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A simple alternative to the conjugate gradient (CG) method is presented; this method is developed as a special case of the more general iterated Ritz method (IRM) for solving a system of linear equations. This novel algorithm is not based on conjugacy; i.e., it is not necessary to maintain overall orthogonalities between various vectors from distant steps. This method is more stable than CG, and restarting techniques are not required. As in CG, only one matrix-vector multiplication is required per step with appropriate transformations. The algorithm is easily explained by energy considerations without appealing to the

Let

The main idea here is to present the solution increment by the discretised Ritz method:

The solution is used to find the increment in (

Obviously, IRM represents an iterative procedure, where a discrete Ritz method is applied at each step and a suitable set of coordinate vectors which span a subspace are generated. A local energy minimum is sought within that subspace (therefore, equation (

while

generate

The central problem involves quickly generating a small and efficient subspace, such that the energy reduction per step is as large as possible and the number of steps is extremely reduced. Usually, one coordinate vector is

It should be noted that the conjugacy property is not explicitly taken into account in IRM, and coordinate vectors may become (almost) linearly dependent. Therefore, routines for subspace generation which prevent such a scenario are preferred and may even change between steps. Nevertheless, if this dependence arises, some pivots approach zero during the decomposition of

IRM can also be considered as a generalisation of some iterative methods [

The algorithm presented here also starts with the steepest descent (SD) step. Other steps are executed using a CG-like algorithm simulated by IRM with two coordinate vectors. The first vector is the current residual

while

This approach has three matrix-vector multiplications per step: one in line 7 and two in line 8. Applying two “induced” recursive relations (“inherent”

Substituting

Second, the frequently used residual recursion

Now, after the line 4 (Algorithm

Due to roundoff errors, as in CG, the residual is periodically (after

The proof of equivalence between CG and IRM-CG is very simple, so it will be discussed only briefly. Initialisation is practically identical for both methods. In other steps, the minimum of the energy function inside the plane spanned by

If exact arithmetic is considered, IRM-CG and CG have an identical sequence of intermediate results. The exact solution is obtained after

During real calculations (with roundoff errors), IRM-CG is more stable and behaves better than CG. First, restarting of this algorithm is not needed because

Consider simple example with diagonal

Stability of CG method: (a) interpretation of disturbance

Notice complexity of the CG solution, even with a diagonal matrix of order two. For better explanation of the expressions, over domain

When large equation systems are considered,

It is possible to interchange methods because the two approaches are equivalent. Each step may be executed by CG or IRM-CG, no matter how the earlier steps were performed. If CG is used as a solution method, it is suggested that one equivalent IRM-CG step be executed after some number of steps but before orthogonality error becomes too large. This may be called “refresh” instead of traditionally “restart.”

The second advantage of this formulation is the natural adoption of the relaxation factor

The coordinate vectors are

Many possibilities to rapidly construct

Consider a simple linear FEM benchmark: a cube discretised by 8-node solid elements, supported by the corner springs of stiffnesses

Behaviour of CG and IRM-CG for well-posed and ill-posed problems.

Although general theorems and proofs about algorithm convergence rate and stability are not given here, according to the results of numerical experiments with exact and floating-point arithmetic, IRM-CG should be an interesting replacement for a standard or preconditioned CG. Recursive

The Wolfram Mathematica data file used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was fully supported by the Croatian Science Foundation under the project IP-2014-09-2899.