On local integro quartic splines

Applied Mathematics and Computation 269 (2015) 301–307

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

On local integro quartic splines T. Zhanlav a, R. Mijiddorj a,b,∗ a b

Institute of Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia Department of Informatics, Mongolian State University of Education, Ulaanbaatar, Mongolia

a r t i c l e

i n f o

Keywords: Integro quartic spline Local construction Quartic B-spline Coefficients of B-representation

a b s t r a c t In this paper, we propose a new local construction of integro quartic splines which successfully works without any additional end conditions and without solving any system of equations. In fact, we obtain analytical formulae for values of the integro spline and up to its third-order derivatives values at the knots. Numerical experiments show our method is easy to implement and effective as well. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Recently, some authors have devoted much attention to construct integro splines [1,2,4–6,8–10]. Such splines have many practical applications in the fields of mathematical statistics, numerical analysis, electricity, climatology, oceanography, and so on (see [1,2,4–6,9,10]). In [4], Lang and Xu discussed the integro quartic spline and its approximation properties. They pointed out that the integro quartic spline possesses superconvergence orders in approximating function values and second order derivatives values at the knots, but a phenomenon of superconvergence remains without answer. We want to explain the reason of the superconvergence approximations. In this paper, we apply the local approximation approach to this end. It should be mentioned that the local approximation approach [6,9] does not need any additional data, and it is easy to implement not solving any system of linear equations. The rest of this paper is organized as follows. In Section 2, we give some preliminary results of integro quartic spline. Section 3 is devoted to the construction of local integro quartic splines. We give the explicit formulae for the quartic spline and its derivatives at the knots and explain the reason of the superconvergence of values of the integro quartic spline and its second derivatives. We also consider B-spline representation of integro quartic spline and present an explicit formula for coefficients of B-representations. In Section 4, we present numerical results which confirm the theoretical analysis. 2. The integro quartic spline interpolation For an interval I = [a, b], divide it into n successive subintervals by the equidistant knots xi = a + ih(i = 0, 1, . . . , n and h = (b − a)/n). In order to use B-spline representation of quartic spline, we extend the interval I to  I = [a − 4h, b + 4h] with the equidistant knots xi = a + ih(i = −4, −3, −2, −1, n + 1, . . . , n + 4). Any quartic spline of class C3 [a, b] can be represented as [3,7]

S(x) =

n+1 

c j B j (x),

j=−2

∗

Corresponding author. Tel.: 97699010363. E-mail addresses: [email protected] (T. Zhanlav), [email protected] (R. Mijiddorj).

http://dx.doi.org/10.1016/j.amc.2015.07.077 0096-3003/© 2015 Elsevier Inc. All rights reserved.

(2.1)

302

T. Zhanlav, R. Mijiddorj / Applied Mathematics and Computation 269 (2015) 301–307

where B j (x)( j = −2, −1, . . . , n + 1) are linearly independent and they form the basis splines of S4 (I) [3]. The spline S(x) ∈ C3 [a, b] satisfying



xi+1

xi

S(x)dx =



xi+1

xi

y(x)dx = Ii ,

i = 0(1)n − 1,

(2.2)

and

S(x0 ) = y0 ,

S(x1 ) = y1 ,

S(xn−1 ) = yn−1 ,

S(xn ) = yn

(2.3)

exits uniquely and is called the integro-quartic spline interpolation [4]. The conditions (2.2) and (2.3) can be rewritten, in terms of cj in (2.1), as follows:

ci−2 + 26ci−1 + 66ci + 26ci+1 + ci+2 =

120 Ii , h

i = 0(1)n − 1

(2.4)

and

c−2 + 11c−1 + 11c0 + c1 = 24y0 , c−1 + 11c0 + 11c1 + c2 = 24y1 ,

(2.5)

cn−3 + 11cn−2 + 11cn−1 + cn = 24yn−1 , cn−2 + 11cn−1 + 11cn + cn+1 = 24yn .

