- Email: [email protected]
The construction of cubature formulae and orthogonal polynomials
THE CONSTRUCTIONOF CUBATURE FORMULAE AND ORTROGONALPOLYNOMIALS*+ I. P.
MYSOVSKIKH
Leningrad 4 January
(Received
CONBIDER the
where sional
formula
x = (~1, . . . . x,,), dx = dxl, . . . . dx,, D is a region in n-dimenEuclidean space. We can assume that the moments of the weight
function
p(x)
following Theorem
cubature
1966)
exist,
theorem
that is
it
is non-negative
in D and
.c D
p(s)&
> 0.
The
obvious.
f
If formula (1) is exact, when f(x) is any polynomial of degree 2k - 1, and its nodes belong to the hypersurface P(x) = 0 of order k (this means that P(x) is a polynomial of degree k), then P(x) is an orthogonal polynomial of region 9 and weight p(r), in other words
for
any polynomial The converse vychis
Q(z)
of degree
of Theorem 1.
Mat.
1 is
not also
*
Zh.
f
A report given to the All-Union matics in Moscow. January 1965.
mat.
Fiz.
7,
greater
than k - 1.
true.
1, 185 - 189. Conference
252
1967.
on Computational
lathe-
Cubature
Theorem
formulae
and
orthogonal
253
polynomials
2
There exist n linearly independent polynomials Pi(x) of degree k, which are orthogonal relative to the region D and of weight p(r) such that the solutions z(j ) = (x,(j ), . . . , z,(j )), ,j = 1, 2, . . . , N. of the system Pi(Z1, . . ., 2,) = 0, can be taken as the exact for polynomials
i = 1, 2, . . . , n,
nodes of the cubature of degree 2k - 1.
formula
for
(2)
D and weight
p(x),
When n = 1, this is the well-known theorem. It has not been proved for the general case. The difficulty lies in the fact that when n 2 2 we can write a system of the form (2) in an infinite number of ways, since there are (n + k - 1) ! (n -
I)! k!
linearly independent orthogonal polynomials of degree k of n variables. As a rule, system (2) comprising n haphazardly chosen linearly independent orthogonal polynomials of degree k will not possess the property described in Theorem 2. The question of connecting cubature formulae and orthogonal polyTheorem 2 is proved nomials is considered in [il and 121. In particular, bounded in [21 for n = 2 and k = 3, when p(z) = 1 and 1) is an arbitrary region on a plane. Suppose there is a cubature formula which is exact for polynomials of degree 2k - 1. It is interesting to show the orthogonal polynomials of degree k whose common roots are nodes of the formula. In this paper we shall indicate the solution of this problem for three specific cubature formulae. We shall then construct cubature formulae for an ndimensional cube and a sphere which are exact for fifth-degree polynomials, the nodes of which are solutions of a system of the form (2). In other words, we shall prove Theorem 2 for these regions and for k = 3. For the square D: -1 < 3c, y < 1 we know the following with weight p(x, y) = 1, which is exact for seventh-degree and has 12 nodes:
ss
f(z,y)dzdy:AIf(a,O)+f(O,a)+f(--a,O)+f(O,-a)l+
cubature formula polynomials
(3)
I.P.
254
Mysovskikh
+B[f(b, q+ f(b, --bj+ f(--b, b)+ f(-b, +
-b)l+
CY(c,c)+ f(c, -c)+
o-c, c)+
f(-c,
-c)l.
where
Let Oj(x) denote a Legendre polynomial the coefficient of xj is equal to 1. Five polynomials of the fourth degree for the using Oj(x) as P(X, Y) = 1 can be written PI-j,j(2, Let
us write
out
the
Y) = (Jk-j(t)aj(Y)9 system
of degree j, normalized so that linearly independent orthogonal square -1~ z, y< 1 and weight follows: j = 0, 1, 2, 3, 4.
(4)
of equations
P40 + Par + o&z = 0,
Pto - PO4= 0,
(5)
the left-hand sides of which are a linear combination with constant coefficients of the polynomials (4) and, therefore, are orthogonal polyFor a = 54/55 the system has twelve solunomials of the fourth degree. tions and these solutions are nodes of the cubature formula (3).
