Numer. Math. (997) 76: 479 488 Numerische Mathematik c Springer-Verlag 997 Electronic Edition Exponential decay of C cubic splines vanishing at two symmetric points in each knot interval Sang Dong Kim,, Seokchan Kim 2, Department of Mathematics, Teachers College, Kyungpook National University, Taegu, Korea 2 Department of Mathematics, Natural Science College, Changwon National University, Changwon, Korea Received May 30, 994 / Revised version received March 3, 996 Summary. In the course of working on the preconditioning of cubic collocation at Gauss points, one has to deal with the exponential decay of certain cubic splines. Such results were obtained by Kim Parter in the case of uniform spacing. In this paper we extend these results to arbitrary grids state its application to the preconditioned cubic collocation method. Mathematics Subject Classification (99): 65N35. Introduction Let {t i } N i=0 be a partition for the unit interval I := [0, ], i.e., 0=t 0 <t < <t N <t N =. Let I i := (t i, t i ), h i := t i t i, let ti be the mid-point of I i, that is, ti 2N + =(t i +t i )/2,i=,2,N. Let {ξ i } i=0 be given by (.) ξ 2i = ti h i 2 θ i, ξ 2i = ti + h i 2 θ i, 0 <θ i <, i=,...,n, with ξ 0 =0, ξ 2N+ =.Note that ξ 2i ξ 2i are symmetric points in I i. If we choose θ i = 3, we have the local Legendre-Gauss[=:LG] points on I i. Consider C -cubic splines {ψ i } 2N i= for the partition {t i } N i=0, i.e., the cubic splines on I with interior knot sequence (t, t, t 2, t 2,...,t N,t N ), satisfying (.2) ψ i (ξ j )=δ ij, j =0,,,2N +. Partially supported by TGRC-KOSEF 95 BSRIP 95, Ministry of Education Partially supported by 96 KOSEF 96-006-04- Correspondence to: S.D. Kim page 479 of Numer. Math. (997) 76: 479 488
480 S.D. Kim, S. Kim 2N + The node sequence {ξ i } i=0 satisfies the Schoenberg-Whitney conditions (see [B], [B4], [SW]) with respect to the space S of C cubic splines for the given partition {t i } N i=0. Therefore, for each i =,...,2N, the condition (.2) determines a unique element of S. In Sect. 2, motivated by the ideas in [B2], [B3], [B4], [B5] we will give a proof that, regardless of the choice of the partition {t i }, these Lagrange splines ψ i decay exponentially away from the point ξ i. As an application of Sect. 2, we state in Sect. 3 a preconditioning result on cubic spline collocation method for nonuniform grid. In the statement of our theorem later, we use i := (i +)/2 for given integer i. Our general theorem on the exponential decay of the {ψ i } for an arbitrary grid can be stated as follows: Theorem. The functions {ψ i } defined by (.2) satisfy (.3) ψ i (t) C(min k θ k, max θ k ) k ( 5 ) i j for t I j, with 5(5 l 2 ) C (l, u) := l( u)(5 u 2 ). For the particular choice θ i = 3, (i =,2,,N), which gives the LG points, we have the stronger result (.4) ψ i (t) C(/ 3, / ( 3) 7 ) i j for t I j. This Theorem shows that ψ i satisfying (.2) with 0 < min k θ k θ i max k θ k < decays exponentially as j moves away from i. 2. Results on exponential decay. In this section, we show first exponential decay of the cubic Lagrange spline ψ i at the knots, second we prove that exponential decay at knots implies exponential decay of ψ i everywhere regardless of partition, following the ideas in [BB], [B2], [B3] [B4]. Throughout this section we use the notation f(x) :=(f(x),f (x)) t. Lemma. Let p be a cubic polynomial on [, ] vanishing at ±θ where 0 θ<. Then (2.) p() = D(θ)p( ) where D(θ) := ( d i,j (θ) ) is the positive matrix whose entries are page 480 of Numer. Math. (997) 76: 479 488
