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Graded-Index Integrated Modulators
Published in Le Nguyen Binh, Optical Modulation, 2017
There are a few major features of the matrix equation above: (i) this type of matrix is often referred to as tridiagonal matrix with fringes. The order of the matrix is N × N, the square of the total number of grid points. Most of terms in the matrix are zeros; (ii) the matrix is non-symmetrical relative to the diagonal term; (iii) the central three diagonal terms always exist and are always non-zero; (iv) the coefficients p, l, r, t, and b make up the five bands of the matrix, with p being the main diagonal, l and r being the sub-diagonal while t and b the super-diagonal; (v) the sub-diagonal diagonal terms are just one term away from the main diagonal while the upper-diagonal terms are NX terms away from the main diagonal. The distance between main diagonal and the last non-zero super-diagonal band is commonly referred to as the half bandwidth of a band matrix; and (vi) terms such as l1, rN, t1−tNX, bN−Nx − bN are missing. This is so since the evaluations of these terms require the E values outside the boundary area, and these values have been assumed zero. Therefore, they need not be represented.
Smoothing Scatterplots
Published in Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka, Exploratory Data Analysis with MATLAB®, 2017
Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka
We now define two band matrices Q and R that will be used in the algorithm for smoothing splines. A band matrix is one whose nonzero entries are contained in a band along the main diagonal. The matrix entries outside this band are zero. First, we let hi = ti+1-ti for i = 1, n - 1. Then Q is the matrix with nonzero entries qij, given by qj-1,j=1hj-1qjj=-1hj-1-1hjqj+1,j=1hj, $$ q_{{j - 1,j}} = \frac{1}{{h_{{j - 1}} }}~q_{{jj}} = - \frac{1}{{h_{{j - 1}} }} - \frac{1}{{h_{j} }}~q_{{j + 1,j}} = \frac{1}{{h_{j} }} , $$
Three-Dimensional Optical Waveguides
Published in Le Nguyen Binh, Wireless And Guided Wave Electromagnetics, 2017
There are a few major features of the matrix equation above: (1) This type of matrix is often referred to as tridiagonal matrix with fringes. The order of the matrix is N × N, the square of the total number of grid points. Most of the terms in the matrix are zeros. (2) The matrix is nonsymmetrical relative to the diagonal term. (3) The central three diagonal terms always exist and are always nonzero. (4) The coefficients p, l, r, t, b make up the five bands of the matrix, with p being the main diagonal, l and r the subdiagonals, and t and b the superdiagonal. (5) The subdiagonal terms are just one term away from the main diagonal, while the superdiagonal terms are NX terms away from the main diagonal. The distance between the main diagonal and the last nonzero superdiagonal band is commonly referred to as the half-bandwidth of a band matrix. (6) Terms such as l1, rN, t1 – tNX, and bN–Nx – bN are missing. This is so because the evaluations of these terms require the E-values outside the boundary area, and these values have been assumed to be zero. Therefore, they need not be represented.
Solution of the symmetric band partial inverse eigenvalue problem for the damped mass spring system
Published in Inverse Problems in Science and Engineering, 2021
Suman Rakshit, Biswa Nath Datta
A band matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side. Let, be a symmetric band matrix of bandwidth p then the matrix A can be expressed as where { : } is a basis of . It can be mentioned that the matrix A has only non-zero distinct variables. It is to be noted that dimension of is .