Capability index

Every public name in gaussx (366 of them), grouped as the API reference groups it, with the first line of its docstring; then the public API of the libraries gaussx builds on, and the private toolkits gaussx's own modules share. Search this page before writing a helper: if what you need is here, compose it; if it is almost here, extend it where it lives. Regenerate with make capabilities after changing a public API (tests/test_capabilities.py checks it is current).

Primitives

Solve, logdet & Cholesky

Name	Kind	What it does
solve	function	Solve A x = b with structural dispatch.
logdet	function	Compute log |det(A)| with structural dispatch.
cholesky	function	Compute Cholesky factor L such that A = L L^T.
cholesky_logdet	function	Compute log|A| from Cholesky factor L where A = L Lᵀ.

Pseudo-determinant

Name	Kind	What it does
pseudo_logdet	function	Log pseudo-determinant: the log of the product of the non-zero eigenvalues.

Trace & diagonal

Name	Kind	What it does
trace	function	Compute the trace of an operator.
diag	function	Extract the diagonal of an operator as a 1D array.
trace_and_diag	function	Jointly estimate the trace and diagonal from one probe pass.

Inverse, square root & spectral decompositions

Name	Kind	What it does
inv	function	Return a lazy inverse operator A^{-1}.
sqrt	function	Compute matrix square root S such that S @ S = A.
eig	function	Compute eigenvalues and eigenvectors.
eigvals	function	Compute eigenvalues only.
svd	function	Compute the singular value decomposition A = U diag(s) V^T.
frobenius_norm	function	Compute the Frobenius norm \|\|A\|\|_F with structural dispatch.
submatrix	function	Extract A[row_idx, col_idx] without forming the full matrix.

Generalised eigenproblems

Name	Kind	What it does
eigh_generalized	function	Solve \(A v = \lambda B v\) for symmetric \(A\) and symmetric PSD \(B\).

Matrix-free square-root products

Name	Kind	What it does
sqrt_matmul	function	Compute \(A^{1/2} b\) via contour-integral quadrature.
sqrt_inv_matmul	function	Compute \(A^{-1/2} b\) via contour-integral quadrature.
estimate_spectral_bounds	function	Estimate \((\lambda_{\min}, \lambda_{\max})\) of a symmetric operator.

Joint inverse-quadratic and log-determinant

Name	Kind	What it does
inv_quad_logdet	function	Joint inverse-quadratic and log-determinant via shared BBMM CG.

Root decompositions

Name	Kind	What it does
root_decomposition	function	Compute a tall factor R such that R Rᵀ ≈ A.
root_inv_decomposition	function	Compute a tall factor R⁻ such that R⁻ (R⁻)ᵀ ≈ A⁻¹.
RootDecomposition	class	Tall factor R with R Rᵀ ≈ A semantics.

Quantiles

Name	Kind	What it does
mixture_quantile	function	Invert a CDF at one or more quantile levels by bracketed root finding.
mixture_quantile_gaussian_approx	function	Quantiles of the Gaussian matched to an ensemble's first two moments.
Chandrupatla	class	Chandrupatla's hybrid inverse-quadratic / bisection root finder.

Support types

Name	Kind	What it does
DenseFallbackWarning	class	Warning emitted when a structured primitive materialises an operator.

Operators & Tags

Structured products & sums

Name	Kind	What it does
Kronecker	class	Kronecker product operator A₁ ⊗ A₂ ⊗ … ⊗ Aₖ.
BlockDiag	class	Block diagonal operator diag(A₁, A₂, …, Aₖ).
KroneckerSum	class	Kronecker sum A \oplus B = A \otimes I_b + I_a \otimes B.
KroneckerSumSqrt	class	Symmetric square root of A \oplus B via per-factor eigenvectors.
SumOfKroneckers	class	Sum of Kronecker products Σ_k A_k \otimes B_k.
SumOfKroneckersSqrt	class	Lazy Lanczos square-root operator for SumOfKroneckers covariances.

Low-rank updates

Name	Kind	What it does
LowRankUpdate	class	Low-rank update operator L + U diag(d) Vᵀ.
low_rank_plus_diag	function	Construct diag(diag) + U diag(d) Vᵀ.
low_rank_plus_identity	function	Construct scale * I + U diag(d) Vᵀ.
svd_low_rank_plus_diag	function	Construct diag(diag) + U diag(S) Vᵀ from a truncated SVD.

Banded & Toeplitz

Name	Kind	What it does
BlockTriDiag	class	Block-tridiagonal operator with a symmetric off-diagonal band.
LowerBlockTriDiag	class	Lower triangular block-bidiagonal Cholesky factor.
UpperBlockTriDiag	class	Upper triangular block-bidiagonal (transpose of LowerBlockTriDiag).
Toeplitz	class	Symmetric Toeplitz matrix from its first column.
ToeplitzCholesky	class	Circulant-embedding sample factor for a symmetric positive Toeplitz matrix.

Fast-diagonalisable operators

Name	Kind	What it does
DiagonalizedOperator	class	Linear operator A x = V⁻¹(Λ ⊙ V x) given by a transform pair.
circulant	function	(Block-)circulant operator from its first column.
circulant_from_symbol	function	(Block-)circulant operator from its DFT symbol.
as_diagonalized	function	Express operator as one DiagonalizedOperator, if possible.
SpectralFunction	class	f(B) for a symmetric B with a factored eigenbasis.

Interpolated & masked operators

Name	Kind	What it does
InterpolatedOperator	class	Structured Kernel Interpolation: K \approx W K_{uu} W^T.
MaskedOperator	class	Row/column-masked view of a base operator.
grid_coupling_indices	function	Flat indices of masked-out grid cells adjacent to the masked-in region.
create_grid	function	Create a regular grid from per-dimension sizes and bounds.
grid_data	function	Expand a grid to the full Cartesian product of grid points.
cubic_interpolation_weights	function	Compute cubic interpolation indices and weights for SKI.

Lazy algebra & sampling

Name	Kind	What it does
sum_operator	function	Lazy sum (A + B + …) v = A v + B v + ….
scaled_operator	function	Lazy scalar multiply (c A) v = c (A v).
product_operator	function	Lazy matmul (A B) v = A (B v).
kronecker_sum_sample	function	Sample from 𝒩(0, A ⊕ B) using per-factor eigendecompositions.
sum_of_kroneckers_sample	function	Sample from 𝒩(0, op) with matrix-free Lanczos square roots.
toeplitz_sample	function	Sample from 𝒩(0, Toeplitz(column)) via FFT circulant embedding.
CapacitanceSolver	class	Solve a base system subject to homogeneous point constraints.

