- Email: [email protected]
Discrete orthogonal polynomials inherent to sampling points at equal intervals
Computer methods in applied mechanics and englneerlng Comput. Methods Appt. Mech. Engrg. 123 (1995) 371-383
ELSEVIER
Discrete orthogonal
polynomials inherent to sampling points at equal intervals Masayuki Okabe
Optopian Inc., 2-8-21 Sakuradai, Hasuda, Saitama 349-01, Japan
Received 9 November 1994
Abstract The discrete orthogonal polynomials in finite series are proposed for arbitrary number of sampling points at equal intervals. Recurrence rules are presented so as to realize recurrent computations in the spectral data smoothing applications
1. Introduction
Polynomial approximations are of great significance in science. Polynomial interpolations, for example, play an important role, not only in numerical simulation [l, 21, but also in surface generation [3]. For the spectral data smoothing that we are concerned with, a variety of powerful methods have been developed [4-61. In case of such smooth spectra without rapid absorption as near infrared ones of the fruit surface refelectance where polynomial interpolations could fit the curve in good accuracy, the polynomial smoothing in least-square sense [7,8] is attractive and to be reconsidered so that much more flexible optimization strategy can be constructed. It is known for the spectral white noise that the noise standard deviation after smoothing could be proportional to 1 /fi where N is the number of sampling points used. To keep the spectral profile accurate, however, rather higher-order polynomials are needed, especially for the large number of sampling points. Although some effective digital filters are proposed by Savitzky and Ziegler [7], we need much more powerful filters which should also be adequate for data smoothing near both ends of the spectral range. To achieve the automatical convergence in data smoothing, furthermore, the so-called p-convergence procedures developed in finite element fields [9, lo] are to be adopted. Here, the polynomial approximation degree is adaptively increased until the sufficient accuracy is attained. It is noted that special polynomial interpolations are required for this purpose. This paper is devoted to develop the orthogonal polynomials tightly connected with the discrete sampling points at equal intervals, which may yield new capabilities in the above-mentioned applications. It is shown that proposed recurrence rules are quite useful in practical computations.
2. Description of the problem In the x system of Fig. 1, consider 2n + 1 points placed at equal intervals in (-l,l). a discrete operator L is defined by
W(x)1 = i--n Ii fG/n) . Oil4578251951 $09.50 0 1995 Elsevier Science S.A. All rights reserved 0045-7825(94)00768-3
SSDZ
For smooth f(x),
(1)
M. Okabe I Comput. Methods Appl. Mech. Engrg. 123 (1995) 371-383
372
-2
-ll
-1
0
1
n
2
x=1
x=0
x=-l
Fig. 1. Placement of 2n + 1 sampling points at equal intervals in the x system.
This paper aims to develop the orthogonal but in finite series so that L[P,P,]
= P&,
polynomials
{P,}z=,
inherent to the discrete operator
.
(2)
Here I and m are non-negative
integers, S,, is the Kronecker
delta and
Pm = L[Pm2]. REMARK
2.1.
L
(3)
For non-negative
integer k, the following relations hold
L[x*~] = Jzk = 212k/n2k ,
(4)
L[X2k+1] = 0 )
(5)
where k=O,
2n+1)/2, C(
(6)
k>O.
For explicit formulations of Z2k, see Appendix A. Noting (5), the orthogonal polynomials P,,, could be even and odd functions corresponding and odd y, respectively. Then P,,, can be expressed as Pm = 5
(-l)iCj,,Xm-2i
to even
(7)
)
i=O
where K,,,= [m/2],
(8)
.
Here, [ .lG denotes the Gauss notation. The cOlm coefficient is fixed to unity (9)
co/m = 1, and the other coefficients in (7) could be determined L]X m-2iPm]=0,
j=I
Then, (3) can be rewritten i), = L[x"P,]
,...,
by the orthogonal
conditions of the form
Km.
(10)
as
.
(11)
3. Strong existence assumption
If the discrete orthogonal polynomials recurrence relation of the form P,=l, P m+l
P,=x, =xPm -A,P,,_,
,
mB1,
P,,,
with the orthogonal
conditions
(10) and (11) satisfy the
(12)
M. Okabe
I Comput. Methods Appl. Mech. Engrg.
123 (1995) 371-383
373
then we call P, strongly existing. THEOREM
3.1.
For strongly existing P,,
the following relation holds
A,,, = P,lPm_l. PROOF.
(13)
From (12), we have
L[x”P,,,]
- A,L[x”-‘P,_,]
THEOREM
3.2.
