**The reduced linear equation method in coupled cluster theory.**

A numerical procedure for expeditiously determination giant systems of linear equations is given. The approach, termed the reduced equation (RLE) methodology, is illustrated by determination the systems of linear equations that arise in linearized versions of coupled‐cluster theory. The nonlinear coupled‐cluster equations are treated with the RLE by presumptuous Associate in Nursing approximate linearization of the nonlinear terms. terribly economical convergence for linear systems and sensible convergence for nonlinear equations area unit found for variety of examples that manifest some degeneracy. These embrace the Be atom, H2 at giant separation, and therefore the N2 molecule.** [1]**

**On sparse and symmetric matrix updating subject to a linear equation**

A procedure for bilateral matrix change subject to a equation and holding any meagreness gift within the original matrix springs. the most feature of this procedure is that the reduction of the matter to the {answer} of an n dimensional distributed system of linear equations. The matrix of this technique is shown to be bilateral and positive definite. the tactic depends on the Frobenius matrix norm. Comments are created on the difficulties of extending the technique in order that it uses a lot of general norms, the most points being shown by a numerical example **[2]**

**Accurate Symmetric Indefinite Linear Equation Solvers**

The Bunch-Kaufman factoring is wide accepted because the formula of alternative for the direct answer of cruciform indefinite linear equations; it’s the formula utilized in each LINPACK and LAPACK. it’s conjointly been tailored to thin cruciform indefinite linear systems. **[3]**

**Quantum annealing for systems of polynomial equations**

Numerous scientific and engineering applications need numerically finding systems of equations. Classically finding a general set of polynomial equations needs unvaried solvers, whereas linear equations is also solved either by direct matrix operation or iteratively with even handed preconditioning. However, the convergence of unvaried algorithms is very variable and depends, in part, on the condition variety. we have a tendency to gift an immediate technique for finding general systems of polynomial equations supported quantum hardening, and that we validate this technique employing a system of second-order polynomial equations solved on a commercially offered quantum annealer. **[4]**

**Comparison of Jacobi and Gauss-Seidel Iterative Methods for the Solution of Systems of Linear Equations**

In this analysis work 2 unvarying ways of finding system of equation has been compared, the unvarying ways ar used for finding thin and dense system of equation and therefore the ways were being thought-about are: Jacobi technique and Gauss-Seidel technique. The results show that Gauss-Seidel technique is additional economical than Jacobi technique by considering most variety of iteration needed to converge and accuracy. **[5]**

**Reference**

**[1]** Purvis III, G.D. and Bartlett, R.J., 1981. The reduced linear equation method in coupled cluster theory. The Journal of Chemical Physics, 75(3), (Web Link)

**[2]** Toint, P.L., 1977. On sparse and symmetric matrix updating subject to a linear equation. Mathematics of Computation, 31(140), (Web Link)

**[3]** Ashcraft, C., Grimes, R.G. and Lewis, J.G., 1998. Accurate symmetric indefinite linear equation solvers. SIAM Journal on Matrix Analysis and Applications, 20(2), (Web Link)

**[4]** Quantum annealing for systems of polynomial equations

Chia Cheng Chang, Arjun Gambhir, Travis S. Humble & Shigetoshi Sota

Scientific Reports volume 9, Article number: 10258 (2019) (Web Link)

**[5]** Bakari, A. I. and Dahiru, I. A. (2018) “Comparison of Jacobi and Gauss-Seidel Iterative Methods for the Solution of Systems of Linear Equations”, Asian Research Journal of Mathematics, 8(3), (Web Link)