Priors¶

A prior is a Gauss–Markov process over the solution and its first q derivatives; it encodes the smoothness assumed before seeing the ODE. See How to choose for picking one.

Prior	Process	Key parameters
`IWP`	`q`-times integrated Wiener process (the default)	`q` (smoothness order), `d` (dimension), `Xi` (structural scale)
`MaternPrior`	Matern process	`q` (smoothness order), `d` (dimension), `length_scale`, `Xi` (structural scale)
`JointPrior`	a state block stacked with a hidden block	exposes `E0_x`, `E0_hidden` for joint state–parameter models
`PrecondIWP`, `PrecondMaternPrior`	preconditioned variants	better conditioning at high `q` / small `h`

q sets the method's order of accuracy; Xi (together with the calibrated sigma^2) sets the size of the uncertainty. Initialize the state from x0 with taylor_mode_initialization(vf, x0, q) (which returns the Taylor-mode mean and a zero/Dirac initial covariance). Symbols are defined in Notation.

Every prior exposes the discrete-time transition A(h), drift b(h), process noise Q(h), and — what the square-root filter actually consumes — its upper-triangular square root Q_sqr(h) (Q = Q_sqr.T @ Q_sqr). For IWP / PrecondIWP this is a closed form that avoids ever factorizing the dense, ill-conditioned Q(h).

Gaussian Markov process prior models.

Modules:

Name	Description
`gmp_priors`

Classes:

Name	Description
`IWP`	Integrated Wiener Process prior model.
`JointPrior`	Joint prior combining independent state (x) and hidden/input (u) priors.
`MaternPrior`	Matern Gaussian process prior model using block matrix exponential.
`PrecondIWP`	Preconditioned Integrated Wiener Process prior.
`PrecondJointPrior`	Joint prior for preconditioned priors combining state (x) and hidden (u).
`PrecondMaternPrior`	Preconditioned Matern Gaussian process prior model.

Functions:

Name	Description
`taylor_mode_initialization`	Return flattened Taylor-mode coefficients produced via JAX Jet.

Classes¶

IWP ¶

IWP(q: int, d: int, Xi: ArrayLike | None = None)

Bases: BasePrior

Integrated Wiener Process prior model.

Methods:

Name	Description
`A`	Return the state transition matrix for step size h.
`Q`	Return the diffusion matrix for step size h.
`Q_sqr`	Closed-form upper-triangular square root of `Q(h)`.
`b`	Return the drift vector for step size h.

Functions¶

A ¶

A(h: ArrayLike) -> Array

Return the state transition matrix for step size h.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike`	Step size.	required

Returns:

Type	Description
`Array`	State transition matrix (shape [(q+1)d, (q+1)d]).

Q ¶

Q(h: ArrayLike) -> Array

Return the diffusion matrix for step size h.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike`	Step size.	required

Returns:

Type	Description
`Array`	Diffusion matrix (shape [(q+1)d, (q+1)d]).

Q_sqr ¶

Q_sqr(h: ArrayLike) -> Array

Closed-form upper-triangular square root of Q(h).

Q(h)_sqr = kron(chol(Q_bar).T @ T(h), chol(xi).T) (see :meth:__init__); never forms or factorizes the dense ill-conditioned Q(h).

b ¶

b(h: ArrayLike) -> Array

Return the drift vector for step size h.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike`	Step size.	required

Returns:

Type	Description
`Array`	Zero drift vector (shape [(q+1)*d]).

JointPrior ¶

JointPrior(prior_x: BasePrior, prior_u: BasePrior)

Bases: BasePrior

Joint prior combining independent state (x) and hidden/input (u) priors.

Creates a block-diagonal prior structure where state and hidden evolution are independent. The resulting matrices are block-diagonal with zeros in the off-diagonal blocks.

For joint state-parameter estimation, the hidden state u can represent unknown parameters that appear in the ODE but evolve according to their own prior (e.g., IWP or Matern).