L[Xmt2
= L[x”-‘P,,,]
= 0.
0
For strongly existing P,,,, the following relation holds
P,l = clim+ZP~
(14)
.
PROOF. L[P,+,P,J THEOREM
= L[x”+2P,]
3.3.
- c,,,+,L[x”P,]
= 0.
q
Suppose P,,, strongly exists, then the coefficients in (7) can recurrently be written as
Ci/m+l =Ci/m +
A,Ci-l/m-1
7
i=l
7
*
*
.
9
Km .
Here we prescribe ‘i/m =O,
i
ASSUMPTION A,
3.4.
m
= m2(2n -t 1 - m)(2n + 1 + m)/4(2m
THEOREM cl,, PROOF.
3.5.
Under Assumption
- 1)(2m + l)n2 .
(16)
3.4, the cl,,,, coefficient is given explicitly as
= m(m - 1){12n(n -I-1) - (m - 2)(m + 1)) /24(2m - l)n* .
(17)
In case of i = 1, (15) is written as
c l/m+1
=
Cl/m
+A,,
(18)
which is surely realized by (17) and (16).
0
Our target is thus to prove the strong existence of the discrete orthogonal explicit form (16) of A,.
polynomials
P, with the
4. Mathematical preparation
In this section we deal with both n and n - 1 cases denoting by (n) and (n - 1) on the shoulder, respectively. LEMMA
4.1.
With respect to JZk in (41, the following relation holds
n2k+2{J $‘+2 - Je} PROOF.
= (n - l)‘k{(n - l)*j$;:’
- n*Jg-‘I}
,
(19)
From the definition of IZk in (6), we have
I$! _ 1(2nk-0= n2k
9
which yields IgJ+2 - n2Z$ = Ifk;:’ - n*lg-l) Thus, we have (19) for nonnegative
.
integer k.
(20) El
374
M. Okabe I Comput. Methods Appl. Mech. Engrg.
LEMMA
4 .2 . Let G9-l) r,m
G!T,-l) = 5
123 (1995) 371-383
and G!:$ be defined by
- 1)2J~~-:;i_zj+z - nzJsnm‘L;i_2i} ,
(-l)‘c,‘;;“{(n
j=O
(21) - Jp;_2i_zi}
G;‘J; = n2 2 (-l)‘~;;;{j&~_~~+~
,
i=O,...,K,.
j=O
Then we have go (-l)‘(n
- 1)2’n2m-2icj;,-1)Gj$
PROOF.
Substitution
LEMMA
4.3.
Gjxl)
= I$o (-l)‘(n
of (21)
GiyG’) and Gi:A
0
can be expressed as
)
i = 0,
. . . ) Km
(23)
.
Suppose the following orthogonal conditions hold
b m-2jp,]
= 0,
L(n)[xm-2j P,,,]=o, then
(22)
= (n _ 1)2L(n-1)[Xm-2i+2pm] _ n2L(n-1)[Xm-2ipm],
4.4.
LQ-1)
.
of (21) and (19) into (22) completes the proof.
G!‘$ = n2{L(“)[x m-2i+2pm] - L’“‘[x”-“iPm]} LEMMA
- ,)2”-2in2ic;;;Gj?;1)
j=l,.
..
(24) ,Km,
have
we
n2m{n2L@jxm+2 P,] - {n’ + (n - 1)2c1;,1’}L(“)[x”P,]} = (n - l)‘“{(n PROOF.
Lemmas 4.2 and 4.3 with (24) give (25).
LEMMA 4.5. such that LWl)
- l)2L(n-1)[~m+2Pm] - n’(1 + ci’f~)L(“-l)[xmPm]}
(25)
.
0
Suppose (13), (16) and (24) hold for both n - 1 and n and suppose (14) holds for n - 1 (n-1)
]x m+2en1 = Cl/m+2
)
p-u m
(26)
.
(27)
then we have L’“‘[x”+2P,] PROOF.
= c;;~+,a~’
From (13) and (16), we have
(n _ 1)2-p:-‘)
/n2”PE’
= (2n - m)(2n - m - 1)/(2n
+ m)(2n + m + 1) .
On the other hand, (17) yields n2cin/k+2 - n2 - (n - l)“cl’f,”
= -(m’
+ m - 1)(2n - m)(2n - m - 1)/2(2m
(n - 1)2c(,:,:), - n2(1 + c(,l)m)= -(m” + m - 1)(2n + m)(2n + m + 1)/2(2m
Lemma 4.4 then gives (27).
+ 3)(2m - 1) ,
+ 3)(2m - 1) .