(2.6)

Thus, the construction of integro-quartic spline leads to solving the system of Eq. (2.4)–(2.6). For simplicity, we will use the following notations:

yi = y(xi ), Si = S(xi ), mi = S (xi ), Mi = S (xi ), Ti = S(3) (xi ). The proof of the following theorem is given in [4]. Theorem 1. Let y(x) be a function of class C∞ [a, b] and S(x) be the integro-interpolating quartic spline obtained by (2.2) and (2.3). For i = 0, 1, . . . , n, we have

h6 (6) y (xi ) + O(h8 ), 5040

(2.7)

mi = y (xi ) +

h4 (5) y (xi ) + O(h6 ), 720

(2.8)

Mi = y (xi ) −

h4 (6) y (xi ) + O(h6 ), 240

(2.9)

Ti = y (xi ) −

h2 (5) y (xi ) + O(h4 ). 12

Si = y(xi ) +

(2.10)

Moreover, from (2.10) we immediately obtain (4) (4) Si+0 − Si−0 = hy(i 5) + O(h3 ),

i = 1, 2, . . . , n − 1.

(2.11)

3. Construction of local integro-quartic splines By the definition of quartic spline S(x), we have





(4) (4) Ti+1 − Ti−1 = h Si+0 + Si−0 ,



(3.1)



(4) (4) Ti+1 − 2Ti + Ti−1 = h Si+0 − Si−0 ,

mi−1 − 2mi + mi+1 = h2 Ti + mi+1 − mi−1 = 2hMi +

(3.2)





h3 (4) (4) Si+0 − Si−0 , 6



(3.3)



h3 (4) (4) Si+0 + Si−0 . 6

(3.4)

If we take (3.1) into account, then from (3.4) we obtain

Mi =

h mi+1 − mi−1 − (Ti+1 − Ti−1 ). 2h 12

(3.5)

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303

The Taylor expansion of S(x) at the point x = xi is

S(x) = Si + mi (x − xi ) +

S(4) Mi T (x − xi )2 + i (x − xi )3 + i+0 (x − xi )4 , 2 6 24

Using the last expansion and (2.2), we find





Ii + Ii−1 = 2hSi +

h3 h5 (4) Mi + S(4) + Si+0 , 3 120 i−0

Ii − Ii−1 = h2 mi +

h4 h5 (4) Ti + S(4) − Si−0 . 12 120 i+0

(4)

x ∈ [xi , xi+1 ].



(3.6)



(3.7)

(4)

Substituting (Si+0 − Si−0 ) obtained from (3.2) into (3.7), we have

h2 I −I (Ti−1 + 8Ti + Ti+1 ) = i 2i−1 . 120 h

mi +

(4)

(3.8)

(4)

Analogously, substituting (Si+0 − Si−0 ) obtained from (3.3) into (3.7), we have

Ti +

3 30 (mi−1 + 18mi + mi+1 ) = 4 (Ii − Ii−1 ). 2h2 h

(3.9)

Also, from (3.2) and (3.3) it follows that

6(mi−1 − 2mi + mi+1 ) = h2 (Ti−1 + 4Ti + Ti+1 ).

(3.10)

Using (2.11), (3.7), and (3.9), we get

mi−1 + 10mi + mi+1 =

12 (Ii − Ii−1 ) + O(h4 ). h2

(3.11)

From (3.9) and (3.11) it immediately follows that

h2 Ii − Ii−1 Ti = + O(h4 ). 12 h2

mi +

(3.12)

Relations (3.1)–(3.12) hold for i = 1, 2, . . . , n − 1. Substituting mi obtained from (3.12) into (3.11), we get

Ti−1 + 10Ti + Ti+1 =

12 (−Ii−2 + 3Ii−1 − 3Ii + Ii+1 ) + O(h2 ), h4

i = 2(1)n − 2.

(3.13)

If we take (2.11) into account, then subtracting (3.13) and (3.2) it follows

Ti =

1 (−Ii−2 + 3Ii−1 − 3Ii + Ii+1 ) + O(h2 ), h4

i = 2(1)n − 2.

