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# The Mathematics of DIIS

Suppose that we have a set of trial vectors which have been generated during the iterative solution of a problem. Now let us form a set of residual'' vectors defined as

 (1)

The DIIS method assumes that a good approximation to the final solution pf can be obtained as a linear combination of the previous guess vectors

 (2)

where m is the number of previous vectors (in practice, only the most recent few vectors are used). The coefficients ci are obtained by requiring that the associated residual vector

 (3)

approximates the zero vector in a least-squares sense. Furthermore, the coefficients are required to add to one,

 (4)

The motivation for the latter requirement can be seen as follows. Each of our trial solutions pi can be written as the exact solution plus an error term, pf + ei. Then, the DIIS approximate solution is given by
 p = (5) =

Hence, we wish to minimize the actual error, which is the second term in the equation above (of course, in practice, we don't know ei, only ); doing so would make the second term vanish, leaving only the first term. For p = pf, we must have .

Thus, we wish to minimize the norm of the residuum vector

 (6)

subject to the constraint (4). These requirements can be satisfied by minimizing the following function with Lagrangian multiplier

 (7)

where B is the matrix of overlaps

 (8)

We can minimize with respect to a coefficient ck to obtain (assuming real quantities)
 = (9) =

We can absorb the factor of 2 into to obtain the following matrix equation, which is eq. (6) of Pulay [3]:

 (10)

Next: Programming DIIS Up: The DIIS Method Previous: Introduction
C. David Sherrill
2000-04-18