Journal articles: 'Condition numbers of matrices'

1

Szarek, Stanislaw J. "Condition numbers of random matrices." Journal of Complexity 7, no. 2 (June 1991): 131–49. http://dx.doi.org/10.1016/0885-064x(91)90002-f.

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Chen, Zizhong, and Jack J. Dongarra. "Condition Numbers of Gaussian Random Matrices." SIAM Journal on Matrix Analysis and Applications 27, no. 3 (January 2005): 603–20. http://dx.doi.org/10.1137/040616413.

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Andersen, L. S., and J. L. Volakis. "Condition Numbers for Various Fem Matrices." Journal of Electromagnetic Waves and Applications 13, no. 12 (January 1999): 1663–79. http://dx.doi.org/10.1163/156939399x00105.

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Viswanath, D., and L. N. Trefethen. "Condition Numbers of Random Triangular Matrices." SIAM Journal on Matrix Analysis and Applications 19, no. 2 (April 1998): 564–81. http://dx.doi.org/10.1137/s0895479896312869.

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Gohberg, I., and I. Koltracht. "Condition numbers for functions of matrices." Applied Numerical Mathematics 12, no. 1-3 (May 1993): 107–17. http://dx.doi.org/10.1016/0168-9274(93)90114-7.

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6

Edelman, Alan. "Eigenvalues and Condition Numbers of Random Matrices." SIAM Journal on Matrix Analysis and Applications 9, no. 4 (October 1988): 543–60. http://dx.doi.org/10.1137/0609045.

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Cheung, Dennis, and Felipe Cucker. "Componentwise Condition Numbers of Random Sparse Matrices." SIAM Journal on Matrix Analysis and Applications 31, no. 2 (January 2009): 721–31. http://dx.doi.org/10.1137/080729463.

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Charpentier, S., K. Fouchet, O. Szehr, and R. Zarouf. "Condition numbers of matrices with given spectrum." Analysis and Mathematical Physics 9, no. 3 (May 21, 2019): 971–90. http://dx.doi.org/10.1007/s13324-019-00328-4.

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Böttcher, A., and S. M. Grudsky. "Structured condition numbers of large Toeplitz matrices are rarely better than usual condition numbers." Numerical Linear Algebra with Applications 12, no. 2-3 (September 29, 2004): 95–102. http://dx.doi.org/10.1002/nla.401.

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De Terán, Fernando, Froilán M. Dopico, and Javier Pérez. "Condition numbers for inversion of Fiedler companion matrices." Linear Algebra and its Applications 439, no. 4 (August 2013): 944–81. http://dx.doi.org/10.1016/j.laa.2012.09.020.

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Rohn, J. "New condition numbers for matrices and linear systems." Computing 41, no. 1-2 (March 1989): 167–69. http://dx.doi.org/10.1007/bf02238741.

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De Terán, Fernando, Froilán M. Dopico, and Javier Pérez. "Eigenvalue condition numbers and pseudospectra of Fiedler matrices." Calcolo 54, no. 1 (April 18, 2016): 319–65. http://dx.doi.org/10.1007/s10092-016-0189-9.

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Ratnarajah, T., R. Vaillancourt, and M. Alvo. "Eigenvalues and Condition Numbers of Complex Random Matrices." SIAM Journal on Matrix Analysis and Applications 26, no. 2 (January 2004): 441–56. http://dx.doi.org/10.1137/s089547980342204x.

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14

Lubinsky, D. S. "Condition numbers of Hankel matrices for exponential weights." Journal of Mathematical Analysis and Applications 314, no. 1 (February 2006): 266–85. http://dx.doi.org/10.1016/j.jmaa.2005.03.101.

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Dopico, Froilán M., and Kenet Pomés. "Structured eigenvalue condition numbers for parameterized quasiseparable matrices." Numerische Mathematik 134, no. 3 (November 16, 2015): 473–512. http://dx.doi.org/10.1007/s00211-015-0779-5.

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Singull, Martin, Denise Uwamariya, and Xiangfeng Yang. "Large-deviation asymptotics of condition numbers of random matrices." Journal of Applied Probability 58, no. 4 (November 22, 2021): 1114–30. http://dx.doi.org/10.1017/jpr.2021.13.

