--- title: krylov_scale: Matrix-Free Linear Systems & Eigenvalues | SpinForge url: https://spin-forge.com/forge15/krylov-scale description: Matrix-free linear-system and extreme-eigenvalue computation with independent true-residual checking. --- FORGE15 / 09 / NUMERICAL COMPUTING [The full offering](https://spin-forge.com/platform) # krylov_scale. Matrix-free linear-system and extreme-eigenvalue computation with independent true-residual checking. [Discuss krylov_scale](https://spin-forge.com/contact?product=forge15%2Fkrylov-scale) Customer outcome [Design and discover](https://spin-forge.com/platform/discovery-design) Available now CLI MCP Headless Choose your access mode THE PRODUCT, IN PRACTICE ## Solve the system without storing the whole operator. Some scientific workloads can apply a very large operator without storing its full matrix. krylov_scale is built for that setting: it solves supported linear systems and extreme-eigenvalue problems using the operator interface, then recomputes the true residual to check the result. The customer receives a numerical component for a specific operator, not a claim of limitless matrix capacity. BUILT FOR Simulation, computational-science and engineering teams with matrix-free numerical operators. CHALLENGES PRODUCT RESPONSE ## The problem you face. What krylov_scale changes. 01 ### The full matrix is too large to materialize An application may know how to apply an operator even when storing every matrix entry is impractical. krylov_scale accepts the supported matrix-free operation rather than requiring the full operator to be retained as a dense matrix. 02 ### A solver’s progress indicator is not enough Internal convergence reporting can differ from checking the final candidate against the actual problem. The independent result check recomputes the true residual for the returned numerical candidate. 03 ### Iteration speed is not the whole workload Setup and execution overhead can matter alongside numerical convergence. Evaluate the operator path, memory and end-to-end time to the required tolerance. The separate Krylov upgrade targets the execution overhead identified in the campaign. WHAT YOU GET ## A defined capability. A useful output. The product scope below describes the capability and outputs to confirm for your release. Custom integration, operated delivery and additional engineering are separately agreed. 01 ### A matrix-free solver path Supported linear-system computation through operator application. 02 ### Extreme-eigenvalue scope The additional supported eigenvalue computation path. 03 ### A numerical candidate The solution or eigen-result returned for the selected problem. 04 ### True-residual checking Independent recomputation of the final numerical residual. AVAILABLE NOW ## One product. Three ways to work. krylov_scale is fully available with CLI, MCP and headless access. Use the interface that fits your terminal, your agents or your automated workflow. 01 / CLI ### For your terminal. Run the product from a terminal or script, alongside the tools your team already uses. [Get CLI access](https://spin-forge.com/contact?product=forge15%2Fkrylov-scale&interface=cli&brief=I+would+like+krylov_scale+access+through+CLI.+Please+share+the+setup+documentation+and+configuration+for+the+selected+product.+We+will+confirm+the+environment%2C+license+scope+and+workflow+during+onboarding.) 02 / MCP ### For your agents. Connect MCP-compatible clients and agents to the product’s exposed capabilities. [Get MCP access](https://spin-forge.com/contact?product=forge15%2Fkrylov-scale&interface=mcp&brief=I+would+like+krylov_scale+access+through+MCP.+Please+share+the+setup+documentation+and+configuration+for+the+selected+product.+We+will+confirm+the+environment%2C+license+scope+and+workflow+during+onboarding.) 03 / HEADLESS ### For your workflows. Operate without a graphical interface in unattended, scheduled or pipeline-driven work. [Set up headless access](https://spin-forge.com/contact?product=forge15%2Fkrylov-scale&interface=headless&brief=I+would+like+krylov_scale+access+through+headless.+Please+share+the+setup+documentation+and+configuration+for+the+selected+product.+We+will+confirm+the+environment%2C+license+scope+and+workflow+during+onboarding.) Product-specific inputs, hardware requirements and result guarantees still apply. An access mode does not change your license or deployment scope. [Explore all product interfaces](https://spin-forge.com/developers?section=access-directory) EXAMPLE CUSTOMER WORKFLOW ## Solve a discretized scientific operator A simulation team can apply its supported operator but does not want to materialize the entire matrix. Illustrative use of the supported scope, not a customer case study or a measured result. 1. 01 Provide the operator, right-hand side or eigenproblem and tolerance. 2. 02 Run the supported matrix-free numerical computation. 3. 03 Recompute the true residual and compare total cost against the existing solve. IN YOUR WORKFLOW ## Start with your task. Agree the useful outcome. Simulation, computational-science and engineering teams with matrix-free numerical operators. What you bring The supported operator application, problem data, precision, tolerance and available execution resources. What a useful evaluation measures - True residual at the requested accuracy - Memory needed for operator and state - End-to-end time including execution overhead Where it fits Use the selected product in a qualified customer environment, or agree a defined implementation with SpinForge. A hosted or managed service is quoted for its actual operating responsibilities; it is not automatically included with software access. QUESTIONS BEFORE YOU BUY ## A clear scope. No hidden assumptions. **How can we use krylov_scale?