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Methods using transfer matrices of higher dimensionality, that is 3×3, 4×4, and 6×6, are also used in optical analysis. [9] [10] [11] In particular, 4×4 propagation matrices are used in the design and analysis of prism sequences for pulse compression in femtosecond lasers .
The transfer-matrix method is a method used in optics and acoustics to analyze the propagation of electromagnetic or acoustic waves through a stratified medium; a stack of thin films. [1] [2] This is, for example, relevant for the design of anti-reflective coatings and dielectric mirrors . The reflection of light from a single interface between ...
After the algorithm has converged, the singular value decomposition = is recovered as follows: the matrix is the accumulation of Jacobi rotation matrices, the matrix is given by normalising the columns of the transformed matrix , and the singular values are given as the norms of the columns of the transformed matrix .
The direct stiffness method is the most common implementation of the finite element method (FEM). In applying the method, the system must be modeled as a set of simpler, idealized elements interconnected at the nodes. The material stiffness properties of these elements are then, through matrix mathematics, compiled into a single matrix equation ...
In chemical analysis, matrix refers to the components of a sample other than the analyte [1] of interest. The matrix can have a considerable effect on the way the analysis is conducted and the quality of the results are obtained; such effects are called matrix effects. [2] For example, the ionic strength of the solution can have an effect on ...
The following tables list the computational complexity of various algorithms for common mathematical operations . Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. [1] See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, below ...
where R 1 is an n×n upper triangular matrix, 0 is an (m − n)×n zero matrix, Q 1 is m×n, Q 2 is m×(m − n), and Q 1 and Q 2 both have orthogonal columns. Golub & Van Loan (1996, §5.2) call Q 1 R 1 the thin QR factorization of A; Trefethen and Bau call this the reduced QR factorization. [1]
Solute – the sample components in partition chromatography. Solvent – any substance capable of solubilizing another substance, and especially the liquid mobile phase in liquid chromatography. Stationary phase – the substance fixed in place for the chromatography procedure. Examples include the silica layer in thin-layer chromatography