• Journal of the Institute of Convergence Signal Processing
• /
• v.12 no.3
• /
• pp.163-168
• /
• 2011

In this paper, we proposed non-linear gamma curve algorithm for gamma correction. The previous non-linear gamma curve algorithm is generated by the least square polynomial using the Gauss-Jordan inverse matrix. However, the previous algorithm has some weak points. When calculating coefficients using inverse matrix of higher degree, occurred truncation errors. Also, only if input sample points are existed regular interval on 10-bit scale, the least square polynomial is accurately works. To compensate weak-points, we calculated accurate coefficients of polynomial using eigenvalue and orthogonal value of mat11x from singular value decomposition (SVD) and QR decomposition of vandemond matrix. Also, we used input data part segmentation, then we performed polynomial curve fitting and merged curve fitting results. When compared the previous method and proposed method using the mean square error (MSE) and the standard deviation (STD), the proposed segmented polynomial curve fitting is highly accuracy that MSE under the least significant bit (LSB) error range is approximately $10^{-9}$ and STD is about $10^{-5}$.

• Journal of the Korean Mathematical Society
• /
• v.50 no.4
• /
• pp.755-770
• /
• 2013

One of the interesting nonlinear matrix equations is the quadratic matrix equation defined by $$Q(X)=AX^2+BX+C=0$$, where X is a $n{\times}n$ unknown real matrix, and A, B and C are $n{\times}n$ given matrices with real elements. Another one is the matrix polynomial $$P(X)=A_0X^m+A_1X^{m-1}+{\cdots}+A_m=0,\;X,\;A_i{\in}\mathbb{R}^{n{\times}n}$$. Newton's method is used to find the symmetric and bisymmetric solvents of the nonlinear matrix equations Q(X) and P(X). The method does not depend on the singularity of the Fr$\acute{e}$chet derivative. Finally, we give some numerical examples.

• Journal of the Chungcheong Mathematical Society
• /
• v.33 no.4
• /
• pp.413-427
• /
• 2020

In this work we study 0-commuting matrix C(0) under relationships with the Pascal matrix C(1). Like kth power C(1)k is an arithmetic matrix of (kx+y)n with yx = xy (k ≥ 1), we express C(0)k as an arithmetic matrix of certain polynomial with yx = 0.

• Journal of applied mathematics & informatics
• /
• v.26 no.3_4
• /
• pp.751-755
• /
• 2008

Formanek([2]) proved that $M_n(K)$, the matrix algebra has a nontrivial central polynomial when char K = 0. Also Razmyslov([3]) showed the same result using the essential weak identity. In this article we explicitly compute Formanek's central polynomial for $M_2(\mathbb{C})$ and $M_3(\mathbb{C})$ and classify the coefficients of the central polynomial.

• Bulletin of the Korean Mathematical Society
• /
• v.35 no.2
• /
• pp.235-258
• /
• 1998

We define a crossing of a link without referring to a specific projection of the link and describe a construction of a non-normalized Alexander polynomial associated to collections of such crossings of oriented links under an equivalence relation, called homology relation. The polynomial is computed from a special Seifert surface of the link. We prove that the polynomial is well-defined for the homology equivalence classes, investigate its relationship with the combinatorially defined Alexander polynomials and study some of its properties.

• Nuclear Engineering and Technology
• /
• v.31 no.2
• /
• pp.172-181
• /
• 1999

Series expansion methods to compute the exponential of a matrix have been compared by applying them to fuel depletion calculations. Specifically, Taylor, Pade, Chebyshev, and rational Chebyshev approximations have been investigated by approximating the exponentials of bum matrices by truncated series of each method with the scaling and squaring algorithm. The accuracy and efficiency of these methods have been tested by performing various numerical tests using one thermal reactor and two fast reactor depletion problems. The results indicate that all the four series methods are accurate enough to be used for fuel depletion calculations although the rational Chebyshev approximation is relatively less accurate. They also show that the rational approximations are more efficient than the polynomial approximations. Considering the computational accuracy and efficiency, the Pade approximation appears to be better than the other methods. Its accuracy is better than the rational Chebyshev approximation, while being comparable to the polynomial approximations. On the other hand, its efficiency is better than the polynomial approximations and is similar to the rational Chebyshev approximation. In particular, for fast reactor depletion calculations, it is faster than the polynomial approximations by a factor of ∼ 1.7.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.6
• /
• pp.1615-1623
• /
• 2014

Let $G{\leq}S_n$ and ${\chi}$ be any nonzero complex valued function on G. We first study the irreducibility of the generalized matrix polynomial $d^G_{\chi}(X)$, where $X=(x_{ij})$ is an n-by-n matrix whose entries are $n^2$ commuting independent indeterminates over $\mathbb{C}$. In particular, we show that if $\mathcal{X}$ is an irreducible character of G, then $d^G_{\chi}(X)$ is an irreducible polynomial, where either $G=S_n$ or $G=A_n$ and $n{\neq}2$. We then give a necessary and sufficient condition for the equality of two generalized matrix functions on the set of the so-called ${\chi}$-singular (${\chi}$-nonsingular) matrices.

• Korean Journal of Mathematics
• /
• v.31 no.2
• /
• pp.145-152
• /
• 2023

Let $P(z):=\sum\limits^{n}_{j=0}A_jz^j$, A_j ∈ ℂ^m×m, 0 ≤ j ≤ n be a matrix polynomial of degree n, such that A_n ≥ A_n-1 ≥ . . . ≥ A₀ ≥ 0, A_n > 0. Then the eigenvalues of P(z) lie in the closed unit disk. This theorem proved by Dirr and Wimmer [IEEE Trans. Automat. Control 52(2007), 2151-2153] is infact a matrix extension of a famous and elegant result on the distribution of zeros of polynomials known as Eneström-Kakeya theorem. In this paper, we prove a more general result which inter alia includes the above result as a special case. We also prove an improvement of a result due to Lê, Du, Nguyên [Oper. Matrices, 13(2019), 937-954] besides a matrix extention of a result proved by Mohammad [Amer. Math. Monthly, vol.74, No.3, March 1967].

• Journal of applied mathematics & informatics
• /
• v.33 no.3_4
• /
• pp.247-260
• /
• 2015

The special form of Schur complement is extended to have a Schur's formula to obtains the explicit formula of determinant, inverse, and eigenvector formula of the doubly Leslie matrix which is the generalized forms of the Leslie matrix. It is also a generalized form of the doubly companion matrix, and the companion matrix, respectively. The doubly Leslie matrix is a nonderogatory matrix.

• The KIPS Transactions:PartD
• /
• v.10D no.7
• /
• pp.1145-1148
• /
• 2003

We show the mixed finite element method which induces solutions that has the same order of errors for both the gradient of the solution and the solution itself. The technique to construct an efficient reusable matrix is suggested. Two families of mixed finite element methods are introduced with an automatic generating technique for matrix with my order of basis. The generated matrix by this technique has more accurate values and is a sparse matrix. This new technique is applied to solve a minimal surface problem.