Jacobi method for symmetric 4 × 4 matrices converges for every cyclic pivot strategy

Kovač, Erna Begović; Hari, Vjeran

doi:10.1007/s11075-017-0396-8

Cited by 2 publications

(13 citation statements)

References 15 publications

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“…Since it immediately follows from (1.1) that θ (0) = φ (1) , θ (1) = φ (0) . Thus, after completing the first two steps in each of the two processes, we have A (2) = P T R T 23 (θ (1) )R T 14 (θ (0) )AR 14 (θ (0) )R 23 (θ (1) )P = P T A (2) P. This shows that the relation (A.1) holds if A (0) and A (0) are replaced by A (2) and A (2) = P T A (2) P , respectively. Checking the angle formula (1.1) we find that θ (2) (2) and therefore A (4) = P T A (4) P .…”

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confidence: 79%

“…Thus, after completing the first two steps in each of the two processes, we have A (2) = P T R T 23 (θ (1) )R T 14 (θ (0) )AR 14 (θ (0) )R 23 (θ (1) )P = P T A (2) P. This shows that the relation (A.1) holds if A (0) and A (0) are replaced by A (2) and A (2) = P T A (2) P , respectively. Checking the angle formula (1.1) we find that θ (2) (2) and therefore A (4) = P T A (4) P . The last check is the easiest one since the denominators in (1.1) for the angles θ (4) and θ (5) are opposite to those for the angles φ (4) and φ (5) .…”

mentioning

confidence: 79%

“…Let us apply the Jacobi method defined by the strategy I O ′′ 1 to a symmetric matrix A of order 4, thus generating the sequence of matrices A (0) = A, A (1) , A (2) , . .…”

Section: Basic Concepts and Notationmentioning

confidence: 99%

“…Hence the global convergence consideration for the general cyclic Jacobi method should scrutinize more than one cycle of the process. Second, the presented result covers the most difficult part in the proof that every cyclic Jacobi method for symmetric matrices of order 4 is globally convergent [1,2].…”

Section: Introductionmentioning

confidence: 99%

“…Hence, for the global convergence proof one has to consider two or three adjacent cycles. It is proved that for any symmetric matrix A of order 4 the inequality S(A [2] ) ≤ (1 − 10 −5 )S(A) holds, where A [2] results from A by applying two cycles of a particular parallel method. Here S(A) stands for the Frobenius norm of the strictly upper-triangular part of A.…”

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confidence: 99%

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On the global convergence of the Jacobi method for symmetric matrices of order 4 under parallel strategies

Kovač

Hari

2017

Linear Algebra and its Applications

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Abstract. The paper analyzes special cyclic Jacobi methods for symmetric matrices of order 4. Only those cyclic pivot strategies that enable full parallelization of the method are considered. These strategies, unlike the serial pivot strategies, can force the method to be very slow or very fast within one cycle, depending on the underlying matrix. Hence, for the global convergence proof one has to consider two or three adjacent cycles. It is proved that for any symmetric matrix A of order 4 the inequality S(A [2] ) ≤ (1 − 10 −5 )S(A) holds, where A [2] results from A by applying two cycles of a particular parallel method. Here S(A) stands for the Frobenius norm of the strictly upper-triangular part of A. The result holds for two special parallel strategies and implies the global convergence of the method under all possible fully parallel strategies. It is also proved that for every ǫ > 0 and n ≥ 4 there exist a symmetric matrix A(ǫ) of order n and a cyclic strategy, such that upon completion of the first cycle of the appropriate Jacobi method the inequality S(A [1] ) > (1 − ǫ)S(A(ǫ)) holds.

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confidence: 79%

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confidence: 79%

“…Let us apply the Jacobi method defined by the strategy I O ′′ 1 to a symmetric matrix A of order 4, thus generating the sequence of matrices A (0) = A, A (1) , A (2) , . .…”

Section: Basic Concepts and Notationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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On the global convergence of the Jacobi method for symmetric matrices of order 4 under parallel strategies

Kovač

Hari

2017

Linear Algebra and its Applications

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Research on Feature Extraction of Performance Degradation for Flexible Material R2R Processing Roller Based on PCA

Deng

Zhou

Yao

et al. 2020

International Journal of Rotating Machinery

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Performance feature extraction is the primary problem in equipment performance degradation assessment. To handle the problem of high-dimensional performance characterization and complexity of calculating the performance indicators in flexible material roll-to-roll processing, this paper proposes a PCA method for extracting the degradation characteristic of roll shaft. Based on the analysis of the performance influencing factors of flexible material roll-to-roll processing roller, a principal component analysis extraction model was constructed. The original feature parameter matrix composed of 10-dimensional feature parameters such as time domain, frequency domain, and time-frequency domain vibration signal of the roll shaft was established; then, we obtained a new feature parameter matrix Z org ∗ by normalizing the original feature parameter matrix. The correlation measure between every two parameters in the matrix Z org ∗ was used as the eigenvalue to establish the covariance matrix of the performance degradation feature parameters. The Jacobi iteration method was introduced to derive the algorithm for solving eigenvalue and eigenvector of the covariance matrix. Finally, using the eigenvalue cumulative contribution rate as the screening rule, we linearly weighted and fused the eigenvectors and derived the feature principal component matrix F of the processing roller vibration signal. Experiments showed that the initially obtained, 10-dimensional features of the processing rollers’ vibration signals, such as average, root mean square, kurtosis index, centroid frequency, root mean square of frequency, standard deviation of frequency, and energy of the intrinsic mode function component, can be expressed by 3-dimensional principal components F 1 , F 2 , and F 3 . The vibration signal features reduction dimension was realized, and F 1 , F 2 , and F 3 contain 98.9% of the original vibration signal data, further illustrating that the method has high precision in feature parameters’ extraction and the advantage of eliminating the correlation between feature parameters and reducing the workload selecting feature parameters.

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Jacobi method for symmetric 4 × 4 matrices converges for every cyclic pivot strategy

Cited by 2 publications

References 15 publications

On the global convergence of the Jacobi method for symmetric matrices of order 4 under parallel strategies

On the global convergence of the Jacobi method for symmetric matrices of order 4 under parallel strategies

Research on Feature Extraction of Performance Degradation for Flexible Material R2R Processing Roller Based on PCA

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