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Condition number estimation of preconditioned matrices.

ABSTRACT: The present paper introduces a condition number estimation method for preconditioned matrices. The newly developed method provides reasonable results, while the conventional method which is based on the Lanczos connection gives meaningless results. The Lanczos connection based method provides the condition numbers of coefficient matrices of systems of linear equations with information obtained through the preconditioned conjugate gradient method. Estimating the condition number of preconditioned matrices is sometimes important when describing the effectiveness of new preconditionerers or selecting adequate preconditioners. Operating a preconditioner on a coefficient matrix is the simplest method of estimation. However, this is not possible for large-scale computing, especially if computation is performed on distributed memory parallel computers. This is because, the preconditioned matrices become dense, even if the original matrices are sparse. Although the Lanczos connection method can be used to calculate the condition number of preconditioned matrices, it is not considered to be applicable to large-scale problems because of its weakness with respect to numerical errors. Therefore, we have developed a robust and parallelizable method based on Hager's method. The feasibility studies are curried out for the diagonal scaling preconditioner and the SSOR preconditioner with a diagonal matrix, a tri-daigonal matrix and Pei's matrix. As a result, the Lanczos connection method contains around 10% error in the results even with a simple problem. On the other hand, the new method contains negligible errors. In addition, the newly developed method returns reasonable solutions when the Lanczos connection method fails with Pei's matrix, and matrices generated with the finite element method.

SUBMITTER: Kushida N

PROVIDER: S-EPMC4376946 | biostudies-other | 2015

REPOSITORIES: biostudies-other

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Condition number estimation of preconditioned matrices.

Kushida Noriyuki N

PloS one 20150327 3

The present paper introduces a condition number estimation method for preconditioned matrices. The newly developed method provides reasonable results, while the conventional method which is based on the Lanczos connection gives meaningless results. The Lanczos connection based method provides the condition numbers of coefficient matrices of systems of linear equations with information obtained through the preconditioned conjugate gradient method. Estimating the condition number of preconditioned ...[more]

PMID: 25816331

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Project description:IntroductionThe wave condition number (WCN) is a non-dimensional number that determines the state of arterial wave reflections. WCN is equal to HR × L eff /PWV where HR, L eff , and PWV are the heart rate, effective length, and pulse wave velocity, respectively. It has been shown that a value of WCN = 0.1 indicates the optimum state of arterial wave reflection in which left ventricle workload is minimized. The pressure wave, flow wave, and PWV are all required to compute WCN, which may limit the potential clinical utility of WCN. The aims of this study are as follows: (1) to assess the feasibility of approximating WCN from the pressure waveform alone (WCN Pinf ), and (2) to provide the proof-of-concept that WCN Pinf can capture age related differences in arterial wave reflection among healthy women and men.MethodsPreviously published retrospective data composed of seventeen patients (age 19-54 years; 34.3 ± 9.6) were used to assess the accuracy of WCN Pinf . The exact value of WCN was computed from PWV (measured by foot-to-foot method), HR, and L eff . A quarter wavelength relationship with minimum impedance modulus were used to compute L eff . WCN Pinf was calculated using HR and the reflected wave arrival time. Previously published analyses from a healthy subset of the Anglo-Cardiff Collaborative Trial (ACCT) study population were used to investigate if non-invasive WCN Pinf captures age related differences in arterial wave reflection among healthy women and men.ResultsA strong correlation (r = 0.83, p-value <0.0001) between WCN Pinf and WCN was observed. The accuracy of WCN Pinf was independent from relevant physiological parameters such as PWV, pulse pressure (PP), and HR. Similar changes in WCN Pinf with advancing age were observed in both healthy men and healthy women. In young, healthy individuals (women and men) the WCN Pinf was around 0.1 (the optimum value), and reduced with aging.ConclusionWCN can be approximated from a single pressure waveform and can capture age related arterial wave reflection alteration. These results are clinically significant since WCN can be extracted from a single non-invasive pressure waveform. Future studies will focus on investigating if WCN is associated with risk for onset of cardiovascular disease events.

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Condition number estimation of preconditioned matrices.

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Condition number estimation of preconditioned matrices.

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