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Using this result and similar ideas, it is shown that at least one of<jats:italic>S<\/jats:italic><jats:sub>1<\/jats:sub>or<jats:italic>TLS<\/jats:italic>does not prove the Matiyasevich-Robinson-Davis-Putnam Theorem. It is also established that<jats:italic>TLS<\/jats:italic>does not prove a statement that roughly means nondeterministic linear time is equal to co-nondeterministic linear time. Here<jats:italic>S<\/jats:italic><jats:sub>1<\/jats:sub>is a conservative extension of the well-studied theory<jats:italic>I<\/jats:italic>\u0394<jats:sub>0<\/jats:sub>and<jats:italic>TLS<\/jats:italic>is a theory for LOGSPACE reasoning.<\/jats:p>","DOI":"10.2178\/jsl\/1067620174","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:13:33Z","timestamp":1109798013000},"page":"1082-1090","source":"Crossref","is-referenced-by-count":0,"title":["A theory for Log-Space and NLIN versus co-NLIN"],"prefix":"10.1017","volume":"68","author":[{"given":"Chris","family":"Pollett","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200008227_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(82)90076-7"},{"key":"S0022481200008227_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-2566-9_6"},{"key":"S0022481200008227_ref019","first-page":"709","volume":"63","author":"Takeuti","year":"1998","journal-title":"Frege proof system and TNC\u00b0"},{"key":"S0022481200008227_ref014","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(99)00008-1"},{"key":"S0022481200008227_ref003","unstructured":"Cook S. , A survey of complexity classes and their associated propositional proof systems and theories, and a proof system for Log Space, Slides for Edinburgh talk, presented at the ICMS Workshop \u201cCircuit and Proof Complexity\u201d, Edinburgh, 10, 2001."},{"key":"S0022481200008227_ref017","doi-asserted-by":"publisher","DOI":"10.1002\/1521-3870(200202)48:2<221::AID-MALQ221>3.0.CO;2-V"},{"key":"S0022481200008227_ref012","first-page":"282","article-title":"Rudimentary predicates and Turing computations","volume":"195","author":"Nepom","year":"1970","journal-title":"Doklady Akademii Nauk"},{"key":"S0022481200008227_ref001","volume-title":"Bounded arithmetic","author":"Buss","year":"1986"},{"key":"S0022481200008227_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-22156-3"},{"key":"S0022481200008227_ref016","article-title":"Sk,exp does not prove NP = co-NP uniformly","volume":"304","author":"Pollett","year":"2003","journal-title":"Zapiski Nauchnykh Seminarov. Dom 1"},{"key":"S0022481200008227_ref008","first-page":"269","volume-title":"Logic colloquium 1998","volume":"13","author":"Johannsen","year":"2000"},{"key":"S0022481200008227_ref018","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1093\/oso\/9780198536901.003.0016","volume-title":"Arithmetic, proof theory, and computational complexity","volume":"23","author":"Takeuti","year":"1993"},{"key":"S0022481200008227_ref020","doi-asserted-by":"publisher","DOI":"10.1137\/0207018"},{"key":"S0022481200008227_ref004","unstructured":"Ferreira F. , A proof that LOGSPACE \u2260 NLIN, Unpublished notes, 1998."},{"key":"S0022481200008227_ref015","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(00)00015-4"},{"key":"S0022481200008227_ref013","volume-title":"Technical Report TR02-051","author":"Pollett"},{"key":"S0022481200008227_ref010","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511529948"},{"key":"S0022481200008227_ref011","first-page":"279","article-title":"Enumerable sets are Diophantine","volume":"191","author":"Matiyasevich","year":"1970","journal-title":"Doklady Akademii Nauk"},{"key":"S0022481200008227_ref006","first-page":"187","article-title":"Fragments of Peano's arithmetic and the MRDP theorem","volume":"30","author":"Gaifman","year":"1982","journal-title":"Monographic de L'Enseignement