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172 Polynomial Approximation of Differential Equations three well-structured matrices, so that the global cost of the computation ¯p → R −1 ¯p is proportional to n (see bertoluzza and funaro(1991)). Slight modifications to (8.4.4) lead to a suitable preconditioner for the matrix of system (7.4.16). Finite-difference preconditioning matrices for fourth-order problems have been in- vestigated in funaro and heinrichs(1990), both for the boundary-value problems (7.4.19) and (7.4.23). The analysis has been carried out in the Chebyshev case, but similar conclusions also apply to other situations. A little care has to be payed when dealing with the finite-difference schemes at the boundary points. Before preconditioning, the ratio between the largest and the smallest eigenvalues is proportional to n 8 . The preconditioned eigenvalues satisfy relation (8.4.2). In particular, in the Chebyshev case, we can take c = π4 16 ≈ 6.08 . For matrices related to problems defined in unbounded domains, we can use the same arguments. However, since it is not advisable to work with high degree Laguerre or Hermite polynomials (see sections 1.6 and 1.7), the matrices are not very large in practical applications. Therefore, we suggest direct methods to solve these systems. 8.5 Preconditioners for first-order operators Denote by D the (n + 1) × (n + 1) matrix corresponding to the system (7.4.1). Here, the idea of using finite-differences to construct a preconditioner for D does not offer the same features emphasized in section 8.4. For instance, let R := {rij} 0≤i≤n 0≤j≤n defined as follows: −1 ⎪⎨ (8.5.1) rij := ⎧ 1 if i = j = 0 , h (n) i if 1 ≤ i = j + 1 ≤ n, 1 ⎪⎩ h (n) if 1 ≤ i = j ≤ n, i 0 elsewhere. be
Eigenvalue Analysis 173 Now, we have a bidiagonal preconditioner. Though the spread of eigenvalues of the preconditioned matrix is reduced, we are still far from the results of the previous section. The eigenvalues of R −1 D are complex with positive real part, but they approach the imaginary axis with increasing n. A more effective preconditioner was proposed in funaro(1987). First introduce the linear operator T : R n → R n . For any polynomial p ∈ Pn−1, the operator T maps the vector {p(ξ (n) j )}1≤j≤n into the vector {p(η (n) i )}1≤i≤n. The way to perform this transformation is described by formula (4.4.1). The coefficients l (n) j (η (n) i ), 1 ≤ i ≤ n, 1 ≤ j ≤ n, are the entries of the n × n matrix relative to the mapping T. Then, we define Z := {zij} 0≤i≤n , to be the (n + 1) × (n + 1) matrix 0≤j≤n ⎧ 1 ⎪⎨ if i = j = 0, (8.5.2) zij := ⎪⎩ 0 elsewhere. l (n) j (η (n) i ) if 1 ≤ i ≤ n, 1 ≤ j ≤ n, We claim that ZR is a good preconditioning matrix for D. We remark that Z is a full matrix. On the other hand, we can find explicitly the entries of Z −1 . Indeed, one easy checks the relation (8.5.3) p(ξ (n) i ) = n j=1 which is true for any p ∈ Pn−1. p(η (n) j ) 1 + η(n) j 1 + ξ (n) i ˜ l (n) j (ξ (n) i ), 1 ≤ i ≤ n, Clearly, (8.5.3) is the inverse of the transformation T. Noting that (ZR) −1 = R −1 Z −1 , we can evaluate ¯p → (ZR) −1 ¯p with a cost proportional to n 2 . The computing time is further reduced, in the Chebyshev case, by using the FFT (see section 4.4). Let us examine the eigenvalues Λn,m, 0 ≤ m ≤ n, of (ZR) −1 D. We have a precise answer in the Chebyshev case. Actually, one has (8.5.4) Λn,0 = 1, Λn,m = m Therefore, we get 1 ≤ Λn,m < π 2 , 0 ≤ m ≤ n. π sin 2n sin mπ , 1 ≤ m ≤ n. 2n
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PREFACE This book is devoted to the
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CONTENTS 1. Special families of pol
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Contents ix 7. Derivative Matrices
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1 SPECIAL FAMILIES OF POLYNOMIALS A
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Special Families of Polynomials 3 n
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Special Families of Polynomials 5 B
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Special Families of Polynomials 7 1
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Special Families of Polynomials 9 (
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Special Families of Polynomials 11
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2 ORTHOGONALITY Inner products and
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Orthogonality 23 The function w is
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Orthogonality 25 As we promised, we
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Orthogonality 27 Let us now define
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Orthogonality 29 Therefore, one obt
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Orthogonality 31 From (2.3.8), we g
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Orthogonality 33 only mention the r
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3 NUMERICAL INTEGRATION As we shall
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Numerical Integration 37 In the cas
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Numerical Integration 39 (3.1.9) z
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Numerical Integration 41 We give th
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Numerical Integration 43 where 1
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Numerical Integration 45 (3.2.10)
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Numerical Integration 47 This means
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Numerical Integration 49 (3.4.2) w
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Numerical Integration 51 3.5 Gauss-
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Numerical Integration 53 The proof
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Numerical Integration 55 3.7 Clensh
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Numerical Integration 57 the first
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Numerical Integration 59 (3.8.14) p
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Numerical Integration 61 (3.9.5) p
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Numerical Integration 63 (3.10.5)
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66 Polynomial Approximation of Diff
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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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