Matrix spectral radius

Sometimes in the literature the spectral radius p is referred to as the spectral norm, but it is not a norm as the term is used here. Thus it is easy to construct a matrix A 0 for which p(A) = 0, so that Mx is violated. However, for any given matrix A, and any e > 0, there exists a norm v such that... 

It is easily shown that the ground state energy of H equals - X, where X is the spectral radius of the matrix A. Therefore (50) gives the ground state with a minimal value of total spin. It is of interest, because the first-PT-order terms for such lattices lead usually to the ferromagnetic ground state. 

The traditional statement in stability theory that all of the eigenvalues have negative real part becomes 5( A) < 0. The spectral radius of a matrix, denoted u A), is defined by... 

The set of eigenvalues is called the spectrum and is denoted as A (A). The maximum eigenvalue of matrix A is called the spectral radius of A ... 

The spectral radius of a square matrix is the absolute value of its largest eigenvalue. 

We can get some handle on this value by noting that any norm is an upper bound on the spectral radius of a matrix. If, during the tending of to to oo, it is found that the matrix given by one of the product sequences has a norm equal to its spectral radius, then that will be the joint spectral radius of the A and B. 

The loo norm of such a matrix is much cheaper to compute than the eigenvalues, and as to increases the nth root of the lowest norm so far of the product sequences converges to the joint spectral radius, as does (by definition) the nth root of the largest eigenvalue so far. 

In particular, the joint spectral radius of any matrix with itself is just the spectral radius of that matrix. 

In the above proposition, p(A) is the spectral radius of matrix A and it is equal to the largest absolute value of.d s eigenvalues. 

The stability of the method therefore depends solely on the spectral radius of the matrix X. Denoting rj = then we compute... 

The relaxation factor o>, by virtue of its definition, satisfies 1 a> < 2, and while it has been first demonstrated by Young [52] that overestimating 0)6 is not as detrimental as underestimating o), it has nevertheless been a difficult problem, from a numerical point of view, to estimate o) [33 55]. Clearly, estimating o) is equivalent, from (4.15), to estimating p[M], Now, we can make use of the fact that the matrix M is non-negative and irreducible, and we express its spectral radius /l[Jf] as a minimax, in the following way. If is a vector with positive components Xi, and Mx = y, then [49]... 

The above considerations have also been used to construct a simple method for the iterative solution of the linear block system (2.9). Let A denote the Jacobian approximation in (2.9) and A the associated matrix with Qy replaced by 0. Then the system Ax = h can be solved easily because of its nearly upper triangular block structure. On this basis, we constructed a fixed point iteration which is known to converge with contraction rate not greater than the spectral radius p(I — A A), Obviously, since y = 0 in the starting point xo, we will have p < 1 in some neighborhood, which can be monitored. 

A proof, relying upon the concepts of matrix norm and spectral radius introduced in the next section, is provided in the supplemental material in the accompanying website. Figure 3.3 demonstrates the application of Gershgorin s theorem to a 3 x 3 matrix. test-Gershgorin.m generates this plot for any input matrix. 

Matrix norm, spectral radius, and condition number... 

The optimum over-relaxation parameter is known only for a small class of linear problems and for select boundary conditions. The iteration matrix has eigenvalues each one of which reflects the factor by which the amplitude of an eigenmode of undesired residual is suppressed for each iterative step. Obviously the modulus of all these modes must be less than 1. The modulus of the factor with the largest amplitude is called the spectral radius and determines the overall long term convergence of the procedure for many iterative steps. If Pj is the spectral radius of the Jacobi iteration then the optimum value of X is known to be ... 

To discuss these more practical iterative matrix solutions, it is convenient to have a problem for which both the exact solution is known and the Jacobi spectral radius is known. One such problem is the following form of Poisson s equation ... 

A few other bookkeeping sections in the code convert the input solution array from an x-y labeled two index matrix into a single column array if needed on lines 454 through 459 and a final loop converts the single column solution array into a j,i two index matrix of solution values on lines 486 through 488. The type of solution desired may be specified as a table of values in order to pass additional parameters to the PDE solver. If a table is specified for the typsola parameter, two additional parameters, rx and ry, can be extracted from the input list as seen on line 444. For a direct sparse matrix solution with the SPM parameter these are not used. For the COE and SOR methods, this feature can be used to specify a spectral radius value with only the rx parameter used. For the ADI method these two values can be used to specify wx and wy parameters. In all cases, default values of these parameters are available, if values are not input to the various functions. 

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