Enum StepFactor 

pub enum StepFactor {
    Ne {
        h: Mat<f64>,
    },
    Svd {
        u: Mat<f64>,
        s: Vec<f64>,
        v: Mat<f64>,
    },
}

A once-per-outer-iteration factorization of the (column-scaled) Jacobian, reused across every λ trial and by geodesic acceleration. This is the key to keeping the inner λ search cheap: the O(m·p²) work (forming JᵀJ, or the thin SVD of J) happens once; each λ trial is then only O(p³) (Cholesky) or O(p²) (closed form).

Both J and the column scaling D = diag(diag) are fixed across the inner loop (only λ changes), so this is sound.

Variants

Ne

Normal equations: stores H = JᵀJ (p×p).

Fields

h: Mat<f64>

The Gauss–Newton Hessian H = JᵀJ (p×p).

Svd

Thin SVD of the column-scaled J̃ = J·diag(1/D): U, s, V.

Fields

u: Mat<f64>

Left singular vectors U (m×k).

s: Vec<f64>

Singular values s (length k), k = min(m, p).

v: Mat<f64>

Right singular vectors V (p×k).

Implementations

impl StepFactor

pub fn solve( &self, g: MatRef<'_, f64>, r: MatRef<'_, f64>, diag: &[f64], lambda: f64, ) -> Result<StepOutput, StepError>

Solve (JᵀJ + λD²) δ = −g for the trial step δ and its predicted reduction −gᵀδ − ½‖Jδ‖². r is the residual (used by the SVD path).

Errors

Returns StepError::NotPositiveDefinite on the StepKind::NormalEqLlt path when H + λD² fails its Cholesky factorization (not positive definite for the current damping λ). The StepKind::SvdSecular path cannot fail here — its factorization risk was already taken in factorize.

pub fn solve_rhs( &self, diag: &[f64], lambda: f64, rhs: MatRef<'_, f64>, ) -> Result<Mat<f64>, StepError>

Solve (JᵀJ + λD²) x = rhs for an arbitrary right-hand side (geodesic acceleration: rhs = −Jᵀr_vv). Reuses this factorization — no re-forming.

Errors

Returns StepError::NotPositiveDefinite on the StepKind::NormalEqLlt path when H + λD² fails its Cholesky factorization, exactly as in StepFactor::solve. The StepKind::SvdSecular path cannot fail here.