Stats Types
Krylov.KrylovStats
— TypeAbstract type for statistics returned by a solver
Krylov.SimpleStats
— TypeType for statistics returned by the majority of Krylov solvers, the attributes are:
- niter
- solved
- inconsistent
- residuals
- Aresiduals
- Acond
- timer
- status
Krylov.LanczosStats
— TypeType for statistics returned by CG-LANCZOS, the attributes are:
- niter
- solved
- residuals
- indefinite
- Anorm
- Acond
- timer
- status
Krylov.LanczosShiftStats
— TypeType for statistics returned by CG-LANCZOS with shifts, the attributes are:
- niter
- solved
- residuals
- indefinite
- Anorm
- Acond
- timer
- status
Krylov.SymmlqStats
— TypeType for statistics returned by SYMMLQ, the attributes are:
- niter
- solved
- residuals
- residualscg
- errors
- errorscg
- Anorm
- Acond
- timer
- status
Krylov.AdjointStats
— TypeType for statistics returned by adjoint systems solvers BiLQR and TriLQR, the attributes are:
- niter
- solved_primal
- solved_dual
- residuals_primal
- residuals_dual
- timer
- status
Krylov.LNLQStats
— TypeType for statistics returned by the LNLQ method, the attributes are:
- niter
- solved
- residuals
- errorwithbnd
- errorbndx
- errorbndy
- timer
- status
Krylov.LSLQStats
— TypeType for statistics returned by the LSLQ method, the attributes are:
- niter
- solved
- inconsistent
- residuals
- Aresiduals
- err_lbnds
- errorwithbnd
- errubndslq
- errubndscg
- timer
- status
Krylov.LsmrStats
— TypeType for statistics returned by LSMR. The attributes are:
- niter
- solved
- inconsistent
- residuals
- Aresiduals
- Acond
- Anorm
- xNorm
- timer
- status
Solver Types
Krylov.KrylovSolver
— TypeAbstract type for using Krylov solvers in-place
Krylov.BlockKrylovSolver
— TypeAbstract type for using block Krylov solvers in-place
Krylov.MinresSolver
— TypeType for storing the vectors required by the in-place version of MINRES.
The outer constructors
solver = MinresSolver(m, n, S; window :: Int=5)
solver = MinresSolver(A, b; window :: Int=5)
may be used in order to create these vectors.
Krylov.MinaresSolver
— TypeType for storing the vectors required by the in-place version of MINARES.
The outer constructors
solver = MinaresSolver(m, n, S)
solver = MinaresSolver(A, b)
may be used in order to create these vectors.
Krylov.CgSolver
— TypeType for storing the vectors required by the in-place version of CG.
The outer constructors
solver = CgSolver(m, n, S)
solver = CgSolver(A, b)
may be used in order to create these vectors.
Krylov.CrSolver
— TypeType for storing the vectors required by the in-place version of CR.
The outer constructors
solver = CrSolver(m, n, S)
solver = CrSolver(A, b)
may be used in order to create these vectors.
Krylov.CarSolver
— TypeType for storing the vectors required by the in-place version of CAR.
The outer constructors
solver = CarSolver(m, n, S)
solver = CarSolver(A, b)
may be used in order to create these vectors.
Krylov.SymmlqSolver
— TypeType for storing the vectors required by the in-place version of SYMMLQ.
The outer constructors
solver = SymmlqSolver(m, n, S)
solver = SymmlqSolver(A, b)
may be used in order to create these vectors.
Krylov.CgLanczosSolver
— TypeType for storing the vectors required by the in-place version of CG-LANCZOS.
The outer constructors
solver = CgLanczosSolver(m, n, S)
solver = CgLanczosSolver(A, b)
may be used in order to create these vectors.
Krylov.CgLanczosShiftSolver
— TypeType for storing the vectors required by the in-place version of CG-LANCZOS-SHIFT.
The outer constructors
solver = CgLanczosShiftSolver(m, n, nshifts, S)
solver = CgLanczosShiftSolver(A, b, nshifts)
may be used in order to create these vectors.
Krylov.MinresQlpSolver
— TypeType for storing the vectors required by the in-place version of MINRES-QLP.
The outer constructors
solver = MinresQlpSolver(m, n, S)
solver = MinresQlpSolver(A, b)
may be used in order to create these vectors.
Krylov.DiomSolver
— TypeType for storing the vectors required by the in-place version of DIOM.
The outer constructors
solver = DiomSolver(m, n, memory, S)
solver = DiomSolver(A, b, memory = 20)
may be used in order to create these vectors. memory
is set to n
if the value given is larger than n
.