Exponential decay of C cubic splines 48 4 d (θ) =d 22 (θ) =+ θ 2, d 2 (θ) =2, d 2 (θ) = 4(3 θ2 ) ( θ 2 ) 2. Moreover the minimum value of d (θ) =d 22 (θ) is 5. Proof. Since p vanishes at θ θ, we have p(s) =(s θ)(s + θ)(as + b) p (s) =2s(as + b)+(s θ)(s + θ)a. Then we can represent a b in terms of p( ) p ( ) as follows: a = 2 ( θ 2 ) 2 p( ) + θ 2 p ( ), b = 3 θ2 ( θ 2 ) 2 p( ) + θ 2 p ( ),, from this, express p() in terms of p( ). By the linear change of variables t = t j +(h j /2)(s + ), ( s ), for any f S we have the following: Corollary. If f S vanishes at the two points t j ± hj 2 θ j where 0 θ j <, then (2.2) f(t j )=D(θ j,h j )f(t j ) where (2.3) D(θ j, h j ):=A(h j )D(θ j )A(h j ), A(h):= [ ] 0. 0 2/h Since a cubic polynomial p on I j is uniquely determined by the two vectors p(t j ) p(t j ), it follows that the space S contains a unique element, ϕ, with the property that ϕ vanishes at {ξ i } 2N i=0 ϕ (0) =. Furthermore, for any i {,...,2N}, (2.4) ψ i = ψ i (0)ϕ on [0, t i ]. Thus, the exponential decay of ψ i as one moves away from ξ i toward the left is proved once such decay is established for ϕ. Since ϕ(0)=(0,) t, the positivity of D(θ j, h j ) implies that ϕ(t j ) ϕ (t j ) are strictly positive for all j > 0. Therefore, by corollary, we have (2.5) ϕ(t j ) T (θ j )ϕ(t j ), ϕ (t j ) T(θ j )ϕ (t j ) page 48 of Numer. Math. (997) 76: 479 488
482 S.D. Kim, S. Kim with (2.6) T (θ) :=+ Therefore 0 ϕ (r) (t j ) 4 θ 2 5. i l=j + T (θ l) ϕ(r) (t i ), j < i, (r =0,). In particular, by (2.4), we have (2.7) ψ i (t i )ψ i (t i ) > 0, by (2.5), we have (2.8) ψ (r) i (t j ) i l=j+ T (θ l) ψ(r) i (t i ), j < i, (r =0,). The change of variables t ( t) in(2.7) (2.8) implies that (2.9) ψ i (t i )ψ i (t i ) < 0, (2.0) ψ (r) i (t j ) j l=i + T (θ l) ψ(r) i (t i ), j i, (r =0,). With these (2.8) (2.0), the exponential decay of ψ i following two estimates. follows from the () A bound for ψ i on I i. (2) A bound for p(s) on [, ] in terms of p( ) p() in case p is a cubic polynomial vanishing at ±θ. The linear change of variables s =(2/h i )(t t i ) converts ψ i on I i to a polynomial p, on[,], with the following properties: (2.) p( )p ( ) > 0, p( θ) =, p(θ)=0, p()p () < 0 for some point θ (, ). Thus, the following Lemma is relevant. Lemma 2. Let p be a cubic polynomial satisfying (2.). Then (2.2) max{ p( ), p() } Q( θ ), with (2.3) Q(u) := 2( + u) u(5 u 2 ). page 482 of Numer. Math. (997) 76: 479 488
Exponential decay of C cubic splines 483 Proof. θ>0. After a change of variables t t if necessary, we may assume that By the Budan-Fourier theorem (see, e.g., [BE]), the number of zeros (counting multiplicities) in (, ) of the cubic polynomial p is bounded by σ(p; ) σ(p; ), with σ(p; t) the number of strong sign changes in the sequence (p(t), p (t), p (t), p (t)). For our particular p, σ(p; ) 2 while σ(p;), hence σ(p; ) σ(p;) 2 =. This implies that p can have no zeros in (, ) other than the one at θ. Therefore < θ <θ< p( ) > 0 > p(). One readily computes that ( (2.4) p(t) =(t θ) at 2 + bt θ 2 a + θb ) 2θ for certain a b. Define ( (2.5) F(a, b) :=p( ) = ( + θ) ( θ 2 )a +(θ )b ). 2θ We are to extremize the affine function F(a, b) over all choices of (a, b) for which (2.6) p( ), p ( ), p(), p () 0. Let E := E E 2 E 3 E 4 where E := {(a, b) : p( ) 0} = {(a, b) : 2θ( θ 2 )a 2θ( θ)b 0}, E 2 := {(a, b) : p ( ) 0} = {(a, b) : 2θ(3 θ)(θ +)a 4θb 0}, E 3 := {(a, b) : p() 0} = {(a, b) : 2θ( θ 2 )a +2θ( + θ)b 0}, E 4 := {(a, b) : p () 0} = {(a, b) : 2θ(3 + θ)( θ)a +4θb 0}. Then the set E is nonempty bounded its corner points are given by ( ) ( ) K 3 := 2θ( θ 2 ), 0 3 θ, K 4 := 2θ( θ)(5 θ 2 ), θ(5 θ 2 ( ) ( ) 3+θ K 23 := 2θ( + θ)(5 θ 2 ), θ(5 θ 2, K 24 := ) 2θ(3 θ 2 ), 2(3 θ 2 ) where K ij denotes the corner point of the sets E i E j. page 483 of Numer. Math. (997) 76: 479 488
