Linear matrix condition numbers

Thus, there are infinite (weD- and ill-) matrix condition numbers for the same linear system that depend on the system formulation. Conversely, there is one single standard form for each linear system and a unique system conditioning. 

The solution of this linear system is completely wrong if it is not preventively written in its standard form the matrix condition number is 1467, while the system conditioning is 37.8. The incorrect solution is obtained with all the factorizations, not just the Gauss one. 

The matrix A is known as the preconditioner and has to be chosen such that the condition number of the transformed linear system is smaller than that of the original system. 

The condition number of a matrix A is intimately connected with the sensitivity of the solution of the linear system of equations A x = b. When solving this equation, the error in the solution can be magnified by an amount as large as cortd A) times the norm of the error in A and b due to the presence of the error in the data. 

The condition number of the Hessian matrix of the objective function is an important measure of difficulty in unconstrained optimization. By definition, the smallest a condition number can be is 1.0. A condition number of 105 is moderately large, 109 is large, and 1014 is extremely large. Recall that, if Newton s method is used to minimize a function/, the Newton search direction s is found by solving the linear equations... 

Because of large scale disparity the numerical solution to the locally linear problem at every iteration (Eq. 15.46) is highly sensitive to small errors. In other words very small variations in the trial solution y(m> or the Jacobian J cause very large variations in the correction vector Ay(m). From the linear algebra perspective, scale disparity can be measured by the condition number1 of the Jacobian matrix. As the condition number increases the... 

The new coefficient matrix is symmetric as M lA can be written as M 1/2AM 1/Z. Preconditioning aims to produce a more clustered eigenvalue structure for M A and/or lower condition number than for A to improve the relevant convergence ratio however, preconditioning also adds to the computational effort by requiring that a linear system involving M (namely, Mz = r) be solved at every step. Thus, it is essential for efficiency of the method that M be factored very rapidly in relation to the original A. This can be achieved, for example, if M is a sparse component of the dense A. Whereas the solution of an n X n dense linear system requires order of 3 operations, the work for sparse systems can be as low as order n.13-14... 

In cases where iterative methods are employed to solve large, sparse linear systems, both the efficiency and robustness of these methods can be significantly improved by use of preconditioners. A preconditioner of a matrix A is a matrix such that has a smaller condition number than A. The... 

In principle, this equation could be solved by directly inverting the matrix This, however, will only work if no linear dependences are valid and the system is, in a mathematical sense, well conditioned. The conditioning of the system is given by the condition number ... 

Condition (condition number) Product of the norms of a matrix and of its inverse condition of the coefficient matrix characterizes the sensitivity of the solution of the linear system to input errors. 

Thus, in that example the linear system is ill-conditioned (its coefficient matrix A has a large condition number) and is sensitive to the input errors even if the computation is performed with infinite precision. Systems like Eq. (1) are not very sensitive to the input errors if the condition number of A is not large (then the system is called well conditioned). The latter fact follows from the next perturbation theorem, which bounds the output errors depending on the perturbation of inputs (on the input error) and on cond(A). 

A linear system can be written in infinite equivalent forms in classical analysis, where round-off errors do not exist. This is also true for numerical analysis on computers, where round-off errors must be accounted for. In fact, it is possible to multiply or divide a row of the system by a nonzero coefficient (for common compilers by a power of 2 to preserve all the significant digits) without introducing any error into the system coefficients, on condition that no overflow or underflow conditions are achieved. All these systems are equivalent for the classical analysis and for the numerical computations on computers, but each of them has a different condition number for the coefficient matrix. 

Let us have verified the condition (8.2.29) at some special values of the Oj then, at least in some neighbourhood of the special values, there exists a subset of columns of G t that are linearly independent, in number Ly. = rankG. Rearranging appropriately the columns of G, thus the components of vector (8.2.31) on which the matrix operates, thus in fact the splitters j e S, if... 

Step 2 is repeated using a quadratic model (in case of several independent variables) or a polynomial model (one independent variable). The condition number of the normal matrix is checked. If it is not much larger than that of the linear model, finish. Otherwise proceed to step 4. 

When the f. spectral vectors of the F matrix are linearly independent, the condition number is 1, and increases as the F matrix becomes ill conditioned. In terms of bits of accuracy ... 

This result is the motivation behind the use of a preconditioner, a transformation of the linear system into a related one whose condition number is closer to 1. This reduces the number of iterations necessary for iterative methods to converge to a solution, and is common practice in the numerical solution of BVPs. Let us choose some nonsingular upper-triangular matrix Mi defining a coordinate transformation,... 

Note that if g is invertible, then G will be full rank. The rank of (ez Z) will thus determine the rank of (e 0) and the number of linearly independent scalars.104 The conditional joint scalar dissipation rate matrix is given by105... 

The basic idea is very simple In many scenarios the construction of an explicit kinetic model of a metabolic pathway is not necessary. For example, as detailed in Section IX, to determine under which conditions a steady state loses its stability, only a local linear approximation of the system at this respective state is needed, that is, we only need to know the eigenvalues of the associated Jacobian matrix. Similar, a large number of other dynamic properties, including control coefficients or time-scale analysis, are accessible solely based on a local linear description of the system. 

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