Cl
LEMMA 4.6. For the polynomials P,,, defined uniquely by (12) with (16), suppose P,,, and P,,,+ 1 satisfy the orthogonal conditions (10) and suppose (14) holds, then we have k+, PROOF.
= A,+$‘,,,
.
From (14), we have
Gw
M. Okabe
The orthogonal
I Comput. Methods Appl. Mech. Engrg.
375
123 (1995) 371-383
conditions with respect to P,,, and P,,,+l give
L]x m-2iPm+J = L[x~-~~+‘P,,,+J
- A,+1L[xm-2iP,]
= 0,
i = 1, . . . , K, .
Hence 0 = L[x”P,+,] = L[xm+l
pm+11 - 4n+,wRI
which completes the proof.
9
0
5. Strong existence of the discrete orthogonal polynomials THEOREM, 5.1. The discrete orthogonal polynomials recurrence rule holds
P,,, are strongly existing so that the following
P1=x,
PO=l,
P m+l=xP,,,-A,P,_l,
(29)
mZ1,
where A,
= m2(2n + 1 - m)(2n + 1 + m)/4(2m
- 1)(2m + l)n2.
PROOF.
(30)
It suffices to prove that the polynomials uniquely determined orthogonal conditions (10) and (13) or (14). For m =0 and 1, we have
by (29) with (30) realize the
P0 = 2n + 1 , L[x2P,] 8, = A& L[x3P,]
(31)
= c1,2i)0 = (2n + l)(n + 1)/3n = J2 , = c~,~~)O, = c1,3i)l = (2n + l)(n + 1){3n(n
(32)
+ 1) - 1}/15n3 = J4.
Suppose here that P,,, by (29) with (30) satisfies the orthogonal evidently we have
conditions
and (13) upto m, then
L]x m+1-2jPm+l] = L[X”+2-2jP,] - A,L[Xm+‘-2jp,_,] =o, j=l,..., Km+l. Next, suppose that (14) holds upto n - 1. Then from Lemma 4.5, (14) holds for n. Noting (32) guarantees (14) for m = 0 and 1, (14) is thus proved. Cl Lemma 4.6 further guarantees (28), and hence strong existence of P,,, is proved. THEOREM
5.2.
For the discrete orthogonal polynomials P,,,, the following relations hold
i), = 2n + 1 ) p,,,=A,p,,_,
,
mE1.
(33)
6. Extended discrete orthogonal polynomials
Let N be the number of sampling points by N=2n+l.
(34)
376
M. Okabe / Comput. Methods Appl. Mech. Engrg.
123 (1995) 371-383
Substitution of (34) into Theorem 5.1 then gives the extended discrete orthogonal are also adequate for even N (Appendix B). THEOREM 6.1. The extended discrete orthogonal polynomials recurrence rule of the form
{P,},“::
polynomials
are strongly existing with the
P1=x,
P()=l,
Pm+l =xP,,, -A,P,,_,
,
which
(35)
mZ1,
where A,
= m2(N - m)(N + m)/(2m
THEOREM
6.2.
- 1)(2m + l)(N - 1)” .
For the extended discrete orthogonal polynomials P,,,, the following relations hold
P()=N, P,,,=A,i),_I, THEOREM Ci/m+l
=
REMARK
(37)
m21.
6.3. Cilm
(36)
Coefficients in (7) of the extended discrete orthogonal polynomials satisfy +
A,c~_~,,,_~
,
i = 1, . . . ,
K,
(38)
.
6.4.
Roots of (N - l)‘“P* with respect to N are N = 0 and N = *i (i = 1, . . . , m). That is, orthogonal polynomials are not in infinite series. For N points, only P,, to PN_I are
the discrete adequate. THEOREM
6.5.
Let
D=d/dx,
(39)
then (35) gives DP, = 1,
DP,=O,
DP,+,
=xDP,,,-A,DP,,_,+P,,,, D2P,=0,
D*P,,=O, D2P,+*
(40)
mZ1,
=xD*P,,,-A,D2P,,_1
+2DP,,,
,
mZ1.
(41)
7. Application examples and concluding remarks
We have thus developed significant applications. 7.1.
One-dimensional
the discrete orthogonal
polynomials in finite series. Here we refer to some
data smoothing (curve fitting)
Within (-1, I), let the trial function be expressed as f(x) = C,
a,P,(x)
.
(42)
Here a,,, denotes the coefficient to be determined, and M is the polynomial degree of our trial function. We then introduce a least-square functional for 2n + 1 sampling points by *=+,jt
where f:
i--n
{f(iln)-ft>*,
denotes the measured data at the sampling point of x = i/n.
