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==Algorithmic steps==
===Unpreconditioned BiCGSTAB===
In the following sections, {{math|1=('''<var>x</var>''','''<var>y</var>''') = '''<var>x</var>'''<sup>T</sup> '''<var>y</var>'''}} denotes the [[dot product]] of vectors. To solve a linear system {{math|'''<var>Ax</var>''' {{=}} '''<var>b</var>'''}}, BiCGSTAB starts with an initial guess {{math|'''<var>x</var>'''<sub>0</sub>}} and proceeds as follows:
# {{math|'''<var>r</var>'''<sub>0</sub> {{=}} '''<var>b</var>''' − '''<var>Ax</var>'''<sub>0</sub>}}
# Choose an arbitrary vector {{math|'''<var>r̂</var>'''<sub>0</sub>}} such that {{math|('''<var>r̂</var>'''<sub>0</sub>, '''<var>r</var>'''<sub>0</sub>) ≠ 0}}, e.g., {{math|'''<var>r̂</var>'''<sub>0</sub> {{=}} '''<var>r</var>'''<sub>0</sub>
# {{math|<var>ρ</var><sub>0</sub> {{=}} ('''<var>r̂</var>'''<sub>0</sub>, '''<var>r</var>'''<sub>0</sub>) }}
# {{math|'''<var>p</var>'''<sub>0</sub> {{=}} '''<var>r</var>'''<sub>0</sub>}}
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