By G. W. Stewart

During this follow-up to Afternotes on Numerical research (SIAM, 1996) the writer keeps to carry the immediacy of the school room to the broadcast web page. just like the unique undergraduate quantity, Afternotes is going to Graduate tuition is the results of the writer writing down his notes instantly after giving every one lecture; therefore 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 figuring out than these within the undergraduate ebook, and their implementations usually are not trivial. Stewart makes use of a clean presentation that's transparent and intuitive as he covers themes akin to discrete and non-stop approximation, linear and quadratic splines, eigensystems, and Krylov series equipment. He concludes with lectures on classical iterative equipment and nonlinear equations.

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We will consider an extremely simple one called economization of power series. 4. Approximation 25 Chebyshev polynomials 5. ) 6. To see that these functions are actually polynomials, we will show that they satisfy a simple recurrence. To get us started, note that po(t) = I and Pi(t) = t. Now let t = cos#, so that pk(t) — cosk9. 5) From this it is obvious that Ck is a polynomial of degree k and that its leading term is 2k~ltk. 7. 3). 8. Since Pk(t) = cos(kcos~lt)on [—1,1], the absolute value of Pk(t]cannot be greater than one on that interval.

Instead we will give enough for the applications to follow. 7. Two polynomials p and q are orthogonal on the interval [a, b] with respect to the weight function w(t) > 0 if The intervals may be improper. For example, either a or b or both may be infinite, in which case the weight function must decrease rapidly enough to make the resulting integrals meaningful. 8. 3) of the Chebyshev polynomials. 9. 6 we showed that Chebyshev polynomials satisfy the simple recurrence relation A recurrence like this allows one to generate values of a sequence of Chebyshev polynomials with the same order of work required to evaluate the very last one from its coefficients.

This problem is known as Chebyshev approximation, after the Russian mathematician who first considered it. It is also known as best uniform approximation. Our first order of business will be to establish a characterization of the solution. 2. 14 is prototypical. 1) shows that the maximum error is attained at three points and the errors at those points alternate in sign. It turns out that the general Chebyshev approximation of degree k will assume its maximum error at k + 2 points and the errors will alternate in sign.