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AbstractLet $\mathbf{X}$ be a $p\times n$ random matrix whose entries are independent and identically distributed real random variables with zero mean and unit variance. We study the limiting behaviors of the 2-normal condition number k(p,n) of $\mathbf{X}$ in terms of large deviations for large n, with p being fixed or $p=p(n)\rightarrow\infty$ with $p(n)=o(n)$ . We propose two main ingredients: (i) to relate the large-deviation probabilities of k(p,n) to those involving n independent and identically distributed random variables, which enables us to consider a quite general distribution of the entries (namely the sub-Gaussian distribution), and (ii) to control, for standard normal entries, the upper tail of k(p,n) using the upper tails of ratios of two independent $\chi^2$ random variables, which enables us to establish an application in statistical inference.

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Böttcher, Albrecht, and Sergei M. Grudsky. "On the condition numbers of large semidefinite Toeplitz matrices." Linear Algebra and its Applications 279, no. 1-3 (August 1998): 285–301. http://dx.doi.org/10.1016/s0024-3795(98)00015-9.

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18

Movassagh, Ramis, and Alan Edelman. "Condition numbers of indefinite rank 2 ghost Wishart matrices." Linear Algebra and its Applications 483 (October 2015): 342–51. http://dx.doi.org/10.1016/j.laa.2015.05.027.

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Cucker, Felipe, and Huaian Diao. "Mixed and componentwise condition numbers for rectangular structured matrices." Calcolo 44, no. 2 (June 2007): 89–115. http://dx.doi.org/10.1007/s10092-007-0130-3.

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20

Diao, Huai-An, and Qing-Le Meng. "Structured generalized eigenvalue condition numbers for parameterized quasiseparable matrices." BIT Numerical Mathematics 59, no. 3 (April 6, 2019): 695–720. http://dx.doi.org/10.1007/s10543-019-00748-5.

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21

Cheung, D., and F. Cucker. "Smoothed analysis of componentwise condition numbers for sparse matrices." IMA Journal of Numerical Analysis 35, no. 1 (February 26, 2014): 74–88. http://dx.doi.org/10.1093/imanum/drt068.

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Xiang, Hua, and Yimin Wei. "Structured mixed and componentwise condition numbers of some structured matrices." Journal of Computational and Applied Mathematics 202, no. 2 (May 2007): 217–29. http://dx.doi.org/10.1016/j.cam.2006.02.026.

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23

Li, Hanyu. "Structured condition numbers for some matrix factorizations of structured matrices." Journal of Computational and Applied Mathematics 336 (July 2018): 219–34. http://dx.doi.org/10.1016/j.cam.2017.12.010.

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Diao, Huaian. "On componentwise condition numbers for eigenvalue problems with structured matrices." Numerical Linear Algebra with Applications 16, no. 2 (February 2009): 87–107. http://dx.doi.org/10.1002/nla.607.

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25

Milinazzo, F., C. Zala, and I. Barrodale. "On the rate of growth of condition numbers for convolution matrices." IEEE Transactions on Acoustics, Speech, and Signal Processing 35, no. 4 (April 1987): 471–75. http://dx.doi.org/10.1109/tassp.1987.1165145.

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Dopico, Froilán M., and Kenet Pomés. "Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices." Numerical Algorithms 73, no. 4 (April 14, 2016): 1131–58. http://dx.doi.org/10.1007/s11075-016-0133-8.

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27

Diederichs, Benedikt, and Armin Iske. "Improved estimates for condition numbers of radial basis function interpolation matrices." Journal of Approximation Theory 238 (February 2019): 38–51. http://dx.doi.org/10.1016/j.jat.2017.10.004.

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Sankar, Arvind, Daniel A. Spielman, and Shang-Hua Teng. "Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices." SIAM Journal on Matrix Analysis and Applications 28, no. 2 (January 2006): 446–76. http://dx.doi.org/10.1137/s0895479803436202.

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29

Kushida, Noriyuki. "Condition Number Estimation of Preconditioned Matrices." PLOS ONE 10, no. 3 (March 27, 2015): e0122331. http://dx.doi.org/10.1371/journal.pone.0122331.

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30

Černá, Dana, and Václav Finěk. "Wavelet basis of cubic splines on the hypercube satisfying homogeneous boundary conditions." International Journal of Wavelets, Multiresolution and Information Processing 13, no. 03 (May 2015): 1550014. http://dx.doi.org/10.1142/s0219691315500149.

