A CLASS OF EVEN DEGREE SPLINES OBTAINED THROUGH A MINIMUM CONDITION

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1 STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Number 3, September 2003 A CLASS OF EVEN DEGREE SPLINES OBTAINED THROUGH A MINIMUM CONDITION GH. MICULA, E. SANTI, AND M. G. CIMORONI Dedicated to Professor George Micula at is 60 t anniversary Abstract. A class of splines minimizing a special functional is investigated. Tis class is determined by te solution of quadratic programming problem. Convergence results and some numerical examples are given.. Introduction Te construction of splines, verifying minimum conditions as been proposed among oters in [2], [5], [6]. In suc papers te splines are interpolating te approximated function in te nodes and wile in [5] te constructive metod can be applied, in teory, for a spline of an arbitrary degree m, minimizing te integral I [g (x)] 2 dx, in [2] e [6] a cubic polynomial interpolating splines are considered satisfying some minimum conditions. In particular, in [6], te considered splines ave been applied for constructing quadrature sums approximating te Caucy principal value integrals I(wf; t) = w(x) f(x) dx. (.) x t In tis paper, utilizing te metod proposed in [2], we construct te spline of even degree minimizing te functional F (f) := [f (3) (x)] 2 dx f W2 3 (I) (.2) I were, denoting AC(I) te set of absolute continuous functions on I, { } W2 3 (I) := f : I IR, f (0) AC(I) and f (3) L 2 (I). (.3) Tis class of splines, called interpolating-derivative splines of degree 2m, m 2, as been determined in [3] by solving a linear system of m + n + equations, were n is te number of internal knots of te partition, and ten te autors proved tat te constructed spline solves problem (.2). Te convergence is proved by supposing f W m+ 2. Received by te editors:

2 GH. MICULA, E. SANTI, AND M. G. CIMORONI In tis paper, exploiting te different form tat we use for defining te interpolating - derivative spline, we can obtain convergence results under weaker conditions on f, tat gives more flexibility in te applications, as for example, wen we consider te numerical evaluation of Caucy singular integrals [8]. In Section 2 we give te details of te construction of te interpolatingderivative spline. In Section 3 we give some convergence results. Finally, in Section 4, some numerical experiments on test functions f are reported. In Appendix we prove some propositions wose results are necessary for proving teorem 2.5 and proposition 2.7 in Section 2 and teorems 3.2, 3.3 in Section Construction of derivative-interpolating spline Let m, n m two given integer positive numbers and Y IR n+,y := {y 0, y,..., y n} a given vector and n := {a = x 0 < x <... < x n < x n+ = b} a given partition of I [a, b] in n + subintervals I k := [x k, x k+ ), k = 0,,..., n, limiting ourselves, for te sake of simplicity, to consider an uniform partition n, wit = x i+ x i, i = 0,,..., n. We denote by IP k te set of polynomials of degree k. Consider te space of polynomial splines of degree 2m S 2m ( n ) = { s : s(x) = si (x) IP 2m, x I i, i = 0,,..., n; D j s i (x i ) = D j s i (x i ), j = 0,,..., 2m, i =, 2,..., n } (2.) wit simple knots x, x 2,..., x n. Te space S 2m ( n ) C 2m (I). A function s f S 2m ( n ) is called derivative-interpolating if s f (x 0 ) = y 0, s f (x i ) = y i, i =, 2,..., n; y 0 = f (x 0 ), y i = f(x i ). (2.2) Limiting ourselves to consider m = 2, if we set by successive integrations, we obtain M i = s (2m ) f (x i ), i = 0,,..., n +, s f (x) Ii = [M i+ (x x i ) 4 M i (x x i+ ) 4 ]/(4!)+ +a i (x x i ) 2 /2 + b i (x x i ) + c i, i = 0,,..., n. (2.3) By imposing te conditions (2.) and (2.2), we obtain 94

3 A CLASS OF EVEN DEGREE SPLINES system a i = y i+ y i 6 (M i+ M i ) i =,..., n b 0 = y 2 6 M a 0 b i = y i 2 6 M i i =,..., n c 0 = y M 0 c = c 0 + y 3 2 M 2 2 a 0 c i = c i + (y i + y i ) M i i = 2,..., n M i = a i a i i =,..., n. (2.4) Substituting te first equations of (2.4) in te last ones, we obtain a linear were à = à M = b (a 0, a n ) (2.5), M = M... M n, b = 6 [ y 2 y a 0,..., y i+ y i y i y i,..., y n y ] T n + a n. Te spline function s f (x) will be determined by solving te following problem wit M = [M 0,..., M n+ ] T, were { min M T ĀM à M = b (a 0, a n ) Ā = = A = e T 0 e A e n 0 e T n 2 (2.6), (2.7) (2.8) 95