Structural tags & predicates

Name	Kind	What it does
is_kronecker	function	Check whether operator carries the Kronecker tag.
is_kronecker_sum	function	Check whether operator carries the Kronecker sum tag.
is_block_diagonal	function	Check whether operator carries the block-diagonal tag.
is_block_tridiagonal	function	Check whether operator carries the block-tridiagonal tag.
is_low_rank	function	Check whether operator carries the low-rank tag.
is_eigen_reducible	function	Whether a sum of Kronecker products has the exact two-term reduction.
is_diagonal	function	Re-exported from lineax.
is_symmetric	function	Re-exported from lineax.
is_positive_semidefinite	function	Re-exported from lineax.
is_negative_semidefinite	function	Re-exported from lineax.
is_lower_triangular	function	Re-exported from lineax.
is_upper_triangular	function	Re-exported from lineax.
kronecker_tag	constant	Operator is a Kronecker product.
kronecker_sum_tag	constant	Operator is a Kronecker sum A (+) B = A (x) I_b + I_a (x) B.
block_diagonal_tag	constant	Operator is block diagonal.
block_tridiagonal_tag	constant	Operator is block tridiagonal.
low_rank_tag	constant	Operator has low-rank structure (e.g. L + U D V^T).
diagonal_tag	constant	Re-exported from lineax.
symmetric_tag	constant	Re-exported from lineax.
positive_semidefinite_tag	constant	Re-exported from lineax.
negative_semidefinite_tag	constant	Re-exported from lineax.
lower_triangular_tag	constant	Re-exported from lineax.
upper_triangular_tag	constant	Re-exported from lineax.
tridiagonal_tag	constant	Re-exported from lineax.
unit_diagonal_tag	constant	Re-exported from lineax.

Deprecated aliases

Name	Kind	What it does
SumKronecker	class	Deprecated alias for SumOfKroneckers.
SumKroneckerSqrt	class	Deprecated alias for SumOfKroneckersSqrt (gh-297).
sumkronecker_sample	function	Deprecated alias for sum_of_kroneckers_sample (gh-297).
SVDLowRankUpdate	class	Deprecated subclass of LowRankUpdate with orthonormal=True.

Sparse Operators

Sparse Cholesky

Name	Kind	What it does
SparseOperator	class	Sparse matrix with a static SparsityPattern and traced values.
SparsityPattern	class	Static, hashable sparsity pattern of a (m, n) matrix.
symbolic_cholesky	function	Symbolic analysis of a sparse Cholesky factorisation (host, cached).
SymbolicCholesky	class	Symbolic Cholesky factor of a symmetric sparsity pattern.
sparse_cholesky	function	Sparse Cholesky factorisation of a symmetric positive-definite operator.
SparseCholeskyFactor	class	Sparse Cholesky factor P Q Pᵀ = L Lᵀ on a static symbolic pattern.

GMRF Precision Builders

Name	Kind	What it does
iid_precision	function	Precision τ I of n independent effects with variance 1/τ.
rw1_structure	function	Structure matrix R = D₁ᵀ W D₁ of a first-order random walk.
rw2_structure	function	Structure matrix R = D₂ᵀ D₂ of a second-order random walk.
ar1_precision	function	Precision of a stationary AR(1) process with marginal precision τ.
besag_structure	function	Validate a graph Laplacian as a Besag (ICAR) structure matrix.
generalized_variance_scale	function	Generalized variance of an intrinsic GMRF (Sørbye & Rue, 2014).
bym2_precision	function	Joint precision of the BYM2 pair (b, u*) (Riebler et al., 2016).
spde_precision	function	SPDE precision Q_α = τ² K (C̃⁻¹K)^{α−1} with K = κ²C̃ + G.
spde_precision_grid	function	SPDE precision on a regular grid: a function of a Kronecker sum.
matern_spde_params	function	SPDE parameters (κ, τ, α) of a Matérn field.
fem_matrices	function	Lumped mass C̃ and stiffness G of P1 elements on a triangle mesh.
fem_projector	function	Observation matrix A of P1 interpolation: (A w)_k = Σ_i ψ_i(s_k) w_i.

Solvers & Preconditioners

The front door

Name	Kind	What it does
linear_solve	function	Solve A x = b through the unified front door.
as_linear_operator	function	Wrap a raw matvec callable as a tagged lineax operator.

Abstract interfaces

Name	Kind	What it does
AbstractSolveStrategy	class	Protocol for linear solve strategies.
AbstractLogdetStrategy	class	Protocol for log-determinant strategies.
AbstractSolverStrategy	class	Protocol for solver strategies that pair solve + logdet.

Direct & iterative solvers

Name	Kind	What it does
DenseSolver	class	Dense solver strategy using gaussx structural dispatch.
AutoSolver	class	Automatic solver selection based on operator type and size.
CGSolver	class	Iterative CG solver with stochastic log-determinant.
PreconditionedCGSolver	class	CG solver with pivoted partial Cholesky preconditioner.
MINRESSolver	class	MINRES solver for symmetric (possibly indefinite) systems.
LSMRSolver	class	LSMR iterative least-squares solver (Fong & Saunders 2011).
BBMMSolver	class	CG solve and SLQ logdet with BBMM's defaults (Gardner et al. 2018).
ComposedSolver	class	Mix-and-match solve and logdet from different strategies.
KeyedSolver	class	Bind a PRNG key to a strategy's stochastic log-determinant (gh-384).
LineaxSolver	class	A lineax solver as a gaussx solve strategy.
SparseCholeskySolver	class	Exact solve / logdet / diag_inv through a sparse Cholesky factor.

Sketch-and-precondition least squares

Name	Kind	What it does
SketchAndPrecondLSMR	class	Sketch-and-precondition LSMR for tall least squares (Rokhlin & Tygert, 2008).
sketch_and_solve	function	Approximate least squares from a sketch.

Logdet strategies

Name	Kind	What it does
DenseLogdet	class	Dense log-determinant via gaussx structural dispatch.
SLQLogdet	class	Stochastic log-determinant via Lanczos quadrature (SLQ).
IndefiniteSLQLogdet	class	Stochastic log\|det(A)\| for symmetric (possibly indefinite) operators.
NystromLogdet	class	Nyström-preconditioned log-determinant of \(A + \mu I\) (Wenger et al., 2022).

Preconditioners

Name	Kind	What it does
AbstractPreconditioner	class	Protocol for preconditioners producing an approximate inverse.
JacobiPreconditioner	class	Diagonal preconditioner M^{-1} = diag(1 / \|diag(A)\|).
OperatorPreconditioner	class	Use an externally supplied approximate inverse as a preconditioner.
NystromPreconditioner	class	Randomized Nyström preconditioner for A + μ I.
PartialCholeskyPreconditioner	class	Preconditioner from a pivoted partial Cholesky factor.

