Tangential Schur verification map

The tangential-Schur layer is intentionally conservative: it implements finite right-tangential Pick certificates, constant-solution sanity paths, and rank-one Potapov/J-inner diagnostics. This page records the mathematical invariants used by the examples and tests so the feature can be audited without reading the implementation first.

Problem and certificate

For right tangential data

\[S(z_i)U_i = V_i, \qquad S : \mathbb{D}\to\mathbb{C}^{p\times q}, \qquad \lVert S\rVert_\infty \le 1,\]

with \(U_i\in\mathbb{C}^{q\times r_i}\) and \(V_i\in\mathbb{C}^{p\times r_i}\), the definite Pick matrix is

\[P_{ij} = \frac{U_i^H U_j - V_i^H V_j} {1-\overline{z_i}z_j}.\]

The implemented finite feasibility certificate is

\[P \succeq 0.\]

This is the central invariant. All tests either construct data where this certificate is known to hold, construct data where it must fail, or verify that algebraic transformations preserve it.

Verification cases

Tangential-Schur verification checklist

Claim

Mathematical check

Where it is tested or shown

Scalar reduction equals classical Pick interpolation

With \(u_i=1\), \(v_i=w_i\), recover \((1-\overline{w_i}w_j)/(1-\overline{z_i}z_j)\).

test_right_tangential_pick_matrix_scalar_reduces_to_classical_pick

One-node feasibility is the norm condition

For one rank-one datum, \(P_{11}\ge0 \Longleftrightarrow \lVert v\rVert\le\lVert u\rVert\).

test_single_node_norm_condition_is_exact_for_rank_one_data

Constant MIMO contractions are feasible

If \(V_i=S_0U_i\) and \(\lVert S_0\rVert_2\le1\), then \(P\succeq0\); when the directions span the input space, constant_schur_solution recovers \(S_0\).

test_random_constant_mimo_data_are_recovered_across_shapes

Nonconstant Schur data need not be constant-compatible

Data from \(S(z)=c b_a(z)\) are feasible for \(|c|<1\), but generally fail the constant-solution helper.

test_scalar_blaschke_data_are_feasible_but_not_constant

Tangential-column basis changes are congruences

Replacing \(U_i,V_i\) by \(U_iQ_i,V_iQ_i\) gives \(P' = Q^H P Q\) with block-diagonal unitary \(Q\).

test_pick_matrix_is_hermitian_and_unitary_mixing_is_congruent

Diagonal MIMO decomposes into scalar Pick blocks

Coordinate directions and diagonal values make the MIMO Pick matrix a direct sum of scalar Pick matrices.

test_diagonal_mimo_tangential_data_decompose_into_scalar_pick_blocks and Diagonal tangential Schur equals independent scalar Pick problems

Near-boundary data remain explicit diagnostics

Points close to \(|z|=1\) can make \(P\) ill-conditioned; the eigenvalues should remain finite and visible rather than hidden.

test_near_boundary_data_remain_finite_and_report_conditioning

Graph-vector strictness is checked before Potapov factors

For \(\xi=[v;u]\) and \(J=\operatorname{diag}(I_p,-I_q)\), require \(\xi^H J\xi<0\).

test_elementary_potapov_factor_rejects_non_strict_data

Elementary factors are J-inner on the boundary

\(\Theta(e^{j\omega})^HJ\Theta(e^{j\omega})=J\) and \(\Theta(z_i)\xi_i=0\).

test_j_inner_factor_preserves_j_form_on_many_boundary_points and Tangential Schur Pick and J-inner diagnostics

Products of compatible factors remain J-inner

Products share the same \(J\) signature and preserve the boundary identity up to numerical tolerance.

test_potapov_product_is_j_inner_on_boundary and test_potapov_product_rejects_mixed_j_signatures

J-inner arithmetic

For a strict rank-one datum, define

\[\begin{split}\xi = \begin{bmatrix} v \\ u \end{bmatrix}, \qquad J = \begin{bmatrix} I_p & 0 \\ 0 & -I_q \end{bmatrix}, \qquad \xi^H J\xi < 0.\end{split}\]

The implemented elementary factor is

\[\Theta(z)=I+(b_a(z)-1)P_\xi, \qquad P_\xi = \xi(\xi^H J\xi)^{-1}\xi^HJ.\]

The projection is checked through

\[P_\xi^2=P_\xi, \qquad P_\xi^H J = JP_\xi,\]

and the boundary identity

\[\Theta(e^{j\omega})^HJ\Theta(e^{j\omega})=J.\]

Because the scalar Blaschke factor satisfies \(|b_a(e^{j\omega})|=1\), the J-orthogonal projection formula gives a direct finite-dimensional diagnostic for J-inner arithmetic. The package tests this on individual factors, products, and vectorized frequency grids.

RKHS and algorithm-lineage checks

The Pick matrix is also documented and tested as a finite RKHS Gram matrix:

\[P_{ij}=U_i^H K_S(z_i,z_j)U_j, \qquad K_S(z,w)=\frac{I-S(z)^HS(w)}{1-\overline z w}.\]

This is the same positivity principle used in tangential Schur interpolation, but the package stops at finite certificates and elementary factors. The test suite therefore verifies the algebraic invariants needed for a credible baseline; it does not test a recursive chart algorithm because no such algorithm is claimed in this release.

In particular, the Hanzon–Olivi–Peeters/Marmorat algorithms use recursive linear-fractional Schur steps, chart data, and balanced state-space realizations to parametrize manifolds of fixed-degree stable all-pass/lossless systems. The lattice-dsp tangential-Schur layer is intentionally narrower: it checks Pick positivity, constant-compatible data, diagonal/scalar reductions, and elementary Potapov–Blaschke J-inner identities.

What is deliberately not tested as solved

The verification suite does not claim a full tangential-Schur synthesis algorithm for arbitrary feasible data. In particular, it does not claim:

  • generalized indefinite Schur interpolation;

  • singular Pick-matrix parametrization;

  • boundary interpolation;

  • automatic construction of all rational Schur solutions;

  • a complete linear-fractional parametrization of the solution set.

Those are natural future directions, but this release treats them as out of scope. The tested baseline is the finite definite Pick/J-inner layer that supports the package’s MIMO lattice diagnostics.

Reproducibility and tolerance policy

The deep tests use deterministic random seeds and small dimensions. Tolerances are chosen around double-precision residuals, not benchmark speed. Near-boundary cases are included to expose conditioning rather than to guarantee uniform accuracy for nearly singular interpolation problems.