484 S.D. Kim, S. Kim Since p(±), p (±) are affine functions of (a, b), these restrict (a, b) tothe convex set E bounded by straight lines, hence the maximum of F(a, b) is taken on at a point where two of the four inequalities are equalities. Since we are maximizing p( ), such a point would involve having equality in two of the other three inequalities. The point p()=0=p () leads to p( )p ( ) < 0, hence p ( ) = 0 is one of the active constraints. The choice p() = 0 leads to while p () = 0 leads to p( ) Q(θ) =F K23, p( ) (2 θ2 )( + θ) 2θ(3 θ 2 ) which is less than Q(θ). Therefore we have (2.7) 0 p( ) Q(θ). By a similar argument we have (2.8) Q(θ) p() 0. = F(a, b) K24, Then (2.7) (2.8) complete the proof. By the linear change of variables, we have the following: Corollary 2. For ψ i on I i, we have (2.9) max{ ψ i (t i ), ψ i (t i ) } Q(θ i ), 0 <θ i <. Combining Corollary 2 with (2.8) (2.0), we have exponential decay of ψ i at the knots {t j }. From this, we get the exponential decay of ψ i (t) with the aid of the following lemma. Lemma 3. Let p be a cubic polynomial on [, ] vanishing at ±θ. Then (2.20) p(s) T(θ) max{ p( ), p() }, <θ<, 2 with T (θ) defined in (2.6). Proof. Using the Hermite cubic functions on [, ], we have 4p(s) = p( )(s ) 2 (s +2)+p()(s +) 2 (2 s) + p ( )(s + )(5 s 2 )+p ()(s )(s +) 2. By Lemma, we have 2p ( ) = T (θ)p( ) + p() 2p () = p( ) + T (θ)p(). page 484 of Numer. Math. (997) 76: 479 488
Exponential decay of C cubic splines 485 Then an easy calculation yields ] 8p(s) = p( ) [(s )(s 2 5) T (θ)(s + )(s ) 2 ] + p() [(s + )(5 s 2 )+T(θ)(s )(s +) 2. Since (s )(s 2 5) + (s + )(5 s 2 ) = 2(5 s 2 ) (s + )(s ) 2 + (s )(s +) 2 = 2( s 2 ) for s [, ], it follows that p(s) (0+2T(θ)) max{ p( ), p() }, 8 which proves (2.20) since T (θ) 5 (for θ [, ]). The linear change of variables gives: Corollary 3. For ψ j on I i j /= i, we have (2.2) ψ i (t) 2 T(θ j) max{ ψ i (t j ), ψ i (t j ) }, 0 θ j <. Now let us prove exponential decay of ψ i. For the convenience, define θ := min θ k, θ := max θ k. k k Theorem. The functions {ψ i } defined by (.2) satisfy (2.22) ( ) i j ψ i (t) C(θ,θ) for 5 t I j, with (2.23) 5(5 l 2 ) C (l, u) := l( u)(5 u 2 ). Proof. Case. For t I j (j < i ), we have ψ i (t) 2 T(θ j) max{ ψ i (t j ), ψ i (t j ) } by (2.2) 2 T (θ j ) i l=j + T (θ l) ψ i (t i ) by (2.8) 2 Q(θ i )T (θ j ) i l=j + T (θ by (2.9). l) Now let us estimate 2 Q(θ i )T (θ j ) in terms of θ θ. By(2.6) (2.3), page 485 of Numer. Math. (997) 76: 479 488
486 S.D. Kim, S. Kim 2 Q(θ i )T (θ (5 θ 2 ) j ) θ( θ)(5 θ 2 ) =: C (θ, θ). Since T (θ) 5, we have ( ) i j (2.24) ψ i (t) C(θ, θ), 5 where C (θ, θ) =5C (θ, θ). Case 2. For t I j (j i + ), a similar argument yields ( ) j i (2.25) ψ i (t) C(θ, θ) j l=i + T (θ j ) C (θ, θ). 5 Case 3. For t I i, an easy calculation shows (2.26) ψ i (t) Q(θ i ) 2( + θ) θ(5 θ 2 ) =: C 2(θ, θ). Since C 2 (θ, θ) C (θ, θ), (2.24 2.26) complete the proof. Remark. The proof of Theorem indicates that the cubic Lagrange splines satisfying (.2) with ( 0 <θ θ θ< ) decay exponentially by a factor at least /5 toward the boundary points 0. Further, the analysis given can be extended to provide sufficient conditions for the exponential decay of the cubic Lagrange splines satisfying (.2) when ξ 2i ξ 2i are arbitrary points in I i, (i =,...,N). We close this section by noting the stronger result for LG-points. Theorem 2. For the functions {ψ i } satisfying (.2) LG points, ψ i (t) C(/ 3, / ( 3) 7 ) i j for t I j. Proof. Let θ i =/ 3inT(θ i ) for all i =,2,,N. Then the modification of the proof of Theorem completes the proof. 3. Appendix The main goal in this section is to state briefly the result on the preconditioned cubic collocation method as an application of Sect. 2 for a nonuniform grid following [KP]. The interested reader should consult [KP]. Let Ω be the unit square. We assume that the grid is quasi-uniform, more precisely, satisfying (3.) γ := max i,j where γ is finite. Accordingly, Ω has a quasi-uniform grid. h i h j page 486 of Numer. Math. (997) 76: 479 488
