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Advanced Watermarking Techniques
Published in Borra Surekha, Thanki Rohit, Dey Nilanjan, Digital Image Watermarking, 2018
Borra Surekha, Thanki Rohit, Dey Nilanjan
The Hessenberg decomposition (Van Loan, 1996) is a matrix decomposition method. A matrix B is decomposed into a unitary matrixU and a Hessenberg matrix H such that U × H × UT = B, where UT is the conjugate transpose of the unitary matrix. This decomposition requires about 14n3/3 computational operations that are higher than the computational operations required in the Schur decomposition but less than those of SVD decomposition. Hessenberg matrix factorization is used in watermarking to ensure that the randomization process is perfectly invertible (Su and Chen, 2017; Singh et al., 2017; Su, 2016; Bhatnagar and Wu, 2013; Bhatnagar et al., 2010). Watermarking using this decomposition is more complex to implement and is less explored by researchers.
Lapack
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Zhaojun Bai, James Demmel, Jack Dongarra, Julien Langou, Jenny Wang
Let us show how to use the simple driver SGEEV to solve the NEP (75.7). SGEEV first reduces A to an upper Hessenberg form (a Hessenberg matrix is a matrix where all entries below the first lower subdiagonal are zeros), and then uses the implicit QR algorithm to compute the Schur decomposition, and finally computes eigenvectors of the upper quasi-triangular matrix. SGEEV has the following calling sequence: CALL SGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
Linear Systems
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
If A is square then it is possible to write it as A=UTU−1 where U is unitary (U−1=UT) and T is upper triangular with the eigenvalues of A on its main diagonal. (This is called the Schur decomposition.) Note that U and T may be complex-valued even though A is real. Use the MATLAB commands [U1,T1]=schur(A);[U,T]=rsf2csf(U1,T1) to find the Schur decomposition of A=[010;−100;001]. (The rsf2csf command converts the output of schur to the form described above.)If you had the Schur decomposition of A pre-computed, how would you solve Ax=b? Assume that complex arithmetic is implemented automatically. (Note: The Schur decomposition is not in fact used to solve linear systems; it is used in numerical eigenvalue computations.)
Monte Carlo Solution of k-Eigenvalue Problem Using Subspace Iteration Method
Published in Nuclear Science and Engineering, 2020
The orthonormalization procedure in step 2b is computationally expensive due to factorization; hence, it may be useful to perform several applications of A (step 2a) before the orthonormalization procedure. It has been demonstrated16 that Xk will essentially converge in direction to the Schur vectors associated with the m dominant eigenvalues of A and the convergence will be governed by the i’th dominance ratio where λi, i = 1, …, m are the m dominant eigenvalues of A with |λi| > | λi+1|. The Schur decomposition of A is defined as follows. An n × n square matrix A can be expressed as A = QUQ*, where Q is a unitary matrix (hence, Q−1 = Q*, the conjugate transpose of Q) and U is an upper triangular matrix referred to as the Schur form. The Schur decomposition of a matrix is computed using the QR algorithm or its variants. The diagonal entries of U are exactly the eigenvalues of A.