4 GH. MICULA, E. SANTI, AND M. G. CIMORONI and e, e n are te vectors [, 0,..., 0] T, [0, 0,..., ] T respectively. We can write b = b e ã 0 + e n ã n wit b = 6 [ y 2 y,..., y i+ y i y i y i,..., y n y ] T n (2.9) and ã 0 = 6 a 0, ã n = 6 a n. Considering tat à is a symmetric positive definite and ten, non singular matrix, from (2.5) we get tus: M = à (b e ã 0 + e n ã n ) (2.0) ] min M T ĀM = min [M 0 M T M n+ 2 et 0 e A e n [ ] T M 0 M T M n+ 0 e T. (2.) n 2 Using (2.0), te problem amounts to find out firstly te vector solution of te linear system N = [ã 0, ã n, M 0, M n+ ] T, were, by setting C = à A Ã, [ ] B B B= 2 e T, B B 2 2I =[ Ce e T Ce n 2 e T n Ce e T n Ce n BN = P (2.2) ] [, B 2 = e T à e e T n à e e T à e n e T n à e n ]. (2.3) I 2 is te second order identity matrix and [ T P = e T Cb, et n Cb, et à b, e T n b] Ã. (2.4) Once determined N, we sall determine s f (x) by solving te system (2.5). Before proving te below teorem 2.5, we need to investigate some properties of matrices Ã, à and C. Proposition 2.. Te matrix à = (a ij) n i,j=, is: (a) symmetric, positive definite; (b) persymmetric, i.e. a ij = a n i+,n j+, i, j =,..., n; (c) totally positive (T.P.), i.e. all te minors are 0; (d) oscillatory, ten all te eigenvalues of à are distinct, real and positive. Proof. It is straigtforward to verify (a), (b), (c). Te property (d) follows by considering tat a non singular T.P. matrix aving te entries a ik 0, i k is oscillatory [4]. 96

5 A CLASS OF EVEN DEGREE SPLINES Proposition 2.2. Te infinitive norm of à satisfies te following relation 6 à 2. (2.5) Proof. (For te proof, see Appendix). Proposition 2.3. Te entries a j, j =,..., n of à ave decreasing absolute values, te sign of ( ) j, in particular, te following inequalities: old. 5 a = et à e 4, (2.6) a e n = T à e n Proof. (For te proof, see Appendix ). 24 if n = 2, 90 if n 3 (2.7) Proposition 2.4. Let C = à A Ã. For te entries c = e T Ce, c n = e T Ce n we ave: 0 < c < (2.8) Proof. ( For te proof, see Appendix). c n < c. (2.9) Now we prove te following: Teorem 2.5. Te system (2.2) is determined and te solution is [[ ] ] T N = e T à b, e T n à b B2, 0, 0. (2.20) Proof. Considering tat e T à e n = e T n à e and for te properties of à and te definition of te symmetric positive matrix A, one as e T Ce n = e T n Ce, (2.) can be written in te form [ ] [ ] [ ] B B 2 x t =, (2.2) B 2 2I 2 y t2 were x = [ã 0, ã n ], y = [ M 0, M n+ ], t = [ e T Cb, e T n Cb ] [ T T, t2 = e T à b, e T n b] Ã. Since A = à ( e e T + e n et n ), and ten, à A à = à à ( e e T + e n et n ) à tere results B = B 2 (I 2 B 2 ) (2.22) and te system (2.2) reduces to: [ B2 (I 2 B 2 ) B 2 B 2 2I 2 ] [ x y Te system (2.2) as unique solution. t = (I 2 B 2 ) t 2, (2.23) ] = [ ] (I2 B 2 ) t 2. (2.24) t 2 97