Exponential decay of C cubic splines 487 Consider a uniformly elliptic, invertible operator given by (3.2) Au := u(x, y)+a (x,y)u x +a 2 (x,y)u y +a 0 (x,y)u,(x,y) Ω with the homogeneous Dirichlet boundary conditions. Let {Â N 2} be the cubic collocation matrix (with numerical weights) of A based on the cubic Lagrange spline basis the LG-points as collocation points. Consider the system of linear equations, (3.3) Â N 2U = F, which arises in the numerical solution of the problem Au = f corresponding to (3.2). The preconditioning of (3.3) proposed in [K] [KP] is given by (3.4) β N 2 where the matrix β N 2 is the stiffness matrix, based on the piecewise bilinear functions using the LG-points, which define subrectangles of Ω, of any symmetric positive definite operator of the form (3.5) Bv = v + bv in Ω with homogeneous Dirichlet boundary conditions. Applying the exponential decay of the cubic Lagrange spline following the same line of arguments developed for the uniform meshing in [KP], one can verify the following preconditioning result for the quasi-uniform grid as follows: Theorem. There are two positive constants D (γ) D 2 (γ), independent of N, depending only on γ, such that for all nonzero vectors U we have D (γ)( β N 2U, U ) l2 ( β N 2[ β N 2 ÂN 2U ], β N 2 ÂN 2U ) l 2 D 2 (γ)( β N 2U, U ) l2. This theorem asserts that one can use the conjugate gradient method with inner product given by β N 2 to solve (3.3) preconditioned by (3.4), corresponding to (3.2) on a quasiuniform grid (see [KP] for more detail). Acknowledgement. The authors deeply thank Prof. Carl de Boor Prof. Seymour V. Parter for their valuable suggestions. Moreover, the authors also thank anonymous referee who recommended this revision. References [BB] Birkhoff, G., de Boor, C. (964): Error bounds for spline interpolation. Jour. Math. Mech. 3, 827 835 [B] de Boor, C. (978): A Practical Guide to Splines. Applied Mathematical Sciences 27, Springer- Verlag [B2] de Boor, C. (974): Bounding the error in spline interpolation. SIAM Rev. 6 page 487 of Numer. Math. (997) 76: 479 488
488 S.D. Kim, S. Kim [B3] de Boor, C. (976): Odd-degree spline interpolation at a biinfinite knot sequence. In: Dodd, A., Eckmann, B. (eds.) Approximation Theory, Lecture Notes in Mathematics, Vol. 556, Springer-Verlag [B4] de Boor, C. (976): On cubic spline functions that vanish at all knots. Advances in Math. 20 [B5] de Boor, C. (976): Total Positivity of the Spline Collocation Matrix. Ind. University J. Math. 25 [BE] Borwein, P., Erdélyi, T. (995): Polynomials Polynomial Inequalities, Graduate Texts in Mathematics 27, Springer-Verlag [K] Kim, S.D. (993): Preconditioning Collocation Method by Finite Element Method. Ph.D. Thesis, University of Wisconsin, Madison, WI [KP] Kim, S.D., Parter, S.V. (995): Preconditioning Cubic Spline Collocation Discretizations of Elliptic Equations. Numer. Math. 72, 39 72 [SW] Schoenberg, J., Whitney, A. (953): On Pólya frequency functions, III: The Positivity of Translation Determinants with Application to The Interpolation Problem by Spline Curves. Trans. Amer. Math. Soc. 74, 246 259 This article was processed by the author using the LaT E X style file pljourm from Springer-Verlag. page 488 of Numer. Math. (997) 76: 479 488