--- title: Krylov: Matrix-Free Linear Systems with Residual Certificate | SpinForge url: https://spin-forge.com/solvers/krylov description: Solve matrix-free linear systems with a recomputed true-residual certificate. --- Garmo Solvers SPINFORGE / A GARMO LABS COMPANY # Krylov Solve matrix-free linear systems with a recomputed true-residual certificate. [Explore the family](https://spin-forge.com/solvers) [Talk about your workload](https://spin-forge.com/contact) Customer outcome [Operate compute and data workflows](https://spin-forge.com/platform/compute-data-operations) THE PRODUCT, IN PRACTICE ## Solve matrix-free linear systems with a recomputed true-residual certificate Solve matrix-free linear systems with a recomputed true-residual certificate. It is an available module in Garmo Solvers 1.0.2, accessed through the common garmo CLI and garmo-mcp server. Every returned answer is rechecked against the original input; mathematical claims are certified or explicitly labeled uncertified. BUILT FOR Engineering, optimization and software teams with numerical methods workloads. CHALLENGES PRODUCT RESPONSE ## The problem you face. What the product changes. YOUR CHALLENGE WITH KRYLOV 01 ### The problem needs the right operation The supported operator, right-hand side and convergence requirements. Solve matrix-free linear systems with a recomputed true-residual certificate. 02 ### A result must answer the original question An answer detached from its original constraints or input is difficult to rely on. The suite rechecks the answer against the original input and identifies the applicable result state. 03 ### The comparison needs a defined scope A capability or benchmark result is not a universal promise for a different workload. Evaluate the selected input, resource budget, complete cost and required certificate against your baseline. WHAT YOU GET ## A defined capability. A useful result. 01 ### Selected solving capability Solve matrix-free linear systems with a recomputed true-residual certificate. 02 ### Original-input result A conjugate-gradient solution and true-residual evidence. 03 ### Common access Use the released garmo CLI and garmo-mcp server; headless execution fits scripts and automated workflows. 04 ### Explicit certificate status Optimality and infeasibility claims carry a certificate or are labeled uncertified. GARMO SOLVERS 1.0.2 / REPORTED STUDY ## Comparison incomplete Compared with CuPy, Torch and the checksummed Forge15 old path; the summarized improvement pass was not started. Source: owner-supplied release account. These are retained study results, not tests rerun for this publication or universal competitor rankings. [Read the full comparative evidence](https://spin-forge.com/results/garmo-solvers) EXAMPLE CUSTOMER WORKFLOW ## Use Krylov in a defined workflow A customer has a supported model and wants an accepted result without adopting every other module. Illustrative customer use—not a new benchmark or completed customer deployment. 1. 01 Provide the original input and define the useful outcome. 2. 02 Run the selected module under its supported resource and interface scope. 3. 03 Review the returned answer, original-input checks and completion state. FIT & ACCEPTANCE ## Start with the task. Agree the useful outcome. What you bring The supported operator, right-hand side and convergence requirements. What a useful evaluation measures - + Accepted original-model result - + Appropriate certificate or disclosed uncertified state - + Total time, memory and operating cost Delivery License the selected capability, commission an implementation, or agree an operated service. Infrastructure and additional engineering are specified separately. QUESTIONS BEFORE BUYING ## A clear scope. A clear relationship. **What is the important boundary? − **A residual is not a universal solution-error bound without the appropriate conditioning assumptions. **How is it priced? + **$15,000/year is the retained standalone price proposal for its named scope. The platform plans buy qualified workflows and capacity, not a second charge every time a compatible method is selected. Existing bundle inclusions do not expand automatically. **How do we get access? + **Available through garmo CLI and garmo-mcp, including headless execution. Product-specific inputs, runtime dependencies, budgets and result checks remain explicit. PUT THE PRODUCT TO WORK ## Tell us what needs to improve. Bring a task, an accepted outcome and the environment in which the product needs to work. [Discuss this workflow ](https://spin-forge.com/contact?product=solvers%2Fkrylov&brief=Product%3A+Krylov%0AOur+challenge%3A%0AOur+workload%2Finput%3A%0AExpected+result%3A%0AAcceptance%3A+Accepted+original-model+result%3B+Appropriate+certificate+or+disclosed+uncertified+state%3B+Total+time%2C+memory+and+operating+cost%0AEnvironment+and+delivery+preference%3A%0A) PRODUCT RELATIONSHIPS ## Related, not interchangeable. [RELATED PRODUCT ### UNISOLVE One headless solver platform for compatible method selection, mixed-capability solving and source-bound result checks.](https://spin-forge.com/unisolve) ONE PLATFORM ## Choose a qualified workflow. Select the capability you need directly or include its compatible role in a qualified platform workflow. Production scope identifies release, environment, capacity and support; a product count is not an entitlement. [Explore monthly plans](https://spin-forge.com/plans) [See the system connections](https://spin-forge.com/system-synergies/product-map) START WITH THIS PRODUCT Use the software directly or ask us to scope the implementation. [Software access](https://spin-forge.com/plans?product=solvers/krylov) [Have us do the work](https://spin-forge.com/contact?product=solvers/krylov&delivery=engineering)