Paper Citation Record · LEDGER

Randomized biorthogonalization through a two-sided Gram-Schmidt process

As of 21 August 2026, this Paper Citation Record lists 38 of 38 outbound references and 0 inbound Pith citation observations for arXiv:2509.04386.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2509.04386 v1

Coverage vector

measured 38 of 38 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-05T10:20:08.598762Z

measured 38 of 38 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-21T06:32:19.484+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: cited_works

Reference resolution

38 of 38 outbound references displayed

verified exact2
verified fuzzy18
unresolved18
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 016ae6ae-8fe8-4b92-a6e5-2b294a324b21 · outbound

This paper cites Abramowitz and I.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Abramowitz and I

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:09.130979Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation b9a858b4-76cf-42ec-9bc8-dd8b8c995835 · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 2

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:09.121505Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation c1319408-f85b-4d16-8ae5-8ac7b01ae375 · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 3

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:09.111631Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 326f527a-0ee9-41c7-9afe-d1bd02edb910 · outbound

This paper cites Balabanov and L.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Balabanov and L

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:09.102273Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 0ca5b12e-375b-4fc4-a3e0-a23e4bcdf0fe · outbound

This paper cites A553–A5 85.

Randomized biorthogonalization through a two-sided Gram-Schmidt process A553–A5 85

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:09.092359Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation ad7b1e53-b810-4406-bc7e-846c08f30caa · outbound

This paper cites Estimation of spectral gaps for sparse symmetric matrices.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Estimation of spectral gaps for sparse symmetric matrices

Reference 6

Resolution: unresolved; no resolver link, observed 2026-08-05T10:20:08.480920Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 24b85e3d-736c-416c-b282-875320045188 · outbound

This paper cites Bj ¨orck, Solving linear least squares problems by Gram-Schmidt orth ogonalization, BIT Numerical Mathe- matics, 7 (1967), pp.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Bj ¨orck, Solving linear least squares problems by Gram-Schmidt orth ogonalization, BIT Numerical Mathe- matics, 7 (1967), pp

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:09.081023Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 74b8e2c2-f508-4c88-84b7-bd25510dd156 · outbound

This paper cites Bj ¨orck and C.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Bj ¨orck and C

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:09.068591Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 27d47be5-99b1-4b1f-9e0d-f9e83d0df89e · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 9

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:09.058471Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation d1771c75-1624-417e-ae87-8703f72ea2bb · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 10

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:09.048310Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation f523dc95-4283-4043-973b-6e0f770ec7f4 · outbound

This paper cites Randomized Implicitly Restarted Arnoldi method for the non-symmetric eigenvalue problem.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Randomized Implicitly Restarted Arnoldi method for the non-symmetric eigenvalue problem

Reference 11

Resolution: verified exact; local_arxiv, observed 2026-08-05T10:20:08.668607Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 950bcc35-7d2b-40fb-a696-80ce7452cf08 · outbound

This paper cites Randomized Krylov-Schur eigensolver with deflation.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Randomized Krylov-Schur eigensolver with deflation

Reference 12

Resolution: unresolved; no resolver link, observed 2026-08-05T10:20:08.503918Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8abc845b-6588-4a6f-8f3c-d0d4d71b5fa9 · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 13

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:09.037069Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation c746799a-e29c-474b-801e-7ec4cb3a9d0c · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 14

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:09.025097Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation c1b2523b-4868-472a-be04-9981a85dabdf · outbound

This paper cites Randomized Householder QR.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Randomized Householder QR

Reference 15

Resolution: verified exact; local_arxiv, observed 2026-08-05T10:20:08.643023Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation e554f959-a451-478d-9f95-f19b2c551e09 · outbound

This paper cites Randomized strong rank-revealing QR for column subset selection and low-rank matrix approximation.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Randomized strong rank-revealing QR for column subset selection and low-rank matrix approximation

Reference 16

Resolution: unresolved; no resolver link, observed 2026-08-05T10:20:08.519906Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c1146e6a-f008-4898-a477-5358a5649457 · outbound

This paper cites G ¨uttel and M.