6 GH. MICULA, E. SANTI, AND M. G. CIMORONI In fact [4], since 2B 2 I 2 = 2I 2 B 2, det [ ] B2 (I 2 B 2 ) B 2 = det(b B 2 2I 2 ) det(2i 2 3B 2 ) 2 and using te results of propositions 2.2 and 2.3, it is straigtforward to verify tat det(b 2 ) 0 and det(2i 2 3B 2 ) 0. From te second equation of (2.24) we obtain x = B 2 (t 2 2I 2 y), and, substituting in te first one, tere results: tat implicates B 2 (I 2 B 2 ) B 2 (t 2 2y) + B 2y = (I 2 B 2 ) t 2, (3B 2 2I 2 )y = 0. (2.25) Terefore we obtain te solution y = 0, x = B2 t 2 and teorem 2.5 is proved. Corollary 2.6. For te spline s f (x) te following property olds. M = s (2m ) f (x ) = M n = s (2m ) f (x n ) = 0 (2.26) Proof. Taking into account tat, from te relation B 2 x + 2y = t 2, we obtain [ ] [ ] [ ] [ M0 = e T à b M n+ 2 e T + ] e T à e e T à e n ã0 n à b 2 e T n à e e T, (2.27) n à e n ã n we get te tesis considering te first and te last equation in (2.0), tat is: [ ] [ ] [ ] [ ] M e = T à b e M n e T T à e e T à e n ã0 n à b e T n à e e T. (2.28) n à e n ã n From teorem 2.5 and corollary 2.6 we can deduce tat, as aspected, te obtained spline s f (x) reduces to a polynomial of second degree in te subintervals I 0 and I n. Remark 2.. In [3, teorem 2] te construction of suc spline is obtained by solving a linear system of m + n + equations tat can ave an increasing condition number wen n increases. Our metod is based on te solution of two linear systems, aving matrix à and B respectively. ) By proposition 2.2, for eac n, for te condition number of Ã, we ave K (à 3; now we prove te following Proposition 2.7. For te condition number K (B) te inequality olds. 98 K (B) , if n 3, (2.29)

7 A CLASS OF EVEN DEGREE SPLINES [ ] B B Proof. B = 2, ten B 2 2I 2 [ B (2I2 3B = 2 ) 0 0 (2I 2 3B 2 ) ] [ 2B 2 I 2 I 2 I 2 B 2 Let n 3. Using te results obtained in te propositions 2.3 and 2.4, one obtains B 2 + a + a 407 n 80, (2.30) B 2 = a + a n 47 (2.3) 80 and B 2 = a a 5. n Besides: B (2I 2 3B 2 ) ( 2 B + ). (2.32) Using te results in [7], since for all vector x suc tat x =, one as and ten Terefore and we get te tesis. 3. Convergence results 2 (2I 2 3B 2 ) x = 73 60, (2.33) ]. B (2.34) K (B) , if n Consider I = [a, b] and te set W2 3. In [3] as been proved te following Teorem 3.. Let f W2 3 (I) and let s f be te derivative-interpolating spline, ten { f (k) s (k) C f 2 2 f (3) 2 if k = 0, C 2 2 k+ 2 f (3) (3.) 2 if k =, 2, were C = 2 (b a) and C 2 = 2. We sall prove a new convergence teorem under weaker ypotesis on function f. For all g C (I), we denote by ω (g ; ; I) = max x,x+δ I, 0 δ g (x + δ) g (x) te modulus of continuity of g. Supposing f C (I), from (2.9), we obtain ten, using (2.5), (2.20): b 2 2 ω (f ; ; I) (3.2) 99

8 GH. MICULA, E. SANTI, AND M. G. CIMORONI a 0, a n 5 ω (f ; ; I). (3.3) and consequently, from (2.0), we obtain: M = M 2 2 ω (f ; ; I). (3.4) If we consider tat f (x i ) = y i and by (2.4): a i = y i+ y i 6 (M i+ M i ) i =,..., n, we can write Terefore, a i 8 ω (f ; ; I). (3.5) a 8 ω (f ; ; I) (3.6) were a = [a 0, a,..., a n ] T. Teorem 3.2. Let f C (I) and s f (x) te interpolating-derivative spline quoted in Section 2 for a given partition n. Ten were C is a constant independent of. ω ( s f ; ; I ) Cω (f ; ; I) (3.7) Proof. It suffices to sow tat for u, v I, u < v : s f (v) s f (u) Cω (f ; v u; I). Firstly consider u, v [x i, x i+ ], i = 0,..., n; using te mean value teorem, we can derive: s f (v) s f (u) = s f (ξ) v u ξ (u, v), were v u. Since for any ξ (u, v), from (3.3), (3.4), (3.5), tere results s f (ξ) 29 ω (f ; ; I) if u, v [x i, x i+ ] t, i =, 2,..., n and s f (ξ) 5 ω (f ; ; I) if u, v [x 0, x ] or u, v [x n, x n+ ], recalling tat [9], v u ω (f ; ; I) 2ω (f ; v u ; I), we get s f (v) s f (u) C ω (f ; v u ; I), C = 58. (3.8) If u [x i, x i+ ], v [x j, x j+ ], i+ j, ten using (3.8) and te smootness of modulus of continuity and, since being x i+ and x j internal nodes, s f (x i+) = f (x i+ ), s f (x j) = f (x j ) : s f (v) s f (u) s f (v) s f (x j ) + s f (x j ) s f (x i+ ) + s f (x i+ ) s f (u) = s f (v) s f (x j ) + f (x j ) f (x i+ ) + s f (x i+ ) s f (u) (2C + ) ω (f ; v u ; I). Tis proves te teorem wit C = (2C + ). 00