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In this paper, we propose a construction of a new cubic spline-wavelet basis on the hypercube satisfying homogeneous Dirichlet boundary conditions. Wavelets have two vanishing moments. Stiffness matrices arising from discretization of elliptic problems using a constructed wavelet basis have uniformly bounded condition numbers and we show that these condition numbers are small. We present quantitative properties of the constructed basis and we provide a numerical example to show the efficiency of the Galerkin method using the constructed basis.

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31

Narcowich, Francis J., and Joseph D. Ward. "Norms of inverses and condition numbers for matrices associated with scattered data." Journal of Approximation Theory 64, no. 1 (January 1991): 69–94. http://dx.doi.org/10.1016/0021-9045(91)90087-q.

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32

Stewart, G. W., and G. Zhang. "Eigenvalues of graded matrices and the condition numbers of a multiple eigenvalue." Numerische Mathematik 58, no. 1 (December 1990): 703–12. http://dx.doi.org/10.1007/bf01385650.

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33

Al-homidan, S., and M. Alshahrani. "Positive Definite Hankel Matrices Using Cholesky Factorization." Computational Methods in Applied Mathematics 9, no. 3 (2009): 221–25. http://dx.doi.org/10.2478/cmam-2009-0013.

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AbstractReal positive definite Hankel matrices arise in many important applications. They have spectral condition numbers which exponentially increase with their orders. We give a structural algorithm for finding positive definite Hankel matrices using the Cholesky factorization, compute it for orders less than or equal to 30, and compare our result with earlier results.

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34

Gil’, Michael. "Perturbation Bounds for Eigenvalues and Determinants of Matrices. A Survey." Axioms 10, no. 2 (May 21, 2021): 99. http://dx.doi.org/10.3390/axioms10020099.

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The paper is a survey of the recent results of the author on the perturbations of matrices. A part of the results presented in the paper is new. In particular, we suggest a bound for the difference of the determinants of two matrices which refines the well-known Bhatia inequality. We also derive new estimates for the spectral variation of a perturbed matrix with respect to a given one, as well as estimates for the Hausdorff and matching distances between the spectra of two matrices. These estimates are formulated in the terms of the entries of matrices and via so called departure from normality. In appropriate situations they improve the well-known results. We also suggest a bound for the angular sectors containing the spectra of matrices. In addition, we suggest a new bound for the similarity condition numbers of diagonalizable matrices. The paper also contains a generalization of the famous Kahan inequality on perturbations of Hermitian matrices by non-normal matrices. Finally, taking into account that any matrix having more than one eigenvalue is similar to a block-diagonal matrix, we obtain a bound for the condition numbers in the case of non-diagonalizable matrices, and discuss applications of that bound to matrix functions and spectrum perturbations. The main methodology presented in the paper is based on a combined usage of the recent norm estimates for matrix-valued functions with the traditional methods and results.

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35

Kushida, Noriyuki. "Correction: Condition Number Estimation of Preconditioned Matrices." PLOS ONE 10, no. 6 (June 17, 2015): e0130920. http://dx.doi.org/10.1371/journal.pone.0130920.

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36

Beasley, LeRoy. "Preservers of term ranks and star cover numbers of symmetric matrices." Electronic Journal of Linear Algebra 31 (February 5, 2016): 549–64. http://dx.doi.org/10.13001/1081-3810.3231.

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Let $\S$ denote the set of symmetric matrices over some semiring, $\s$. A line of $A\in\S$ is a row or a column of $A$. A star of $A$ is the submatrix of $A$ consisting of a row and the corresponding column of $A$. The term rank of $A$ is the minimum number of lines that contain all the nonzero entries of $A$. The star cover number is the minimum number of stars that contain all the nonzero entries of $A$. This paper investigates linear operators that preserve sets of symmetric matrices of specified term rank and sets of symmetric matrices of specific star cover numbers. Several equivalences to the condition that $T$ preserves the term rank of any matrix are given along with characterizations of a couple of types of linear operators that preserve certain sets of matrices defined by the star cover number that do not preserve all term ranks.

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37

Shakil, M., and M. Ahsanullah. "A note on the characterizations of the distributions of the condition numbers of real Gaussian matrices." Special Matrices 6, no. 1 (July 1, 2018): 282–96. http://dx.doi.org/10.1515/spma-2018-0022.

