Table of Contents |
| List of Contributors | xi |
1. | Preface | 1 |
pt. I | Walter Gautschi | |
2. | Biography of Walter Gautschi / Ahmed Sameh | 5 |
3. | A Brief Summary of My Scientific Work and Highlights of My Career / Walter Gautschi | 9 |
4. | Publications / Walter Gautschi | 19 |
pt. II | Commentaries | |
5. | Numerical Conditioning / Nicholas J. Higham | 37 |
5.1. | Conditioning of Vandermonde Matrices | 37 |
5.2. | Conditioning of Polynomials | 39 |
6. | Special Functions / Javier Segura | 41 |
6.1. | Computation of Special Functions | 42 |
6.1.1. | Exponential Integrals, Incomplete Gamma Functions, and the Error Function | 42 |
6.1.2. | Computing Special Functions by Gaussian Quadrature | 44 |
6.2. | Inequalities | 45 |
6.2.1. | Orthogonal Polynomials and Their Zeros | 45 |
6.2.2. | Gamma Functions | 46 |
7. | Interpolation and Approximation / Miodrag M. Spalevic | 49 |
7.1. | Attenuation Factors in Practical Fourier Analysis | 49 |
7.2. | Pade Approximants Associated with Hamburger Series | 50 |
7.3. | Convergence Behavior of Continued Fractions with Real Elements | 51 |
7.4. | Moment-preserving Spline Approximation | 52 |
7.5. | Convergence of Extended Lagrange Interpolation | 53 |
7.6. | Experimental Mathematics Involving Orthogonal Polynomials | 53 |
7.6.1. | Jacobi Polynomials | 53 |
7.6.2. | Quadrature Formulae | 54 |
7.7. | Exotic Weight Functions | 54 |
pt. III | Reprints | |
8. | Papers on Numerical Conditioning | 59 |
8.1. | [16] On Inverses of Vandermonde and Confluent Vandermonde Matrices, Numer. Math. 4, 117--123 (1962) | 60 |
8.2. | [19] On Inverses of Vandermonde and Confluent Vandermonde Matrices. II, Numer. Math. 5, 425--430 (1963) | 68 |
8.3. | [43] The Condition of Orthogonal Polynomials, Math. Comp. 26, 923--924 (1972) | 75 |
8.4. | [45] On the Condition of Algebraic Equations, Numer. Math. 21, 405--424 (1973) | 78 |
8.5. | [51] Norm Estimates for Inverses of Vandermonde Matrices, Numer. Math. 23, 337--347 (1975) | 99 |
8.6. | [62] On Inverses of Vandermonde and Confluent Vandermonde Matrices III, Numer. Math. 29, 445--450 (1978) | 111 |
8.7. | [64] Questions of Numerical Condition Related to Polynomials, in Symposium on Recent Advances in Numerical Analysis (C. de Boor and G. H. Golub, eds.), 45--72 (1978) [Revised and reprinted in MAA Studies in Mathematics 24: Studies in Numerical Analysis (G. H. Golub, ed.), 140--177, Math. Assoc. America, Washington, DC, [ect.] | 118 |
8.8. | [66] The Condition of Polynomials in Power Form, Math. Comp. 33, 343--352 (1979) | 157 |
8.9. | [83] The Condition of Vandermonde-like Matrices Involving Orthogonal Polynomials, Linear Algebra Appl. 52/53, 293--300 (1983) | 168 |
8.10. | [110] (with G. Inglese) Lower Bounds for the Condition Number of Vandermonde Matrices, Numer. Math. 52, 241--250 (1988) | 177 |
8.11. | [118] How (Un)stable Are Vandermonde Systems?, in Asymptotic and Computational Analysis (R. Wong, ed.), 193--210, Lecture Notes Pure Appl. Math. 124 (1990) | 188 |
8.12. | [120] (with A. Cordova and S. Ruscheweyh) Vandermonde Matrices on the Circle: Spectral Properties and Conditioning, Numer. Math. 57, 577--591 (1990) | 207 |
8.13. | [200] Optimally Scaled and Optimally Conditioned Vandermonde and Vandermonde-like Matrices, BIT Numer. Math. 51, 103--125 (2011) | 223 |
9. | Papers on Special Functions | 247 |
9.1. | [9] Some Elementary Inequalities Relating to the Gamma and Incomplete Gamma Function, J. Math. and Phys. 38, 77--81 (1959) | 249 |
9.2. | [10] Exponential Integral [∫][∞]1 e-xt t-n dt for Large Values of n, J. Res. Nat. Bur. Standards 62, 123--125 (1959) | 255 |
9.3. | [13] Recursive Computation of the Repeated Integrals of the Error Function, Math. Comp. 15, 227--232 (1961) | 259 |
9.4. | [39] Efficient Computation of the Complex Error Function, SIAM J. Numer. Anal. 7, 187--198 (1970) | 266 |