9 A CLASS OF EVEN DEGREE SPLINES Supposing f C (I), we define r (x) = f (x) s f (x) and r (x) = f (x) s f (x), were s f (x) is te interpolating-derivative spline quoted in Section 2. For x I i, i = 0,..., n we can write { r r (x) = (x ) + (x x ) [x x] r if x I 0, r (x i ) + (x x i ) [x i x] r (3.9) if x I i, i =,..., n. were [x i x] r, i =,..., n, denotes te first divise difference of r. Terefore, from (2.2) and teorem 3.2: r (x) Ii (C + ) ω (f ; ; I), i = 0,,..., n. (3.0) We are ready to prove te following convergence result. Teorem 3.3. Let f C (I) and s f (x) te interpolating-derivative spline. Tere results f s f (b a) (C + ) ω (f ; ; I). (3.) Proof. We can write, for x I i, i = 0,,..., n x r (x) = r (t) dt max x x I r (x) x x 0 (b a) (C + ) ω (f ; ; I) (3.2) 0 and (3.) is proved. We remark tat te above teorems old even wen te partition n is quasiuniform, i.e. suc tat: is bounded for n were i = x i+ x i and max 0 i n i is te norm of te partition. [E.Santi, M.G.Cimoroni: Some new convergence results and applications of a class of interpolating-derivative splines. In preparation]. We add now a property of te splines considered in tis paper. Te derivativeinterpolating spline s f (x) defined in (2.3), considering a uniform partition n, reproduces any f IP 2. In fact, for f =, x, x 2 it is straigtforward to verify tat, b (a 0, a n ) = M = 0. Terefore te coefficients of s f (x) Ii are: and tus, for x I i, i = 0,..., n : 4. Numerical results a i = 0, b i = 0, c i =, if f(x) =, a i = 0, b i =, c i = x i, if f(x) = x, a i = 2, b i = 2x i, c i = x 2 i, if f(x) = x2, s f (x) =, if f(x) =, s f (x) = (x x i ) + x i = x, if f(x) = x, s f (x) = 2(x xi) x i (x x i ) + x 2 i = x2, if f(x) = x 2. We present now, some numerical results obtained by approximating some test functions by te spline considered in tis paper. We denote r n (x) = f (x) s f (x) te error at x obtained by using a uniform partition of te interval [, ] in n + subintervals. In table we report te results relative to a test function f(x) aving only f (x) C[, ] and considering different uniform partition wit 0

10 GH. MICULA, E. SANTI, AND M. G. CIMORONI n = 4, 9, 39, 99. In table 2 we report, for confirming te polynomial reproducibility, te results relative to a function f(x) IP 2 considering a uniform partition wit n = 4. Table 3 contains te results relative to a more regular function f(x) C [, ] wit different uniform partition, taking n = 4, 9, 39, 79. Table f(x) = sign(x)x 2 /2 + e x x r 4 (x) r 9 (x) r 39 (x) r 99 (x) (-3).2 (-4).4 (-5) 8.6 (-7) (-3).8 (-4).5 (-5) 8.6 (-7) (-2).7 (-3) 4.3 (-4) 6.8 (-5) (-2).8 (-3) 4.3 (-4) 6.8 (-5) 5.6 (-3) 2.5 (-3) 5.2 (-4) 7.4 (-5) Table 2 f(x) = x 2 + 2x 5 x r 4 (x) (-6) (-5) (-6) 8.9 (-6) Table 3 f(x) = /(x ) x r 4 (x) r 9 (x) r 39 (x) r 79 (x) (-5) 3.4 (-7) 4.4 (-8) 5.6 (-9) (-5) 3.4 (-7) 4.4 (-8) 5.6 (-9) (-5) 3.4 (-7) 4.4 (-8) 5.6 (-9) (-5) 3.4 (-7) 4.4 (-8) 5.6 (-9) (-7) Appendix Proposition 2.2. Te infinitive norm of à satisfies te following relation 6 à 2. (5.) Proof. It is straigtforward to verify tat à = 6 and ten, being à Ã, we obtain te left inequality in (5.). For proving tat à 2, we write à = 4I + H, were 02 H =