Math\u00e9matique"},{"key":"S0022481200008227_ref005","doi-asserted-by":"publisher","DOI":"10.1006\/jcss.1999.1671"},{"key":"S0022481200008227_ref021","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(97)00026-2"}],"container-title":["Journal of Symbolic Logic"],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200008227","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,23]],"date-time":"2024-01-23T22:45:51Z","timestamp":1706049951000},"score":21.286455,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200008227\/type\/journal_article"}},"issued":{"date-parts":[[2003,12]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2003,12]]}},"alternative-id":["S0022481200008227"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1067620174","ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"published":{"date-parts":[[2003,12]]}},{"institution":[{"name":"Chulalongkorn University","acronym":["CU"],"place":["Bangkok, Thailand"],"department":["Animal Physiology"]}],"indexed":{"date-parts":[[2025,3,20]],"date-time":"2025-03-20T13:40:30Z","timestamp":1742478030275,"version":"3.40.1"},"reference-count":0,"publisher":"Office of Academic Resources, Chulalongkorn University","content-domain":{"domain":[],"crossmark-restriction":false},"abstract":"<jats:p>The objective of this study is to study the effects of cereal supplementation, which contains many nutritional factors including amino acids on renal functions and morphological alteration in 5\/6 nephrectomized rats. Spraque-Dawley rats, 250-300 g, were use in this study. Rats were divided into 4 groups. The first group has been undergone sham operation (SOR). Group 2, 3 and 4 undergone 5\/6 nephrectomy by removed the right kidney and ligated 2\/3 of renal artery. After surgery rats were kept in the individual cage for 5 weeks. Rats in group 1 and 2 (CNR) fed with normal protein diet only. Rats in group 3 were fed with normal protein diet and supplemented with 0.03 g\/rat\/day cereal (LNR). Rats in group 4 were fed with normal protein diet supplemented with 0.06 g\/rat\/day cereal (HNR). The cereal supplement was started 1 day after surgery. After 5 weeks rats were anesthetized and subjected to renal function study. At the end of the experiments, left kidney was isolated for the measurement of morphological alteration. There was no significant difference of body weight, blood pH, blood pressure and heart rate among all groups of rat post-treatment. However, rats with cereal supplementation had significant higher (P&lt;0.05) plasma urea and plasma creatinine concentration than rat without cereal supplementation. Rats in CNR had slightly higher in GFR and RPF compared to the left kidney from SOR indicating functional compensatory. Rats in LNR groups had similar GFR and slightly lower RPF compared to CNR causing higher filtration fraction. Rats in HNR group had higher GFR and RPF than CNR group. The higher GFR caused urinary sodium excretion in HNR higher than in CNR while fractional excretion of sodium was similar. The higher urinary flow rate caused urinary potassium excretion slightly higher in HNR compare to CNR while fractional excretion of potassium was similar. The structural changes that observed were similar among all groups of nephrectomized rats. In conclusion, 0.03 mg\/rat\/day cereal supplement did not improve renal function while 0.06 mg\/rat\/day cereal supplement increase GFR and ERPF in 5\/6 nephrectomized rats 5 weeks after treatment. Different levels of cereal supplement did not show any differnce on renal structural alteration compared with nephrectomized alone.