimated max

$\lambda_{\text{max}}$ = c * estimated min + d * estimated max

Example:

# setup
mesh = Mesh1D(1.0);
diffusion = GalleryOperator("diffusion", 3, 3, mesh);

# preconditioner
chebyshev = Chebyshev(diffusion)

# set eigenvalue estimate scaling
# PETSc default is to use 1.1, 0.1 of max eigenvalue estimate
#   https://www.mcs.anl.gov/petsc/petsc-3.3/docs/manualpages/KSP/KSPChebyshevSetEstimateEigenvalues.html
chebyshev.eigenvaluescaling = [0.0, 0.1, 0.0, 1.1];
println(chebyshev)

# output

chebyshev preconditioner:
1st-kind Chebyshev
eigenvalue estimates:
  estimated minimum 0.0000
  estimated maximum 2.1429
estimate scaling:
  λ_min = a * estimated min + b * estimated max
  λ_max = c * estimated min + d * estimated max
  a = 0.0000
  b = 0.1000
  c = 0.0000
  d = 1.1000

geteigenvalueestimates(chebyshev)

Compute or retrieve the eigenvalue estimates for a Chebyshev preconditioner

Returns:

• eigenvalue estimates for the operator

Example:

# setup
mesh = Mesh1D(1.0);
diffusion = GalleryOperator("diffusion", 3, 3, mesh);

# preconditioner
chebyshev = Chebyshev(diffusion);

# verify eigenvalue estimates
@assert chebyshev.eigenvalueestimates ≈ [0, 15 / 7]

# output

getbetas(k)

Compute beta coefficients associated with fourth/optimal fourth-kind Chebyshev polynomials

Returns:

• beta coefficients associated with fourth/optimal fourth-kind Chebyshev polynomials