Projection matrices

E0: Extracts [x, u] - zeroth derivatives of both (shape [d_x + d_u, D]) E0_x: Extracts x only - zeroth derivative of state (shape [d_x, D]) E0_hidden: Extracts u only - zeroth derivative of hidden (shape [d_u, D]) E1: Extracts dx/dt - first derivative of state x (shape [d_x, D]) E2: Extracts d^2x/dt^2 - second derivative of x (shape [d_x, D]), if q >= 2

Parameters:

Name	Type	Description	Default
`prior_x`	`BasePrior`	BasePrior instance for state evolution.	required
`prior_u`	`BasePrior`	BasePrior instance for hidden/input evolution.	required

Methods:

Name	Description
`A`	Return the block-diagonal state transition matrix.
`Q`	Return the block-diagonal diffusion matrix.
`Q_sqr`	Block-diagonal upper-triangular square root of `Q(h)`.
`apply_state_sigma_sqr`	Scale only the state block of `Q_sqr` by `sqrt(sigma_sqr)`.
`apply_state_sigma_to_cov_sqr`	Scale only the state block of a full-state covariance sqrt.
`b`	Return the concatenated drift vector.

Attributes:

Name	Type	Description
`E0_hidden`	`Array`	Hidden state extraction matrix for u only (shape [d_u, D]).
`E0_state`	`Array`	State-only value projection (Bosch et al. 2022 sec. 3).
`E0_x`	`Array`	State extraction matrix for x only (shape [d_x, D]).
`xi_state`	`Array`	State-prior component-correlation matrix (shape `[d_x, d_x]`).

Attributes¶

E0_hidden `property` ¶

E0_hidden: Array

Hidden state extraction matrix for u only (shape [d_u, D]).

Use this as E0_hidden in the measurement model.

E0_state `property` ¶

E0_state: Array

State-only value projection (Bosch et al. 2022 sec. 3).

E0_x `property` ¶

E0_x: Array

State extraction matrix for x only (shape [d_x, D]).

Use this as E0 in the measurement model when you have hidden states.

xi_state `property` ¶

xi_state: Array

State-prior component-correlation matrix (shape [d_x, d_x]).

Diagonal-mode calibration only requires the state block to be per-component diagonal; the hidden block's xi can be arbitrary.

Functions¶

A ¶

A(h: ArrayLike) -> Array

Return the block-diagonal state transition matrix.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike`	Step size.	required

Returns:

Type	Description
`Array`	Block-diagonal transition matrix with state and input blocks.

Q ¶

Q(h: ArrayLike) -> Array

Return the block-diagonal diffusion matrix.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike`	Step size.	required

Returns:

Type	Description
`Array`	Block-diagonal diffusion matrix with state and input blocks.

Q_sqr ¶

Q_sqr(h: ArrayLike) -> Array

Block-diagonal upper-triangular square root of Q(h).

Q is block-diagonal, so its square root is the block-diagonal of the sub-priors' square roots -- this delegates to each sub-prior's :meth:Q_sqr (closed-form for IWP) rather than factorizing the dense joint Q(h).

apply_state_sigma_sqr ¶

apply_state_sigma_sqr(
    Q_sqr: Array, sigma_sqr: ArrayLike
) -> Array

Scale only the state block of Q_sqr by sqrt(sigma_sqr).

Accepts a scalar (uniform state-block scaling) or a length-d_x vector (per-component state-block scaling used by calibration="diagonal"). In both cases the input block is untouched. Q is block-diagonal, so its upper-triangular sqrt is also block-diagonal: column-scaling the state block of the sqrt by sqrt(sigma_sqr_tiled) yields blkdiag(diag(sigma) Q_x diag(sigma), Q_u) (where diag(sigma) collapses to a scalar in the scalar case).

apply_state_sigma_to_cov_sqr ¶

apply_state_sigma_to_cov_sqr(
    P_sqr: Array, sigma_sqr: ArrayLike
) -> Array

Scale only the state block of a full-state covariance sqrt.

P = P_sqr.T @ P_sqr. Multiplying columns [:, :D_x] of the upper-triangular sqrt by sqrt(sigma_sqr) scales rows and columns :D_x of P by sigma_sqr and leaves the input block alone.

b ¶

b(h: ArrayLike) -> Array

Return the concatenated drift vector.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike`	Step size.	required

Returns:

Type	Description
`Array`	Drift vector concatenating state and input drifts.