Krylov.FomSolver
— TypeType for storing the vectors required by the in-place version of FOM.
The outer constructors
solver = FomSolver(m, n, memory, S)
solver = FomSolver(A, b, memory = 20)
may be used in order to create these vectors. memory
is set to n
if the value given is larger than n
.
Krylov.DqgmresSolver
— TypeType for storing the vectors required by the in-place version of DQGMRES.
The outer constructors
solver = DqgmresSolver(m, n, memory, S)
solver = DqgmresSolver(A, b, memory = 20)
may be used in order to create these vectors. memory
is set to n
if the value given is larger than n
.
Krylov.GmresSolver
— TypeType for storing the vectors required by the in-place version of GMRES.
The outer constructors
solver = GmresSolver(m, n, memory, S)
solver = GmresSolver(A, b, memory = 20)
may be used in order to create these vectors. memory
is set to n
if the value given is larger than n
.
Krylov.UsymlqSolver
— TypeType for storing the vectors required by the in-place version of USYMLQ.
The outer constructors
solver = UsymlqSolver(m, n, S)
solver = UsymlqSolver(A, b)
may be used in order to create these vectors.
Krylov.UsymqrSolver
— TypeType for storing the vectors required by the in-place version of USYMQR.
The outer constructors
solver = UsymqrSolver(m, n, S)
solver = UsymqrSolver(A, b)
may be used in order to create these vectors.
Krylov.TricgSolver
— TypeType for storing the vectors required by the in-place version of TRICG.
The outer constructors
solver = TricgSolver(m, n, S)
solver = TricgSolver(A, b)
may be used in order to create these vectors.
Krylov.TrimrSolver
— TypeType for storing the vectors required by the in-place version of TRIMR.
The outer constructors
solver = TrimrSolver(m, n, S)
solver = TrimrSolver(A, b)
may be used in order to create these vectors.
Krylov.TrilqrSolver
— TypeType for storing the vectors required by the in-place version of TRILQR.
The outer constructors
solver = TrilqrSolver(m, n, S)
solver = TrilqrSolver(A, b)
may be used in order to create these vectors.
Krylov.CgsSolver
— TypeType for storing the vectors required by the in-place version of CGS.
The outer constructorss
solver = CgsSolver(m, n, S)
solver = CgsSolver(A, b)
may be used in order to create these vectors.
Krylov.BicgstabSolver
— TypeType for storing the vectors required by the in-place version of BICGSTAB.
The outer constructors
solver = BicgstabSolver(m, n, S)
solver = BicgstabSolver(A, b)
may be used in order to create these vectors.
Krylov.BilqSolver
— TypeType for storing the vectors required by the in-place version of BILQ.
The outer constructors
solver = BilqSolver(m, n, S)
solver = BilqSolver(A, b)
may be used in order to create these vectors.
Krylov.QmrSolver
— TypeType for storing the vectors required by the in-place version of QMR.
The outer constructors
solver = QmrSolver(m, n, S)
solver = QmrSolver(A, b)
may be used in order to create these vectors.
Krylov.BilqrSolver
— TypeType for storing the vectors required by the in-place version of BILQR.
The outer constructors
solver = BilqrSolver(m, n, S)
solver = BilqrSolver(A, b)
may be used in order to create these vectors.
Krylov.CglsSolver
— TypeType for storing the vectors required by the in-place version of CGLS.
The outer constructors
solver = CglsSolver(m, n, S)
solver = CglsSolver(A, b)
may be used in order to create these vectors.
Krylov.CrlsSolver
— TypeType for storing the vectors required by the in-place version of CRLS.
The outer constructors
solver = CrlsSolver(m, n, S)
solver = CrlsSolver(A, b)
may be used in order to create these vectors.
Krylov.CgneSolver
— TypeType for storing the vectors required by the in-place version of CGNE.
The outer constructors
solver = CgneSolver(m, n, S)
solver = CgneSolver(A, b)
may be used in order to create these vectors.
Krylov.CrmrSolver
— TypeType for storing the vectors required by the in-place version of CRMR.
The outer constructors
solver = CrmrSolver(m, n, S)
solver = CrmrSolver(A, b)
may be used in order to create these vectors.
Krylov.LslqSolver
— TypeType for storing the vectors required by the in-place version of LSLQ.
The outer constructors
solver = LslqSolver(m, n, S)
solver = LslqSolver(A, b)
may be used in order to create these vectors.
Krylov.LsqrSolver
— TypeType for storing the vectors required by the in-place version of LSQR.