Linear-Algebra Utilities

Robust factorization & hygiene

Name	Kind	What it does
safe_cholesky	function	Cholesky decomposition with adaptive jitter for near-singular matrices.
symmetrize	function	Symmetrize the trailing two axes: 0.5 * (X + X^T).

Matrix identities

Name	Kind	What it does
woodbury_solve	function	Standalone Woodbury identity solve: (L + U diag(D) U^T)^{-1} b.
schur_complement	function	Schur complement: K_XX - K_XZ @ K_ZZ^{-1} @ K_ZX.
conditional_variance	function	Predictive variance: Schur complement diagonal plus optional variational correction.
diag_conditional_variance	function	Diagonal of Schur complement: diag(K_XX - A K_ZX).
cov_transform	function	Covariance propagation through a linear map: J @ Sigma @ J^T.
sandwich	function	Return A @ P @ A.T exploiting compatible operator structure.
trace_product	function	Trace of a matrix product: tr(A @ B) with structural dispatch.

Marginal variances & selected inverses

Name	Kind	What it does
diag_inv	function	Compute the diagonal of the inverse of a linear operator.
selected_inverse	function	The band of Σ = Q⁻¹ for a symmetric positive-definite BlockTriDiag.

Matrix-RHS & batched solves

Name	Kind	What it does
solve_matrix	function	Solve A X = B with a single factorization on the matrix RHS.
solve_columns	function	Solve A X = B column-by-column via vmap.
solve_rows	function	Solve A x = bᵢ for each row bᵢ of a matrix via vmap.
tridiagonal_solve	function	Solve a tridiagonal system A x = d.
tridiagonal_solve_batched	function	Solve independent tridiagonal systems over leading batch dimensions.

Matrix diagonalization & shifted Kronecker-sum solves

Name	Kind	What it does
EigenDecomposition	class	Precomputed eigendecomposition A = V diag(λ) V⁻¹ of a square matrix.
kronecker_sum_solve	function	Solve a shifted Kronecker-sum system in tensor (unvectorized) form.

Stable distances & Lyapunov

Name	Kind	What it does
stable_squared_distances	function	Squared Euclidean distances with mixed-precision stability.
discrete_lyapunov_solve	function	Solve the discrete Lyapunov equation P - G P G^T = Q.

Sketching

Abstract interface

Name	Kind	What it does
AbstractSketch	class	A random sketching matrix \(S \in \mathbb{R}^{d \times m}\), sampled once.

Sketches

Name	Kind	What it does
GaussianSketch	class	Gaussian sketch: i.i.d. entries \(S_{ij} \sim \mathcal{N}(0, 1/d)\).
OrthonormalSketch	class	Gaussian sketch with orthonormal rows, \(S S^\top = I_d\).
SparseSignSketch	class	Sparse sign sketch (sparse Johnson-Lindenstrauss transform).
SRHTSketch	class	Subsampled randomized Hadamard transform sketch.
RowSamplingSketch	class	Weighted row-sampling sketch.

Fast transforms

Name	Kind	What it does
hadamard_transform	function	Unnormalized Walsh-Hadamard transform along the last axis.

Randomized Linear Algebra

Range finder, QB, SVD and eigh

Name	Kind	What it does
range_finder	function	Orthonormal basis \(Q\) for the dominant range of \(A\).
qb	function	Randomized QB factorisation \(A \approx QB\) with \(B = Q^\top A\).
randomized_svd	function	Truncated SVD \(A \approx U \operatorname{diag}(s) V^\top\) by randomized QB.
randomized_eigh	function	Partial eigendecomposition of a symmetric operator by Rayleigh-Ritz.

Randomized Nyström

Name	Kind	What it does
randomized_nystrom	function	Randomized Nyström approximation \(\hat A = U\hat\Lambda U^\top\) of a PSD \(A\).

Randomly pivoted Cholesky

Name	Kind	What it does
rp_cholesky	function	Randomly pivoted partial Cholesky of a PSD matrix A.

Interpolative and CUR decompositions

Name	Kind	What it does
column_id	function	Randomized column interpolative decomposition (Voronin & Martinsson, 2017).
cur	function	Randomized CUR decomposition (Voronin & Martinsson, 2017, §4).
ColumnID	class	Column interpolative decomposition \(A \approx A_{:,J}\,X\).
CUR	class	CUR decomposition \(A \approx C\,U\,R\), \(C = A_{:,J}\), \(R = A_{I,:}\).

Distributions & Exponential Family

Multivariate normal distributions

Name	Kind	What it does
AbstractMultivariateNormal	class	Common surface of MultivariateNormal and MultivariateNormalPrecision.
MultivariateNormal	class	Multivariate normal parameterized by a lineax linear operator.
MultivariateNormalPrecision	class	Multivariate normal parameterized by a precision (inverse covariance) operator.

Gaussian Markov random fields

Name	Kind	What it does
GaussianMRF	class	Gaussian Markov random field \(x \sim \mathcal N(\mu, Q^{-1})\).
IntrinsicGMRF	class	Improper \(\mathcal N(\mu, (\tau R)^+)\) on the complement of \(\ker R\).
ConstrainedGMRF	class	A GaussianMRF conditioned on hard linear constraints \(A_cx = e\).
BYM2GMRF	class	The BYM2 pair \((b, u^\ast)\) (Riebler et al., 2016), with its exact density.

Sequential distributions

Name	Kind	What it does
LGSSM	class	Linear-Gaussian state-space model as a density over (T, M).
MaskedLGSSM	class	LGSSM whose log_prob marginalises unobserved channels exactly.
LGSSMFactory	class	Callable obs_mask -> MaskedLGSSM, for conditional use.
MarkovGaussian	class	Chain-Markov Gaussian over \(x_{0:T-1}\) as a density over (T, d).

Gaussian sugar ops

Name	Kind	What it does
gaussian_log_prob	function	Multivariate normal log-probability.
gaussian_entropy	function	Entropy of a multivariate normal N(mu, Sigma).
quadratic_form	function	Compute x^T A^{-1} x via a single solve.
kl_standard_normal	function	KL(first ‖ second): KL(N(m, S) \|\| N(0, I)).
gaussian_kl	function	KL(first ‖ second): KL(p \|\| q) between two multivariate normals.
conditional	function	Compute p(x_A \| x_B = b) from a joint Gaussian p(x_A, x_B).
joseph_update	function	Numerically stable Joseph-form covariance update.
add_jitter	function	Add diagonal jitter for numerical stability: A + eps * I.
project	function	Compute A_X = K_XZ @ K_ZZ^{-1} via Cholesky solve.