MaternPrior ¶

MaternPrior(
    q: int,
    d: int,
    length_scale: float,
    Xi: ArrayLike | None = None,
)

Bases: BasePrior

Matern Gaussian process prior model using block matrix exponential.

Note

Unlike IWP, the Matern process noise Q(h) is derived from a block matrix exponential and factorized via a (jittered) Cholesky rather than a closed-form square root. At high smoothness (roughly q >= 6) that dense Q loses positive-definiteness in float64 and Q_sqr can return NaN. Prefer q <= 4 for Matern priors, or use IWP / PrecondIWP (closed-form Q_sqr) when you need higher orders.

Initialize the Matern prior.

This creates a Matern process prior for a d-dimensional process whose dimensions share the same smoothness q and length scale and are coupled by the component-correlation matrix Xi: the process noise is kron(Q_scalar(h), Xi). Xi may be any positive-(semi)definite matrix -- a diagonal Xi gives independent per-dimension output scales, off-diagonal entries add cross-dimension correlation. (The per-component "diagonal" calibration modes additionally require Xi diagonal; that is checked at solve time.)

Parameters:

Name	Type	Description	Default
`q`	`int`	Smoothness order.	required
`d`	`int`	State dimension.	required
`length_scale`	`float`	Length scale of the process.	required
`Xi`	`ArrayLike \| None`	Optional component-correlation matrix (shape [d, d]).	`None`

Methods:

Name	Description
`A`	Return the state transition matrix for step size h.
`A_and_Q`	Compute both A(h) and Q(h) efficiently in a single expm call.
`Q`	Return the diffusion matrix for step size h.
`b`	Return the drift vector for step size h.

Functions¶

A ¶

A(h: ArrayLike) -> Array

Return the state transition matrix for step size h.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike`	Step size.	required

Returns:

Type	Description
`Array`	State transition matrix (shape [n, n]).

A_and_Q ¶

A_and_Q(h: ArrayLike) -> tuple[Array, Array]

Compute both A(h) and Q(h) efficiently in a single expm call. This is sometimes called matrix fraction decomposition (MFD)

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike`	Step size.	required

Returns:

Type	Description
`Array`	Tuple of (A_h, Q_h) where:
`Array`	A_h: State transition matrix (shape [n, n])
`tuple[Array, Array]`	Q_h: Diffusion matrix (shape [n, n])

Q ¶

Q(h: ArrayLike) -> Array

Return the diffusion matrix for step size h.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike`	Step size.	required

Returns:

Type	Description
`Array`	Diffusion matrix (shape [(q+1)d, (q+1)d]).

b ¶

b(h: ArrayLike) -> Array

Return the drift vector for step size h.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike`	Step size.	required

Returns:

Type	Description
`Array`	Zero drift vector (shape [(q+1)*d]).

PrecondIWP ¶

PrecondIWP(q: int, d: int, Xi: ArrayLike | None = None)

Bases: BasePrior

Preconditioned Integrated Wiener Process prior.

Uses a preconditioning transformation T(h) to make matrices stepsize-independent. The transformation matrices A() and Q() are constant (independent of h), while the stepsize dependence is absorbed into T(h).

Methods:

Name	Description
`A`	Return the constant preconditioning transition matrix.
`Q`	Return the constant preconditioning diffusion matrix.
`Q_sqr`	Return the constant upper-triangular square root of `Q`.
`T`	Return the stepsize-dependent preconditioning transformation.
`b`	Return the zero drift vector.

Functions¶

A ¶

A(h: ArrayLike | None = None) -> Array

Return the constant preconditioning transition matrix.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike \| None`	Step size (unused, accepted for interface compatibility).	`None`

Returns:

Type	Description
`Array`	Constant transition matrix (shape [(q+1)d, (q+1)d]).

Q ¶

Q(h: ArrayLike | None = None) -> Array

Return the constant preconditioning diffusion matrix.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike \| None`	Step size (unused, accepted for interface compatibility).	`None`

Returns:

Type	Description
`Array`	Constant diffusion matrix (shape [(q+1)d, (q+1)d]).