(3.14)

From (3.12) and (3.14) it follows

mi =

1 (Ii−2 − 15Ii−1 + 15Ii − Ii+1 ) + O(h4 ), 12h2

i = 2(1)n − 2.

(3.15)

Thus, we have explicit and approximate formulae for Ti and mi for i = 2(1)n − 2. From (3.9) for i = 2 we find m1 and from (3.12) T1 is found. In a similar way, from (3.9) for i = 1 we find m0 and from (3.10) for i = 1 we find T0 . Likewise, mi and Ti for i = n − 1, n are found. Thus, we find all the Ti and mi for i = 0, 1, . . . , n. If we take (3.1) and (3.5) into account, then from (3.6), we obtain

Si =

Ii + Ii−1 7h3 h (mi+1 − mi−1 ) + (Ti+1 − Ti−1 ), i = 1(1)n − 1. − 2h 12 720

(3.16)

In our opinion, the forms of formulae (3.5) and (3.16) are the main reason for superconvergence of Si and Mi . That is, using the Taylor formulae, we have O(h6 ).

1 (y 2h i+1

− yi−1 ) −

h  12 (yi+1

− y ) = yi + O(h4 ) and i−1

Ii +Ii−1 2h

−

h  12 (yi+1

− yi−1 ) +

7h3  720 (yi+1

− y ) = yi + i−1

Moreover, one can find explicit formulae for Mi and Si . In fact, substituting mi and Ti given by (3.14) and (3.15) into (3.5) and (3.16), we obtain

Mi =

1 (−Ii−3 + 7Ii−2 − 6Ii−1 − 6Ii + 7Ii+1 − Ii+2 ), 8h3

i = 3(1)n − 3,

(3.17)

Si =

1 (Ii−3 − 8Ii−2 + 37Ii−1 + 37Ii − 8Ii+1 + Ii+2 ), 60h

i = 3(1)n − 3

(3.18)

and

in which we neglect the small terms.

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Suppose Sˆ(x) is given by

ˆi Sˆ(4) M Tˆ (x − xi )2 + i (x − xi )3 + i+0 (x − xi )4 , 2 6 24 (4) ˆ ˆ ˆ i = 0, 1, . . . , n − 1, S = (Ti+1 − Ti )/h,

ˆ i (x − xi ) + Sˆ(x) = Sˆi + m x ∈ [xi , xi+1 ],

(3.19)

i+0

then Sˆ(x) is called a local integro-quartic spline. Here approximations without the small terms in (3.14)–(3.18) are denoted by Sˆi , ˆ i , and Tˆi , respectively. We shall now proceed to error analysis of the local integro-quartic spline. ˆ i, M m Theorem 2. Let y(x) be a function of class C8 [a, b] and Sˆ(x) be the local integro-quartic spline (3.19). Then, for i = 3, 4, . . . , n − 3, we have

h6 (6) y + O(h8 ), 140 i

(3.20)

ˆ i = yi − m

h4 (5) y + O(h6 ), 90 i

(3.21)

ˆ i = y − M i

7 4 (6) h yi + O(h6 ), 120

(3.22)

Tˆi = y i +

h2 (5) y + O(h4 ). 6 i

(3.23)

Sˆi = yi +

Proof. Using Taylor expansion of function y(x) ∈ C8 [a, b] we obtain

Ii + Ii−1 = 2

Ii − Ii−1 = 2

3 

y(i 2k)

(2k + 1)! k=0 3  y(i 2k+1) k=0

(2k + 2)!

h2k+1 + O(h9 ),

(3.24)

h2k+2 + O(h9 ).

