Fixed-point iteration

Numerical Solution of Nonlinear Equations of One Variable

Published in Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer, Classical and Modern Numerical Analysis, 2009

Azmy S. Ackleh, Edward James Allen, Ralph Baker Hearfott, Padmanabhan Seshaiyer

An important question is: when does {xk}k=0∞ converge to z, a fixed point of g? Fixed-point iteration does not always converge. Consider g(x) = x2, whose fixed points are x = 0 and x = 1. If x0 = 2, then xk+1=xk2, so x1 = 4, x2 = 16, x3 = 256,…

On the origin of bonding in metals: lithium as a case study

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Journal Information

Published in Molecular Physics, 2022

Inge Røeggen, Bin Gao

The next step is to solve the Hartree-Fock equations for atom A: and The orbitals are expressed in terms of the modified large basis sets, i.e. and A fixed-point iteration procedure does not necessarily converge. Hence, we introduce a UHF functional where Let denote the spin orbitals of the previous iteration, and the spin orbitals of the last one. Then we minimise the functional with respect to the length of the orbital correction: The orbitals have to be properly symmetrised. The procedure is described in Appendix. We denote the symmetrised orbitals .

Fixed-point iteration

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Numerical Solution of Nonlinear Equations of One Variable

On the origin of bonding in metals: lithium as a case study