The outer constructors
solver = LsqrSolver(m, n, S)
solver = LsqrSolver(A, b)
may be used in order to create these vectors.
Krylov.LsmrSolver
— TypeType for storing the vectors required by the in-place version of LSMR.
The outer constructors
solver = LsmrSolver(m, n, S)
solver = LsmrSolver(A, b)
may be used in order to create these vectors.
Krylov.LnlqSolver
— TypeType for storing the vectors required by the in-place version of LNLQ.
The outer constructors
solver = LnlqSolver(m, n, S)
solver = LnlqSolver(A, b)
may be used in order to create these vectors.
Krylov.CraigSolver
— TypeType for storing the vectors required by the in-place version of CRAIG.
The outer constructors
solver = CraigSolver(m, n, S)
solver = CraigSolver(A, b)
may be used in order to create these vectors.
Krylov.CraigmrSolver
— TypeType for storing the vectors required by the in-place version of CRAIGMR.
The outer constructors
solver = CraigmrSolver(m, n, S)
solver = CraigmrSolver(A, b)
may be used in order to create these vectors.
Krylov.GpmrSolver
— TypeType for storing the vectors required by the in-place version of GPMR.
The outer constructors
solver = GpmrSolver(m, n, memory, S)
solver = GpmrSolver(A, b, memory = 20)
may be used in order to create these vectors. memory
is set to n + m
if the value given is larger than n + m
.
Krylov.FgmresSolver
— TypeType for storing the vectors required by the in-place version of FGMRES.
The outer constructors
solver = FgmresSolver(m, n, memory, S)
solver = FgmresSolver(A, b, memory = 20)
may be used in order to create these vectors. memory
is set to n
if the value given is larger than n
.
Krylov.BlockGmresSolver
— TypeType for storing the vectors required by the in-place version of BLOCK-GMRES.
The outer constructors
solver = BlockGmresSolver(m, n, p, memory, SV, SM)
solver = BlockGmresSolver(A, B; memory=5)
may be used in order to create these vectors. memory
is set to div(n,p)
if the value given is larger than div(n,p)
.
Utilities
Krylov.roots_quadratic
— Functionroots = roots_quadratic(q₂, q₁, q₀; nitref)
Find the real roots of the quadratic
q(x) = q₂ x² + q₁ x + q₀,
where q₂, q₁ and q₀ are real. Care is taken to avoid numerical cancellation. Optionally, nitref
steps of iterative refinement may be performed to improve accuracy. By default, nitref=1
.
Krylov.sym_givens
— Function(c, s, ρ) = sym_givens(a, b)
Numerically stable symmetric Givens reflection. Given a
and b
reals, return (c, s, ρ)
such that
[ c s ] [ a ] = [ ρ ]
[ s -c ] [ b ] = [ 0 ].
Numerically stable symmetric Givens reflection. Given a
and b
complexes, return (c, s, ρ)
with c real and (s, ρ) complexes such that
[ c s ] [ a ] = [ ρ ]
[ s̅ -c ] [ b ] = [ 0 ].
Krylov.to_boundary
— Functionroots = to_boundary(n, x, d, radius; flip, xNorm2, dNorm2)
Given a trust-region radius radius
, a vector x
lying inside the trust-region and a direction d
, return σ1
and σ2
such that
‖x + σi d‖ = radius, i = 1, 2
in the Euclidean norm. n
is the length of vectors x
and d
. If known, ‖x‖² and ‖d‖² may be supplied with xNorm2
and dNorm2
.
If flip
is set to true
, σ1
and σ2
are computed such that
‖x - σi d‖ = radius, i = 1, 2.
Krylov.vec2str
— Functions = vec2str(x; ndisp)
Display an array in the form
[ -3.0e-01 -5.1e-01 1.9e-01 ... -2.3e-01 -4.4e-01 2.4e-01 ]
with (ndisp - 1)/2 elements on each side.
Krylov.ktypeof
— FunctionS = ktypeof(v)
Return the most relevant storage type S
based on the type of v
.
Krylov.kzeros
— Functionv = kzeros(S, n)
Create a vector of storage type S
of length n
only composed of zero.
Krylov.kones
— Functionv = kones(S, n)
Create a vector of storage type S
of length n
only composed of one.
Krylov.vector_to_matrix
— FunctionM = vector_to_matrix(S)
Return the dense matrix storage type M
related to the dense vector storage type S
.
Krylov.matrix_to_vector
— FunctionS = matrix_to_vector(M)
Return the dense vector storage type S
related to the dense matrix storage type M
.