## Stats Types

Krylov.AdjointStatsType

Type for statistics returned by adjoint systems solvers BiLQR and TriLQR, the attributes are:

• niter
• solved_primal
• solved_dual
• residuals_primal
• residuals_dual
• status
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Krylov.LSLQStatsType

Type for statistics returned by the LSLQ method, the attributes are:

• niter
• solved
• inconsistent
• residuals
• Aresiduals
• err_lbnds
• errorwithbnd
• errubndslq
• errubndscg
• status
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## Solver Types

Krylov.MinresSolverType

Type for storing the vectors required by the in-place version of MINRES.

The outer constructors

solver = MinresSolver(n, m, S; window :: Int=5)
solver = MinresSolver(A, b; window :: Int=5)

may be used in order to create these vectors.

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Krylov.CgSolverType

Type for storing the vectors required by the in-place version of CG.

The outer constructors

solver = CgSolver(n, m, S)
solver = CgSolver(A, b)

may be used in order to create these vectors.

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Krylov.CrSolverType

Type for storing the vectors required by the in-place version of CR.

The outer constructors

solver = CrSolver(n, m, S)
solver = CrSolver(A, b)

may be used in order to create these vectors.

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Krylov.SymmlqSolverType

Type for storing the vectors required by the in-place version of SYMMLQ.

The outer constructors

solver = SymmlqSolver(n, m, S)
solver = SymmlqSolver(A, b)

may be used in order to create these vectors.

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Krylov.CgLanczosSolverType

Type for storing the vectors required by the in-place version of CG-LANCZOS.

The outer constructors

solver = CgLanczosSolver(n, m, S)
solver = CgLanczosSolver(A, b)

may be used in order to create these vectors.

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Krylov.CgLanczosShiftSolverType

Type for storing the vectors required by the in-place version of CG-LANCZOS-SHIFT.

The outer constructors

solver = CgLanczosShiftSolver(n, m, nshifts, S)
solver = CgLanczosShiftSolver(A, b, nshifts)

may be used in order to create these vectors.

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Krylov.MinresQlpSolverType

Type for storing the vectors required by the in-place version of MINRES-QLP.

The outer constructors

solver = MinresQlpSolver(n, m, S)
solver = MinresQlpSolver(A, b)

may be used in order to create these vectors.

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Krylov.DiomSolverType

Type for storing the vectors required by the in-place version of DIOM.

The outer constructors

solver = DiomSolver(n, m, 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.

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Krylov.FomSolverType

Type for storing the vectors required by the in-place version of FOM.

The outer constructors

solver = FomSolver(n, m, 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.

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Krylov.DqgmresSolverType

Type for storing the vectors required by the in-place version of DQGMRES.

The outer constructors

solver = DqgmresSolver(n, m, 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.

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Krylov.GmresSolverType

Type for storing the vectors required by the in-place version of GMRES.

The outer constructors

solver = GmresSolver(n, m, 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.

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Krylov.UsymlqSolverType

Type for storing the vectors required by the in-place version of USYMLQ.

The outer constructors

solver = UsymlqSolver(n, m, S)
solver = UsymlqSolver(A, b)

may be used in order to create these vectors.

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Krylov.UsymqrSolverType

Type for storing the vectors required by the in-place version of USYMQR.

The outer constructors

solver = UsymqrSolver(n, m, S)
solver = UsymqrSolver(A, b)

may be used in order to create these vectors.

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Krylov.TricgSolverType

Type for storing the vectors required by the in-place version of TRICG.

The outer constructors

solver = TricgSolver(n, m, S)
solver = TricgSolver(A, b)

may be used in order to create these vectors.

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Krylov.TrimrSolverType

Type for storing the vectors required by the in-place version of TRIMR.

The outer constructors

solver = TrimrSolver(n, m, S)
solver = TrimrSolver(A, b)

may be used in order to create these vectors.

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Krylov.TrilqrSolverType

Type for storing the vectors required by the in-place version of TRILQR.

The outer constructors

solver = TrilqrSolver(n, m, S)
solver = TrilqrSolver(A, b)

may be used in order to create these vectors.

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Krylov.CgsSolverType

Type for storing the vectors required by the in-place version of CGS.

solver = CgsSolver(n, m, S)
solver = CgsSolver(A, b)

may be used in order to create these vectors.