FIG.
The graphs of the of system (5) for intersection. Thus we are
2.
fourth-order algebraic curves defined by the equations a = 54/55 are given in Fig. 1, with their points of
correct
in calling
formula
(3)
a cubature
formula
of the
formulae
Cubature
and
orthogonal
255
polynomials
Gauss type: its nodes are the roots of orthogonal polynomials. We note that such a formula is not unique. For example, there is a cubature formula which is exact for seventh-degree polynomials and has 16 nodes. It can be obtained by a double application of the Gauss auadrature formula, and its nodes are the solutions of the system p40
This
system
is
equivalent
=
to
PO4 =
0,
system
(5)
0.
with
a = 0.
For the circle x2 + y2 < 1 and weight p(x, y) = 1 there is a cubature formula of form (3) which is exact for seventh-degree polynomials. Its twelve nodes are the solutions of p40
where ‘iS(X’
PO6
+
&2
o = “1, and P&O= Pto(z) = +
of the
for
+
y*)
+
l/68
algebraic
Let us also the sphere
are
X4
=
-
orthogonal
curves
defined
0,
3/4X2
p60
+
-
'/ia,
PO4
PO6
polynomials by (6)
consider the cubature D: x2 + y2 + z2< 1:
=
=
P40(y),
for
are
the
similar
formula
(6)
0,
&
circle.
to those
given
by V.A.
=
X2yz
-
The graphs of Fig. Ditkin
1. [31
12
1 S1 f(x,Y,z)dxdYdZr~f(O,O,O)+~~ f(xi,Yi,zi), i=1
D
this is exact for of the icosahedron radius
lx
and the
x(22 + y2 + 22 -
fifth degree polynomials. Its nodes are the vertices inscribed in a sphere with centre at the origin and coordinate S/7) = 0,
origin. Y(z” -
has as its solutions the nodes of tions are orthogonal third-degree first equation defines the surface The two other equations define six of the surfaces defined by equations Suppose now that . . . ) n. We construct exact for fifth-degree
The set
of equations
[(z+~x)~--5/,](x-222)
‘h) = 0,
= 0
(6)
(7). The left-hand sides of the eauapolynomials for the sphere n. The of the sphere and the plane x = 0. planes. The points of intersection (8) are shown in Fig. 2.
D is an n-dimensional cube: -1 < xi < 1, i = 1, 2, a cubature formula with weight p(n) = 1, which is polynomials. Consider the system 5,
r12+...+xnz-p
5n+4
15
(3)
256
I.P.
where the left-hand sides for D and for weight p(x) The solutions
of
(9)
of the
Wysovskikh
equations
are
orthogonal
polynomials
= 1.
can be written
as follows:
(0, 0, .-., 01, 0,_ 0, .. . . .
t
i-l
$*pqT *&, 5&,.‘.&&}~
as the nodes Let us choose these Zn+l - 1 points and determine its coefficients with the condition be exact for fifth-degree polynomials. We have
c
of the cubature formula, that the formula must
2n+2
--ff(0, f(s)dr =
.
i = 1,2, . . ., n.
o)+
.‘.,
5n + 4
D
2”+i--i 5.2if’ +i
The inside different
5i + 4
25 * f (0,.
iJ5w5i+4;
1/ e
.., 0, f
,f---
15
1
_I es., *’
13
1/g )
*
sum here contains 2 nf1-i terms which can be obtained for coordinates. combinations of the + and - signs of the non-zero
formula for the sphere D: xl2 + . . . + x,2 g 1 with weight The cubature in exactly the same way, and is exact for fifth= 1 is constructed degree polynomials. From orthogonal third-degree polynomials for the sphere we form the system P(X)
5,
n+2
Z,24..*fXn2---(
nt4
>
1 “j-1
5j2--
=()
n+4
>
i
= 0,
3,4, . . . , n,,
=
’
2i(212 - 3s7.2). = 0. It
has
Zntl
- 1 solutions: (4 4 ..*, 91,
i
. .. . .. ?,& o,o* i-l
v--i+2 -n+4
WI..., ’ *l/n%+4
*
=,
1/nf+4
J
i = 1, 2,
, It,
Cubature
which we shall form
s
f(z)dx
r
D
take
faraulae
and
nodes.