Q_sqr ¶

Q_sqr(h: ArrayLike | None = None) -> Array

Return the constant upper-triangular square root of Q.

Stepsize-independent in preconditioned space (see :meth:__init__).

T ¶

T(h: ArrayLike) -> Array

Return the stepsize-dependent preconditioning transformation.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike`	Step size.	required

Returns:

Type	Description
`Array`	Preconditioning transformation matrix (shape [(q+1)d, (q+1)d]).

b ¶

b(h: ArrayLike | None = None) -> Array

Return the zero drift vector.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike \| None`	Step size (unused, accepted for interface compatibility).	`None`

Returns:

Type	Description
`Array`	Zero drift vector (shape [(q+1)*d]).

PrecondJointPrior ¶

PrecondJointPrior(
    prior_x: PrecondIWP | PrecondMaternPrior,
    prior_u: PrecondIWP | PrecondMaternPrior,
)

Joint prior for preconditioned priors combining state (x) and hidden (u).

This is the preconditioned version of JointPrior, designed specifically for PrecondIWP priors. The matrices A(), b(), Q() are constant (stepsize-independent), with the stepsize dependence absorbed into T(h).

Note: Mixing preconditioned and non-preconditioned priors is not supported and would lead to inconsistent filter behavior.

Projection matrices

E0: Extracts [x, u] - zeroth derivatives of both (shape [d_x + d_u, D]) E0_x: Extracts x only - zeroth derivative of state (shape [d_x, D]) E0_hidden: Extracts u only - zeroth derivative of hidden (shape [d_u, D]) E1: Extracts dx/dt - first derivative of state x (shape [d_x, D]) E2: Extracts d^2x/dt^2 - second derivative of x (shape [d_x, D]), if q >= 2

Parameters:

Name	Type	Description	Default
`prior_x`	`PrecondIWP \| PrecondMaternPrior`	PrecondIWP instance for state evolution.	required
`prior_u`	`PrecondIWP \| PrecondMaternPrior`	PrecondIWP instance for hidden/input evolution.	required

Example

prior_x = PrecondIWP(q=2, d=2) prior_u = PrecondIWP(q=1, d=1) joint = PrecondJointPrior(prior_x, prior_u) A = joint.A() # Constant transition matrix T_h = joint.T(0.1) # Stepsize-dependent transformation

Methods:

Name	Description
`A`	Return the block-diagonal transition matrix.
`Q`	Return the block-diagonal diffusion matrix.
`Q_sqr`	Block-diagonal upper-triangular square root of `Q`.
`T`	Return the stepsize-dependent block-diagonal transformation matrix.
`apply_state_sigma_sqr`	Scale only the state block of `Q_sqr` by `sqrt(sigma_sqr)`.
`apply_state_sigma_to_cov_sqr`	Scale only the state block of a full-state covariance sqrt.
`b`	Return the zero drift vector.

Attributes:

Name	Type	Description
`E0`	`Array`	Combined state extraction matrix (shape [d_x + d_u, D]).
`E0_hidden`	`Array`	Hidden state extraction matrix for u only (shape [d_u, D]).
`E0_state`	`Array`	State-only value projection (Bosch et al. 2022 sec. 3).
`E0_x`	`Array`	State extraction matrix for x only (shape [d_x, D]).
`E1`	`Array`	First derivative extraction matrix for x (shape [d_x, D]).
`E2`	`Array \| None`	Second derivative extraction matrix for x (shape [d_x, D]), or None.
`xi_state`	`Array`	State-prior component-correlation matrix (shape `[d_x, d_x]`).

Attributes¶

E0 `property` ¶

E0: Array

Combined state extraction matrix (shape [d_x + d_u, D]).

E0_hidden `property` ¶

E0_hidden: Array

Hidden state extraction matrix for u only (shape [d_u, D]).

Use this as E0_hidden in the measurement model.

E0_state `property` ¶

E0_state: Array

State-only value projection (Bosch et al. 2022 sec. 3).

E0_x `property` ¶

E0_x: Array

State extraction matrix for x only (shape [d_x, D]).