**CLI, MCP and headless access are all available. Run from a terminal or script, connect an MCP-compatible client, or use the product without a graphical interface. Choose the mode during onboarding and receive its setup instructions. Runtime requirements and supported operations remain product-specific. **What is the important product boundary? **The billion-unknown Poisson result is a specific reported operator and H100 configuration. It does not establish capacity for arbitrary stored matrices. A small residual does not alone guarantee a small solution error for an ill-conditioned problem. **What comes with the selected license? **The agreed krylov_scale release, its supported interfaces and outputs, the applicable checks and the support stated in your order. The plan defines users, workflow, environment and capacity. Source-code transfer, unrestricted redistribution and bespoke engineering are not automatically included. **Can SpinForge implement or operate it for us? **Request a scoped engineering or operated-service proposal. It identifies what our team delivers, where the workload runs, infrastructure costs, acceptance requirements and ongoing support. Your software subscription alone does not activate a hosted runtime. [Explore engineering](https://spin-forge.com/engineering) **What evidence is available for review? **P42 / Company-reported. The supplied campaign reports a billion-unknown Poisson workload fitting on an H100. The new Krylov track targets execution overhead identified in that campaign; no unmeasured successor speedup is inferred. These descriptions come from the supplied product and campaign material. Website checks are not an independent rerun of the engine. Qualification identifies the actual release, workload and available artifacts. [Read the evidence summary](https://spin-forge.com/results) SOFTWARE / OPERATED DELIVERY By proposal Customer-hosted access and a service operated by SpinForge are different scopes. Confirm the release, deployment and included support before activation. [Plans & delivery](https://spin-forge.com/pricing) PUT KRYLOV_SCALE TO WORK ## Tell us what needs to improve. Bring a representative workload and the result your team needs. We’ll define the product, deployment and engineering scope around that task. [Discuss this workflow](https://spin-forge.com/contact?product=forge15%2Fkrylov-scale&brief=Product%3A+krylov_scale%0AOur+challenge%3A+The+full+matrix+is+too+large+to+materialize%0AWorkload+%2F+baseline%3A+%0AEnvironment%3A+%0ARequired+output%3A+%0AAcceptance+measures%3A+True+residual+at+the+requested+accuracy%3B+Memory+needed+for+operator+and+state%3B+End-to-end+time+including+execution+overhead%0ADelivery+preference+%28software+%2F+engineering+%2F+managed%29%3A+) **Related products, workflows and system connections + **RELATED CAPABILITIES ## Connect the right products. Distinct responsibilities. Explore each scope before choosing a combined deployment. [### Krylov A matrix-free numerical solver track for supported linear systems and eigenproblems, with residual-based evaluation.](https://spin-forge.com/solvers/krylov) [### SpectralForge Large FFT, convolution and supported spectral-PDE computation with independent consistency checks.](https://spin-forge.com/forge15/spectralforge) [### TomoForge Signal and image reconstruction from noisy linear measurements, with a checked data-fit residual.](https://spin-forge.com/forge15/tomoforge) ONE PLATFORM / OR A DIRECT PURCHASE ## Checked Decisions with krylov_scale. Activate this capability inside a qualified checked decisions workflow. Compatible methods and required shared checking are included once within that workflow; the product remains independently purchasable. Capacity, release, environment, result contract and operating responsibilities are specified before activation. Choosing an additional method is not permission to add an unrelated workload or unqualified deployment. [Explore Checked Decisions](https://spin-forge.com/platform/checked-decisions) [Choose workflow capacity](https://spin-forge.com/plans?configuration=checked-decisions&product=forge15/krylov-scale) CONNECTED WORKFLOWS ## Use krylov_scale in your workflow. The product remains independently available. These customer-specific solution paths use the relevant supported capabilities, with integration and delivery rights agreed separately. [05 ### Performance & release checks Make performance and quality evaluation a repeatable part of your release process. Compare an agreed workload before a change is accepted into production.](https://spin-forge.com/solutions/release-checks) [Explore all 17 solution workflows](https://spin-forge.com/workflows) FROM PRODUCT TO CUSTOMER WORKFLOW ## A useful outcome. A defined delivery. Commission the model, integration and acceptance work needed for your task. Existing products stay independently available. [03 / Scoped customer program ### Improve a computational workflow Commission a bounded performance improvement with source identity, numerical acceptance and the appropriate checking evidence.](https://spin-forge.com/programs/compute-improvement) [Explore decision programs](https://spin-forge.com/decision-systems) THE WHOLE-SYSTEM ROLE ## krylov_scale in a wider workflow. Refine only decision-relevant numerical quantities where residual-to-error bounds exist. POTENTIAL SYSTEM CONTRIBUTION Conditioning matters; a residual alone does not bound solution error. [S2 · Decision-sensitive computation](https://spin-forge.com/system-synergies/decision-sensitive-compute) [See this product in the map](https://spin-forge.com/system-synergies/product-map?q=krylov_scale) [Define an integration evaluation](https://spin-forge.com/system-synergies/evaluation) The product is independently available through CLI, MCP and headless access. The joins above are integration candidates; compatibility, authority, data rights and end-to-end benefit must be established for the selected workflow. START WITH THIS PRODUCT Use the software directly or ask us to scope the implementation. [Software access](https://spin-forge.com/plans?product=forge15/krylov-scale) [Have us do the work](https://spin-forge.com/contact?product=forge15/krylov-scale&delivery=engineering)