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Additional info for Periodic solutions of nonlinear wave equations with general nonlinearities
All the eigenvalues of a 50-by-50 matrix can be computed for less than a 1970 dollar. The situation is rather different for very large matrices (n > 10,000). Roundoff error does not dominate the accuracy of the results, and the iterative methods in current use sometimes converge quite slowly. There is a high premium on those methods which are capable of delivering the desired accuracy and no more! Here again it often happens that if 20 eigenvalues are wanted, then the first 15 will be correct to working precision by the iterative step at which the twentieth is accurate to four significant decimals.
That is the gist of the next fact. 6 (Sylvester's inertia theorem). Each A is congruent to a matrix diag^, — 11,, O^), where the number triple (TT, z/, £) depends only on A and is called A's inertia. Moreover TT, v, £ are the number of positive, negative, and zero eigenvalues of A. 6 shows that two self-adjoint matrices are congruent if and only if they have the same inertia. 9. It is overkill to derive the inertia theorem from the spectral theorem. Among quadratic forms are those which are strictly positive for all nonzero vectors: i^(x) > 0 if x 7^ o.
A thorough and elementary presentation of error analysis is given in [Wilkinson, 1964], and so we hope that the reader will be content with an informal discussion of a few important issues. 5. Cancellation Genuine subtraction produces the difference of two numbers with the same sign, say plus. There is a widespread misconception that subtraction of numbers which are very close is an inherently inaccurate process in which the relative error is greatly enhanced. 5 As with most misconceptions there is a grain of truth in the saying, but the simple fact is that the error in the subtraction of normalized floating point numbers with the same exponent is zero.