(43)
M. Okabe
Differentiation Q/da,
I Comput. Methods Appl. Mech. Engrg.
123 (1995) 371-383
371
of (43) with respect to a, gives
= 5 a,L[P,P,] I=0 =a,i),-
-
i: P,(iln)f: i=-n
i P,(iln)fT i=-n
=o.
Immediately a, = [ i: P,(iln)fT] i=_n
(44)
Ii), .
The trial function (42) at the sampling point j of x = j/n takes the value of (45) If we express f( j/n)
in convolution
form by
f(jln) = i: wi.fT,
(46)
1=--n
then the weight wi can be written as P,(i/n)P,( m=O
w{ = 2
j/n)li),
.
(47)
At the origin of x = 0, (47) is written as wp = 5
m=o
Pm(i/n)Pm(0)/Pm
For non-negative P2k(O) = (-l)Lz, Px+1(0)
.
(48)
integer k, we have 7
= 0.
(49) (50)
It is obvious in (50) that (46) with (48) has the precision of order 2k + 1 even if our trial function (42) is of degree 2k. This feature is realized only at the origin. Savitzky and Ziegler [7] present explicit values of WYfor several odd points (N = 2n + 1) which have been efficiently used, for example, in spectroscopic data smoothing with some modification by Bromba and Ziegler [8]. At the other sampling points except for the origin, however, no formulas are proposed and consequently data smoothing has never been performed near ends of the spectroscopic range. It is emphasized that the proposed recurrent structure of the discrete orthogonal polynomials enable us in least-square data smoothing to adopt the so-called p-convergence procedures originally developed in finite element fields [9, lo]. Here the approximation degree M in our trial function (42) is automatically increased until sufficient smoothness is obtained with required accuracy. REMARK
7.1. In computations, recurrence rules (35) and (37) with (36) are needed without explicit formulations of Pm. For explicit form of P,,, upto 5, see Appendix C. First and second derivatives of f(x) can also be computed by using the recurrence relations (40) and (41), respectively. REMARK 7.2. For N points, our polynomial trial space is to be restricted This feature is clearly observed in Remark 6.4.
to degree N - 1 at most.
378
M. Okabe
REMARK
Legendre
I Comput. Methods Appl. Mech. Engrg.
123 (1995) 371-383
7.3. Our discrete orthogonal polynomials can thus be recognized as a discrete version of the polynomials.
7.2. Multi-dimensional data smoothing (surface fitting) Extension to multi dimensions is straightforward. can be written as fb,
Y) = 7
aijpi(x)pj(Y> .
7
Then we have in least-square aij
where
=
f
&
[C
ZZ
(51)
sense
pi(x,,P,(Yq)fzq]
P 4
In two dimensions, for example, our trial function
lpi’j
(52)
9
denotes the measured data at the sampling point (x,, y,).
7.3. Numerical integration Integration
of the discrete orthogonal
’ PZkdx = 2 i
I -1
(-l)i~i,2k/(2k
polynomials gives
- 2i + 1) ,
(53)
i=O
1
P 2k+l
dx
=
o
(54)
3
I -1
where k designates non-negative integer. For the F(x) function, we then have the following numerical integration formula in Newton type with N sampling points at equal intervals 1’ F(x)d_x=f -1
21”zG{i p=l
k=O
(-l)‘ci,,,/(2k-2i+l) i=O
1
.
{p2k
@2k>
1 F(x,)
)
(55)
where the abscissa is given by x,=(2p-N-l)/(N-I),
p=l,...,
N.
(56)
For N points, the formula due to the trial function of degree M = N - 1 is undoubtedly effective. REMARK
7.4.
the most
Noting (54), the integral formula utilizing upto P2k has the precision of order 2k + 1.
REMARK
7.5. For this integral use, the coefficient recurrence explicit form of those coefficients are given in Appendix C.
rule (38) is needed with (36). Some
7.4. Optimization example through the multiple regression analysis The problem we consider is to determine the calibration for peach brix (sugar content) t by the near infrared spectroscopic data from the fruit surface. Typical spectrum of the peach reflectance is shown in Fig. 2 which is composed of data at equal intervals by 2 nm from 650 nm to 1050 nm. It can be seen that the near infrared spectrum of fruits are so smooth that the polynomial interpolations could work well. The calibration model we adopt is t = a, + i
ai In ri _
(57)
i=l
Here, ri denotes the measured reflectance
at wave length Ai (A, = 850 nm, A, = 880 nm, A, = 910 nm,
M. Okabe
I Comput. Methods Appl. Mech. Engrg.
