Hessenberg form

The method of Hessenberg, which reduces the matrix to Hessenberg form, replaces the matrix A by... 

A Hessenberg form H (the same form but not the same matrix) can also be obtained by a sequence of orthogonal transformations, either by plane rotations (the method of Givens), each rotation annihilating an individual element, or by using unitary hermitians, I — 2wiwf, wfwt = 1 (the method of Householder), each of which annihilates ill possible elements in a column. Thus, at the first step, if A = A, and... 

Hermitian operators for electric and magnetic field intensities, 561 Herzfeld, C. M., 768 Hessenberg form, 73 Hessenberg method, 75 Heteroperiodic oscillation, 372 Hilbert space abstract, 426... 

Banachiewicz method, 67 characteristic roots, 67 characteristic vectors, 67 Cholesky method, 67 Danilevskii method, 74 deflation, 71 derogatory form, 73 "equations of motion, 418 Givens method, 75 Hessenberg form, 73 Hessenberg method, 75 Householder method, 75 Jacobi method, 71 Krylov method, 73 Lanczos form, 78 method of modification, 67 method of relaxation, 62 method of successive displacements,... 

The determination of eigenvalues and eigenvectors of the matrix A is based on a routine by Grad and Brebner (1968). The matrix is first scaled by a sequence of similarity transformations and then normalized to have the Euclidian norm equal to one. The matrix is reduced to an upper Hessenberg form by Householder s method. Then the QR double-step iterative process is performed on the Hessenberg matrix to compute the eigenvalues. The eigenvectors are obtained by inverse iteration. 

In the determination of eigenvalues, it is desirable to reduce matrix A to a super-triangular matrix of upper Hessenberg form ... 

Estimation of the shift factor Yj is relatively easy when the matrix has been reduced to upper Hessenberg form ... 

It may be shown that the DAE system corresponding to the discrete form of the compressible stagnation-flow equations is of the so-called Hessenberg-index-two structure [46], which is represented by Eq. 17.29. The constraints g do not depend on x, and the matrix... 

The matrix elements /i, y of the upper Hessenberg representation of L are thus automatically generated during the construction of the vectors vy. The essence of the short-iterative-Arnoldi propagator is to form an explicit representation of the exponential operator in the n-dimen-sional Krylov space based on the initial density matrix, cr t). 

The DAE form (2.22) is known as Hessenberg index-2 form. This form can be directly numerically solved by the DASKP 3.0 solver [43, 44] or the MEBDFI [45] solver available in Maple [46], Both solvers use the BDF method. ... 

Equations 4.25a, 4.25b are known as Hessenberg index-2 form. Codes suitable for numerical solution of such DAE systems are, e.g. DASPK 3.1 [19, 20] (based on the BDF method) and Radau5 [21] that uses an implicit Runge-Kutta method of order 5. 

If the engagement of a clutch is considered instantaneously and is modelled by means of a residual effort sink that provides a moment M as long as the plates of the clutch are in contact, then the resulting model is also of Hessenberg index 2 form. ... 

Since Avj span(Vj+i), and v,- G spanCVj+i) only if z < j +1, A hsis the important property that Aij = 0 for z > j +1. This kind of matrix, known as upper-Hessenberg, is usually much easier to manipulate than A because A is almost upper triangular. Thus, the philosophy behind the Krylov space based methods (KSM) is to transform the original linear system into the simpler form Ax = b where x = V x and b = Vjb, which can be solved more easily. The computational effort is usually dominated by the construction and orthogonalization of the transformation matrix Vp, which requires a matrix vector multiply at each iteration. 

The elementary similarity transformation to produce an upper Hessenberg matrix in formula form is as follows ... 

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