Use this as E0 in the measurement model when you have hidden states.

E1 `property` ¶

E1: Array

First derivative extraction matrix for x (shape [d_x, D]).

E2 `property` ¶

E2: Array | None

Second derivative extraction matrix for x (shape [d_x, D]), or None.

xi_state `property` ¶

xi_state: Array

State-prior component-correlation matrix (shape [d_x, d_x]).

Functions¶

A ¶

A(h: ArrayLike | None = None) -> Array

Return the block-diagonal transition matrix.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike \| None`	Step size (passed through to sub-priors).	`None`

Returns:

Type	Description
`Array`	Block-diagonal transition matrix with state and hidden blocks.

Q ¶

Q(h: ArrayLike | None = None) -> Array

Return the block-diagonal diffusion matrix.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike \| None`	Step size (passed through to sub-priors).	`None`

Returns:

Type	Description
`Array`	Block-diagonal diffusion matrix with state and hidden blocks.

Q_sqr ¶

Q_sqr(h: ArrayLike | None = None) -> Array

Block-diagonal upper-triangular square root of Q.

Delegates to each sub-prior's :meth:Q_sqr (constant for PrecondIWP) rather than factorizing the dense joint Q.

T ¶

T(h: ArrayLike) -> Array

Return the stepsize-dependent block-diagonal transformation matrix.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike`	Step size.	required

Returns:

Type	Description
`Array`	Block-diagonal preconditioning transformation matrix.

apply_state_sigma_sqr ¶

apply_state_sigma_sqr(
    Q_sqr: Array, sigma_sqr: ArrayLike
) -> Array

Scale only the state block of Q_sqr by sqrt(sigma_sqr).

Accepts a scalar or a length-d_x vector. See :meth:JointPrior.apply_state_sigma_sqr for the rationale.

apply_state_sigma_to_cov_sqr ¶

apply_state_sigma_to_cov_sqr(
    P_sqr: Array, sigma_sqr: ArrayLike
) -> Array

Scale only the state block of a full-state covariance sqrt.

See :meth:JointPrior.apply_state_sigma_to_cov_sqr for the rationale.

b ¶

b(h: ArrayLike | None = None) -> Array

Return the zero drift vector.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike \| None`	Step size (passed through to sub-priors).	`None`

Returns:

Type	Description
`Array`	Zero drift vector (shape [D_x + D_u]).

PrecondMaternPrior ¶

PrecondMaternPrior(
    q: int,
    d: int,
    length_scale: float,
    Xi: ArrayLike | None = None,
)

Bases: BasePrior

Preconditioned Matern Gaussian process prior model.

Uses the same diagonal preconditioner D(h) as PrecondIWP, but the preconditioned transition A_bar(h) and diffusion Q_bar(h) are stepsize-dependent (they converge to the IWP constants as h -> 0).

Note

Like MaternPrior, the underlying expm-derived Q is Cholesky- factorized and can return NaN at high smoothness (roughly q >= 6); preconditioning does not rescue this. Prefer q <= 4, or PrecondIWP for higher orders.

Parameters:

Name	Type	Description	Default
`q`	`int`	Smoothness order (nu = q + 1/2).	required
`d`	`int`	State dimension.	required
`length_scale`	`float`	Length scale of the Matern process.	required
`Xi`	`ArrayLike \| None`	Optional scaling matrix (shape [d, d]).	`None`

Methods:

Name	Description
`A`	Return the preconditioned transition matrix for step size h.
`Q`	Return the preconditioned diffusion matrix for step size h.
`Q_sqr`	Upper-triangular square root of the preconditioned `Q(h)`.
`T`	Return the stepsize-dependent preconditioning transformation.
`b`	Return the zero drift vector.

Functions¶

A ¶

A(h: ArrayLike | None = None) -> Array

Return the preconditioned transition matrix for step size h.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike \| None`	Step size (required; Matern preconditioning is stepsize-dependent).	`None`

Returns:

Type	Description
`Array`	Preconditioned transition matrix (shape [(q+1)d, (q+1)d]).