379
123 (1995) 371-383
Fig. 2. Typical measured spectrum of the peach reflectance.
through the mulitple A, = 950nm, A, = 970 nm). The coefficients a, and ui are to be determined regression analysis. From the spectral curve of Fig. 2, for simplicity, we prescribe the approximation degree to be M = 3. Our optimization problem is now written as Find the half width n of sampling points yielding the maximum multiple correlation due to the calibration model (57) with approximation degree of M = 3.
factor
It is emphasized that our optimization is not dependent apparently on the smoothing accuracy but directly on the statistical result. Fig. 3 shows the obtained correlation factor y vs. II. Here, 72 peach data are used. By the best
.88
0
.86 OQ 0
Q
0
o”
0 0
0 0
0
.82 0
.80
L-__
0
--
--A_
10
20
-s
n 30
Fig. 3. Multiple correlation factor due to the polynomial smoothing of sampling points 2n + 1 with approximation
degree M = 3.
380
M. Okabe
I Comput. Methods Appl. Mech. Engrg.
123 (1995) 371-383
Fig. 4. Smoothed result of Fig. 2 by the polynomial smoothing according to the best solution of n = 24 and M = 3.
solution of n = 24 (i.e. sampling points of N = 49) with A4 = 3, the measured spectrum of Fig. 2 is smoothed as shown in Fig. 4. Our solution is thus adequate also in the smoothing accuracy sense. It is especially remarked that the smoothed curve is quite natural even near both ends of the spectral range. We further note in these practical applications that the optimization strategy could be constructed to find the best combination of hi, M and IZ. REMARK 7.6. Strictly speaking, the orthogonal polynomial system is always problem-oriented so long as the full or partial orthogonality is aimed. To the finite element method related to some typical differential equation in one dimension, for example, the so-called sparse polynomials are designed so that the partial orthogonality in the finite element matrix is realized [lo]. From the view-point of the least-square finite element method, our discrete orthogonal polynomials can be regarded as the specific system designed for only one element of Fig. 1 having several nodes at equal intervals. Here the full orthogonality is realized with the sequential use in finite difference manner. For spectra with sharp absorption, on the other hand, other discrete orthogonal polynomials should be designed with some partial orthogonality under the least-square finite element method with several finite elements, details of which will be reported in due course. Here only Co continuity is aimed at the connecting node between two adjacent finite elements which corresponds to the relevant absorption.
Acknowledgment
This work is financially supported by Maki Manufacturing
Appendix
A. Summation
Co., Ltd., to which the author is grateful.
rule
We present here some recurrence
rules related to
(A.1)
i=O
Notice r;’
that _ z;-”
THEOREM
A.1.
= nk
The following even recurrence
(A.21 relation holds
M. Okabe
I Comput. Methods Appl. Mech. Engrg.
2(2m + l)Z*, = (2n + l)nrn(n + 1)” - 2 i
381
123 (1995) 371-383
(mC2k + 2 mCZk+l)Z*m_*k- {1+ (-l)“]Z,
7 nZ?l.
k=l 64.3)
Here . C. denotes the combination and (Y = [(m - 1)/2], PROOF.
.
64.4)
In case of m = 1, (A.3) holds such that
I, = (2n + l)n(n + 1)/6.
(A-5)
Suppose (A.3) holds upto m - 1. Then we have 2(2m + l)[ZFi - Zri”] - 2 2
= (2n + l)nm(n + 1)” - (2n - l)(n - l)mnm
(mC2k + 2 mCZk+l)n2m-2k - (1 + (-l)“}n”
= n2m ,
k=l
which completes the proof. REMARK
AL’.
0
For example, we have explicitly
Z4= (2n + l)n’(n + l)*/lO -Z2/5, Z, = (2n + l)n’(n + 1)3/14 - 51,/7,
(A.61
Z8= (2n + l)n4(n + 1)4/18 - 141,/9 - Z4/9. THEOREM
A.3.
2mZ2,_,
The following odd recurrence
= nm(n + 1)” - 2 i
relation holds
mC2k+IZzm-2k_1 .
(A-7)
k=l
PROOF.
In case of m = 1, (A.7) holds such that
Z,=n(n+1)/2.
G4.8)
Suppose (A.7) holds upto m - 1. Then we have 2m[Z&_,
- ZriIi]
= nm(n + 1)” - (n - l)Yzm - 2 i:
mC2k+1n2m-2k-1
k=l
=n
which completes the proof. REMARK
A.4.