<\/jats:p>","DOI":"10.58837\/chula.the.2001.1657","type":"dissertation","created":{"date-parts":[[2025,3,20]],"date-time":"2025-03-20T13:17:51Z","timestamp":1742476671000},"approved":{"date-parts":[[2001]]},"source":"Crossref","is-referenced-by-count":0,"title":["Effects of cereal supplement on renal functions and structural alterations in 5\/6 nephrectomized rats"],"prefix":"10.58837","author":[{"sequence":"additional","affiliation":[]},{"given":"Nlin","family":"Arya","sequence":"first","affiliation":[]}],"member":"37481","deposited":{"date-parts":[[2025,3,20]],"date-time":"2025-03-20T13:17:52Z","timestamp":1742476672000},"score":19.54114,"degree":["Master of Science"],"resource":{"primary":{"URL":"https:\/\/digiverse.chula.ac.th\/Info\/item\/dc:54835"}},"issued":{"date-parts":[[null]]},"references-count":0,"URL":"https:\/\/doi.org\/10.58837\/chula.the.2001.1657"},{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T15:23:33Z","timestamp":1759332213529},"reference-count":0,"publisher":"IGI Global","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2009]]},"abstract":"<jats:p>This chapter delivers general format of Higher Order Neural Networks (HONNs) for nonlinear data analysis and six different HONN models. This chapter mathematically proves that HONN models could converge and have mean squared errors close to zero. This chapter illustrates the learning algorithm with update formulas. HONN models are compared with SAS Nonlinear (NLIN) models and results show that HONN models are 3 to 12% better than SAS Nonlinear models. 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To address this we have developed a novel online tool called PrimerROC (<jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" ext-link-type=\"uri\" xlink:href=\"http:\/\/www.primer-dimer.com\/roc\/\">www.primer-dimer.com\/roc\/<\/jats:ext-link>), which uses epidemiologically-based Receiver Operating Characteristic (ROC) curves to assess dimer prediction accuracy. Moreover, by integrating PrimerROC with our PrimerDimer prediction software we can determine a \u0394G-based dimer-free threshold above which dimer formation is predicted unlikely to occur. Notably, PrimerROC determines this cut-off without any additional information such as salt concentration or annealing temperature, meaning that our PrimerROC method is an assay and condition independent prediction tool. To demonstrate the broad utility of PrimerROC we assessed the performance of seven publically available primer design and dimer analysis tools using a dataset of over 300 primer pairs. We found that our PrimerROC\/PrimerDimer software consistently outperforms these other tools and can achieve predictive accuracies greater than 92%. To illustrate its predictive power this method was used in multiplex PCR design to successfully generate four resequencing assays containing up to 126 primers with no observable primer-primer amplification artefacts.<\/jats:p>","DOI":"10.1038\/s41598-018-36612-9","type":"journal-article","created":{"date-parts":[[2019,1,14]],"date-time":"2019-01-14T05:43:19Z","timestamp":1547444599000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["PrimerROC: accurate condition-independent dimer prediction using ROC analysis"],"prefix":"10.1038","volume":"9","author":[{"given":"Andrew D.","family":"Johnston","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1777-4788","authenticated-orcid":false,"given":"Jennifer","family":"Lu","sequence":"additional","affiliation":[]},{"given":"Ke-lin","family":"Ru","sequence":"additional","affiliation":[]},{"given":"Darren","family":"Korbie","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5516-1280","authenticated-orcid":false,"given":"Matt","family":"Trau","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,1,18]]},"reference":[{"key":"36612_CR1","doi-asserted-by":"crossref","unstructured":"Rychlik, W. 