Pith Number

pith:QQEI464Z

pith:2026:QQEI464ZWK7MWJXSN47CNTLNSX

not attested not anchored not stored refs pending

Towards Universal Convergence of Backward Error in Linear System Solvers

Elizaveta Rebrova, Micha{\l} Derezi\'nski, Yuji Nakatsukasa

Richardson iteration achieves at most 1/k relative backward error after k steps on any positive semidefinite linear system, independent of condition number.

arxiv:2604.16075 v2 · 2026-04-17 · math.NA · cs.DS · cs.LG · cs.NA · math.OC

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\pithnumber{QQEI464ZWK7MWJXSN47CNTLNSX}

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3 Author claim open · sign in to claim

4 Citations open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

the classical and simple Richardson iteration incurs at most 1/k (relative) backward error after k iterations on any positive semidefinite (PSD) linear system, irrespective of its condition number

C2weakest assumption

The matrix must be positive semidefinite for the 1/k backward error guarantee to hold unconditionally; the extension to general matrices relies on normal equations and is only observed empirically.

C3one line summary

Richardson iteration achieves universal 1/k backward error on PSD systems, enabling O(n²/ε) solvers; MINBERR reaches O(1/k²) rate and O(n²/√ε) complexity, with empirical O(1/k) extension to general systems.

Cited by

1 paper in Pith

Well-Conditioned Oblivious Perturbations in Linear Space

Receipt and verification

First computed	2026-05-26T01:02:34.186019Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

84088e7b99b2becb26f26f3e26cd6d95dc1d3c7f143a8d2fb60b57387cea6b52

Aliases

arxiv: 2604.16075 · arxiv_version: 2604.16075v2 · doi: 10.48550/arxiv.2604.16075 · pith_short_12: QQEI464ZWK7M · pith_short_16: QQEI464ZWK7MWJXS · pith_short_8: QQEI464Z

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/QQEI464ZWK7MWJXSN47CNTLNSX \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 84088e7b99b2becb26f26f3e26cd6d95dc1d3c7f143a8d2fb60b57387cea6b52

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "96fd09a4d6d1860cc73bdb393e31792d8b1857de15b7f4995665d2e2abd840cf",
    "cross_cats_sorted": [
      "cs.DS",
      "cs.LG",
      "cs.NA",
      "math.OC"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NA",
    "submitted_at": "2026-04-17T14:00:57Z",
    "title_canon_sha256": "f1409633cca170c8478e9b18a5f533ee865e53d2ef6777d22a2ff12a86c0f296"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.16075",
    "kind": "arxiv",
    "version": 2
  }
}