By Henri Paul Cartan

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Breakdowns aside, there are other problems with Bi-CG. A transposevector product is needed, which at the least will require additional programming and may not be possible at all. The performance of the algorithm can be erratic or even unstable with residuals increasing by several orders of magnitude from one iteration to the next. Moreover, the eﬀort in computing rˆ at each iteration is wasted in that rˆ makes no contribution to x. However, Bi-CG sometimes performs extremely well and the remaining algorithms in this section represent attempts to capture this good performance and damp the erratic behavior when it occurs.

Let Λ be a diagonal matrix with Λii = λi and let p be a polynomial. Prove that p(Λ) = maxi |p(λi )| where · is any induced matrix norm. 4. 3. 5. 2). 6) for the number of CG iterations required to reduce the A norm of the error by a factor of 10−3 and for the number of CG iterations required to reduce the residual by a factor of 10−3 . 6. For the matrix A in problem 5, assume that the cost of a matrix vector multiply is 4N ﬂoating-point multiplies. Estimate the number of ﬂoatingpoint operations reduce the A norm of the error by a factor of 10−3 using CG iteration.

The savings are slight if k is small relative to N , which is often the case for large problems, and the simple one-line MATLAB approach can be eﬃcient for such problems. A more serious problem with the implementation proposed in Algorithm gmresa is that the vectors vj may become nonorthogonal as a result of cancellation errors. 9), which depends on this orthogonality, will not hold and the residual and approximate solution could be inaccurate. A partial remedy is to replace the classical Gram–Schmidt orthogonalization in Algorithm gmresa with modiﬁed Gram–Schmidt orthogonalization.

### Differential Forms by Henri Paul Cartan

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