---

name: nystrom_preconditioner description: Build and apply the inverse action of the Nyström preconditioner for regularized PSD linear systems.

Nyström Preconditioner

Use this skill after obtaining U, lambda_hat from a randomized Nyström factorization. It provides the inverse preconditioner action used by Nyström PCG and dense diagnostics for small recovery experiments.

Inputs

• U: orthonormal Nyström basis.
• lambda_hat: nonnegative approximate eigenvalues.
• mu > 0: regularization parameter.
• Residual vector or right-hand-side matrix.

Outputs

• apply_inverse_preconditioner(U, lambda_hat, mu, r) result.
• Preconditioner metadata, including rank and baseline scale.
• Optional condition-number comparison for A + mu I versus the symmetrically preconditioned matrix.

Workflow

1. Validate dimensions, mu, and approximate orthonormality.
2. Sort eigenpairs by decreasing lambda_hat if necessary.
3. Set sigma = lambda_hat[-1] + mu.
4. Apply P^{-1} by scaling captured components with sigma / (lambda_hat_i + mu) and leaving orthogonal components unchanged.
5. For dense recovery checks, construct the SPD square-root action and estimate condition numbers.

Validation

Run:

python scripts/preconditioner.py --self-test
python tests/test_preconditioner.py

The tests verify inverse-action shape, dimension checks, and condition-number reduction on a diagonal PSD example.

Limitations

Dense condition-number diagnostics are intended for small experiments. Large systems should only use the vector action.