Structured sampling

Name	Kind	What it does
sample_mvn	function	Draw samples from \(\mathcal{N}(\mu, K)\), dispatching on \(K\)'s structure.
sample_joint_conditional	function	Joint draws of a partitioned Gaussian, and conditional draws from them.

Exponential family

Name	Kind	What it does
GaussianExpFam	class	Gaussian in natural (exponential family) parameters.
to_mean_cov	function	Convert natural parameters to the mean and covariance.
mean_cov_to_natural	function	Convert mean/covariance to natural parameters (operator form).
natural_to_mean_cov	function	Convert natural parameters to mean/covariance (operator form).
mean_chol_to_natural	function	Convert mean/variance (Cholesky) to natural parameters.
natural_to_mean_chol	function	Convert natural parameters to mean/variance (Cholesky).
mean_chol_to_expectation	function	Convert mean/variance (Cholesky) to expectation parameters.
expectation_to_mean_chol	function	Convert expectation parameters to mean/variance (Cholesky).
expectation_to_natural	function	Convert expectation parameters to natural parameters.
natural_to_expectation	function	Convert natural parameters to expectation parameters.
log_partition	function	Log-partition function A(eta).
fisher_info	function	Fisher information with respect to the mean, \(F_\mu = \Sigma^{-1}\).
sufficient_stats	function	Compute sufficient statistics T(x) = [x, x x^T].
kl_divergence	function	KL(first ‖ second): KL(q \|\| p) via the Bregman form on natural parameters.

Deprecated aliases

Name	Kind	What it does
to_natural	function	Deprecated alias of gaussx.mean_cov_to_natural.
to_expectation	function	Deprecated alias of to_mean_cov.

Gaussian Processes

Conditioning & prediction

Name	Kind	What it does
sparse_conditional	function	Sparse-GP conditional q(f_x) = \int p(f_x \| u) q(u) du.
build_prediction_cache	function	Solve K_y alpha = y and cache the result.
PredictionCache	class	Cached training solve (and factor) for amortized predictions.
predict_mean	function	Predictive mean: mu* = K_*f @ alpha.
predict_variance	function	Predictive variance sigma^2_* = k_** - diag(K_*f K_y^{-1} K_f*).
conditional_interpolate	function	Two-filter fusion at t from a filtered state and a backward message.
rts_interpolate	function	Smoothed marginal at an unobserved t from filtered and smoothed states.
kronecker_posterior_predictive	function	Posterior mean and variance for a Kronecker GP at test points.
kronecker_mll	function	Exact marginal log-likelihood for a Kronecker-structured GP.

Pathwise sampling

Name	Kind	What it does
matheron_update	function	Posterior samples via Matheron's-rule correction.

Whitening

Name	Kind	What it does
unwhiten	function	Unwhiten variational mean: m = L @ m_tilde.
unwhiten_covariance	function	Unwhiten variational covariance: S = L S̃ Lᵀ.
whiten_covariance	function	Unwhiten variational covariance: S = L S̃ Lᵀ.
whitened_svgp_predict	function	Whitened SVGP prediction: mean and variance at test points.
svgp_variance_adjustment	function	Compute the SVGP variance adjustment operator.

Variational bounds & KL

Name	Kind	What it does
variational_elbo_gaussian	function	Uncollapsed (SVGP) ELBO for a Gaussian likelihood.
variational_elbo_mc	function	Monte Carlo ELBO for non-conjugate likelihoods.
collapsed_elbo	function	Collapsed ELBO (Titsias bound) for sparse GP regression.
gauss_kl	function	KL(first ‖ second): KL[q(u) \|\| p(u)] with p(u) = N(0, K).

Cross-validation & diagnostics

Name	Kind	What it does
love_cache	function	Precompute a Lanczos approximation of K^{-1} for fast variance.
love_variance	function	Approximate k_*^\top K^{-1} k_* from a LOVE cache.
love_residual	function	Relative solve residual of a LOVE cache for one cross-covariance.
love_variance_error_bound	function	Upper bound on the variance over-estimate of a LOVE cache.
LOVECache	class	Cached rank-k Lanczos approximation of K^{-1}.
leave_one_out_cv	function	LOO-CV via the bordered-system identity.
LOOResult	class	Result of leave-one-out cross-validation.

Multi-output projections

Name	Kind	What it does
oilmm_project	function	Project multi-output data to independent latent GPs via OILMM.
oilmm_back_project	function	Back-project latent GP predictions to the observation space.

Deprecated aliases

Name	Kind	What it does
base_conditional	function	Deprecated: use sparse_conditional (note K_xz = K_mn.T).

Quadrature & Moment Matching

State & integrators

Name	Kind	What it does
GaussianState	class	Gaussian distribution as (mean, covariance operator) pair.
PropagationResult	class	Output of uncertainty propagation through a nonlinear function.
AbstractIntegrator	class	Protocol for Gaussian integral approximation.
moment_transform	function	Moment triple of (x, fn(x)) for x Gaussian.
GaussHermiteIntegrator	class	Gauss-Hermite quadrature integrator.
UnscentedIntegrator	class	Unscented transform: deterministic sigma points.
CubatureIntegrator	class	Third-order spherical-radial cubature.
FifthOrderCubatureIntegrator	class	Fifth-order fully symmetric cubature (McNamee & Stenger).
TaylorIntegrator	class	1st or 2nd order Taylor expansion for uncertainty propagation.
MonteCarloIntegrator	class	Monte Carlo moment matching: sample, propagate, compute moments.
AssumedDensityFilter	class	KL-optimal Gaussian projection via moment matching.

Quadrature rules

Name	Kind	What it does
gauss_hermite_points	function	Gauss-Hermite quadrature points and weights.
cubature_points	function	Spherical-radial cubature points and weights.
fifth_order_cubature_points	function	Fifth-order fully symmetric cubature points and weights.
sigma_points	function	Unscented transform sigma points and weights.

Likelihoods

Name	Kind	What it does
AbstractLikelihood	class	Base class for likelihood functions with optional analytical ELL.
GaussianLikelihood	class	Gaussian likelihood log N(y \| f, noise_var * I).
HeteroscedasticGaussianLikelihood	class	Heteroscedastic Gaussian likelihood with input-dependent noise.
BernoulliLikelihood	class	Bernoulli likelihood with logit link.
BinomialLikelihood	class	Binomial likelihood with logit link.
PoissonLikelihood	class	Poisson likelihood with log link.
NegativeBinomialLikelihood	class	Negative-binomial likelihood with log link (NB2, a gamma-Poisson mixture).
SoftmaxLikelihood	class	Softmax (categorical) likelihood for multi-class classification.
StudentTLikelihood	class	Student-t likelihood for robust regression.