(3.25)

Again, using Taylor expansions of y(x) ∈ C8 [a, b] we get

Ii−2 = Ii−1 − h2 yi + h3 y(i 2) − Ii+1 = Ii + h2 yi + h3 y(i 2) +

7h4 (3) h5 (4) 31 6 (5) h7 (6) 127h8 (7) yi + yi − h yi + yi − y + O(h9 ), 12 4 360 40 20, 160 i

7h4 (3) h5 (4) 31 6 (5) h7 (6) 127h8 (7) yi + yi + h yi + yi + y + O(h9 ). 12 4 360 40 20, 160 i

(3.26)

(3.27)

Substituting (3.25)–(3.27) into (3.14) and (3.15) we arrive at (3.21) and (3.23) for i = 3(1)n − 3. By virtue of (3.21) and (3.23) we have

ˆ i−1 = 2hyi + ˆ i+1 − m m

h3 (4) h5 (6) y − y + O(h7 ), 3 i 180 i

(3.28)

Tˆi+1 − Tˆi−1 = 2hy(i 4) +

2h3 (6) y + O(h5 ). 3 i

(3.29)

Substituting (3.24), (3.28), and (3.29) into (3.5) and (3.16) we get (3.20) and (3.22) for i = 3(1)n − 3. The proof of Theorem 2 is completed.  ˆ i and Tˆi at the end points in terms of Ij ’s values are The expressions of m

ˆ 0 = (−45I0 + 109I1 − 105I2 + 51I3 − 10I4 )/(12h2 ), m ˆ 1 = (−10I0 + 5I1 + 9I2 − 5I3 + I4 )/(12h2 ), m ˆ n−1 = (−In−5 + 5In−4 − 9In−3 − 5In−2 + 10In−1 )/(12h2 ), m ˆ n = (10In−5 − 51In−4 + 105In−3 − 109In−2 + 45In−1 )/(12h2 ), m

(3.30)

Tˆ0 = (−3I0 + 11I1 − 15I2 + 9I3 − 2I4 )/h4 , Tˆ1 = (−2I0 + 7I1 − 9I2 + 5I3 − I4 )/h4 , Tˆn−1 = (In−5 − 5In−4 + 9In−3 − 7In−2 + 2In−1 )/h4 , Tˆn = (2In−5 − 9In−4 + 15In−3 − 11In−2 + 3In−1 )/h4 .

(3.31)

and

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305

ˆ i at the end points we cannot use (3.5) and (3.16) because of the inaccuracy. So at the end points In order to determine Sˆi and M values we use the same technique as in [4], we get

Sˆ0 = (147I0 − 213I1 + 237I2 − 163I3 + 62I4 − 10I5 )/(60 h), Sˆ1 = (10I0 + 87I1 − 63I2 + 37I3 − 13I4 + 2I5 )/(60 h), Sˆ2 = (−2I0 + 22I1 + 57I2 − 23I3 + 7I4 − I5 )/(60 h), Sˆn−2 = (−In−6 + 7In−5 − 23In−4 + 57In−3 + 22In−2 − 2In−1 )/(60 h), Sˆn−1 = (2In−6 − 13In−5 + 37In−4 − 63In−3 + 87In−2 + 10In−1 )/(60 h), Sˆn = (−10In−6 + 62In−5 − 163In−4 + 237In−3 − 213In−2 + 147In−1 )/(60 h),

(3.32)

ˆ 0 = (49I0 − 183I1 + 278I2 − 218I3 + 89I4 − 15I5 )/(8h3 ), M ˆ 1 = (15I0 − 41I1 + 42I2 − 22I3 + 7I4 − I5 )/(8h3 ), M ˆ 2 = (I0 + 9I1 − 26I2 + 22I3 − 7I4 + I5 )/(8h3 ), M ˆ n−2 = (In−6 − 7In−5 + 22In−4 − 26In−3 + 9In−2 + In−1 )/(8h3 ), M ˆ n−1 = (−In−6 + 7In−5 − 22In−4 + 42In−3 − 41In−2 + 15In−1 )/(8h3 ), M ˆ n = (−15In−6 + 89In−5 − 218In−4 + 278In−3 − 183In−2 + 49In−1 )/(8h3 ). M

