compute_sigma

t3toolbox.backend.probing.compute_sigma(var_tt_cores, right_tt_cores, down_tt_cores, xis, dxis, mus)

def compute_sigma(
        var_tt_cores:       typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(rLi,nUi,rR(i+1))
        right_tt_cores:     typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(rRi,nUi,rR(i+1))
        down_tt_cores:     typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(rLi,nOi,rR(i+1))
        xis:                typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(...,nUi),
        dxis:               typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(...,nOi)
        mus:                typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(...,rLi)
) -> typ.Union[typ.Sequence[NDArray], NDArray]: # sigmas. len=d, elm_shape=(...,rR(i+1))

Compute var-leftward edge variables sigma. Used for probing a tangent vector.

See Section 6.2.2, particularly Algorithm 7, in:

Alger, N., Christierson, B., Chen, P., & Ghattas, O. (2026). “Tucker Tensor Train Taylor Series.” arXiv preprint arXiv:2603.21141. https://arxiv.org/abs/2603.21141

See also

compute_dxi, compute_tau, compute_deta, assemble_tangent_z, tv_probe

Parameters:

• var_tt_cores (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

• right_tt_cores (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

• down_tt_cores (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

• xis (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

• dxis (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

• mus (t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray])

Return type:

t3toolbox.backend.common.typ.Union[t3toolbox.backend.common.typ.Sequence[NDArray], NDArray]