Q ¶

Q(h: ArrayLike | None = None) -> Array

Return the preconditioned diffusion matrix for step size h.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike \| None`	Step size (required; Matern preconditioning is stepsize-dependent).	`None`

Returns:

Type	Description
`Array`	Preconditioned diffusion matrix (shape [(q+1)d, (q+1)d]).

Q_sqr ¶

Q_sqr(h: ArrayLike | None = None) -> Array

Upper-triangular square root of the preconditioned Q(h).

The preconditioned Q_bar(h) is well-conditioned (it converges to the IWP constant as h -> 0), so a symmetrized Cholesky is stable here. Accepts h=None for interface symmetry with the other preconditioned priors, but Q itself requires h.

T ¶

T(h: ArrayLike) -> Array

Return the stepsize-dependent preconditioning transformation.

Uses the same preconditioner as PrecondIWP.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike`	Step size.	required

Returns:

Type	Description
`Array`	Preconditioning transformation matrix (shape [(q+1)d, (q+1)d]).

b ¶

b(h: ArrayLike | None = None) -> Array

Return the zero drift vector.

Parameters:

Name	Type	Description	Default
`h`	`ArrayLike \| None`	Step size (unused, accepted for interface compatibility).	`None`

Returns:

Type	Description
`Array`	Zero drift vector (shape [(q+1)*d]).

Functions¶

taylor_mode_initialization ¶

taylor_mode_initialization(
    vf: VectorField,
    inits: ArrayLike | tuple[ArrayLike, ...],
    q: int,
    t0: float = 0.0,
    order: int = 1,
) -> tuple[Array, Array]

Return flattened Taylor-mode coefficients produced via JAX Jet.

Parameters:

Name	Type	Description	Default
`vf`	`VectorField`	Vector field whose Taylor coefficients are required. For order=1: vf(x, , t) -> dx/dt For order=2: vf(x, dx, , t) -> d²x/dt²	required
`inits`	`ArrayLike \| tuple[ArrayLike, ...]`	Initial value(s) around which the expansion takes place. For order=1: x0 (initial state) For order=2: (x0, dx0) tuple of initial state and velocity	required
`q`	`int`	Number of higher-order coefficients to compute.	required
`t0`	`float`	Initial time for Taylor expansion (default 0.0).	`0.0`
`order`	`int`	ODE order (1 or 2, default 1).	`1`

Returns:

Type	Description
`Array`	Tuple of (coefficients, covariance) where:
`Array`	coefficients are the flattened Taylor coefficients as numpy array
`tuple[Array, Array]`	covariance is a zero covariance matrix as numpy array

Priors¶

Classes¶

IWP ¶

Functions¶

A ¶

Q ¶

Q_sqr ¶

b ¶

JointPrior ¶

Attributes¶

E0_hidden property ¶

E0_state property ¶

E0_x property ¶

xi_state property ¶

Functions¶

A ¶

Q ¶

Q_sqr ¶

apply_state_sigma_sqr ¶

apply_state_sigma_to_cov_sqr ¶

b ¶

MaternPrior ¶

Functions¶

A ¶

A_and_Q ¶

Q ¶

b ¶

PrecondIWP ¶

Functions¶

A ¶

Q ¶

Q_sqr ¶

T ¶

b ¶

PrecondJointPrior ¶

Attributes¶

E0 property ¶

E0_hidden property ¶

E0_state property ¶

E0_x property ¶

E1 property ¶

E2 property ¶

xi_state property ¶

Functions¶

A ¶

Q ¶

Q_sqr ¶

T ¶

apply_state_sigma_sqr ¶

apply_state_sigma_to_cov_sqr ¶

b ¶

PrecondMaternPrior ¶

Functions¶

A ¶

Q ¶

Q_sqr ¶

T ¶

b ¶

Functions¶

taylor_mode_initialization ¶

E0_hidden `property` ¶

E0_state `property` ¶

E0_x `property` ¶

xi_state `property` ¶

E0 `property` ¶

E0_hidden `property` ¶

E0_state `property` ¶

E0_x `property` ¶

E1 `property` ¶

E2 `property` ¶

xi_state `property` ¶