Singular Value Decomposition

This page documents verified SVD computation and related tools.

Rigorous SVD

The rigorous_svd function computes verified singular value decomposition with guaranteed error bounds on the singular values and vectors. It returns a RigorousSVDResult containing the factors and bounds.

Adaptive Ogita SVD

The adaptive_ogita_svd function uses iterative refinement to achieve tighter singular value bounds. This is particularly useful when high precision is needed. The refinement algorithm is based on Ref. [6].

Related functions:

• ogita_svd_refine - Single refinement step
• OgitaSVDRefinementResult - Result type for refinement
• AdaptiveSVDResult - Result type for adaptive SVD

Block Gershgorin enclosure in the Schur basis

The schur_gershgorin_enclosure function encloses the spectrum by inflated Gershgorin discs in a unitary basis of the midpoint matrix, grouped into clusters, and returns a SchurGershgorinResult. The basis is permuted, not transformed, so the remainder is the off-block part of Z*AZ; for a basis that decouples the blocks see miyajima2014a_schurnewton.

Singular Value Bounds

Low-level functions for computing rigorous bounds on singular values:

• svdbox - Box enclosure for singular values
• collatz_upper_bound_L2_opnorm - Collatz bound for L2 operator norm
• upper_bound_L2_opnorm - Upper bound on L2 operator norm