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in the rigorous screening of small exchange interactions bellow the two-electron integral threshold τ$_{2e}$ [13]. The consequence of these linear scaling approximations is an inexact linear algebra that challenges Krylov solvers due to nested error accumulation, a subject of recent formal interest [14, 15]. Consistent with this view, TINC found that matrix perturbation (a truncation proxy) disrupts convergence of Krylov solvers with slow convergence, i.e. Lanczos and Arnoldi for the RPA, but has less impact on solvers with rapid convergence, i.e. generically for the TDA or Davidson for the RPA. Relative to semi- empirical Hamiltonians, the impact of incompleteness on subspace iteration may be amplified with first principles models and large basis sets (ill-conditioning). 

An alternative is Rayleigh Quotient Iteration (RQI), which poses the eigenproblem as non-linear optimiza- tion and is variational with respect to matrix pertur- bation. Narita and Shibuya [10] considered optimiza- tion of the quotient ω $^{2}$[ ⃗ p, ⃗ q ] with second order methods, but these are beyond the capabilities of current linear scaling technologies and also, convergence is disadvan- taged by a power of 1 / 2 . For semi-empirical Hamiltoni- ans, TINC found that optimization of the Thouless func- tional ω [ ⃗ v ] = ⃗ v · L · ⃗ v/ | ⃗ v · ⃗ v | , corresponding to the so- lution of Eq. (1), was significantly slower for the RPA relative to the TDA, and also compared to subspace solvers. For first principles models and non-trivial basis sets, this naive RQI can become pathologically slow as shown in Fig. 1. On the other hand, the Tsiper formula- tion exposes the underlying pseudo-Hermitian structure of the TD-SCF equations. Here, this structure is ex- ploited with QUasi-Independent Rayleigh Quotient Iter- ation (QUIRQI), involving dual channel optimization of the Tsiper quotients coupled only weakly through line search. 

Our development begins with a brief review of the rep- resentation independent formulation developed by TINC, which avoids the O ( N $^{3}$) cost of rotating into an ex- plicit p-h , h-p symmetry. Instead, this symmetry is maintained implicitly via annihilation, x ← f$_{a}$ ( x ) = P · x · Q + Q · x · P , with P the first order reduced den- sity matrix and Q = I $_{−}$P its compliment. Likewise, the indefinite metric associated with the J -symmetry of Eq. (1) is carried through the generalized norm 〈 x , y 〉 = tr { x T · [ y , P ] } . Introducing the operator equivalents, L [ p ] ⇔ K .⃗ p and L [ q ] ⇔ T .⃗ q , the Tsiper functional becomes ω [ p , q ] = 〈 p , L [ p ] 〉 2 |〈 p , q 〉| + 〈 q , L [ q ] 〉 2 |〈 p , q 〉| . Transformations between the transition density and the dual space in- volves simple manipulations and minimal cost, allowing Fock builds with the transition density and optimization in the dual space. The splitting operation is given by p = f$_{+}$ ( v ) = P · v · Q + [ Q · v · P ] T and q = f$_{−}$ ( v ) = P · v · Q − [ Q · v · P ] $^{T}$. Likewise, L [ p ] = f$_{−}$ ( L [ v ]) and L [ q ] = f$_{+}$ ( L [ v ]) . The back transformation (merge) from dual to density is v = F ( p , q ) = ( p + q + [ p$_{−}$ q ] $^{T}$) / 2 . 

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This framework provides the freedom to work in any or- thogonal representation, and to switch between transi- tion density and oscillator duals with minimal cost. 

QUIRQI is given in Algorithm 1. It begins with a guess for the transition density, which is then split into its dual (lines 2-3). The choice of initial guess is discussed later. Lines 4-24 consist of the non-linear conjugate gra- dient optimization of the nearly independent channels: In each step, the flow of information proceeds from opti- mization of the duals to builds involving the density and back to the duals in a merge-annihilate-truncate-build- split-truncate (MATBST) sequence. For the variables v , p and q this sequence is comprised by lines 22-23 and 5-7, and lines 15-19 for the corresponding conjugate gradients h$_{v}$ , h$_{p}$ and h$_{q}$ . Truncation is carried out with the filter operation as described in Ref. [11], with cost and error determined by the matrix threshold τ$_{mtx}$ . 

The Tsiper functional is the sum of dual quotients ω$_{p}$ and ω$_{q}$ , determined at line 8, followed by the gradients g$_{p}$ and g$_{q}$ computed at line 9. After the first cycle, the cor- responding relative error e$_{rel}$ (10) and maximum matrix element of the gradient g$_{max}$ (11) are computed and used as an exit criterion at line 4, along with non-variational behavior ω > ω $^{old}$. 

Next, the Polak-Ribiere variant of non-linear conjugate gradients yields the search direction in each channel, h$_{p}$ and h$_{q}$ (12-14). The action of L [ · ] on to h$_{p}$ and h$_{q}$ is then computed, again with a MATBST sequence (15- 19), followed by a self-consistent dual channel line search at line 20, as described below. With steps λ$_{p}$ and λ$_{q}$ in hand, minimizing updates are taken along each conjugate direction (22), and the cycle repeats with the MATBST sequence spanning lines 21-23 and 5-7. 

Optimization of the Tsipper functional ω [ λ$_{p}$, λ$_{q}$ ] ≡ 