11 A CLASS OF EVEN DEGREE SPLINES Tus, for all x : x =, tere results Ãx = (4I + H)x 4x Hx 2, and ten [7], (5.) is proved. Proposition 2.3. Te entries a j, j =,..., n of à ave decreasing absolute values, te sign of ( ) j, in particular, te following inequalities: old. a n = 5 a = et à e 4, (5.2) 24 if n = 2, e T à e n 90 if n 3 (5.3) Proof. Using te results in [], for te evaluation of te inverse matrix of a tridiagonal symmetric matrix, we can write and ten, were l ij = 0 for i j. In our case, tere results à = L + uv T (5.4) a ij = l ij + u i v j (5.5) u =, u 2 = 5, u i = 4u i u i 2, i = 3,..., n (5.6) v i = α u n i+, i =,..., n (5.7) wit α = 5u n + u n.te matrix à is symmetric and a i a in a = a i = α u n i+, = a ni = α u i, i =, 2,..., n. Terefore, using proposition 2., we deduce tat a = a nn = u n /α, a j = n,n j+, j =, 2,..., n are decreasing in absolute value and ave te sign of ( )j, and, in particular, a n = /α. For proving (5.2) consider tat a = u v = u n /α = 5u n+u n u n 5 5u n+u n = ( ) 5 un 5u n+u n, and ten, using (5.6), 0 < un 5u n+u n = u n+u n 2 4 5u n+u n < 4, (5.3) follows. We get te inequality (5.3) considering tat a n = 5u n+u n and ten, from (5.6), a n = 24 if n = 2, a n = 90 for n = 3, and a n decreases wen n increases. Terefore te proposition is completely proved. Proposition 2.4. Let C = à A Ã. For te entries c = e T Ce, c n = e T Ce n we ave: 0 < c < (5.8) c n < c. (5.9) 03

12 GH. MICULA, E. SANTI, AND M. G. CIMORONI Proof. For n 3, [ by (2.22), using (5.2) and (5.3), it is straigtforward to verify tat (a 0 < c = a ) 2 ( ) + a 2 ] n < and c n = a ( ) n 2a < c. References [] Bevilacqua, R., Structural and computational properties of band matrices. Complexity of structured computational problems. Applied Matematics monograps. Comitato Nazionale per le Scienze Matematice, C.N.R. (99), [2] Bini, D., Capovani, M., A class of cubic splines obtained troug minimum conditions. Matematics of Computation 46, n.73 (986), [3] Blaga, P., Micula, G. Natural spline functions of even degree. Studia Univ. Babes-Bolyai Cluj-Napoca Matematica, 38, (993) Nr. 2, [4] Gantmacer, F.R., Te teory of matrices, Celsea Publ. Company N.Y [5] Gizzetti, A., Interpolazione con splines verificanti un opportuna condizione. Calcolo 20 (983), [6] Gori, L., Splines and Caucy principal value integrals. Proc. Intern. Worksop on Advanced Mat. Tools in Metrology (Ed. Ciarlini, Cox, Monaco, Pavese) (994), [7] Kersaw, D., A note on te convergence of interpolatory cubic splines. Siam J.Numer. Anal. 8 (97), [8] Rabinowitz, P., Numerical integration based on approximating splines. J. Comp. Appl. Mat. 33 (990), [9] Rabinowitz, P., Application on approximating splines for te solution of Caucy singular integral equations. Appl. Numer. Mat. 5 (994), Dept. of Applied Matematics, Babeş-Bolyai Univ. Cluj-Napoca, Romania address: gmicula@mat.ubbcluj.ro Dipartimento di Energetica, Universita degli Studi de l Aquila, Italy address: esanti@dsiaqi.ing.univaq.it Dipartimento di Energetica, Universita degli Studi de l Aquila, Italy 04