(3.33)

and

ˆ i , and Tˆi are defined by (3.30)–(3.33) then the function values and its derivatives can ˆ i, M Proposition 1. If y(x) ∈ C8 [a, b] and Sˆi , m ˆ i , and Tˆi for i = 0, 1, 2, n − 2, n − 1, n with errors O(h6 ), O(h4 ), O(h4 ), and O(h2 ), respectively. ˆ i, M locally be approximated by Sˆi , m ˆ i for i = 0, 1, . . . , n are determined without loss of accuracy and without solving any linear Thus, all the values of Sˆi and M systems. It should be mentioned that the relations (2.7)–(2.10) were proved in [4] under the strong restriction y ∈ C∞ [a, b], while the relations (3.20)–(3.23) hold under y ∈ C8 [a, b]. The expressions (3.20) and (3.22) show that the local integro-quartic spline possesses superconvergence orders, like the integro-quartic one, in approximating function values and second-order derivative values at the knots. Remark 1. As concerning the global approximation error for Sˆ(x), the analogy of Theorem 4.2 in [4] holds also too. ˆ i , and Tˆi (i = 0(1)n) are some linear combination of Ij approximating yi , y , y , and y with errors O(h6 ), O(h4 ), O(h4 ), ˆ i, M Sˆi , m i i i and O(h2 ), respectively. Using these approximating values, we can represent various local integro-quartic splines. Now we will give the explicit formula for the coefficients ci in (2.1). For this purpose we use

ci−2 + 11ci−1 + 11ci + ci+1 , 24 −ci−2 − 3ci−1 + 3ci + ci+1 , mi = 6h ci−2 − ci−1 − ci + ci+1 Mi = , 2h2 −ci−2 + 3ci−1 − 3ci + ci+1 Ti = , h3 Si =

(3.34)

which are immediately followed from the properties of B-spline. The coefficients ci−2 , ci−1 , ci , and ci+1 are expressed through Si , mi , Mi , and Ti in a unique way as

ci = Si +

h h2 h3 mi − Mi − Ti , 2 12 12

i = 0(1)n.

(3.35)

The B-representation of the local integro-quartic spline is,

Sˆ(x) =

n+1 

cˆ j B j (x).

(3.36)

j=−2

Substituting (3.14), (3.15), (3.17), and (3.18) into (3.35) we obtain

cˆi =

13Ii−3 − 39Ii−2 − 94Ii−1 + 746Ii − 159Ii+1 + 13Ii+2 , 480h

i = 3(1)n − 3,

(3.37)

in which we neglect the small term. The remainder coefficients cˆi for i = −2, −1, 0, 1, 2 and i = n − 2, n − 1, n, n + 1 are determined from (2.4), (3.35), and (3.37) uniquely. Thus, all the coefficients cˆi are determined by the explicit way without solving system (2.4) and without any additional end conditions, as required in [4]. Local integro-quartic splines based on the Hermite–Birkhoff interpolation polynomial can variously be chosen. For example, from a simple computation, in each subinterval [xi , xi+1 ], i = 0, 1, . . . , n − 1, Sˆ(x) can be defined by

ˆ i + ψ5 (x)M ˆ i+1 , ˆ i + ψ4 (x)M Sˆ(x) = ψ1 (x)Sˆi + ψ2 (x)Sˆi+1 + ψ3 (x)m

(3.38)

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T. Zhanlav, R. Mijiddorj / Applied Mathematics and Computation 269 (2015) 301–307 Table 1 The maximum absolute errors Rr (n) for y1 (x) = ex . n