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Centre","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"published-print":{"date-parts":[[2023,5,31]]},"abstract":"<jats:p>\u0421\u043e\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u044b\u0435 \u0442\u0440\u0435\u0431\u043e\u0432\u0430\u043d\u0438\u044f \u043a \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u0438 \u0432 \u0438\u0437\u043c\u0435\u0440\u0435\u043d\u0438\u044f\u0445 \u0438 \u0438\u043d\u0436\u0435\u043d\u0435\u0440\u043d\u044b\u0445 \u043c\u043e\u0434\u0435\u043b\u044f\u0445 \u0443\u0432\u0435\u043b\u0438\u0447\u0438\u0432\u0430\u044e\u0442 \u0447\u0438\u0441\u043b\u043e \u043e\u0431\u0443\u0441\u043b\u043e\u0432\u043b\u0435\u043d\u043d\u043e\u0441\u0442\u0438 \u0437\u0430\u0434\u0430\u0447. \u0412 \u0442\u043e \u0432\u0440\u0435\u043c\u044f \u043a\u0430\u043a \u0443\u0432\u0435\u043b\u0438\u0447\u0435\u043d\u0438\u0435 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u0438 \u0447\u0438\u0441\u0435\u043b \u0441 \u043f\u043b\u0430\u0432\u0430\u044e\u0449\u0435\u0439 \u0437\u0430\u043f\u044f\u0442\u043e\u0439 \u043f\u0440\u0438\u0432\u0435\u043b\u043e \u043a \u0441\u0442\u0430\u0431\u0438\u043b\u044c\u043d\u044b\u043c \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f\u043c, \u0443\u0432\u0435\u043b\u0438\u0447\u0438\u043b\u0430\u0441\u044c \u043d\u0435\u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0441\u0442\u044c \u0432 \u0432\u043e\u043f\u0440\u043e\u0441\u0435 \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438 \u043f\u0440\u0438 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0438 \u043d\u0435\u0432\u044f\u0437\u043a\u0438 \u0432 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435 \u043a\u0440\u0438\u0442\u0435\u0440\u0438\u044f \u043e\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0438. \u041c\u044b \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u043c \u0430\u043d\u0430\u043b\u0438\u0437 \u043d\u0435\u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0441\u0442\u0438 \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u0438 \u043f\u0440\u0438 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0438 \u043e\u0442\u043d\u043e\u0441\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043d\u0435\u0432\u044f\u0437\u043a\u0438 \u0432 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435 \u043a\u0440\u0438\u0442\u0435\u0440\u0438\u044f \u043e\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0438 \u0434\u043b\u044f \u0438\u0442\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u043e\u0433\u043e \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043b\u0438\u043d\u0435\u0439\u043d\u044b\u0445 \u0441\u0438\u0441\u0442\u0435\u043c, \u0430 \u0442\u0430\u043a\u0436\u0435 \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c\u043e\u0435 \u0443\u0432\u0435\u043b\u0438\u0447\u0435\u043d\u0438\u0435\/\u0443\u043c\u0435\u043d\u044c\u0448\u0435\u043d\u0438\u0435 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0439 \u043f\u0440\u0438 \u0437\u0430\u0434\u0430\u043d\u043d\u043e\u0439 \u0434\u043e\u043f\u0443\u0441\u0442\u0438\u043c\u043e\u0439 \u043f\u043e\u0433\u0440\u0435\u0448\u043d\u043e\u0441\u0442\u0438. \u042d\u0442\u043e \u043f\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442, \u0447\u0442\u043e \u043e\u0446\u0435\u043d\u043a\u0430 \u043e\u0448\u0438\u0431\u043a\u0438 \u0432\u0430\u0436\u043d\u0430 \u0434\u043b\u044f \u044d\u0444\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u043e\u0433\u043e \u0438\u043b\u0438 \u0442\u043e\u0447\u043d\u043e\u0433\u043e \u0440\u0435\u0448\u0435\u043d\u0438\u044f, \u0434\u0430\u0436\u0435 \u043a\u043e\u0433\u0434\u0430 \u0447\u0438\u0441\u043b\u043e \u043e\u0431\u0443\u0441\u043b\u043e\u0432\u043b\u0435\u043d\u043d\u043e\u0441\u0442\u0438 \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u043d\u0435\u0432\u0435\u043b\u0438\u043a\u043e. \u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u043e\u0446\u0435\u043d\u043a\u0438 \u043e\u0448\u0438\u0431\u043a\u0438 \u039f(1) \u0434\u043b\u044f \u0438\u0442\u0435\u0440\u0430\u0446\u0438\u0439 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430 CG \u0431\u044b\u043b\u0430 \u043f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0430 \u0431\u043e\u043b\u0435\u0435 \u0434\u0432\u0443\u0445 \u0434\u0435\u0441\u044f\u0442\u0438\u043b\u0435\u0442\u0438\u0439 \u043d\u0430\u0437\u0430\u0434. \u041d\u0435\u0434\u0430\u0432\u043d\u043e \u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u043e\u0446\u0435\u043d\u043a\u0438 \u043e\u0448\u0438\u0431\u043a\u0438 \u039f(\u03ba2) \u0431\u044b\u043b\u0430 \u043e\u043f\u0438\u0441\u0430\u043d\u0430 \u0434\u043b\u044f \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0430 GMRES, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u0434\u043e\u043f\u0443\u0441\u043a\u0430\u0435\u0442 \u0442\u0430\u043a\u0436\u0435 \u043d\u0435\u0441\u0438\u043c\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u044b\u0435 \u043b\u0438\u043d\u0435\u0439\u043d\u044b\u0435 \u0441\u0438\u0441\u0442\u0435\u043c\u044b, \u0433\u0434\u0435 \u03ba - \u043d\u043e\u043c\u0435\u0440 \u0438\u0442\u0435\u0440\u0430\u0446\u0438\u0438. \u041c\u044b \u043f\u0440\u0435\u0434\u043b\u0430\u0433\u0430\u0435\u043c \u043d\u0435\u0431\u043e\u043b\u044c\u0448\u0443\u044e \u043c\u043e\u0434\u0438\u0444\u0438\u043a\u0430\u0446\u0438\u044e \u044d\u0442\u043e\u0439 \u043e\u0446\u0435\u043d\u043a\u0438 \u043e\u0448\u0438\u0431\u043a\u0438 GMRES \u0434\u043b\u044f \u043f\u043e\u0432\u044b\u0448\u0435\u043d\u0438\u044f \u0443\u0441\u0442\u043e\u0439\u0447\u0438\u0432\u043e\u0441\u0442\u0438. \u0412 \u0434\u0430\u043d\u043d\u043e\u0439 \u0440\u0430\u0431\u043e\u0442\u0435 \u043c\u044b \u0442\u0430\u043a\u0436\u0435 \u043f\u0440\u0435\u0434\u043b\u0430\u0433\u0430\u0435\u043c \u0444\u043e\u0440\u043c\u0443\u043b\u0443 \u043e\u0446\u0435\u043d\u043a\u0438 \u043e\u0448\u0438\u0431\u043a\u0438 \u039f(n) \u0434\u043b\u044f A-\u043d\u043e\u0440\u043c\u044b \u0438 l2-\u043d\u043e\u0440\u043c\u044b \u0432\u0435\u043a\u0442\u043e\u0440\u0430 \u043e\u0448\u0438\u0431\u043a\u0438 \u0432 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0435 Bi-CG. \u041d\u0430\u0434\u0435\u0436\u043d\u0430\u044f \u0440\u0430\u0431\u043e\u0442\u0430 \u044d\u0442\u0438\u0445 \u043e\u0446\u0435\u043d\u043e\u043a \u0432 \u043a\u0430\u0447\u0435\u0441\u0442\u0432\u0435 \u043a\u0440\u0438\u0442\u0435\u0440\u0438\u044f \u043e\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0438 \u0443\u0432\u0435\u043b\u0438\u0447\u0438\u0432\u0430\u0435\u0442 \u044d\u043a\u043e\u043d\u043e\u043c\u0438\u044e \u0438 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0439 \u043f\u043e \u043c\u0435\u0440\u0435 \u0443\u0432\u0435\u043b\u0438\u0447\u0435\u043d\u0438\u044f \u0447\u0438\u0441\u043b\u0430 \u043e\u0431\u0443\u0441\u043b\u043e\u0432\u043b\u0435\u043d\u043d\u043e\u0441\u0442\u0438 \u0438 \u0440\u0430\u0437\u043c\u0435\u0440\u0430 \u0437\u0430\u0434\u0430\u0447.<\/jats:p>\n                                                                                            <jats:p>The demands of accuracy in measurements and engineering models today render the condition number of problems larger. While a corresponding increase in the precision of floating point numbers ensured a stable computing, the uncertainty in convergence when using residue as a stopping criterion has increased. We present an analysis of the uncertainty in convergence when using relative residue as a stopping criterion for iterative solution of linear systems, and the resulting over\/under computation for a given tolerance in error. This shows that error estimation is significant for an efficient or accurate solution even when the condition number of the matrix is not large. An \u039f(1) error estimator for iterations of the CG algorithm was proposed more than two decades ago. Recently, an \u039f(\u03ba2) error estimator was described for the GMRES algorithm which allows for non-symmetric linear systems as well, where \u03ba is the iteration number. We suggest a minor modification in this GMRES error estimation for increased stability. In this work, we also propose an \u039f(n) error estimator for A-norm and l2-norm of the error vector in Bi-CG algorithm. 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