Expectations & EP moments

Name	Kind	What it does
elbo	function	Evidence lower bound (ELBO).
expected_log_likelihood	function	Unified expected log-likelihood with analytical dispatch.
log_likelihood_expectation	function	Compute E[log p(y_{obs} \| f(x))] where x ~ N(mu, Sigma).
mean_expectation	function	Compute E[f(x)] where x ~ N(mu, Sigma).
gradient_expectation	function	Compute E[nabla f(x)] via Stein's lemma.
cost_expectation	function	Compute E[Cost(f(x), target)] where x ~ N(mu, Sigma).
ep_tilted_moments	function	Moments of the scalar EP tilted distribution, elementwise over a batch.

Moment matching & linearisation

Name	Kind	What it does
moment_match	function	EP moment matching: tilted log-normaliser and its first two derivatives.
MomentMatchResult	class	Tilted log-normaliser and its derivatives w.r.t. the cavity mean.
statistical_linear_regression	function	Moment-match a nonlinear conditional to a linear-Gaussian surrogate.
SLRResult	class	Linear-Gaussian surrogate produced by statistical linear regression.

Hyperparameter designs (INLA)

Name	Kind	What it does
theta_design	function	Integration points and log-weights over hyperparameters \(\theta\).

Kernel expectations & uncertain-input GPs

Name	Kind	What it does
kernel_expectations	function	Compute kernel expectations Psi_0, Psi_1, Psi_2 for uncertain inputs.
compute_psi_statistics	function	Compute Ψ statistics, dispatching to analytical or numerical.
AnalyticalPsiStatistics	class	Protocol for kernels with closed-form Ψ statistics.
uncertain_gp_predict	function	Predictive mean and variance for GP with uncertain inputs.
uncertain_gp_predict_mc	function	Monte Carlo GP prediction with uncertain inputs.
uncertain_svgp_predict	function	Predictive mean and variance for SVGP with uncertain inputs.
uncertain_vgp_predict	function	Predictive mean and variance for dense VGP with uncertain inputs.
uncertain_bgplvm_predict	function	Multi-output uncertain GP prediction for BGPLVM.

State-Space Models & Kalman

SDE kernels

Name	Kind	What it does
SDEKernel	class	Abstract base class for state-space kernel representations.
SDEParams	class	Continuous-time SDE parameters for a linear SDE.
ConstantSDE	class	State-space representation of a constant kernel.
MaternSDE	class	State-space representation of the Matern kernel.
PeriodicSDE	class	State-space representation of the periodic (MacKay) kernel.
QuasiPeriodicSDE	class	Quasi-periodic kernel: product of Matern and Periodic SDE.
CosineSDE	class	State-space representation of the cosine kernel.
IntegratedWienerSDE	class	State-space representation of an integrated Wiener process.
ProductSDE	class	Product of two SDE kernels via Kronecker composition.
SumSDE	class	Sum of SDE kernels via block-diagonal composition.
sde_autocovariance	function	Compute the stationary autocovariance of an SDE kernel.
discretize_mfd	function	Discretise a linear SDE by matrix-fraction decomposition.
discretize_mfd_sequence	function	Vectorised gaussx.discretize_mfd over a vector of time steps.

Nonlinear filters

Name	Kind	What it does
nonlinear_kalman_filter	function	Moment-matched nonlinear Kalman filter.
nonlinear_rts_smoother	function	Moment-matched nonlinear Rauch-Tung-Striebel smoother.
nonlinear_kalman_predict	function	One moment-matched predict step.
nonlinear_kalman_update	function	One moment-matched update step.
nonlinear_rts_step	function	One moment-matched RTS backward step.
masked_moment_inputs	function	Make masked observation channels inert in a moment-matched update.

Kalman filtering & smoothing

Name	Kind	What it does
EmissionModel	class	Observation (emission) model wrapping a linear observation matrix.
FilterState	class	Output of kalman_filter.
kalman_filter	function	Kalman filter forward pass via jax.lax.scan.
rts_smoother	function	Rauch-Tung-Striebel backward smoother.
kalman_gain	function	Compute Kalman gain K = P @ H^T @ (H @ P @ H^T + R)^{-1}.
parallel_kalman_filter	function	Parallel Kalman filter via jax.lax.associative_scan.
parallel_rts_smoother	function	Parallel RTS smoother via reverse jax.lax.associative_scan.
meanfield_kalman_filter	function	Mean-field Kalman filter over L = D // block_size state blocks.
meanfield_rts_smoother	function	Mean-field RTS smoother over L = D // block_size state blocks.
infinite_horizon_filter	function	Infinite-horizon Kalman filter with fixed steady-state gain.
infinite_horizon_smoother	function	Infinite-horizon RTS smoother with fixed steady-state gain.
dare	function	Steady-state Kalman filter covariance and gain (the filtering DARE).
DAREResult	class	Result of DARE solver.
pairwise_marginals	function	Joint p(x_k, x_{k+1}) for each consecutive pair.

SpInGP

Name	Kind	What it does
spingp_log_likelihood	function	Log marginal likelihood via sparse inverse GP formulation.
spingp_posterior	function	Posterior mean and precision via SpInGP.

Precision-form chains

Name	Kind	What it does
UDLDecomposition	class	Banded factorisation \(\Lambda = U \tilde{D} U^{\top}\) of a BlockTriDiag.
udl_decomposition	function	Factorise a symmetric block-tridiagonal precision as \(U \tilde{D} U^{\top}\).
udl_to_ssm_params	function	Read the equivalent Gauss-Markov chain off a UDLDecomposition.
udl_from_ssm_params	function	Build the UDLDecomposition of a Gauss-Markov chain's precision.

Sites & natural parameters

Name	Kind	What it does
GaussianSites	class	Time-varying Gaussian likelihood sites in natural parameterization.
cvi_update_sites	function	Natural gradient update for CVI sites.
sites_to_precision	function	Convert Gaussian sites to a block-tridiagonal precision.
cavity_from_marginal	function	Compute cavity distribution by removing a site from the marginal.
site_natural_from_tilted	function	Compute site natural parameters from tilted and cavity moments.
site_mean_var_from_natural	function	Convert per-site natural parameters to mean/variance.
expectations_to_ssm	function	Convert expectation parameters back to SSM marginals.
naturals_to_ssm	function	Convert natural parameters back to SSM parameters.
ssm_to_expectations	function	Convert SSM marginals to expectation parameters.
ssm_to_naturals	function	Convert SSM parameters to natural parameters.

Variational SDE linearisation

Name	Kind	What it does
LinearizedSDE	class	Linear surrogate \(dx = (A_t x + b_t)\,dt + Q_t^{1/2}\,dW\) of an SDE.
linearize_sde	function	Optimal linear drift of a nonlinear SDE along a Gaussian path.
sde_kl_divergence	function	Drift path-KL between a linear and a nonlinear SDE along a Gaussian path.