R0

R1

R2

R3

320 640 1280

3.587 × 10−16 5.628 × 10−18 8.811 × 10−20

1.960 × 10−10 1.229 × 10−11 7.695 × 10−13

4.973 × 10−10 3.120 × 10−11 1.954 × 10−12

7.480 × 10−5 1.875 × 10−5 4.694 × 10−6

βr

5.99

4.00

3.99

2.00

Table 2 The maximum absolute errors Rr (n) for y2 (x) = sin (π x). n

R0

R1 −15

R2 −8

R3 −9

320 640 1280

6.395 × 10 9.993 × 10−17 1.561 × 10−18

2.221 × 10 1.388 × 10−9 8.677 × 10−11

5.348 × 10 3.343 × 10−10 2.089 × 10−11

8.466 × 10−3 2.117 × 10−3 5.292 × 10−4

βr

6.00

4.00

4.00

2.00

Table 3 The maximum absolute errors Rr (n) for y3 (x) = cos (π x). n

R0

R1 −13

R2 −10

R3 −7

320 640 1280

1.279 × 10 1.998 × 10−15 3.123 × 10−17

4.536 × 10 2.027 × 10−11 1.267 × 10−12

1.772 × 10 1.108 × 10−8 6.924 × 10−10

4.981 × 10−4 1.245 × 10−4 3.113 × 10−5

βr

6.00

4.51

4.00

2.00

Table 4 The maximum absolute errors Rr (n) for y4 (x) = n

R0

R1 −16

1 . x+2

R2 −10

R3 −9

320 640 1280

7.273 × 10 1.153 × 10−17 1.814 × 10−19

1.335 × 10 8.424 × 10−12 5.290 × 10−13

1.009 × 10 6.395 × 10−11 4.024 × 10−12

5.103 × 10−5 1.286 × 10−5 3.229 × 10−6

βr

5.98

3.99

3.98

1.99

where

ψ1 (x) =

1 [ h4

ψ2 (x) =

1 [− h4

(x − xi+1 )4 + 2h(x − xi+1 )3 − 2h3 (x − xi+1 )],

ψ3 (x) =

1 [ h3

ψ4 (x) =

1 [2 6h2

ψ5 (x) =

1 [ 6h2

(x − xi )4 + 2h(x − xi )3 ],

(x − xi+1 )4 + 2h(x − xi+1 )3 − h3 (x − xi+1 )],

(3.39)

(x − xi+1 ) + 3h(x − xi+1 ) − h (x − xi+1 )], 4

3

3

(x − xi )4 − h(x − xi )3 ].

4. Numerical examples In this section we present some numerical results to test the accuracy and efficiency of our method. For [a, b] = [0, 1], we take 1 as examples and compute in 64 decimal digits of precision. y1 (x) = exp (x), y2 (x) = sin (π x), y3 (x) = cos (π x), and y4 (x) = x+2 The respective maximum absolute errors are given in Tables 1–4 where

Rr (n) = max |Sˆi(r) − y(i r) |. 0≤i≤n

In order to verify the convergence rate we calculate the Runge coefficient β r :

βr = log2 |

Rr (h) − Rr (h/2) |, Rr (h/2) − Rr (h/4)

for each r from 0 to 3. Theoretically, the rate of convergence of our method should be βr = 6 − r (r = 0, 2), β1 = 4, and β3 = 2. From the numerical experiments (Tables 1–4) we can see that the values of β r correspond to the theoretical ones. To make the comparison of the proposed local integro-quartic splines, we present some global uniform errors in Table 5. From Tables 1–5 we can see that the results are more reliable and better capture the asymptotic behavior of the error.

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307

Table 5 The maximum errors E(n) for y3 (x) and y4 (x), where E (n) = maxx∈[0,1] |Sˆ(x) − y(x)|. n

y3 (x)

y4 (x)

(3.19)

(3.36) −12

(3.38) −12

(3.19) −13

(3.36) −13

(3.38) −13

320 640 1280

2.995×10 8.223×10−14 2.598×10−15

3.520×10 7.777×10−14 2.430×10−15

6.151×10 1.119×10−14 3.496×10−16

6.104×10 1.901×10−14 5.583×10−16

7.388×10 2.340×10−14 7.362×10−16

1.295×10−13 4.103×10−15 1.291×10−16

βr

5.19

5.51

5.80

5.00

4.98

4.98

5. Conclusions In this paper, we constructed the local integro-quartic splines without solving systems of equations. This construction does not require any end conditions and it is easy to implement. It possesses good approximation properties like the integro-quartic one. The superconvergence orders of the function values and second-order derivatives values at knots are also confirmed by numerical experiments. Acknowledgments We appreciate the reviewers for their careful reading, valuable suggestions and comments. The work was partially supported by Foundation of Science and Technology of Mongolia. References [1] [2] [3] [4] [5] [6] [7] [8]

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