2m-1
7
0
For example, we have explicitly
Z3= n’(n + 1)2/4, Z,=n3(n+1)3/6-Z3/3,
(A-9)
Z,=n4(n+l)4/8-Z,.
Appendix
B. Extension to arbitrary
number
of sampling
points
In case of N = 2n, (4) is replaced by l(2n - 1)2k ,
(B-1)
382
M. Okabe I Comput. Methods Appl. Mech. Engrg.
which can explicitly be obtained through Theorems (2n - 1)2k+2{J&;2 -Jc’}
123 (1995) 371-383
A.1 and A.3. Then we have immediately
= (2n - 3)2k{(2n - 3)*./g;;’
- (2ti - l)*Jg-“}
.
(B.2)
Now (19) and (B.2) can unifiedly be expressed as (N - 1)2k+2{J~~+2- Jk)} = (N - 3)2k{(N - 3)25g7;’ - (N - l)‘J$_“}
)
03.3)
with notation Iz = [N/2],
.
(B-4)
Unified expressions related to (25) and (17) can further be written as (N - 1)2”{(N - l)zL’“‘[X”+2P,]
- {(N - 1)2 + (ZV- 3)*c’,“,,“}L’“‘[X”P,]}
= (N - 3)2”{(N - 3)2L(n-*)[Xm+2Pm] - (N - 1)2(1 + c’,“,jJL’“-‘qx”P,]} clim = m(m - 1){3(N* - 1) - (m - 2)(m + 1)}/6(2m
- l)(N - l)* .
)
(B.5) (B.6)
Theorem 6.1 can thus be proved. The discrete operator of (1) is now unifiedly expressed as
W~41 =jl f&J 7
(B.7)
with the abscissa of (56).
Appendix C. Explicit formulas We present here explicit formulas of our discrete orthogonal that coefficients in Table 1 are to be examined by (B.6) and
polynomials upto degree 5. It is noted
360(2m - 1)(2m - 3)(N - 1)4c2,, = m(m - l)(m - 2)(m - 3)[45(N2 - l)* - 30m(m - 1)(N2 - 1) + (m - 4)(m + 1)(5m2 - 7m + 6)] . Table 1 Explicit formulas of Pm upto degree 5. PO= 1 p0 = N P, =x P,=N(N+1)/3(N-1) P* =x2 - cl,* p, = 4N(N + l)(N’ - 4)/45(N - 1)3 cl12 = (N + 1)/3(N - 1) P3=x3-cI13x p3 = 4N(N + 1)(N2 - 4)(N* - 9)/175(N - 1)5 cl13 = (3N2- 7)/5(N - 1)2 P, =x4 - c,,4x2 + czi4 pd = 64N(N + l)(N* - 4)(NZ - 9)(N* - 16)/11025(N - 1)’ c,,~ = 2(3N* - 13)/7(N - 1)’ c~,~ = 3(N + l)(N’ - 9)/35(N - I)’ Ps =x5 - CIISX3+ c,,5x p, = 64N(N + l)(N’ - 4)(N2 - 9)(N* - 16)(N* - 25)/43659(N c,,~ = lO(N’ - 7)/9(N - 1)2 c*,~ = (15N4 - 230N’ + 407)/63(N - 1)4
- 1)’
cc.11
M. Okabe I Comput. Methods Appl. Mech. Engrg. 123 (199.5) 371-383
383
References [l] B.M. Irons, Engineering application of numerical integration in stiffness method, J.A.I.A.A. 14 (1966) 2035-2037. [2] M. Okabe and N. Kikuchu, Some general Lagrange interpolations over simplex finite elements with reference to derivative singularities, Comput. Methods Appl. Mech. Engrg. 28 (1981) l-25. [3] S.A. Coons, Surfaces for computer aided design of space form, MIT. Project MAC, MAC-TR-41, 1967. [4] A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977). [5] S. Haykin, ed., Nonlinear Methods of Spectral Analysis, Topics in Applied Physics, Vol. 34, 2nd edition (Springer-Verlag, Berlin, 1983). [6] R.N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965). [7] A. Savitzky and M.J.E. Golay, Smoothing and differentiation of data by simplified least squares procedures, Anal. Chem. 36 (1964) 1627-1639. [8] M.U.A. Bromba and H. Ziegler, Efficient computation of polynomial smoothing digital fulters, Anal. Chem. 51 (1979) 1760-1762. [9] I. Babuska, B.A. Szabo and IN. Katz, The p-version of the finite element method, SIAM J. Numer. Anal. 18 (1981) 515-545. [lo] M. Okabe, One-dimensional self-adaptive interpolations in the p-convergence procedure, Comput. Methods Appl. Mech. Engrg. 41 (1983) 69-89.