Process noise

Name	Kind	What it does
process_noise_covariance	function	Compute process noise from stationary covariance.

Bayesian Inference & Ensembles

Bayesian learning rule

Name	Kind	What it does
blr_full_update	function	Full-rank natural parameter BLR update step.
blr_diag_update	function	Diagonal natural parameter BLR update step.
log_marginal_likelihood	function	GP log marginal likelihood.
gaussian_expected_log_lik	function	Expected log-likelihood E_q[log N(y \| f, R)].

Newton & natural-gradient updates

Name	Kind	What it does
newton_update	function	Convert a Newton step to natural pseudo-likelihood parameters.
damped_natural_update	function	Damped update in natural parameter space.
gauss_newton_precision	function	Gauss-Newton precision matrix J^T J, optionally plus a prior.
ggn_diagonal	function	Generalized Gauss-Newton diagonal approximation.
hutchinson_hessian_diag	function	Stochastic Hessian diagonal via Hutchinson with Rademacher probes.
riemannian_psd_correction	function	Riemannian gradient correction for PSD precision updates.
cavity_distribution	function	Compute EP cavity distribution by removing a site.
trace_correction	function	Trace term in Titsias collapsed ELBO.

Precision-form Laplace (INLA)

Name	Kind	What it does
laplace_mode	function	Mode, Hessian and log-marginal of the Laplace approximation, in precision form.
LaplaceResult	class	The Gaussian approximation \(\mathcal N(\hat x, H^{-1})\) at the mode.

Low-rank VB mean correction (INLA)

Name	Kind	What it does
vb_mean_correction	function	Low-rank VB correction of the Laplace mean (Van Niekerk & Rue, 2024).

Ensemble covariances, gain & analysis

Name	Kind	What it does
ensemble_covariance	function	Empirical covariance from an ensemble as a low-rank operator.
ensemble_cross_covariance	function	Cross-covariance between two ensemble sets.
ensemble_kalman_gain	function	Kalman gain from an ensemble and its image in observation space.
enkf_analysis	function	Stochastic (perturbed-observation) ensemble Kalman analysis step.
etkf_transform	function	Ensemble Transform Kalman Filter (ETKF) analysis weights.

Ensemble Kalman inversion

Name	Kind	What it does
eki_step	function	One ensemble Kalman inversion (EKI) update.
tikhonov_augment	function	Observation augmentation for Tikhonov-regularised EKI (TEKI).
discrepancy_step_size	function	Adaptive EKI tempering step: the data misfit controller.

Localization & inflation

Name	Kind	What it does
localization_matrix	function	Pairwise Gaspari-Cohn taper rho(dist(a_i, b_j); c).
localized_kalman_gain	function	Ensemble Kalman gain with Hadamard (Schur-product) localization.
gaspari_cohn	function	Gaspari-Cohn (1999) fifth-order compactly-supported taper.
inflate_multiplicative	function	Multiplicative ensemble inflation about the mean.
inflate_rtpp	function	Relaxation to prior perturbations (RTPP; Zhang et al. 2004).
inflate_rtps	function	Relaxation to prior spread (RTPS; Whitaker & Hamill 2012).

Distances

Name	Kind	What it does
euclidean_distance	function	Pairwise Euclidean distances \|\|a_i - b_j\|\|.
haversine_distance	function	Pairwise great-circle (haversine) distances on a sphere.

Listed at lineax 0.1.1, matfree 0.6.2, optimistix 0.1.0.

Upstream: lineax

The operator protocol, tags and solvers every gaussx operator extends. Use these before wrapping or re-implementing an operator or solver.

Name	Kind	What it does
AbstractLinearOperator	class	Abstract base class for all linear operators.
AbstractLinearSolver	class	Abstract base class for all linear solvers.
AddLinearOperator	class	A linear operator formed by adding two other linear operators together.
AutoLinearSolver	class	Automatically determines a good linear solver based on the structure of the operator.
BiCGStab	class	Biconjugate gradient stabilised method for linear systems.
CG	class	Conjugate gradient solver for linear systems.
Cholesky	class	Cholesky solver for linear systems.
ComposedLinearOperator	class	A linear operator formed by composing (matrix-multiplying) two other linear operators together.
Diagonal	class	Diagonal solver for linear systems.
DiagonalLinearOperator	class	A diagonal linear operator, e.g. for a diagonal matrix.
DivLinearOperator	class	A linear operator formed by dividing a linear operator by a scalar.
FunctionLinearOperator	class	Wraps a linear function fn: X -> Y into a linear operator. (So that self.mv(x) is defined by self.mv(x) == fn(x).)
GMRES	class	GMRES solver for linear systems.
IdentityLinearOperator	class	Represents the identity transformation X -> X, where each x in X is some PyTree of floating-point JAX arrays.
JacobianLinearOperator	class	Given a function fn: X -> Y, and a point x in X, then this defines the linear operator (also a function X -> Y) given by the Jacobian (d(fn)/dx)(x).
LSMR	class	LSMR solver for linear systems.
LU	class	LU solver for linear systems.
MatrixLinearOperator	class	Wraps a 2-dimensional JAX array into a linear operator.
MulLinearOperator	class	A linear operator formed by multiplying a linear operator by a scalar.
NegLinearOperator	class	A linear operator formed by computing the negative of a linear operator.
Normal	class	Wrapper for an inner solver of positive (semi)definite systems.
NormalCG	function	Deprecated helper function.
PyTreeLinearOperator	class	Represents a PyTree of floating-point JAX arrays as a linear operator.
QR	class	QR solver for linear systems.
RESULTS	class	An enumeration, with the following entries:
SVD	class	SVD solver for linear systems.
Solution	class	The solution to a linear solve.
TaggedLinearOperator	class	Wraps another linear operator and specifies that it has certain tags, e.g. representing symmetry.
TangentLinearOperator	class	Internal to lineax.
Triangular	class	Triangular solver for linear systems.
Tridiagonal	class	Tridiagonal solver for linear systems, uses the LAPACK/cusparse implementation of Gaussian elimination with partial pivotting (which increases stability). .
TridiagonalLinearOperator	class	As lineax.MatrixLinearOperator, but for specifically a tridiagonal matrix.
conj	function	Elementwise conjugate of a linear operator.
diagonal	function	Extracts the diagonal from a linear operator, and returns a vector.
diagonal_tag	constant	Structural / property tag for operators.
has_unit_diagonal	function	Returns whether an operator is marked as having unit diagonal.
invert	function	Returns a lineax.FunctionLinearOperator representing the (pseudo)inverse of operator.
is_diagonal	function	Returns whether an operator is marked as diagonal.
is_lower_triangular	function	Returns whether an operator is marked as lower triangular.
is_negative_semidefinite	function	Returns whether an operator is marked as negative semidefinite.
is_positive_semidefinite	function	Returns whether an operator is marked as positive semidefinite.
is_symmetric	function	Returns whether an operator is marked as symmetric.
is_tridiagonal	function	Returns whether an operator is marked as tridiagonal.
is_upper_triangular	function	Returns whether an operator is marked as upper triangular.
linear_solve	function	Solves a linear system.
linearise	function	Linearises a linear operator.
lower_triangular_tag	constant	Structural / property tag for operators.
materialise	function	Materialises a linear operator.
negative_semidefinite_tag	constant	Structural / property tag for operators.
positive_semidefinite_tag	constant	Structural / property tag for operators.
symmetric_tag	constant	Structural / property tag for operators.
transpose_tags	function	Lineax uses "tags" to declare that a particular linear operator exhibits some property, e.g. symmetry.
transpose_tags_rules	constant	Re-exported from builtins.
tridiagonal	function	Extracts the diagonal, lower diagonal, and upper diagonal, from a linear operator.
tridiagonal_tag	constant	Structural / property tag for operators.
unit_diagonal_tag	constant	Structural / property tag for operators.
upper_triangular_tag	constant	Structural / property tag for operators.

