By G. W. Stewart

During this follow-up to Afternotes on Numerical research (SIAM, 1996) the writer keeps to carry the immediacy of the study room to the broadcast web page. just like the unique undergraduate quantity, Afternotes is going to Graduate university is the results of the writer writing down his notes instantly after giving each one lecture; subsequently the afternotes are the results of a follow-up graduate direction taught by way of Professor Stewart on the college of Maryland. The algorithms provided during this quantity require deeper mathematical realizing than these within the undergraduate publication, and their implementations are usually not trivial. Stewart makes use of a clean presentation that's transparent and intuitive as he covers themes reminiscent of 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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**Example text**

5. The residual vector y — Xb is orthogonal to the column space of X. 1. Summary of best approximation in an inner-product space. 13. To derive the classical way, note that XTPx = XT. 7) by XT we obtain This is a A; x A; system of linear equations for b. They are called the normal equations. It is worth noting that the normal equations are really a statement that the residual y — Xb must be orthogonal to the column space of X, which is equivalent to saying that XT(y — Xb} = 0. Since the columns of X are linearly independent,thematrixXTXis positive definite.

11. 5): (Pythagorean equality) Now the second term in the last expression is independent of b. Consequently, we can minimize \\y — Xb\\ by minimizing ||Px"2/ ~~ ^^11- But Pxy is in the space spanned by the columns of X. Px2/ — Xb\\ — 0, which is as small as a norm can get. 7) is the unique solution of our approximation problem. 12. 7). There are two ways — one classical (due to Gauss and Legendre) and one modern. 44 Afternotes Goes to Graduate School Let V be an inner-product space. Let X G Vn have linearly independent columns, and let X = QR be the QR factorization of X.

Xk to do the approximating. We look for an approximation in the form The best approximation will minimize \\y — faxi —fax 2 — • • • — faxk\\, where i n9 I X\\* — T XX. 10. To bring matrix techniques into play, let Then we wish to determine b to minimize \\y — Xb\\. 11. 5): (Pythagorean equality) Now the second term in the last expression is independent of b. Consequently, we can minimize \\y — Xb\\ by minimizing ||Px"2/ ~~ ^^11- But Pxy is in the space spanned by the columns of X. Px2/ — Xb\\ — 0, which is as small as a norm can get.