{"id":20500288,"url":"https://github.com/ohno/ill-conditioned","last_synced_at":"2026-03-19T16:07:37.773Z","repository":{"id":156627775,"uuid":"584197117","full_name":"ohno/ill-conditioned","owner":"ohno","description":null,"archived":false,"fork":false,"pushed_at":"2023-01-03T05:56:43.000Z","size":8,"stargazers_count":1,"open_issues_count":0,"forks_count":0,"subscribers_count":1,"default_branch":"main","last_synced_at":"2025-06-05T00:31:46.524Z","etag":null,"topics":[],"latest_commit_sha":null,"homepage":null,"language":"Fortran","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":null,"status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/ohno.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":null,"code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null}},"created_at":"2023-01-01T19:40:39.000Z","updated_at":"2023-04-26T01:33:29.000Z","dependencies_parsed_at":null,"dependency_job_id":"b486dc45-442b-4076-a01c-81c2d94d6e39","html_url":"https://github.com/ohno/ill-conditioned","commit_stats":null,"previous_names":[],"tags_count":0,"template":false,"template_full_name":null,"purl":"pkg:github/ohno/ill-conditioned","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ohno%2Fill-conditioned","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ohno%2Fill-conditioned/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ohno%2Fill-conditioned/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ohno%2Fill-conditioned/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/ohno","download_url":"https://codeload.github.com/ohno/ill-conditioned/tar.gz/refs/heads/main","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ohno%2Fill-conditioned/sbom","scorecard":null,"host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":286080680,"owners_count":28889863,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2026-01-29T21:06:44.224Z","status":"ssl_error","status_checked_at":"2026-01-29T21:06:42.160Z","response_time":59,"last_error":"SSL_read: unexpected eof while reading","robots_txt_status":"success","robots_txt_updated_at":"2025-07-24T06:49:26.215Z","robots_txt_url":"https://github.com/robots.txt","online":false,"can_crawl_api":true,"host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":[],"created_at":"2024-11-15T18:20:24.659Z","updated_at":"2026-01-29T23:03:04.347Z","avatar_url":"https://github.com/ohno.png","language":"Fortran","funding_links":[],"categories":[],"sub_categories":[],"readme":"# Ill-conditioned\n\nExamples of ill-conditioned eigenvalue problems.\n\n## Usage\n\nRun following commands on WSL/Ubuntu. Each eigenvalue problems of [Frank matrix](https://doi.org/10.1137/0106026) and of [Hilbert matrix](https://doi.org/10.1007/BF02418278) will be solved using [SSYEV](https://netlib.org/lapack/explore-html/d3/d88/group__real_s_yeigen_ga63d8d12aef8f2711d711d9e6bd833e46.html) and [DSYEV](https://netlib.org/lapack/explore-html/d2/d8a/group__double_s_yeigen_ga442c43fca5493590f8f26cf42fed4044.html). Results depend on the number of threads.\n\n```\ngit clone https://github.com/ohno/ill-conditioned.git\ncd ill-conditioned\nexport OMP_NUM_THREADS=1\nbash run.sh\n```\n\nGCC, [BLAS](https://netlib.org/blas/) and [LAPACK](https://netlib.org/lapack/) must be installed before use. Run following commands on WSL/Ubuntu to install GCC, BLAS and LAPACK.\n\n```\nsudo apt update\nsudo apt install build-essential\nsudo apt install gfortran\ngcc -v\ng++ -v\ngfortran -v\nsudo apt install libblas-dev\nsudo apt install liblapack-dev\n```\n\nIf you need other BLAS distributions, run following commands on WSL/Ubuntu to install and to choice.\n\n```\nsudo apt update\nsudo apt install libblas-dev\nsudo apt install libatlas-base-dev\nsudo apt install libblis-dev\nsudo apt install intel-mkl\nsudo apt install libopenblas-base\nsudo apt install libopenblas-dev\nsudo update-alternatives --config libblas.so-x86_64-linux-gnu\n```\n\n## Formulation\n\nEigenvalue problems of $n\\times n$ matrix $A$\n\n$$\nAx = \\lambda x\n$$\n\nare solved. For example, $5\\times5$ [Frank matrix](https://doi.org/10.1137/0106026) is\n\n$$\nA = \n  \\left[\n  \\begin{array}{ccccc}\n  5 \u0026 4 \u0026 3 \u0026 2 \u0026 1 \\\\\n  4 \u0026 4 \u0026 3 \u0026 2 \u0026 1 \\\\\n  3 \u0026 3 \u0026 3 \u0026 2 \u0026 1 \\\\\n  2 \u0026 2 \u0026 2 \u0026 2 \u0026 1 \\\\\n  1 \u0026 1 \u0026 1 \u0026 1 \u0026 1 \\\\\n  \\end{array}\n  \\right],\n$$\n\n$5\\times5$ [Hilbert matrix](https://doi.org/10.1007/BF02418278) is\n\n$$\nA = \n  \\left[\n  \\begin{array}{ccccc}\n  1 \u0026 \\frac{1}{2} \u0026 \\frac{1}{3} \u0026 \\frac{1}{4} \u0026 \\frac{1}{5} \\\\\n  \\frac{1}{2} \u0026 \\frac{1}{3} \u0026 \\frac{1}{4} \u0026 \\frac{1}{5} \u0026 \\frac{1}{6} \\\\\n  \\frac{1}{3} \u0026 \\frac{1}{4} \u0026 \\frac{1}{5} \u0026 \\frac{1}{6} \u0026 \\frac{1}{7} \\\\\n  \\frac{1}{4} \u0026 \\frac{1}{5} \u0026 \\frac{1}{6} \u0026 \\frac{1}{7} \u0026 \\frac{1}{8} \\\\\n  \\frac{1}{5} \u0026 \\frac{1}{6} \u0026 \\frac{1}{7} \u0026 \\frac{1}{8} \u0026 \\frac{1}{9} \\\\\n  \\end{array}\n  \\right].\n$$\n\n## Results\n\nMinimum eigenvalues are following.\n\n```\nFrank matrix\nn = 10\n SSYEV  0.255679399\n DSYEV  0.25567956279643672\nn = 100\n SSYEV  0.250060558\n DSYEV  0.25006108272069305\nn = 1000\n SSYEV  0.247511998\n DSYEV  0.25000061623489633\n```\n\n```\nHilbert matrix\nn = 1\n SSYEV   1.00000000\n DSYEV   1.0000000000000000\nn = 2\n SSYEV   6.57414496E-02\n DSYEV   6.5741454089335100E-002\nn = 3\n SSYEV   2.68730382E-03\n DSYEV   2.6873403557734501E-003\nn = 4\n SSYEV   9.66915541E-05\n DSYEV   9.6702304022688464E-005\nn = 5\n SSYEV   3.30206194E-06\n DSYEV   3.2879287721638091E-006\nn = 6\n SSYEV   7.15312751E-08\n DSYEV   1.0827994840427084E-007\n```\n\n## Reference\n\n- W. L. Frank, Jpn J Ind Appl Math, 6, 378 (1958) https://doi.org/10.1137/0106026\n- D. Hilbert, Acta Math., 18, 155 (1894) https://doi.org/10.1007/BF02418278\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fohno%2Fill-conditioned","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fohno%2Fill-conditioned","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fohno%2Fill-conditioned/lists"}