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Amanda2277 (talk | contribs) →Unpreconditioned BiCGSTAB: Added reference to paper showing numerical impact of choosing a different rhat0 |
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{{Short description|Concept in mathematics}}
{{Technical|date=May 2015}}
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## {{math|<var>β</var> {{=}} (<var>ρ<sub>i</sub></var>/<var>ρ</var><sub><var>i</var>−1</sub>)(<var>α</var>/<var>ω</var>)}}
## {{math|<var>'''p'''<sub>i</sub></var> {{=}} '''<var>r</var>'''<sub><var>i</var></sub> + <var>β</var>('''<var>p</var>'''<sub><var>i</var>−1</sub> − <var>ω</var>'''<var>v</var>''')}}
In some cases, choosing the vector {{math|'''<var>r̂</var>'''<sub>0</sub>}} randomly improves numerical stability.<ref>{{Cite journal |last=Schoutrop |first=Chris |last2=Boonkkamp |first2=Jan ten Thije |last3=Dijk |first3=Jan van |date=July 2022
===Preconditioned BiCGSTAB===
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which entails the necessity of a recurrence relation for {{math|<var>Q<sub>i</sub></var>('''<var>A</var>''')<var>T<sub>i</sub></var>('''<var>A</var>''')'''<var>r</var>'''<sub>0</sub>}}. This can also be derived from the BiCG relations:
:{{math|<var>Q<sub>i</sub></var>('''<var>A</var>''')<var>T<sub>i</sub></var>('''<var>A</var>''')'''<var>r</var>'''<sub>0</sub> {{=}} <var>Q<sub>i</sub></var>('''<var>A</var>''')<var>P<sub>i</sub></var>('''<var>A</var>''')'''<var>r</var>'''<sub>0</sub> + <var>β</var><sub><var>i</var>+1</sub>('''<var>I</var>''' − <var>ω<sub>i</sub>'''A'''</var>)<var>Q</var><sub><var>i</var>−1</sub>('''<var>A</var>''')<var>
Similarly to defining {{math|<var>'''r̃'''<sub>i</sub></var>}}, BiCGSTAB defines
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==References==
{{reflist}}
* {{Cite journal | doi = 10.1137/0913035 | title = Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems | year = 1992 | last1 = Van der Vorst | first1 = H. A. | journal = [[SIAM Journal on Scientific Computing|SIAM J. Sci. Stat. Comput.]] | volume = 13 | issue = 2 | pages = 631–644 | hdl = 10338.dmlcz/104566 | hdl-access = free }}
* {{cite book
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