Randomized biorthogonalization through a two-sided Gram-Schmidt process G ¨uttel and M

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:09.014724Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 85daab4f-aabd-4fba-9938-047d092e97e7 · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 18

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:09.002980Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation fee23f12-4bfb-455c-b110-a1ecdf3d8f8e · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 19

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:08.991633Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 71db9018-d54c-4ad8-8f46-81c9a654784e · outbound

This paper cites Kahan, B.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Kahan, B

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:08.981650Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation f863ec9b-4d6d-4233-bd6c-244e8fff8c76 · outbound

This paper cites Lanczos , An iteration method for the solution of the eigenvalue probl em of linear diﬀerential and integral operators, J.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Lanczos , An iteration method for the solution of the eigenvalue probl em of linear diﬀerential and integral operators, J

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:08.971272Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 0d6255e5-4ae1-4a13-b065-abd17298c482 · outbound

This paper cites Martinsson and J.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Martinsson and J

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:08.960649Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 4433344f-f354-47ea-b948-01c84f593b9f · outbound

This paper cites Meng and M.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Meng and M

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:08.950827Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 3d97255b-4dca-495e-8a28-34333eae22ed · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 24

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:08.941208Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 1da33e9c-fe00-45d3-bec2-3548a114756e · outbound

This paper cites Nadarajah and T.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Nadarajah and T

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:08.931118Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 70a76a53-88d2-4581-8d1f-9484c24c93b4 · outbound

This paper cites Nakatsukasa and J.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Nakatsukasa and J

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:08.824951Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation dc2c693c-21fd-4657-9180-f2c224bf791a · outbound

This paper cites Oktay and E.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Oktay and E

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:08.813719Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 73047c33-4d09-40d6-a0d1-6313656fca43 · outbound

This paper cites Palitta, M.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Palitta, M

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:08.802158Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 2208753e-855c-4157-bd6e-7386e21baf60 · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 29

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:08.789688Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 1dec110d-2d92-4e2b-933c-c23d936f600e · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 30

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:08.779715Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation a1c3cf1a-134f-47eb-856f-13764913fa1a · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 31

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:08.768813Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 49f38cdd-4daa-40da-9465-03159c0a1086 · outbound

This paper cites Saad , Krylov subspace methods on supercomputers , SIAM Journal on Scientiﬁc and Statistical Computing, 10 (1989), pp.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Saad , Krylov subspace methods on supercomputers , SIAM Journal on Scientiﬁc and Statistical Computing, 10 (1989), pp

Reference 32

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:08.757841Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 358ac608-19a2-44ea-8943-bda084803475 · outbound

This paper cites Saad , Numerical Methods for Large Eigenvalue Problems , Society for Industrial and Applied Mathematics,.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Saad , Numerical Methods for Large Eigenvalue Problems , Society for Industrial and Applied Mathematics,

Reference 33

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:08.745897Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 2de83490-1f9c-4556-9ab9-ea48170ce862 · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 34

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:08.734002Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation abd4478f-a6a1-465e-a21e-262cd1bc63e8 · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 35

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:08.723255Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation f659037d-67ef-4568-93df-501eb0297100 · outbound

This paper cites Yamazaki, S.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Yamazaki, S

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:08.712937Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 65329c88-850a-4761-8fd8-6e9925924941 · outbound

This paper cites Yamazaki, S.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Yamazaki, S

Reference 37

Resolution: verified fuzzy; raw_fallback, observed 2026-08-05T10:20:08.700661Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation efb91457-1e29-4e13-990e-36e531f8587a · outbound

This paper cites an unresolved cited work.

Randomized biorthogonalization through a two-sided Gram-Schmidt process Unresolved cited work

Reference 38

Resolution: unresolved; raw_fallback, observed 2026-08-05T10:20:08.689680Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Pith citing papers

No inbound Pith citation observations are available.