Abstract type for statistics returned by a solver

Type for statistics returned by the majority of Krylov solvers, the attributes are:

• solved
• inconsistent
• residuals
• Aresiduals
• status

Type for statistics returned by Lanczos solvers CG-LANCZOS and CG-LANCZOS-SHIFT-SEQ, the attributes are:

• solved
• residuals
• flagged
• Anorm
• Acond
• status

Type for statistics returned by SYMMLQ, the attributes are:

• solved
• residuals
• residualscg
• errors
• errorscg
• Anorm
• Acond
• status

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

• solved_primal
• solved_dual
• residuals_primal
• residuals_dual
• status

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.

(c, s, ρ) = sym_givens(a, b)

Numerically stable symmetric Givens reflection. Given a and b, return (c, s, ρ) such that

[ c  s ] [ a ] = [ ρ ]
[ s -c ] [ b ] = [ 0 ].

roots = to_boundary(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. If known, ‖x‖² may be supplied in xNorm2.

If flip is set to true, σ1 and σ2 are computed such that

‖x - σi d‖ = radius, i = 1, 2.

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.

v = kzeros(S, n)

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

v = kones(S, n)

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