8s the
The required
(n +
2nniz --f(O,
cubstnre
257
formula has the
4) .%“I2
. . . . o)+
(h + 2) l’(n/2 + 2)
X T(n/2 + 2)
2n+f-i
n
polynomials
orthogonal
- ,
i+2
0, . . . I 0, f
1 * .-x--, ,‘n + 4
n+4
1
...( i
--
I
w+4
The problem of connecting cubature formulae and orthogonal POSYnomlals is also discussed in i41. In particular, the orthogonal thirddegree polynomials PJO= 8.z~- 122,
Pz, =
wy
-
4y,
PI1 = 8zyZ - 4x,
Pas =
8~” - 12~. (f(l)
with weight emx2- Y* are given for n = 2, when the region of integration 1s the whole plane. It is shown that the solutions of the system P3o=O, PO3 = 0 can be taken as the nodes of the cubature formula with weight e-xz -y 2 , exact for fifth-degree polynomials, and that there is no other pair of polynomials (10) which will form a system with this property. This is true, but it has the disadvantage that the systems for determining the nodes must be constructed from linear combinations of the polynomials (101, and not simply from these polynomials themselves. We would, for example, form the system P30 = 0, P21 + Fe3 = 0, the solutions of which lead to the required cubature formula. Translated
by
R.
Feinstein
1.
APPELL, M.P. Sur une classe de polynomes a deux variables et le calcul approche des integralea doubles. Annls. Foe. Sci. Univ. Toulause. 4. H. 1 - 20, 1890.
2.
RADON,
3.
DITKIN, V.A. Some approximate farmulae grals, I)okZ. Akad. Nauk SSSR, 62, 4,
4.
S.
Zur mechanischen
Kubatur. Mfr. ,ibath. for 445
52, 4. 286 - 300,
calculating triple - 447, 1948.
1948. inte-
STKOUD,A.H. and SECREST, D. Approximate Integration formulae for certain spherically symmetric regions. Uath. Coarput. 17, 82, 105 135. 1963.
MYSOVSKIKH
Leningrad 4 January
(Received
CONBIDER the
where sional
formula
x = (~1, . . . . x,,), dx = dxl, . . . . dx,, D is a region in n-dimenEuclidean space. We can assume that the moments of the weight
function
p(x)
following Theorem
cubature
1966)
exist,
theorem
that is
it
is non-negative
in D and
.c D
p(s)&
> 0.
The
obvious.
f
If formula (1) is exact, when f(x) is any polynomial of degree 2k - 1, and its nodes belong to the hypersurface P(x) = 0 of order k (this means that P(x) is a polynomial of degree k), then P(x) is an orthogonal polynomial of region 9 and weight p(r), in other words
for
any polynomial The converse vychis
Q(z)
of degree
of Theorem 1.
Mat.
1 is
not also
*
Zh.
f
A report given to the All-Union matics in Moscow. January 1965.
mat.
Fiz.
7,
greater
than k - 1.
true.
1, 185 - 189. Conference
252
1967.
on Computational
lathe-
Cubature
Theorem
formulae
and
orthogonal
253
polynomials
2
There exist n linearly independent polynomials Pi(x) of degree k, which are orthogonal relative to the region D and of weight p(r) such that the solutions z(j ) = (x,(j ), . . . , z,(j )), ,j = 1, 2, . . . , N. of the system Pi(Z1, . . ., 2,) = 0, can be taken as the exact for polynomials
i = 1, 2, . . . , n,
nodes of the cubature of degree 2k - 1.
formula
for
(2)
D and weight
p(x),
When n = 1, this is the well-known theorem. It has not been proved for the general case. The difficulty lies in the fact that when n 2 2 we can write a system of the form (2) in an infinite number of ways, since there are (n + k - 1) ! (n -
I)! k!
