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In this paper we make a theoretical analysis of the convergence rates of Kaczmarz and Extended Kaczmarz projection algorithms for some of the most practically used control sequences. We first prove an at least linear convergence rate for the Kaczmarz-Tanabe and its Extended version methods (the one in which a complete set of projections using row/column index is performed in each iteration). Then we apply the main ideas of this analysis in establishing an at least sublinear, respectively linear convergence rate for the Kaczmarz algorithm with almost cyclic and the remotest set control strategies, and their extended versions, respectively. These results complete the existing ones related to the random selection procedures.

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This study aimed to describe the influence of recovery duration during a repeated sprint ability (RSA) test (6 × 40 m) by investigating a number of variables, such as general performance, metabolic demand, and muscular stretch-shortening performance. Seventeen male soccer outfield players (16 ± 0 years, 66 ± 10 kg) performed three field shuttle-running tests with 15, 20, and 25-sec recoveries. In addition to specific shuttle test's variables, blood lactate concentration and vertical jump height were assessed. Resulting measures were highly reliable (intra-class correlation coefficient up to 0.86). 25-sec recovery improved test performance (-3% total time from 15-sec to 25-sec recovery), vertical jump height (+7% post-test height from 15-sec to 25-sec recovery), and decreased blood lactate accumulation (-33% post-test from 15-sec to 25-sec recovery). Study findings suggest that metabolic acidosis plays a role in worsening performance and fatigue development during the shuttle test. A 25-sec recovery duration maximized performance, containing metabolic-anaerobic power involvement and muscular stretch-shortening performance deterioration during a RSA test.

To find the least squares solution of a very large and inconsistent system of equations, one can employ the extended Kaczmarz algorithm. This method simultaneously removes the error term, such that a consistent system is asymptotically obtained, and applies Kaczmarz iterations for the current approximation of this system. For random corrections of the right hand side and Kaczmarz updates selected at random, convergence to the least squares solution has been shown. We consider the deterministic control strategies, and show convergence to a least squares solution when row and column updates are chosen according to the almost-cyclic or maximal-residual choice.Date: April 2, 2015.

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