By James M. Ortega and Werner Rheinboldt (Auth.)
Addresses many of the simple questions in numerical research: convergence theorems for iterative equipment for either linear and nonlinear equations; discretization mistakes, particularly for usual differential equations; rounding mistakes research; sensitivity of eigenvalues; and suggestions of linear equations with admire to alterations within the info
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Additional resources for Numerical Analysis. A Second Course
For if the data of the problem are measured quantities, or otherwise known to only a limited precision, then even an exact solution to the problem (exact with respect to numerical analysis error) is quite likely to be meaningless. It may be much more important simply to detect the instability so that perhaps the problem can be reformulated into a more stable form. Second, there is usually no precise demarcation between stable and unstable problems but, rather, a continuum of possibilities. Some appropriate measure of this instability will usually enter into the error estimates for all three of the fundamental errors—discretization, convergence, and rounding.
Then y is a left eigenvector of A (and in this context χ is a right eigenvector). Assume that ||x|| 2 = ||y|| 2 = 1. T 1 Then I y x | ~ is the condition number of λ. T Since we may take χ as the ith column of Ρ and a scalar multiple of y 1 λ as the ith row of P " , where Ρ is the matrix such that Ρ~ ΑΡ is the Jordan X form of A with λ in the ith diagonal position, it follows from P~ P = / T T H that y x Φ 0. Note that it is y and not y which is used, even if y is complex. 6). 3 as Ε - • 0. 9 Assume that A eL(C ) is similar to a diagonal matrix and let λ be n a simple eigenvalue with left and right eigenvectors y and x.
9 Then | | A - / i | | 2 < | | y | | 2 . 9 τ μ„) , γ = (γί9 T . . , y„) . 3 SPECIAL RESULTS FOR SYMMETRIC MATRICES 59 For the next result, recall that the convex hull of a set of vectors is the smallest convex set which contains them. 4, λ lies in the convex hull of the set of vectors of the form μ + Py9 where Ρ runs over all possible permutation matrices. 3. Suppose that η = 2 and μί = 1, μ2 = 2, y l = —ε, y 2 = 2ε. 3 λ1 - 11 < 2ε, \λ2 - 2 | <2ε. 1). 5 gives a much more precise location of the eigenvalues of A .
Numerical Analysis. A Second Course by James M. Ortega and Werner Rheinboldt (Auth.)