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Abstract Many researchers and authors have studied the distributions of the condition numbers of real Gaussian matrices, which appear in many fields of pure and applied sciences, such as, probability, statistics, multivariate statistics, linear algebra, operator algebra theory, actuarial science, physics, wireless communications, and polarimetric synthetic aperture radar (PolSAR). Motivated by this, in this paper, we first present several new distributional properties of the distributions of the condition numbers of real Gaussian matrices. Since it is important to know the percentage points of a given distribution for any statistical application, we have also computed percentiles of the said distributions of the condition numbers. Before a particular probability distribution model is applied to fit the real world data, it is necessary to confirm whether the given continuous probability distribution satisfies the underlying requirements by its characterizations. Also, the truncated distributions arise in practical statisticswhere the ability of record observations is limited to a given threshold or within a specified range. In view of these facts, some characterizations by truncated first moment are also presented. We hope that the findings of this paper will be quite useful to the researchers in various fields of pure and applied sciences as stated above.

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38

Soto, Ricardo, Ana Julio, and Macarena Collao. "Brauer's theorem and nonnegative matrices with prescribed diagonal entries." Electronic Journal of Linear Algebra 35 (February 1, 2019): 53–64. http://dx.doi.org/10.13001/1081-3810.3886.

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The problem of the existence and construction of nonnegative matrices with prescribed eigenvalues and diagonal entries is an important inverse problem, interesting by itself, but also necessary to apply a perturbation result, which has played an important role in the study of certain nonnegative inverse spectral problems. A number of partial results about the problem have been published by several authors, mainly by H. \v{S}migoc. In this paper, the relevance of a Brauer's result, and its implication for the nonnegative inverse eigenvalue problem with prescribed diagonal entries is emphasized. As a consequence, given a list of complex numbers of \v{S}migoc type, or a list $\Lambda = \left\{\lambda _{1},\ldots ,\lambda _{n} \right \}$ with $\operatorname{Re}\lambda _{i}\leq 0,$ $\lambda _{1}\geq -\sum\limits_{i=2}^{n}\lambda _{i}$, and $\left\{-\sum\limits_{i=2}^{n}\lambda _{i},\lambda _{2},\ldots ,\lambda _{n} \right\}$ being realizable; and given a list of nonnegative real numbers $% \Gamma = \left\{\gamma _{1},\ldots ,\gamma _{n} \right\}$, the remarkably simple condition $\gamma _{1}+\cdots +\gamma _{n} = \lambda _{1}+\cdots +\lambda _{n}$ is necessary and sufficient for the existence and construction of a realizing matrix with diagonal entries $\Gamma .$ Conditions for more general lists of complex numbers are also given.

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Tabeart, Jemima M., Sarah L. Dance, Amos S. Lawless, Nancy K. Nichols, and Joanne A. Waller. "Improving the condition number of estimated covariance matrices." Tellus A: Dynamic Meteorology and Oceanography 72, no. 1 (December 13, 2019): 1–19. http://dx.doi.org/10.1080/16000870.2019.1696646.

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40

Avron, Haim, Alex Druinsky, and Sivan Toledo. "Spectral condition‐number estimation of large sparse matrices." Numerical Linear Algebra with Applications 26, no. 3 (March 12, 2019): e2235. http://dx.doi.org/10.1002/nla.2235.

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41

Guo, Xiaobin, and Dequan Shang. "Approximate Solution of LR Fuzzy Sylvester Matrix Equations." Journal of Applied Mathematics 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/752760.

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The fuzzy Sylvester matrix equationAX~+X~B=C~in whichA,Barem×mandn×ncrisp matrices, respectively, andC~is anm×nLR fuzzy numbers matrix is investigated. Based on the Kronecker product of matrices, we convert the fuzzy Sylvester matrix equation into an LR fuzzy linear system. Then we extend the fuzzy linear system into two systems of linear equations according to the arithmetic operations of LR fuzzy numbers. The fuzzy approximate solution of the original fuzzy matrix equation is obtained by solving the crisp linear systems. The existence condition of the LR fuzzy solution is also discussed. Some examples are given to illustrate the proposed method.

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42

Axelsson, Owe, and Hao Lu. "A survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices." Journal of Computational and Applied Mathematics 80, no. 2 (May 1997): 241–64. http://dx.doi.org/10.1016/s0377-0427(97)00023-x.

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43

Wu, Gang, and Lu Zhang. "New algorithms for approximating $\varphi$-functions and their condition numbers for large sparse matrices." IMA Journal of Numerical Analysis 38, no. 3 (August 13, 2017): 1185–208. http://dx.doi.org/10.1093/imanum/drx035.

