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34 Polynomial Approximation of Differential Equations We note that (2.6.3) has a computational cost proportional to n 2 , while the cost of (2.3.6) is only proportional to n. Generalizations to other bases can be also considered, but they are outside the aims of this book.
3 NUMERICAL INTEGRATION As we shall see in this chapter, orthogonal polynomials allow us to generate in a very elegant way high order integration formulas, known as Gauss formulas. Since the theory is based on the knowledge of the zeroes of the polynomials, the first part is devoted to the characterization of their main properties. 3.1 Zeroes of orthogonal polynomials It is well-known that a polynomial of degree n has at most n distinct complex zeroes. In general, very little can be said about real zeroes. An amazing theorem gives a precise and complete answer in the case of orthogonal polynomials. Theorem 3.1.1 - Let {un} n∈N be a sequence of solutions of (1.1.1), where un is a polynomial of degree n, satisfying the orthogonal relation (2.2.1). Then, for any n ≥ 1, un has exactly n real distinct zeroes in I. Proof - We first note that I un(x)w(x)dx = 0. Therefore un changes sign in I, hence it has at least one real zero ξ1 ∈ I. If n = 1 the proof is ended. If n > 1, then we can find another zero ξ2 ∈ I of un, with ξ2 = ξ1, since if un vanished only at ξ1, then the polynomial (x − ξ1)un would not change sign in I, which is in contradiction with the relation I (x − ξ1)un(x)w(x)dx = 0, obtained by orthogonality. In a similar fashion,
Page 7 and 8: PREFACE This book is devoted to the
Page 9 and 10: CONTENTS 1. Special families of pol
Page 11 and 12: Contents ix 7. Derivative Matrices
Page 13 and 14: 1 SPECIAL FAMILIES OF POLYNOMIALS A
Page 15 and 16: Special Families of Polynomials 3 n
Page 17 and 18: Special Families of Polynomials 5 B
Page 19 and 20: Special Families of Polynomials 7 1
Page 21 and 22: Special Families of Polynomials 9 (
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Page 33 and 34: 2 ORTHOGONALITY Inner products and
Page 35 and 36: Orthogonality 23 The function w is
Page 37 and 38: Orthogonality 25 As we promised, we
Page 39 and 40: Orthogonality 27 Let us now define
Page 41 and 42: Orthogonality 29 Therefore, one obt
Page 43 and 44: Orthogonality 31 From (2.3.8), we g
Page 45: Orthogonality 33 only mention the r
Page 49 and 50: Numerical Integration 37 In the cas
Page 51 and 52: Numerical Integration 39 (3.1.9) z
Page 53 and 54: Numerical Integration 41 We give th
Page 55 and 56: Numerical Integration 43 where 1
Page 57 and 58: Numerical Integration 45 (3.2.10)
Page 59 and 60: Numerical Integration 47 This means
Page 61 and 62: Numerical Integration 49 (3.4.2) w
Page 63 and 64: Numerical Integration 51 3.5 Gauss-
Page 65 and 66: Numerical Integration 53 The proof
Page 67 and 68: Numerical Integration 55 3.7 Clensh
Page 69 and 70: Numerical Integration 57 the first
Page 71 and 72: Numerical Integration 59 (3.8.14) p
Page 73 and 74: Numerical Integration 61 (3.9.5) p
Page 75: Numerical Integration 63 (3.10.5)
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100 Polynomial Approximation of Dif
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REFERENCES adams r.(1975), Sobolev
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References 295 canuto c., hussaini
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References 297 friedman a.(1959), F
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References 299 jackson d.(1930), Th
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References 301 pavoni d.(1988), Sin
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References 303 vainikko g.m.(1964),
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