ELSEVIER
Discrete orthogonal
polynomials inherent to sampling points at equal intervals Masayuki Okabe
Optopian Inc., 2-8-21 Sakuradai, Hasuda, Saitama 349-01, Japan
Received 9 November 1994
Abstract The discrete orthogonal polynomials in finite series are proposed for arbitrary number of sampling points at equal intervals. Recurrence rules are presented so as to realize recurrent computations in the spectral data smoothing applications
1. Introduction
Polynomial approximations are of great significance in science. Polynomial interpolations, for example, play an important role, not only in numerical simulation [l, 21, but also in surface generation [3]. For the spectral data smoothing that we are concerned with, a variety of powerful methods have been developed [4-61. In case of such smooth spectra without rapid absorption as near infrared ones of the fruit surface refelectance where polynomial interpolations could fit the curve in good accuracy, the polynomial smoothing in least-square sense [7,8] is attractive and to be reconsidered so that much more flexible optimization strategy can be constructed. It is known for the spectral white noise that the noise standard deviation after smoothing could be proportional to 1 /fi where N is the number of sampling points used. To keep the spectral profile accurate, however, rather higher-order polynomials are needed, especially for the large number of sampling points. Although some effective digital filters are proposed by Savitzky and Ziegler [7], we need much more powerful filters which should also be adequate for data smoothing near both ends of the spectral range. To achieve the automatical convergence in data smoothing, furthermore, the so-called p-convergence procedures developed in finite element fields [9, lo] are to be adopted. Here, the polynomial approximation degree is adaptively increased until the sufficient accuracy is attained. It is noted that special polynomial interpolations are required for this purpose. This paper is devoted to develop the orthogonal polynomials tightly connected with the discrete sampling points at equal intervals, which may yield new capabilities in the above-mentioned applications. It is shown that proposed recurrence rules are quite useful in practical computations.
2. Description of the problem In the x system of Fig. 1, consider 2n + 1 points placed at equal intervals in (-l,l). a discrete operator L is defined by
W(x)1 = i--n Ii fG/n) . Oil4578251951 $09.50 0 1995 Elsevier Science S.A. All rights reserved 0045-7825(94)00768-3
SSDZ
For smooth f(x),
(1)
M. Okabe I Comput. Methods Appl. Mech. Engrg. 123 (1995) 371-383
372
-2
-ll
-1
0
1
n
2
x=1
x=0
x=-l
Fig. 1. Placement of 2n + 1 sampling points at equal intervals in the x system.
This paper aims to develop the orthogonal but in finite series so that L[P,P,]
= P&,
polynomials
{P,}z=,
inherent to the discrete operator
.
(2)
Here I and m are non-negative
integers, S,, is the Kronecker
delta and
Pm = L[Pm2]. REMARK
2.1.
L
(3)
For non-negative
integer k, the following relations hold
L[x*~] = Jzk = 212k/n2k ,
(4)
L[X2k+1] = 0 )
(5)
where k=O,
2n+1)/2, C(
(6)
k>O.
For explicit formulations of Z2k, see Appendix A. Noting (5), the orthogonal polynomials P,,, could be even and odd functions corresponding and odd y, respectively. Then P,,, can be expressed as Pm = 5
(-l)iCj,,Xm-2i
to even
(7)
)
i=O
where K,,,= [m/2],
(8)
.
Here, [ .lG denotes the Gauss notation. The cOlm coefficient is fixed to unity (9)
co/m = 1, and the other coefficients in (7) could be determined L]X m-2iPm]=0,
j=I
Then, (3) can be rewritten i), = L[x"P,]
,...,
by the orthogonal
conditions of the form
Km.
(10)
as
.
(11)
3. Strong existence assumption
If the discrete orthogonal polynomials recurrence relation of the form P,=l, P m+l
P,=x, =xPm -A,P,,_,
,
mB1,
P,,,
with the orthogonal
conditions
(10) and (11) satisfy the
(12)
M. Okabe
I Comput. Methods Appl. Mech. Engrg.
123 (1995) 371-383
373
then we call P, strongly existing. THEOREM
3.1.
For strongly existing P,,
the following relation holds
A,,, = P,lPm_l. PROOF.
(13)
From (12), we have
L[x”P,,,]
- A,L[x”-‘P,_,]
THEOREM
3.2.
L[Xmt2
= L[x”-‘P,,,]
= 0.