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44

Garloff, Jürgen, Mohammad Adm, Khawla Al Muhtaseb, and Ayed Abedel Ghani. "Relaxing the Nonsingularity Assumption for Intervals of Totally Nonnegative Matrices." Electronic Journal of Linear Algebra 36, no. 36 (March 27, 2020): 106–23. http://dx.doi.org/10.13001/ela.2020.5015.

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Totally nonnegative matrices, i.e., matrices having all their minors nonnegative, and matrix intervals with respect to the checkerboard partial order are considered. It is proven that if the two bound matrices of such a matrix interval are totally nonnegative and satisfy certain conditions, then all matrices from this interval are totally nonnegative and satisfy these conditions, too, hereby relaxing the nonsingularity condition in the former paper [M. Adm and J. Garloff. Intervals of totally nonnegative matrices. Linear Algebra Appl., 439:3796--3806, 2013.].

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45

Benvenuti, Luca. "The inverse eigenvalue problem for Leslie matrices." Electronic Journal of Linear Algebra 35 (February 1, 2019): 319–30. http://dx.doi.org/10.13001/1081-3810.3980.

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The Nonnegative Inverse Eigenvalue Problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of $n$ complex numbers to be the spectrum of an entry--wise nonnegative matrix of dimension $n$. This is a very difficult and long standing problem and has been solved only for $n\leq 4$. In this paper, the NIEP for a particular class of nonnegative matrices, namely Leslie matrices, is considered. Leslie matrices are nonnegative matrices, with a special zero--pattern, arising in the Leslie model, one of the best known and widely used models to describe the growth of populations. The lists of nonzero complex numbers that are subsets of the spectra of Leslie matrices are fully characterized. Moreover, the minimal dimension of a Leslie matrix having a given list of three numbers among its spectrum is provided. This result is partially extended to the case of lists of $n > 2$ real numbers.

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46

Wang, Jinru, Wenhui Shi, and Lin Hu. "Wavelet Numerical Solutions for a Class of Elliptic Equations with Homogeneous Boundary Conditions." Mathematics 9, no. 12 (June 15, 2021): 1381. http://dx.doi.org/10.3390/math9121381.

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This paper focuses on a method to construct wavelet Riesz bases with homogeneous boundary condition and use them to a kind of second-order elliptic equation. First, we construct the splines on the interval [0,1] and consider their approximation properties. Then we define the wavelet bases and illustrate the condition numbers of stiffness matrices are small and bounded. Finally, several numerical examples show that our approach performs efficiently.

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47

Ma, Jiangming, Tao Qiu, and Chengyuan He. "A New Method of Matrix Decomposition to Get the Determinants and Inverses of r -Circulant Matrices with Fibonacci and Lucas Numbers." Journal of Mathematics 2021 (November 18, 2021): 1–9. http://dx.doi.org/10.1155/2021/4782594.

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We use a new method of matrix decomposition for r -circulant matrix to get the determinants of A n = Circ r F 1 , F 2 , … , F n and B n = Circ r L 1 , L 2 , … , L n , where F n is the Fibonacci numbers and L n is the Lucas numbers. Based on these determinants and the nonsingular conditions, inverse matrices are derived. The expressions of the determinants and inverse matrices are represented by Fibonacci and Lucas Numbers. In this study, the formulas of determinants and inverse matrices are much simpler and concise for programming and reduce the computational time.

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48

Wang, Cheng, Dao-Sheng Zheng, Guo-Liang Chen, and Shu-Qin Zhao. "Structures of p-isometric matrices and rectangular matrices with minimum p-norm condition number." Linear Algebra and its Applications 184 (April 1993): 261–78. http://dx.doi.org/10.1016/0024-3795(93)90383-y.

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Zimmermann, R. "On the condition number anomaly of Gaussian correlation matrices." Linear Algebra and its Applications 466 (February 2015): 512–26. http://dx.doi.org/10.1016/j.laa.2014.10.038.

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50

Türkmen, Ramazan, and Zübeyde Ulukök. "On The Frobenius Condition Number of Positive Definite Matrices." Journal of Inequalities and Applications 2010, no. 1 (2010): 897279. http://dx.doi.org/10.1155/2010/897279.

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Journal articles on the topic 'Condition numbers of matrices'

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