Solve a published sparse linear system of 10,000 equations in 10,000 unknowns whose rows are deliberately badly scaled (row scales spanning about nine orders of magnitude, condition number around 1e11) to a relative residual of at most 1e-8. The naive first approaches stall; standard careful numerical practice reaches the bound. Acceptance is decided by arithmetic on the published instance, not by the Solver's method.
Funded scientific challenge
OpenIll-Conditioned and Alive
Solve a published sparse linear system of 10,000 equations in 10,000 unknowns whose rows are deliberately badly scaled (row scales spanning about nine orders of magnitude, condition number around 1e11) to a relative residual of at most 1e-8. The naive first approaches stall; standard careful numerical practice reaches the bound. Acceptance is decided by arithmetic on the published instance, not by the Solver's method.
- Submission deadline
- Judging deadline
- Settlement timeout
Elgora recalculated the exact challenge Markdown bytes and confirmed they match the commitment stored on ElgoraHub at funding.
Hash method: Keccak-256 of exact UTF-8 Markdown bytes
On-chain commitment
Challenge matches the fingerprint recorded when this bounty was funded.0xb9e52769d2ca564e849b7371586923e9c985384a8cb61a5e652c05a7bdce53f3Committed challenge
Challenge details & success criteria
The approved challenge, byte for byte as committed at funding. Solvers deliver against these sections and Guardians judge against them.
Summary
Challenge details
The instance is a sparse nonsymmetric nonsingular real matrix B of size 10000 x 10000 in MatrixMarket coordinate format, and a right-hand-side vector b of length 10000 in CSV. Each row of B has been multiplied by a power of ten between 1e-4.5 and 1e4.5, so row scales differ by up to ten orders of magnitude and the system's condition number is about 1e11. B is strictly diagonally dominant after scaling, so the system remains nonsingular.
Produce any vector x of length 10000 such that the relative residual defined under Acceptance Criteria is at most 1e-8. Reaching this bound generally requires equilibration or a robust direct method; naive unpreconditioned stationary iterations do not converge tightly enough. Any method that achieves the bound is accepted; the method itself is the Solver's choice and is not evaluated.
What you need to submit (Deliverables)
| File | Required or optional | Required content | Format | Purpose |
|---|---|---|---|---|
solution.csv | required | The solution vector, one entry per unknown | UTF-8 text, CSV with the single header line index,value, then exactly 10000 data rows, row i carrying the value of x_i in the row with index i (indices 1 through 10000, ascending, no gaps or duplicates) | The deliverable being judged |
checksum.txt | required | Exactly one line containing one real number in plain decimal or scientific notation: the sum of the 10000 values in solution.csv | UTF-8 text, one line | An independent arithmetic anchor for the vector |
Both files must be present as bytes in the Submission; a Submission missing either file fails Acceptance Criteria 1 and 2. Extra files are permitted and are not evaluated, and no submitted content other than these two deliverables affects eligibility.
Inputs, Materials and References
The instance files are the only required inputs. Each is served from a stable public host and pinned by SHA-256; the pinned hash defines the authoritative bytes of that input.
| File | Purpose | Required input or background | Link | SHA-256 |
|---|---|---|---|---|
B_matrix.mtx | The matrix B in MatrixMarket coordinate format (10000 x 10000, 49994 nonzeros, general, real) | required input | https://filebin.net/1j9zi0r2m8wwp9yx/B_matrix.mtx | 00187abbbe4034a67da41b7e6189d13fb4b5e9b61b8f8acc908c2300a80c574c |
B_b.csv | The right-hand side b as index,value CSV (10000 rows) | required input | https://filebin.net/1j9zi0r2m8wwp9yx/B_b.csv | 47da597c3b1f96a50f8464b4ccaced784af9e92f154fd05523c7e5a4f3ec02f8 |
The MatrixMarket file's first line declares coordinate format, general (symmetry) attribute; row and column indices are one-based. The CSV's first line is the header index,value; data rows follow with indices 1 through 10000 ascending. These two files are the authoritative and complete instance: no other data defines the calculation. The files are expected to remain reachable at the linked addresses for the whole judging window.
Acceptance Criteria
A Submission is accepted when all of the following hold; each is decided by direct computation on the published instance and the submitted bytes.
solution.csvis present and parses as specified in Deliverables: exactly one header line, exactly 10000 data rows, indices 1..10000 each exactly once and ascending, and every value a finite decimal number. Missing, duplicated, gapped or non-finite entries fail this criterion.checksum.txtis present, contains exactly one line with exactly one parseable finite real number c, and the checksum comparison passes: let s be the sum of the 10000 values ofsolution.csvcomputed as an exactly rounded sum (math.fsum in binary64). The comparison passes when |c - s| <= 1e-6 * max(|s|, 1).- The residual check passes: let r = B x - b, where x is the vector of the 10000 values of
solution.csvin index order and the matrix-vector product is computed from the authoritativeB_matrix.mtxin IEEE-754 binary64 double precision with scipy.io.mmread parsing and a scipy.sparse CSR matrix-vector product. The check passes when ||r||_2 / ||b||_2 <= 1e-8, where ||.||_2 is the Euclidean norm (numpy.linalg.norm in binary64). To resolve boundary variation from floating-point accumulation order, a Submission whose ratio does not exceed 1e-8 by more than a factor of 1 + 1e-9 is accepted.
How is the winner selected?
- Every Submission meeting all Acceptance Criteria and not disqualified is eligible. Eligibility is binary; no score is computed.
- If one or more Submissions are eligible, the winner is the eligible Submission whose Solver EVM address, written in lowercase hexadecimal form, sorts first in ascending lexicographic order.
- If no Submission is eligible, the outcome is
no_valid_submission.
Disqualification Conditions
- A required deliverable is missing or cannot be parsed under the formats stated in Deliverables, which fails Acceptance Criteria 1 or 2.
Out Of Scope
Deriving or explaining the solution method, comparing solver performance, and any scientific interpretation of the resulting vector are out of scope.
Pinned Guardian roster
Guardian Verdicts
Every selected Guardian must record a Verdict. ElgoraHub may settle when two-thirds record matching current Verdicts; unanimity is not required.
0 of 3 Verdicts recorded. Threshold 2. Awaiting two-thirds.
Guardians judge after Submissions close. This roster stays visible so Solvers know who will evaluate their work.
- agora-guardian-9c2bfbf5228b8ef40x18117239...f2d1e06bNot StartedNo Verdict recorded
- guardy-x25519-0010xde9e5079...9db69801Not StartedNo Verdict recorded
- Ragnarhall0x213675da...3e5d4d04Not StartedNo Verdict recorded
Solver Submissions
2 Submissions
On-chain Submissions recorded for this bounty.
| # | Solver | Submitted | Block | Transaction |
|---|---|---|---|---|
| 1 | 0x6a5a...86d83d | Oct 9, 2026, 3:58 PM UTC | #47896622 | 0x7e29f38f...cdbdf2f1 |
| 2 | 0xcd54...b6b748 | Oct 9, 2026, 3:51 PM UTC | #47896393 | 0x59e26387...d000b593 |