Eugen Constantinescu. 86KB Jun 04 2011 12:08:39 AM

A CERTAIN CLASS OF QUADRATURES

by

Eugen Constantinescu

Abstract. Our aim is to investigate a quadrature of form:

( )

x dx c f

( )

x c f

( )

x c f

( )

x c f

( )

x c f

( ) ( )

x R f

f = + + + + +

∫

1 1 2 2 3 3 4 4 5 5

1

0

(1)

where f :

[ ]

0,1 →IR is integrable, R(f) is the remainder-term and the distinct knots

x

jan

supposed to be symmetric distributed in [0,1]. Under the additional hypothesis that all

x

_j an of rational type (see(4)), we are interested to find maximum degree of exactness of such quadrature.

1 Introduction

Let

∏

m be the linear space of all real polynomials of degree

≤

m

and denote

( )

t t j N

e_j = j, ∈ . A quadrature of form

( )

( ) ( )

∫

1 =

∑

₌ +

0 0

n

k

k

kf x R f

c dx x

f (2)

has degrees (of exactness) m if R(h) = 0 for any polynomial

∈

∏

m

h

. If R(h) = 0 for all

∈

∏

m

h

and moreover

R

( )

e

m₊₁

≠

0

it is said that (2) has the exact degree m. It is known that if (2) has degree m, then m≤2n−1. Likewise, there exists only one formula

(2) having maximum degree 2n−1.

The aim of this paper is to study the formulas like (2) for n = 5 having some practical properties. Let us note that in this case, the optimal formula having maximum degree m = 9 is

( )

( ) ( )

1

10

1

1

1

0

5

1

=

≤

≤

±

±

=

+

=

∫

f

x

dx

∑

₌

c

f

x

r

f

k

It is clear that not all knots

x

_k are rational numbers.

Definition 1 Formula (1) is said to be of “practical-type”, iff i) the knots

x

_j are of form

1 5

2 4

3 2 2 1

1

,

1

,

1

2

1

,

,

x

r

x

x

r

x

r

r

x

=

=

=

=

−

=

−

(4)

where r₁,r₂distinct rational numbers from











2

1

,

0

.

ii) all coefficients

c

₁

,

c

₂

,

c

₃

,

c

₄

,

c

₅are rational numbers with

c

₁

=

c

₅ and c₂ =c₄. iii) (1) is of order p, with

p

≥

1

. Therefore, in case n = 5 a practical-type formula has the form

( )

x dx A

(

f

( ) (

r f r

)

)

B

(

f

( ) (

r f r

)

)

C f R

( )

f

f +

     ⋅ + − + +

− + =

∫

2

1 1

1 _{2 2}

1

0

1

1 (5)

A, B being rational numbers, C =1−2

(

A+B

)

, and when r₁,r₂ are distinct rational numbers from











2

1

,

0

.

Lemma 1 Let s be a natural number and suppose in (5) we have R

( )

h =0 for all

∏

∈

_s

h

2 Then R(g) = 0 for every g from

∏

2s+1 .

Proof. Let

( )

1 2

2

1

+











 −

=

s

x

x

H

. According to symmetry

∫

( )

=

1

0

0

dx x

H and also

( )

H =0

R . Observe that

e

s

( )

x

x

s

H

( ) ( )

x

h

1

x

1 2 1

2

≡

=

+

+

+ with

h

1

∈

∏

2s . Therefore

( )

e

_2s+1

=

0

R

and supposing

∈

∏

₊

1 2s

g

with

g

( )

x

=

a

₀

x

2s+1

+

...,

, we have

( )

g

a

R

( ) ( )

e

s

R

h

h

R

s

R

=

₀

⋅

_{2 +1}

+

₂

,

₂

∈

₂ , that is R(g) = 0.

