By Nicholas J. Higham
A therapy of the behaviour of numerical algorithms in finite precision mathematics that mixes algorithmic derivations, perturbation idea, and rounding errors research. software program practicalities are emphasised all through, with specific connection with LAPACK and MATLAB.
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This is often the second one quantity in a projected five-volume survey of numerical linear algebra and matrix algorithms. It treats the numerical resolution of dense and large-scale eigenvalue issues of an emphasis on algorithms and the theoretical heritage required to appreciate them. The notes and reference sections include tips to different equipment in addition to historic reviews.
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Extra info for Accuracy and Stability of Numerical Algorithms
Strassen's method for fast matrix multiplication provides another example of the unpredictable relation between the number of arithmetic operations and the error. 2. 12. Instability Without Cancellation It is tempting to assume that calculations free from subtractive cancellation must be accurate and stable, especially if they involve only a small number of operations. The three examples in this section show the fallacy of this assumption. 1. 1. The Need for Pivoting Suppose we wish to compute an LU factorization Clearly, u11 — e, u12 = —1, 121 = e -1, and U22 = 1 — l21u12 = 1 + e -1.
8. 5. Use only well-conditioned transformations of the problem. In matrix computations this amounts to multiplying by orthogonal matrices instead of nonorthogonal, and possibly, ill-conditioned matrices, where possible. 2 for a simple explanation of this advice in terms of norms. 6. 8). Concerning the second point, good advice is to look at the numbers generated during a computation. This was common practice in the early days of electronic computing. On some machines it was unavoidable because the contents of the store were displayed on lights or monitor tubes!
2). 3. Look for different formulations of a computation that are mathematically but not numerically equivalent. 8). 3). 4. It is advantageous to express update formulae as new-value = old_value + small_correction if the small correction can be computed with many correct significant figures. Numerical methods are often naturally expressed in this form; examples include methods for solving ordinary differential equations, where the correction is proportional to a stepsize, and Newton's method for solving a nonlinear system.
Accuracy and Stability of Numerical Algorithms by Nicholas J. Higham