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The Official Journal of the Pan-Pacific Association of Input-Output Studies (PAPAIOS)

Fig. 2 | Journal of Economic Structures

Fig. 2

From: Balancing and reconciling large multi-regional input–output databases using parallel optimisation and high-performance computing

Fig. 2

Visualisation of Kaczmarz’s algorithm. This visualisation shows the concept of Kaczmarz’s algorithm, with two hyperplanes. The initial estimate \({\mathbf {p}}^0\) is projected onto the first hyperplane, yielding \({\mathbf {p}}^1\). The first iterate \({\mathbf {p}}^1\) is then projected onto the second hyperplane and so forth. Contrary to Cimmino’s algorithm, the iterates produced by Kaczmarz’s algorithm will always adhere to at least one of the constraints given by the rows of the matrix \({\mathbf {G}}\). Although the convergence appears to be a lot faster for this simple example, it can be extremely slow for larger problems in higher dimensions. The algorithm will therefore only be used for a certain class of subproblems that occur during the projection step of Cimmino’s algorithm

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