(6)

Proof. We use standard method, namely by considering polynomials

For instance, taking into account that

are finds

and we conclude with

In a similar way are finds coefficients B and C. Taking into account that (5) is symmetric, we give:

Lemma 3 If (5) has order

m

,

m

≥

6

, then r₁,r₂ must be distinct rational numbers from

(

0,1

]

such that

(

)

56

(

)

560

(

1

)

5

0

56

560

1 2 1 2 1 2

2 2 2 1 2

2 2

1

r

+

r

+

r

−

r

+

r

+

r

r

−

r

−

r

+

=

r

(7)

Proof. It is sufficient to impose condition

R

( )

e

₆

=

0

,

e

₆

( )

x

=

x

6 . By considering

[ ] [ ]

a,b = −1,1 , are find

( )

2

2

0

7

1

6

2 6

1

6

=

−

Ar

−

Br

=

e

R

. Using Lemma 2, see (6) we

obtains condition (7).

Corollary 2 Suppose that (5) is of practical-type. If r₁,r₂ are distinct rational numbers from

(

0,1

]

such that equalities (6) and (7) are verified, then (5) has order m = 7.

Let us remark, that from above proposition implies that

Corollary 3 The maximum order of m of practical-type quadratus formula at 5-knots satisfied m≤7.

Proof. Formulas like (7) having order m = 8 does not exist. the reason is that by assuming m≥8, then according to Lemma 1 we must have m = 9. But in this case numbers r₁and r₂ are not rational (see (3)).

Lemma 4 Then does not exist pairs of rational numbers

( )

r₁,r₂ which satisfy

Proof. The case

(

1−2r₁

)(

1−2r₂

)

=0is impossible. Further, consider

(

1−2r₁

)(

1−2r₂

)

≠0

and let

,

1

2

,

,

,

,

,

0

,

0

2

2

1

−

₁

=

−

₂

=

p

q

x

y

∈

Z

q

>

y

>

y

x

r

p

r

with (p; q) = 1, (x; y) = 1.

Because

(

)

[

(

)

]

(

)

[

2

]

1 2 1 2

2

2 1 5 3 7

2 1 7 5 3 2

1

r r r

− −

− − =

− , we obtain

(

2 2

)

2

(

2 2

)

2

7

5

3

5

3

7

x

q

−

p

=

y

q

−

p

. It follows that

x

2

≡

0

(

mod

3

)

or

(

mod

3

)

0

2

≡

Then after dividing by 3, are finds

y

2

(

5

q

2

−

7

p

2

)

=

3

⋅

7

(

3

q

2

−

5

p

2

)

, with means that

5

q

2

−

7

p

2 must be divisible by 3.

From (x; y) = 1 it is clear that y is not divisible by 3. Now

implies

p

2

+

q

2

≡

0

(

mod

3

)

which is impossible unless p≡q≡0

(

mod3

)

, which can’t happen because (p; g) = 1.

Theorem 1 The practical quadratures at five knots, having maximal degree of exactness m = 5 are those of form

( )

[

( ) (

)

]

[

( ) (

)

]

( )

∫

+

     + − + +

− + =

1

0

2 2

1 1

2 1 1

1 r B f r f r Cf R f

f r f A dx x

f (8)

where R

( )

f

is remainder,

r₁,r₂ are distinct rational numbers from (0, 1] and

Let us note that in quadrature formula from (8) we have

If by

[

z

₀

,

z

₁

,...,

z

_k

;

f

]

is denoted the difference of a function f :

[ ]

0,1 →IR at a system of distinct points

{

z

₀

,

z

₁

,...,

z

_k

}

⊂

[ ]

0

,

1

, it may be shown that.

( )

(

r

)

_{r r r r f R}

( )

_f

f r r f dx x f +     _{− −} ⋅ − − + +     ₋ +       =

∫

; 1 , 1 , 2 1 , , 240 2 1 5 3 ; 1 , 2 1 , 12 1 2 1 1 2 2 1 2 1 1 0 1 1 (9)

where r₁,r₂ are distinct rational numbers from

(

0,1

]

. 2 Examples

In the following of R_j

( )

f , j∈N∗, we shall denote the remainders terms in certain quadratures formulas.

