{"paper":{"title":"A new method for obtaining approximate solutions of the hyperbolic Kepler's equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.NA","physics.comp-ph"],"primary_cat":"physics.class-ph","authors_text":"Jorge Ortigas-Galindo, Mart\\'in Avendano, Ver\\'onica Mart\\'in-Molina","submitted_at":"2015-02-20T15:27:16Z","abstract_excerpt":"We provide an approximate zero $\\widetilde{S}(g,L)$ for the hyperbolic Kepler's equation $S-g\\, {\\rm asinh} (S)-L=0$ for $g\\in(0,1)$ and $L\\in[0,\\infty)$. We prove, by using Smale's $\\alpha$-theory, that Newton's method starting at our approximate zero produces a sequence that converges to the actual solution $S(g,L)$ at quadratic speed, i.e. if $S_n$ is the value obtained after $n$ iterations, then $|S_n-S|\\leq 0.5^{2^n-1}|\\widetilde{S}-S|$. The approximate zero $\\widetilde{S}(g,L)$ is a piecewise-defined function involving several linear expressions and one with cubic and square roots. In bo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01641","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}