{"uid":"cap_JVX6ZP9theKW0DDBx8IB_","slug":"bounded-dense-multi-output-least-squares-minimum-norm-solver-7383288b","name":"Bounded Dense Multi-Output Least-Squares Minimum-Norm Solver","description":"Solves a bounded dense multi-output least-squares problem and returns the minimum-norm solution together with the diagnostics that say whether it can be trusted: numerical rank, the full singular value spectrum, the rank threshold actually applied, the residual, and the effective condition number. A rank-deficient system still gets an answer here, but it gets one that announces its own deficiency instead of a plausible vector with no warning attached.","url":"https://stat.halowerk.com/v1/oss-openblas-dense-minimum-norm-solve?utm_source=zero.xyz","method":"POST","headers":{},"bodySchema":{"type":"object","$schema":"https://json-schema.org/draft/2020-12/schema","required":["input"],"properties":{"input":{"type":"object","required":["type","method","bodyType"],"properties":{"type":{"const":"http"},"method":{"const":"POST"},"bodyType":{"const":"json"},"schemaUrl":{"type":"string","format":"uri"},"exampleUrl":{"type":"string","format":"uri"}},"additionalProperties":true},"output":{"type":"object","required":["type"],"properties":{"type":{"const":"json"},"schemaUrl":{"type":"string","format":"uri"},"exampleUrl":{"type":"string","format":"uri"}},"additionalProperties":true}},"additionalProperties":true},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.002","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.002/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.002","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.002","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_t0vQ6xrstGPA_hOTqs18-","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.002","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Solves a bounded dense multi-output least-squares problem and returns the minimum-norm solution along with diagnostics including numerical rank, singular value spectrum, rank threshold, residual, and effective condition number.","exampleAgentPrompt":"Solve this overdetermined system for me: matrix A is [[1,2],[3,4],[5,6]] and B is [[7],[8],[9]] — use an rcond of 1e-10 and give me the minimum-norm solution along with the numerical rank, full singular value spectrum, and condition number so I know whether to trust the result.","exampleUseCases":[{"title":"Rank-deficient system diagnostics in sensor fusion","prompt":"I have a sensor fusion system where my measurement matrix A is 64x32 and my observation matrix B is 64x4 — the system is likely rank-deficient. Can you solve for the minimum-norm solution using rcond=1e-8 and tell me the numerical rank and singular values so I know which directions are numerically degenerate?"},{"title":"Multi-output regression with trust verification","prompt":"I'm fitting a multi-output regression model: my design matrix A is 100x10 and my targets B are 100x5. Solve for the coefficient matrix using minimum-norm least-squares with rcond=1e-6, and give me the residual and effective condition number so I can decide whether the fit is numerically reliable."},{"title":"Ill-conditioned system detection in physics simulation","prompt":"My physics simulation has a stiffness matrix A of shape 80x20 and a force matrix B of shape 80x3 — I suspect it's ill-conditioned. Please solve it with rcond=0.001 and return the full singular value spectrum along with the rank threshold actually applied so I can see where the numerical cutoff lands."}],"resultDescription":"Returns the minimum-norm least-squares solution matrix (X such that AX ≈ B), along with: the numerical rank of A as determined by the rcond threshold, the complete singular value spectrum of A, the rank-condition threshold actually applied, the residual norm of the solution, and the effective condition number. For rank-deficient systems, the response explicitly announces the deficiency rather than silently returning a plausible-looking answer.","failureModes":["Matrix A or B dimensions are mismatched (rows of A must equal rows of B)","rcond value outside allowed range [1e-12, 0.01]","Matrix values outside the bounded range [-1e6, 1e6]","Matrix A exceeds 128x128 or B exceeds 128x32 dimensions","Malformed array structure (non-rectangular or empty arrays)","Payment failure via x402 protocol"],"whenToPreferThis":"Prefer this endpoint when you need not just a least-squares solution but also trustworthiness diagnostics — especially for potentially rank-deficient or ill-conditioned systems where a silent answer would be dangerous. It is ideal for multi-output problems (multiple right-hand side columns), when you need the full singular value spectrum, or when you want the minimum-norm solution among all possible least-squares solutions. Choose it over a simple linear solver when numerical reliability assessment matters as much as the solution itself.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-10-29T00:33:15.458Z","isFirstParty":false,"canonicalSlug":"bounded-dense-multi-output-least-squares-minimum-norm-solver-7383288b"}