Example 1. The closed formulas like (8) are obtained in case r₂ =1, namely

( )

x dx A

[

f

( ) ( )

f

]

C f B

[

f

( ) ( )

r f r

]

R

( )

f

f _{0 0 0 1}

1 0 1 2 1 1

0 + + − +

     + + =

∫

(10)

where

( ) ( )

(

)

6 6 6 1

2

105

6

2

1

14

,

1

,

0

,

⋅

−

−

=

∈

∈

Q

r

R

e

r

r

and

Example 2. For instance, when

( )













=

2

1

;

1

,

₂ 1

r

r

, (10) gives

( )

[

( ) ( )

]

( )

∫

_ + _      +       +       + + = 1 0 2 2 1 25 2 4 3 4 1 25 16 1 0 90 7 f R f f f f f dx x

f (11)

( )

6 7

2

2

21

1

⋅

=

e

R

.

Example 3. In case

( )













=

4

1

;

2

1

,

₂ 1

r

r

are finds

( )

( )

∫

_ + _      +       +       −     _      +       = 1 0 3 2 1 15 107 8 5 8 3 45 224 4 3 4 1 45 86 f R f f f f f dx x

f (12)

3 The remainder term

In order to investigated the remainder we use same methods as in [1]-[6].

Theorem 3 Let _{1 1 2 2 3}

,

₄

1

₂

,

₅

1

₁

2

1

,

,

,

2

1

,

2

1

r

x

r

x

x

r

x

r

x

h

m

=

=

=

=

=

=

−

=

−

.

If

( )

(

)

(

)

.

4

1

2

1

4

1

2

1

₁ 2 2 ₂ 2

2









−

−

⋅









−

−

⋅

=

Ω

t

t

r

t

r

( )

f t

( )

t t r r r r t f dt

R

  

 _{− − − +}

Ω =

∫

−

; 2 1 , 1 , 1 , 2 1 , , , 2 1

1 2 2

1 2

1

2 1

2

.

Proof. Let

( )

∏

(

)

=

−

=

5

1

j

j

x

x

x

ω

. Because our formula (8) is of interpolatory type, it

follows that we have

where

( )

=

∫

( )

[

]

1

0

5 4 3 2

1, , , , ;

,x x x x x f dx

x x f

R

ω

.

But

∫

( )

− =

∫

( )

1

0

1

0

1 x dx f x dx

f and using the symmetry of knots

{

x

₁

,

x

₂

,...,

x

₅

}

we have

Further, the equality

ω

( )

1−x =−

ω

( )

x gives

Therefore the remainder from (8) may be written as

( )

=

∫

( ) ( )

1

; 1

dx x f D x f

In this manner

which is the same with (13).

Further for g∈C

[ ]

0,1 we use the uniform norm

[ ]

g

( )

x

g

b a x ,

max

∈

=

.

Corollary 4 Let us denote

If R

( )

f is the remainder in (8), then for

f

∈

C

6

[ ]

0

,

1

( )

( )

( )6

2 1

,

46080

1

f

r

r

J

f

R

≤

.

References

[1] Brass H., Quadraturverfahren, Vandenhoeck & Ruprecht, G¨ottingen, 1977.

[2] Ghizzetti A., Ossicini A., Quadrature Formulae, Birkh¨auser Verlag Basel, Stuttgart, 1970.

[3] Krylov V.I., Approximate calculation of integrals, Macmillan, New York, 1962. [4] Lupas A., Teoreme de medie pentru transform¢ari liniare ¸si pozitive (Romanian), Rev. de Anal. Num. ¸si Teor. Aprox. 3, 1974, 2, 121-140.

[5] Lupas A., Contributions to the theory of approximation by linear operators, Dissertation, Romanian, Cluj, 1975.

[6] Lupas A., Metode numerice (Romanian), Ed. Constant, Sibiu, 2001. Author:

Constantinescu Eugen Department of Mathematics “Lucian Blaga” University of Sibiu Str. Dr. I. Rat¸iu, nr. 5-7

550012 Sibiu, Romˆania.