DIIS (`‘direct inversion of iterative subspace’') extrapolation. More...

#include <diis.h>

Public Types
typedef diis::traits< D >::element_type	value_type

Public Member Functions
	DIIS (unsigned int strt=1, unsigned int ndi=5, value_type dmp=0, unsigned int ngr=1, unsigned int ngrdiis=1, value_type mf=0)
	Constructor.

void	extrapolate (D &x, D &error, bool extrapolate_error=false)

void	start_extrapolation ()
	calling this function forces the extrapolation to start upon next call to `extrapolate()` even if this object was initialized with start value greater than the current iteration index.

void	reinitialize (const D *data=0)

Detailed Description

template<typename D>
class libint2::DIIS< D >

DIIS (`‘direct inversion of iterative subspace’') extrapolation.

The DIIS class provides DIIS extrapolation to an iterative solver of (systems of) linear or nonlinear equations of the $ f(x) = 0 $ form, where $ f(x) $ is a (non-linear) function of $ x $ (in general, $ x $ is a set of numeric values). Such equations are usually solved iteratively as follows:

given a current guess at the solution, , evaluate the error (`‘residual’') (NOTE that the dimension of and do not need to coincide);
use the error to compute an updated guess $x_{i+1} = x_i + g(e_i)$ ;
proceed until a norm of the error is less than the target precision $\epsilon$ . Another convergence criterion may include $||x_{i+1} - x_i|| < \epsilon$ . \ For example, in the Hartree-Fock method in the density form, one could choose $x \equiv \mathbf{P}$ , the one-electron density matrix, and $f(\mathbf{P}) \equiv [\mathbf{F}, \mathbf{P}]$ , where $\mathbf{F} = \mathbf{F}(\mathbf{P})$ is the Fock matrix, a linear function of the density. Because $\mathbf{F}$ is a linear function of the density and DIIS uses a linear extrapolation, it is possible to just extrapolate the Fock matrix itself, i.e. $x \equiv \mathbf{F}$ and $f(\mathbf{F}) \equiv [\mathbf{F}, \mathbf{P}]$ . \ Similarly, in the Hartree-Fock method in the molecular orbital representation, DIIS is used to extrapolate the Fock matrix, i.e. $x \equiv \mathbf{F}$ and $f(\mathbf{F}) \equiv \{ F_i^a \}$ , where and are the occupied and unoccupied orbitals, respectively. \ Here's a short description of the DIIS method. Given a set of solution guess vectors $\{ x_k \}, k=0..i$ and the corresponding error vectors $\{ e_k \}$ DIIS tries to find a linear combination of that would minimize the error by solving a simple linear system set up from the set of errors. The solution is a vector of coefficients $\{ C_k \}$ that can be used to obtain an improved : $x_{\mathrm{extrap},i+1} = \sum\limits_{k=0}^i C_{k,i} x_{k}$ A more complicated version of DIIS introduces mixing: $x_{\mathrm{extrap},i+1} = \sum\limits_{k=0}^i C_{k,i} ( (1-f) x_{k} + f x_{extrap,k} )$ Note that the mixing is not used in the first iteration. \ The original DIIS reference: P. Pulay, Chem. Phys. Lett. 73, 393 (1980).
Template Parameters

D type of x

Constructor & Destructor Documentation

◆ DIIS()

template<typename D >

libint2::DIIS< D >::DIIS	(	unsigned int	strt = 1,
		unsigned int	ndi = 5,
		value_type	dmp = 0,
		unsigned int	ngr = 1,
		unsigned int	ngrdiis = 1,
		value_type	mf = 0 )

inline

Constructor.

Parameters

strt	The DIIS extrapolation will begin on the iteration given by this integer (default = 1).
ndi	This integer maximum number of data sets to retain (default = 5).
dmp	This nonnegative floating point number is used to dampen the DIIS extrapolation (default = 0.0).
ngr	The number of iterations in a DIIS group. DIIS extrapolation is only used for the first `ngrdiis` of these iterations (default = 1). If `ngr` is 1 and `ngrdiis` is greater than 0, then DIIS will be used on all iterations after and including the start iteration.
ngrdiis	The number of DIIS extrapolations to do at the beginning of an iteration group. See the documentation for `ngr` (default = 1).
mf	This real number in [0,1] is used to dampen the DIIS extrapolation by mixing the input data with the output data for each iteration (default = 0.0), which performs no mixing. The approach described in Kerker, Phys. Rev. B, 23, p3082, 1981.

Member Function Documentation

◆ extrapolate()