linearly independent orthogonal polynomials of degree k of n variables. As a rule, system (2) comprising n haphazardly chosen linearly independent orthogonal polynomials of degree k will not possess the property described in Theorem 2. The question of connecting cubature formulae and orthogonal polyTheorem 2 is proved nomials is considered in [il and 121. In particular, bounded in [21 for n = 2 and k = 3, when p(z) = 1 and 1) is an arbitrary region on a plane. Suppose there is a cubature formula which is exact for polynomials of degree 2k - 1. It is interesting to show the orthogonal polynomials of degree k whose common roots are nodes of the formula. In this paper we shall indicate the solution of this problem for three specific cubature formulae. We shall then construct cubature formulae for an ndimensional cube and a sphere which are exact for fifth-degree polynomials, the nodes of which are solutions of a system of the form (2). In other words, we shall prove Theorem 2 for these regions and for k = 3. For the square D: -1 < 3c, y < 1 we know the following with weight p(x, y) = 1, which is exact for seventh-degree and has 12 nodes:
ss
f(z,y)dzdy:AIf(a,O)+f(O,a)+f(--a,O)+f(O,-a)l+
cubature formula polynomials
(3)
I.P.
254
Mysovskikh
+B[f(b, q+ f(b, --bj+ f(--b, b)+ f(-b, +
-b)l+
CY(c,c)+ f(c, -c)+
o-c, c)+
f(-c,
-c)l.
where
Let Oj(x) denote a Legendre polynomial the coefficient of xj is equal to 1. Five polynomials of the fourth degree for the using Oj(x) as P(X, Y) = 1 can be written PI-j,j(2, Let
us write
out
the
Y) = (Jk-j(t)aj(Y)9 system
of degree j, normalized so that linearly independent orthogonal square -1~ z, y< 1 and weight follows: j = 0, 1, 2, 3, 4.
(4)
of equations
P40 + Par + o&z = 0,
Pto - PO4= 0,
(5)
the left-hand sides of which are a linear combination with constant coefficients of the polynomials (4) and, therefore, are orthogonal polyFor a = 54/55 the system has twelve solunomials of the fourth degree. tions and these solutions are nodes of the cubature formula (3).
FIG.
The graphs of the of system (5) for intersection. Thus we are
2.
fourth-order algebraic curves defined by the equations a = 54/55 are given in Fig. 1, with their points of
correct
in calling
formula
(3)
a cubature
formula
of the
formulae
Cubature
and
orthogonal
255
polynomials
Gauss type: its nodes are the roots of orthogonal polynomials. We note that such a formula is not unique. For example, there is a cubature formula which is exact for seventh-degree polynomials and has 16 nodes. It can be obtained by a double application of the Gauss auadrature formula, and its nodes are the solutions of the system p40
This
system
is
equivalent
=
to
PO4 =
0,
system
(5)
0.
with
a = 0.
For the circle x2 + y2 < 1 and weight p(x, y) = 1 there is a cubature formula of form (3) which is exact for seventh-degree polynomials. Its twelve nodes are the solutions of p40
where ‘iS(X’
PO6
+
&2
o = “1, and P&O= Pto(z) = +
of the
for
+
y*)
+
l/68
algebraic
Let us also the sphere
are
X4
=
-
orthogonal
curves
defined
0,
3/4X2
p60
+
-
'/ia,
PO4
PO6
polynomials by (6)
consider the cubature D: x2 + y2 + z2< 1:
=
=
P40(y),
for
are
the
similar
formula
(6)
0,
&
circle.
to those
given
by V.A.
=
X2yz
-
The graphs of Fig. Ditkin
1. [31
12
1 S1 f(x,Y,z)dxdYdZr~f(O,O,O)+~~ f(xi,Yi,zi), i=1
D
this is exact for of the icosahedron radius
lx
and the
x(22 + y2 + 22 -
fifth degree polynomials. Its nodes are the vertices inscribed in a sphere with centre at the origin and coordinate S/7) = 0,
origin. Y(z” -
has as its solutions the nodes of tions are orthogonal third-degree first equation defines the surface The two other equations define six of the surfaces defined by equations Suppose now that . . . ) n. We construct exact for fifth-degree
The set
of equations
[(z+~x)~--5/,](x-222)
‘h) = 0,
= 0
(6)
(7). The left-hand sides of the eauapolynomials for the sphere n. The of the sphere and the plane x = 0. planes. The points of intersection (8) are shown in Fig. 2.
