By Lisa Lorentzen (auth.), Claude Brezinski, Ahmed Sameh (eds.)
Walter Gautschi has written generally on issues starting from distinct services, quadrature and orthogonal polynomials to distinction and differential equations, software program implementations, and the heritage of arithmetic. he's international popular for his pioneering paintings in numerical research and confident orthogonal polynomials, together with a definitive textbook within the former, and a monograph within the latter region.
This three-volume set, Walter Gautschi: chosen Works with Commentaries, is a compilation of Gautschi’s such a lot influential papers and contains commentaries by means of major specialists. The paintings starts off with a close biographical part and ends with a bit commemorating Walter’s upfront deceased dual brother. This name will attract graduate scholars and researchers in numerical research, in addition to to historians of science.
Selected Works with Commentaries, Vol. 1
Numerical Conditioning
Special Functions
Interpolation and Approximation
Selected Works with Commentaries, Vol. 2
Orthogonal Polynomials at the actual Line
Orthogonal Polynomials at the Semicircle
Chebyshev Quadrature
Kronrod and different Quadratures
Gauss-type Quadrature
Selected Works with Commentaries, Vol. 3
Linear distinction Equations
Ordinary Differential Equations
Software
History and Biography
Miscellanea
Works of Werner Gautschi
Read or Download Walter Gautschi, Volume 3: Selected Works with Commentaries PDF
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Additional info for Walter Gautschi, Volume 3: Selected Works with Commentaries
Sample text
This is further discussed at the end of this paragraph. 6 44 37 THREE-TERM RECURRENCE RELATIONS tained recursively. 2) we now have So 1 L Amfm = T1 (s 00 =T JO m=l JO Aofo), and so s fo =Ao + So · This gives us the initial value of the desired solution. The remaining values ean now be obtained immediately from n = 1, 2, · · · , N. The actual algorithm follows this procedure very closely, except that for the infinite continued fraction, and the infinite series, representing rn-1 and sn, respectively, we now substitute truncated continued fractions, and truncated series.
9). Let g,. : gn = 0. Clearly, for some constants a<•>, b<•>. +l = 0' a<•>f. + b(v) g. = l. f. J /\mOm- • v-+OO Ov+l m=O We have proved the following theorem. 1. 2) holds. 1). 16) is satisfied. , if the X's are uniformly bounded, and t Ov+l - - 7 1, g. 14). 1, in this case, has been noted previously in [16]. , v ---* co' where r,. , s,. 3). 4) and ( 3. 7). The second follows by induction on n. 9), So <•> = S - 'Aofo <•> f 0 <• l ---* S - ~ 'Aofo JO = v ---* co. ) - 'An ---* - - An = Sn v ---* co.
Is faster the better p. approximates r. , and cr. , is clearly vindicated. 16). 19). 2p) 8 ~ 0, where pis some real number. "21]P, L m-n+l 0 ~ n < v. We obtain . 9p) r (v) s. (v) - - ' 0 8 [ = 1 + Sn(v)) (v) )P(' (v) 1\n = [ fn-1 Sn-1 = ' (v) fo 0 + so<•> ] 1/p ' f n (v) - (,) Tn-1 n = v, v - 1, · · ·, 1, 7 J(") n-1, n = 1, 2, · · ·, N. The nonuniqueness off,. 9p) converges as v ~ «> if i = 1, 2, ... 9). The effectiveness of the algorithm is clearly enhanced if good estimates of the initial value of v are available.