Editing DIIS (section)

'''DIIS''' ('''direct inversion in the iterative subspace''' or '''direct inversion of the iterative subspace'''), also known as '''Pulay mixing''', is a technique for [[extrapolation|extrapolating]] the solution to a set of linear equations by directly minimizing an error residual (e.g. a [[Newton's method|Newton–Raphson]] step size) with respect to a linear combination of known sample vectors. DIIS was developed by [[Peter Pulay]] in the field of computational [[quantum chemistry]] with the intent to accelerate and stabilize the [[convergence (mathematics)|convergence]] of the [[Hartree–Fock]] self-consistent field method.<ref>{{cite journal|last=Pulay|first=Péter |year=1980|title=Convergence acceleration of iterative sequences. the case of SCF iteration|journal=Chemical Physics Letters|volume=73|issue=2|pages=393–398|doi=10.1016/0009-2614(80)80396-4|bibcode=1980CPL....73..393P}}</ref><ref>{{cite journal|last=Pulay|first=Péter |year=1982|title=Improved SCF Convergence Acceleration|journal=Journal of Computational Chemistry|volume=3|issue=4|pages=556–560|doi=10.1002/jcc.540030413|s2cid=120876883 }}</ref><ref>{{Cite journal|doi=10.1080/00268970701691611|title=Some comments on the DIIS method|journal=Molecular Physics|volume=105|issue=19–22|pages=2839–2848|year=2010|last1=Shepard|first1=Ron|last2=Minkoff|first2=Michael|bibcode=2007MolPh.105.2839S|s2cid=94014926}}</ref>

At a given iteration, the approach constructs a [[linear combination]] of approximate error vectors from previous iterations. The coefficients of the linear combination are determined so to best approximate, in a [[least squares]] sense, the [[null vector]]. The newly determined coefficients are then used to extrapolate the function variable for the next iteration.