Matrix Hessenberg

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. 

These zeros uk of QK(u) coincide with the eigenvalues of both the evolution matrix U and the corresponding Hessenberg matrix H from Eqs. (131) and (130), respectively. The zeros of Qk(u) are called eigenzeros. The structure of CM is determined by its scalar product for analytic functions of complex variable z or u. For any two regular functions/(m) and g(u) from CM, the scalar product in CM is defined by the generalized Stieltjes integral ... 

Note also that for a generic linear generator A such as that of CODDE (Sec. 2.3), the Lanczos or Arnoldi algorithm results in an upper Hessenberg matrix such that [45, 46, 52]... 

A supertriangular matrix, also called a Hessenberg matrix, is one that ha.s all zero elements below die subdiagonal, such as the upper Hessenberg matrix of Eq. (2.57) ... 

The complete elementary similarity transformation that converts matrix A to the upper Hessenberg matrix H is shown by... 

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

The QR algorithm, which will be discussed next, utilizes the upper Hessenberg matrix H to determine its eigenvalues, which are equivalent to the eigenvalues of matrix A. 

Repeat steps 4 and 5, calculating the transformation matrix and eliminating one element on subdiagonal in each set of steps. The application of steps 4 and 5 for (n - 1) times, with the counter q increasing from 1 to ( -1) and the counter p set at q + 1), will convert the Hessenberg matrix //, to a triangular matrix Ry... 

Use as the new Hessenberg matrix and repeat steps 2-8 until < e, where e... 

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... 

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). 

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. 

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 ... 

---