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Krylov.BicgstabSolverType

Type for storing the vectors required by the in-place version of BICGSTAB.

The outer constructors

solver = BicgstabSolver(n, m, S)
solver = BicgstabSolver(A, b)

may be used in order to create these vectors.

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Krylov.BilqSolverType

Type for storing the vectors required by the in-place version of BILQ.

The outer constructors

solver = BilqSolver(n, m, S)
solver = BilqSolver(A, b)

may be used in order to create these vectors.

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Krylov.QmrSolverType

Type for storing the vectors required by the in-place version of QMR.

The outer constructors

solver = QmrSolver(n, m, S)
solver = QmrSolver(A, b)

may be used in order to create these vectors.

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Krylov.BilqrSolverType

Type for storing the vectors required by the in-place version of BILQR.

The outer constructors

solver = BilqrSolver(n, m, S)
solver = BilqrSolver(A, b)

may be used in order to create these vectors.

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Krylov.CglsSolverType

Type for storing the vectors required by the in-place version of CGLS.

The outer constructors

solver = CglsSolver(n, m, S)
solver = CglsSolver(A, b)

may be used in order to create these vectors.

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Krylov.CrlsSolverType

Type for storing the vectors required by the in-place version of CRLS.

The outer constructors

solver = CrlsSolver(n, m, S)
solver = CrlsSolver(A, b)

may be used in order to create these vectors.

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Krylov.CgneSolverType

Type for storing the vectors required by the in-place version of CGNE.

The outer constructors

solver = CgneSolver(n, m, S)
solver = CgneSolver(A, b)

may be used in order to create these vectors.

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Krylov.CrmrSolverType

Type for storing the vectors required by the in-place version of CRMR.

The outer constructors

solver = CrmrSolver(n, m, S)
solver = CrmrSolver(A, b)

may be used in order to create these vectors.

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Krylov.LslqSolverType

Type for storing the vectors required by the in-place version of LSLQ.

The outer constructors

solver = LslqSolver(n, m, S)
solver = LslqSolver(A, b)

may be used in order to create these vectors.

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Krylov.LsqrSolverType

Type for storing the vectors required by the in-place version of LSQR.

The outer constructors

solver = LsqrSolver(n, m, S)
solver = LsqrSolver(A, b)

may be used in order to create these vectors.

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Krylov.LsmrSolverType

Type for storing the vectors required by the in-place version of LSMR.

The outer constructors

solver = LsmrSolver(n, m, S)
solver = LsmrSolver(A, b)

may be used in order to create these vectors.

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Krylov.LnlqSolverType

Type for storing the vectors required by the in-place version of LNLQ.

The outer constructors

solver = LnlqSolver(n, m, S)
solver = LnlqSolver(A, b)

may be used in order to create these vectors.

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Krylov.CraigSolverType

Type for storing the vectors required by the in-place version of CRAIG.

The outer constructors

solver = CraigSolver(n, m, S)
solver = CraigSolver(A, b)

may be used in order to create these vectors.

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Krylov.CraigmrSolverType

Type for storing the vectors required by the in-place version of CRAIGMR.

The outer constructors

solver = CraigmrSolver(n, m, S)
solver = CraigmrSolver(A, b)

may be used in order to create these vectors.

source
Krylov.GpmrSolverType

Type for storing the vectors required by the in-place version of GPMR.

The outer constructors

solver = GpmrSolver(n, m, 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.

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Krylov.FgmresSolverType

Type for storing the vectors required by the in-place version of FGMRES.

The outer constructors

solver = FgmresSolver(n, m, 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.

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## Utilities

Krylov.roots_quadraticFunction
roots = 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.

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Krylov.sym_givensFunction
(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 ].
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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 ].
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Krylov.to_boundaryFunction
roots = 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.
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Krylov.vec2strFunction
s = 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.

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Krylov.ktypeofFunction
S = ktypeof(v)

Return a dense storage type S based on the type of v.

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Krylov.kzerosFunction
v = kzeros(S, n)

Create an AbstractVector of storage type S of length n only composed of zero.

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Krylov.konesFunction
v = kones(S, n)

Create an AbstractVector of storage type S of length n only composed of one.

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Krylov.vector_to_matrixFunction
M = vector_to_matrix(S)

Return the dense matrix storage type M related to the dense vector storage type S.

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