D is an n-dimensional cube: -1 < xi < 1, i = 1, 2, a cubature formula with weight p(n) = 1, which is polynomials. Consider the system 5,
r12+...+xnz-p
5n+4
15
(3)
256
I.P.
where the left-hand sides for D and for weight p(x) The solutions
of
(9)
of the
Wysovskikh
equations
are
orthogonal
polynomials
= 1.
can be written
as follows:
(0, 0, .-., 01, 0,_ 0, .. . . .
t
i-l
$*pqT *&, 5&,.‘.&&}~
as the nodes Let us choose these Zn+l - 1 points and determine its coefficients with the condition be exact for fifth-degree polynomials. We have
c
of the cubature formula, that the formula must
2n+2
--ff(0, f(s)dr =
.
i = 1,2, . . ., n.
o)+
.‘.,
5n + 4
D
2”+i--i 5.2if’ +i
The inside different
5i + 4
25 * f (0,.
iJ5w5i+4;
1/ e
.., 0, f
,f---
15
1
_I es., *’
13
1/g )
*
sum here contains 2 nf1-i terms which can be obtained for coordinates. combinations of the + and - signs of the non-zero
formula for the sphere D: xl2 + . . . + x,2 g 1 with weight The cubature in exactly the same way, and is exact for fifth= 1 is constructed degree polynomials. From orthogonal third-degree polynomials for the sphere we form the system P(X)
5,
n+2
Z,24..*fXn2---(
nt4
>
1 “j-1
5j2--
=()
n+4
>
i
= 0,
3,4, . . . , n,,
=
’
2i(212 - 3s7.2). = 0. It
has
Zntl
- 1 solutions: (4 4 ..*, 91,
i
. .. . .. ?,& o,o* i-l
v--i+2 -n+4
WI..., ’ *l/n%+4
*
=,
1/nf+4
J
i = 1, 2,
, It,
Cubature
which we shall form
s
f(z)dx
r
D
take
faraulae
and
nodes.
8s the
The required
(n +
2nniz --f(O,
cubstnre
257
formula has the
4) .%“I2
. . . . o)+
(h + 2) l’(n/2 + 2)
X T(n/2 + 2)
2n+f-i
n
polynomials
orthogonal
- ,
i+2
0, . . . I 0, f
1 * .-x--, ,‘n + 4
n+4
1
...( i
--
I
w+4
The problem of connecting cubature formulae and orthogonal POSYnomlals is also discussed in i41. In particular, the orthogonal thirddegree polynomials PJO= 8.z~- 122,
Pz, =
wy
-
4y,
PI1 = 8zyZ - 4x,
Pas =
8~” - 12~. (f(l)
with weight emx2- Y* are given for n = 2, when the region of integration 1s the whole plane. It is shown that the solutions of the system P3o=O, PO3 = 0 can be taken as the nodes of the cubature formula with weight e-xz -y 2 , exact for fifth-degree polynomials, and that there is no other pair of polynomials (10) which will form a system with this property. This is true, but it has the disadvantage that the systems for determining the nodes must be constructed from linear combinations of the polynomials (101, and not simply from these polynomials themselves. We would, for example, form the system P30 = 0, P21 + Fe3 = 0, the solutions of which lead to the required cubature formula. Translated
by
R.
Feinstein
1.
APPELL, M.P. Sur une classe de polynomes a deux variables et le calcul approche des integralea doubles. Annls. Foe. Sci. Univ. Toulause. 4. H. 1 - 20, 1890.
2.
RADON,
3.
DITKIN, V.A. Some approximate farmulae grals, I)okZ. Akad. Nauk SSSR, 62, 4,
4.
S.
Zur mechanischen
Kubatur. Mfr. ,ibath. for 445
52, 4. 286 - 300,
calculating triple - 447, 1948.
1948. inte-
STKOUD,A.H. and SECREST, D. Approximate Integration formulae for certain spherically symmetric regions. Uath. Coarput. 17, 82, 105 135. 1963.