Upstream: matfree

Matrix-free Krylov decompositions, matrix functions and stochastic trace / diagonal estimators. Never hand-roll Lanczos, Arnoldi or Hutchinson.

matfree.decomp

Name	Kind	What it does
bidiag	function	Construct an implementation of bidiagonalisation via the Golub-Kahan algorithm.
hessenberg	function	Construct a Hessenberg-factorisation via the Arnoldi iteration.
tridiag_sym	function	Construct an implementation of tridiagonalisation.

matfree.eig

Name	Kind	What it does
eig_partial	function	Compute a partial eigendecomposition of an arbitrary square matrix via Hessenberg factorisation.
eigh_partial	function	Compute a partial eigendecomposition \(A \approx V \Lambda V^\top\) for symmetric/Hermitian matrices.
svd_partial	function	Compute a partial SVD \(A \approx U \Sigma V^\top\) via bidiagonalisation.

matfree.funm

Name	Kind	What it does
dense_funm_pade_exp	function	Implement dense matrix-exponentials using a Pade approximation.
dense_funm_product_svd	function	Implement dense matrix-functions of a product of matrices via SVDs.
dense_funm_schur	function	Implement dense matrix-functions via symmetric Schur decompositions.
dense_funm_sym_eigh	function	Implement dense matrix-functions via symmetric eigendecompositions.
funm_arnoldi	function	Implement a matrix-function-vector product via the Arnoldi iteration.
funm_chebyshev	function	Compute a matrix-function-vector product via Chebyshev's algorithm.
funm_lanczos_sym	function	Implement a matrix-function-vector product via Lanczos' tridiagonalisation.
monte_carlo_funm_product	function	Construct the integrand for matrix-function-trace estimation.
monte_carlo_funm_product_logdet	function	Construct the integrand for the log-determinant of a matrix-product.
monte_carlo_funm_product_schatten_norm	function	Construct the integrand for the \(p\)-th power of the Schatten-p norm.
monte_carlo_funm_sym	function	Construct the integrand for matrix-function-trace estimation.
monte_carlo_funm_sym_logdet	function	Construct the integrand for the log-determinant.

matfree.low_rank

Name	Kind	What it does
cholesky_partial	function	Compute a partial Cholesky factorisation.
cholesky_partial_pivot	function	Compute a partial Cholesky factorisation with pivoting.
preconditioner	function	Turn a low-rank approximation into a preconditioner.

matfree.lstsq

Name	Kind	What it does
lsmr	function	Construct a matrix-free implementation of LSMR.

matfree.stochtrace

Name	Kind	What it does
estimator_leave_one_out	function	Construct a leave-one-out stochastic estimator.
estimator_leave_one_out_mean_and_sem	function	Construct a LOO estimator that returns mean and standard error.
estimator_monte_carlo	function	Construct a stochastic trace-/diagonal-estimator.
estimator_monte_carlo_mean_and_sem	function	Construct a stochastic estimator that returns mean and standard error.
leave_one_out_xdiag	function	Construct an integrand for estimating the diagonal using the XDiag algorithm (Epperly et al. 2024).
leave_one_out_xnysdiag	function	Construct an integrand for estimating the diagonal of a positive semi-definite operator using the XNysDiag algorithm (Epperly et al. 2025).
leave_one_out_xnystrace	function	Construct an integrand for estimating the trace of a positive semi-definite operator using the XNysTrace algorithm (Epperly et al. 2024).
leave_one_out_xrownorms_squared	function	Construct an integrand for estimating squared row norms using the XRowNorm algorithm (Epperly, 2025).
leave_one_out_xsymrownorms_squared	function	Construct an integrand for estimating squared row norms of a normal operator using the XSymRowNorm algorithm (Epperly, 2025).
leave_one_out_xtrace	function	Construct an integrand for estimating the trace using the XTrace algorithm (Epperly et al. 2024).
monte_carlo_diagonal	function	Construct the integrand for estimating the diagonal.
monte_carlo_frobeniusnorm_squared	function	Construct the integrand for estimating the squared Frobenius norm.
monte_carlo_rownorms_squared	function	Construct the integrand for estimating the squared row norms.
monte_carlo_trace	function	Construct the integrand for estimating the trace.
monte_carlo_trace_and_diagonal	function	Construct the integrand for estimating the trace and diagonal jointly.
nystrom_eigh	function	Construct a Nystrom approximation of an operator using a Hermitian eigendecomposition.
nystrom_shifted_cholesky	function	Construct a Nystrom approximation of a shifted operator using a Cholesky decomposition.
sampler_normal	function	Construct a function that samples from a standard-normal distribution.
sampler_signs	function	Construct a function that samples signs uniformly.
sampler_sphere	function	Construct a function that samples from a unit sphere scaled to have identity covariance.

matfree.bounds

Name	Kind	What it does
baigolub96_logdet_spd	function	Bound the log-determinant of a symmetric, positive definite matrix.

Upstream: optimistix

Root finding, fixed points, minimisation and least squares with implicit-differentiation adjoints. Use it instead of a hand-written while_loop when gradients must flow through the iteration.

