By G. W. Stewart

During this follow-up to Afternotes on Numerical research (SIAM, 1996) the writer keeps to convey the immediacy of the study room to the broadcast web page. just like the unique undergraduate quantity, Afternotes is going to Graduate college is the results of the writer writing down his notes instantly after giving every one lecture; subsequently the afternotes are the results of a follow-up graduate path taught through Professor Stewart on the college of Maryland. The algorithms provided during this quantity require deeper mathematical knowing than these within the undergraduate e-book, and their implementations should not trivial. Stewart makes use of a clean presentation that's transparent and intuitive as he covers themes resembling discrete and non-stop approximation, linear and quadratic splines, eigensystems, and Krylov series tools. He concludes with lectures on classical iterative tools and nonlinear equations.

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We are going to show how to use the QR factorization to compute projections. 6. Let the columns of X G Vk be linearly independent, and let X = QR be the QR factorization of X. Then the columns of Q form a basis for the space X spanned by the columns of X. 5) 42 After-notes Goes to Graduate School The following result shows that these two matrices are related to the geometry of the space X. 5). Then To prove the first of the above results, note that if x G X then x = Qb for some b (because the columns of Q span X}.

4. The Gram-Schmidt algorithm can be used in both continuous and discrete spaces. Unfortunately, in the discrete case it is numerically unstable and can give vectors that are far from orthogonal. The modified version is better, but it too can produce nonorthogonal vectors. We will later give an algorithm for the discrete case that preserves orthogonality. Projections 5. 1 suggests that the best approximation in a subspace to a vector y will be the shadow cast by y at high noon on the subspace. Such shadows are called projections.

12. We will need the following useful fact about basic sequences of polynomials. Let pchPi) • • • be a basic sequence of polynomials, and let q be a polynomial of degree k. Then q can be written uniquely as a linear combination of PQ,PI, ... 14, where we showed how to expand a truncated power series in Chebyshev polynomials. The pi correspond to the Chebyshev polynomials, and q corresponds to the truncated power series. 14 depends only on the fact that the sequence of Chebyshev polynomials is basic.