9.5. | [47] A Harmonic Mean Inequality for the Gamma Function, SIAM J. Math. Anal. 5, 278--281 (1974) | 279 |
9.6. | [48] Some Mean Value Inequalities for the Gamma Function, SIAM J. Math. Anal. 5, 282--292 (1974) | 284 |
9.7. | [49] Computational Methods in Special Functions --- A Survey, in Theory and Applications of Special Functions (R. A. Askey, ed.), 1--98, Math. Res. Center, Univ. Wisconsin Publ. 35 (1975) | 296 |
9.8. | [61] Anomalous Convergence of a Continued Fraction for Ratios of Kummer Functions, Math. Comp. 31, 994--999 (1977) | 395 |
9.9. | [68] A Computational Procedure for Incomplete Gamma Functions, ACM Trans. Math. Software 5, 466--481 (1979) | 402 |
9.10. | [72] (with F. Costabile) Lower Bounds for the Largest Zeros of Orthogonal Polynomials, Boll. Un. Mat. Ital. (5) 17A, 516--522 (1980) (translated from Italian) | 419 |
9.11. | [155] The Incomplete Gamma Functions Since Tricomi, in Tricomi's Ideas and Contemporary Applied Mathematics, 203--237, Atti Convegni Lincei 147 (1998) | 428 |
9.12. | [168] Gauss Quadrature Approximations to Hypergeometric and Confluent Hypergeometric Functions, J. Comput. Appl. Math. 139, 173--187 (2002) | 464 |
9.13. | [169] Computation of Bessel and Airy Functions and of Related Gaussian Quadrature Formulae, BIT 42, 110--118 (2002) | 480 |
9.14. | [178] Numerical Quadrature Computation of the Macdonald Function for Complex Orders, BIT Numer. Math. 45, 593--603 (2005) | 490 |
9.15. | [182] (with P. Leopardi) Conjectured Inequalities for Jacobi Polynomials and Their Largest Zeros, Numer. Algorithms 45, 217--230 (2007) | 502 |
9.16. | [190] On a Conjectured Inequality for the Largest Zero of Jacobi Polynomials, Numer. Algorithms 49, 195--198 (2008) | 517 |
9.17. | [191] On Conjectured Inequalities for Zeros of Jacobi Polynomials, Numer. Algorithms 50, 93--96 (2009) | 522 |
9.18. | [192] New Conjectured Inequalities for Zeros of Jacobi Polynomials, Numer. Algorithms 50, 293--296 (2009) | 527 |
9.19. | [193] How Sharp is Bernstein's Inequality for Jacobi Polynomials?, Electr. Trans. Numer. Anal. 36, 1--8 (2009) | 532 |
9.20. | [199] The Lambert W-functions and Some of Their Integrals: a Case Study of High-precision Computation, Numer. Algorithms 57, 27--34 (2011) | 541 |
9.21. | [203] Remark on "New Conjectured Inequalities for Zeros of Jacobi Polynomials by Walter Gautschi, Numer. Algorithms 50: 293--296 (2009)", Numer. Algorithms 57, 511 (2011) | 550 |
10. | Papers on Interpolation and Approximation | 553 |
10.1. | [41] Attenuation Factors in Practical Fourier Analysis, Numer. Math. 18, 373--400 (1972) | 554 |
10.2. | [86] On Pade Approximants Associated with Hamburger Series, Calcolo 20, 111--127 (1983) | 583 |
10.3. | [87] On the Convergence Behavior of Continued Fractions with Real Elements, Math. Comp. 40, 337--342 (1983) | 601 |
10.4. | [89] Discrete Approximations to Spherically Symmetric Distributions, Numer. Math. 44, 53--60 (1984) | 608 |
10.5. | [100] (with G. V. Milovanovic) Spline Approximations to Spherically Symmetric Distributions, Numer. Math. 49, 111--121 (1986) | 617 |
10.6. | [102] (with M. Frontini and G. V. Milovanovic) Moment-preserving Spline Approximation on Finite Intervals, Numer. Math. 50, 503--518 (1987) | 629 |
10.7. | [132] On Mean Convergence of Extended Lagrange Interpolation, J. Comput. Appl. Math. 43, 19--35 (1992) | 646 |
10.8. | [147] (with S. Li) On Quadrature Convergence of Extended Lagrange Interpolation, Math. Comp. 65, 1249--1256 (1996) | 664 |
10.9. | [165] Remark: "Barycentric Formulae for Cardinal (SINC-) Interpolants" by Jean-Paul Berrut, Numer. Math. 87, 791--792 (2001) | 673 |
10.10. | [202] Experimental Mathematics Involving Orthogonal Polynomials, in Approximation and Computation --- In Honor of Gradimir V. Milovanovic (W. Gautschi, G. Mastroianni, and Th. M. Rassias, eds.), 117--134, Springer Optim. Appl. 42 (2011) | 676 |