Name	Kind	What it does
AbstractAdjoint	class	The abstract base class of all adjoints.
AbstractBFGS	class	Abstract version of the BFGS (Broyden–Fletcher–Goldfarb–Shanno) minimisation algorithm.
AbstractDFP	class	Abstract version of the DFP (Davidon–Fletcher–Powell) minimisation algorithm.
AbstractDescent	class	The abstract base class for descents.
AbstractFixedPointSolver	class	Abstract base class for all fixed point solvers.
AbstractGaussNewton	class	Abstract base class for all Gauss-Newton type methods.
AbstractGradientDescent	class	The gradient descent method for unconstrained minimisation.
AbstractIterativeSolver	class	Abstract base class for all iterative solvers.
AbstractLBFGS	class	Abstract version of the L-BFGS (limited memory Broyden–Fletcher–Goldfarb–Shanno) solver.
AbstractLeastSquaresSolver	class	Abstract base class for all least squares solvers.
AbstractMinimiser	class	Abstract base class for all minimisers.
AbstractQuasiNewton	class	Abstract quasi-Newton minimisation algorithm.
AbstractRootFinder	class	Abstract base class for all root finders.
AbstractSearch	class	The abstract base class for all searches. (Which are our generalisation of line searches, trust regions, and learning rates.)
BFGS	class	BFGS (Broyden–Fletcher–Goldfarb–Shanno) minimisation algorithm.
BacktrackingArmijo	class	Perform a backtracking Armijo line search.
BestSoFarFixedPoint	class	Wraps another fixed-point solver, to return the best-so-far value.
BestSoFarLeastSquares	class	Wraps another least-squares solver, to return the best-so-far value.
BestSoFarMinimiser	class	Wraps another minimiser, to return the best-so-far value.
BestSoFarRootFinder	class	Wraps another root-finder, to return the best-so-far value.
Bisection	class	The bisection method of root finding.
Chord	class	The Chord method of root finding.
ClassicalTrustRegion	class	The classic trust-region update algorithm which uses a quadratic approximation of the objective function to predict reduction.
DFP	class	DFP (Davidon–Fletcher–Powell) minimisation algorithm.
DampedNewtonDescent	class	The damped Newton (Levenberg--Marquardt) descent.
Dogleg	class	Dogleg algorithm.
DoglegDescent	class	The Dogleg descent step, which switches between the Cauchy and the Newton descent directions.
FixedPointIteration	class	Repeatedly calls a function in search of a fixed point.
FunctionInfo	class	Different solvers (BFGS, Levenberg--Marquardt, ...) evaluate different quantities of the objective function.
GaussNewton	class	Gauss-Newton algorithm, for solving nonlinear least-squares problems.
GoldenSearch	class	Golden-section search for finding the minimum of a univariate function in a given interval.
GradientDescent	class	Classic gradient descent with a learning rate learning_rate.
ImplicitAdjoint	class	Backpropagate via the implicit function theorem.
IndirectDampedNewtonDescent	class	The indirect damped Newton (Levenberg--Marquardt) trust-region descent.
IndirectLevenbergMarquardt	class	The Levenberg--Marquardt method as a true trust-region method.
LBFGS	class	L-BFGS (Limited-memory Broyden–Fletcher–Goldfarb–Shanno) minimisation algorithm.
LearningRate	class	Move downhill by taking a step of the fixed size learning_rate.
LevenbergMarquardt	class	The Levenberg--Marquardt method.
LinearTrustRegion	class	The trust-region update algorithm which uses a linear approximation of the objective function to predict reduction.
NelderMead	class	The Nelder-Mead minimisation algorithm. (Downhill simplex derivative-free method.)
Newton	class	Newton's method for root finding.
NewtonDescent	class	Newton descent direction.
NonlinearCG	class	The nonlinear conjugate gradient method.
NonlinearCGDescent	class	The nonlinear conjugate gradient step.
OptaxMinimiser	class	A wrapper to use Optax first-order gradient-based optimisers with optimistix.minimise.
RESULTS	class	An enumeration, with the following entries:
RecursiveCheckpointAdjoint	class	Backpropagate by differentiating through the iterates directly.
Solution	class	The solution to a nonlinear solve.
SteepestDescent	class	The descent direction given by locally following the gradient.
dai_yuan	function	The Dai--Yuan formula for β.
fixed_point	function	Find a fixed-point of a function.
fletcher_reeves	function	The Fletcher--Reeves formula for β.
hestenes_stiefel	function	The Hestenes--Stiefel formula for β.
least_squares	function	Solve a nonlinear least-squares problem.
max_norm	function	Compute the L-infinity norm of a PyTree of arrays.
minimise	function	Minimise a function.
polak_ribiere	function	The Polak--Ribière formula for β.
rms_norm	function	Compute the RMS (root-mean-squared) norm of a PyTree of arrays.
root_find	function	Solve a root-finding problem.
two_norm	function	Computes the L2 norm of a PyTree of arrays.

Shared private toolkits

Not public API: the plumbing gaussx's own modules share. Contributors use these instead of writing new helpers; a helper two modules need is added here, not inline in one caller.

Module	Purpose	Helpers
gaussx._einx	Typed einx adapters for tensor reshaping and contraction.	rearrange, repeat, reduce, einsum
gaussx._testing	Test utilities for gaussx.	default_tolerances, tree_allclose, _resolve_dtype, key_sequence, random_pd_matrix, random_pd_operator, psd_operator, random_kronecker_pd, random_block_diag_pd, random_low_rank_update, random_sum_of_kroneckers_pd, random_spd_block_tridiag, empirical_moments, dense_solve, dense_logdet, dense_inv, dense_diag, dense_trace, assert_sample_moments
gaussx._deprecation	gaussx's own deprecation warning category and helper (gh-332).	GaussxDeprecationWarning, renamed_attribute, warn_deprecated, renamed_kwargs
gaussx._operators._utils	Private utilities shared by operator implementations.	lineax_diagonal, _square_components, lineax_conj, register_lineax_structure_functions, _identity
gaussx._strategies._dispatch	Dispatch helpers: route through solver strategy or fall back to primitives.	dispatch_solve, dispatch_logdet
gaussx._strategies._tolerances	Dtype-aware default tolerances for the iterative strategies (gh-327).	operator_dtype, resolve_tolerance, rhs_scaling
gaussx._strategies._lineax	Adapter from a lineax solver to a gaussx solve strategy (gh-376).	LineaxSolver, as_solve_strategy
gaussx._distributions._utils	Shared helpers for batched distribution reshaping.	_axis_names, _reshape_batch, _reshape_series, _reshape_samples, _unflatten_sample_axis
gaussx._ssm._utils	Shared internal helpers for SSM operators.	_materialise, _matvec, _as_operator, _right_matmul_transpose, _left_matmul, _innovation_covariance, _masked_obs_inputs, _is_operator_input, _normalise_tv_inputs, _warn_unused_process_noise