0
For strongly existing P,,,, the following relation holds
P,l = clim+ZP~
(14)
.
PROOF. L[P,+,P,J THEOREM
= L[x”+2P,]
3.3.
- c,,,+,L[x”P,]
= 0.
q
Suppose P,,, strongly exists, then the coefficients in (7) can recurrently be written as
Ci/m+l =Ci/m +
A,Ci-l/m-1
7
i=l
7
*
*
.
9
Km .
Here we prescribe ‘i/m =O,
i
ASSUMPTION A,
3.4.
m
= m2(2n -t 1 - m)(2n + 1 + m)/4(2m
THEOREM cl,, PROOF.
3.5.
Under Assumption
- 1)(2m + l)n2 .
(16)
3.4, the cl,,,, coefficient is given explicitly as
= m(m - 1){12n(n -I-1) - (m - 2)(m + 1)) /24(2m - l)n* .
(17)
In case of i = 1, (15) is written as
c l/m+1
=
Cl/m
+A,,
(18)
which is surely realized by (17) and (16).
0
Our target is thus to prove the strong existence of the discrete orthogonal explicit form (16) of A,.
polynomials
P, with the
4. Mathematical preparation
In this section we deal with both n and n - 1 cases denoting by (n) and (n - 1) on the shoulder, respectively. LEMMA
4.1.
With respect to JZk in (41, the following relation holds
n2k+2{J $‘+2 - Je} PROOF.
= (n - l)‘k{(n - l)*j$;:’
- n*Jg-‘I}
,
(19)
From the definition of IZk in (6), we have
I$! _ 1(2nk-0= n2k
9
which yields IgJ+2 - n2Z$ = Ifk;:’ - n*lg-l) Thus, we have (19) for nonnegative
.
integer k.
(20) El
374
M. Okabe I Comput. Methods Appl. Mech. Engrg.
LEMMA
4 .2 . Let G9-l) r,m
G!T,-l) = 5
123 (1995) 371-383
and G!:$ be defined by
- 1)2J~~-:;i_zj+z - nzJsnm‘L;i_2i} ,
(-l)‘c,‘;;“{(n
j=O
(21) - Jp;_2i_zi}
G;‘J; = n2 2 (-l)‘~;;;{j&~_~~+~
,
i=O,...,K,.
j=O
Then we have go (-l)‘(n
- 1)2’n2m-2icj;,-1)Gj$
PROOF.
Substitution
LEMMA
4.3.
Gjxl)
= I$o (-l)‘(n
of (21)
GiyG’) and Gi:A
0
can be expressed as
)
i = 0,
. . . ) Km
(23)
.
Suppose the following orthogonal conditions hold
b m-2jp,]
= 0,
L(n)[xm-2j P,,,]=o, then
(22)
= (n _ 1)2L(n-1)[Xm-2i+2pm] _ n2L(n-1)[Xm-2ipm],
4.4.
LQ-1)
.
of (21) and (19) into (22) completes the proof.
G!‘$ = n2{L(“)[x m-2i+2pm] - L’“‘[x”-“iPm]} LEMMA
- ,)2”-2in2ic;;;Gj?;1)
j=l,.
..
(24) ,Km,
have
we
n2m{n2L@jxm+2 P,] - {n’ + (n - 1)2c1;,1’}L(“)[x”P,]} = (n - l)‘“{(n PROOF.
Lemmas 4.2 and 4.3 with (24) give (25).
LEMMA 4.5. such that LWl)
- l)2L(n-1)[~m+2Pm] - n’(1 + ci’f~)L(“-l)[xmPm]}
(25)
.
0
Suppose (13), (16) and (24) hold for both n - 1 and n and suppose (14) holds for n - 1 (n-1)
]x m+2en1 = Cl/m+2
)
p-u m
(26)
.
(27)
then we have L’“‘[x”+2P,] PROOF.
= c;;~+,a~’
From (13) and (16), we have
(n _ 1)2-p:-‘)
/n2”PE’
= (2n - m)(2n - m - 1)/(2n
+ m)(2n + m + 1) .
On the other hand, (17) yields n2cin/k+2 - n2 - (n - l)“cl’f,”
= -(m’
+ m - 1)(2n - m)(2n - m - 1)/2(2m
(n - 1)2c(,:,:), - n2(1 + c(,l)m)= -(m” + m - 1)(2n + m)(2n + m + 1)/2(2m
Lemma 4.4 then gives (27).
+ 3)(2m - 1) ,
+ 3)(2m - 1) .
Cl
LEMMA 4.6. For the polynomials P,,, defined uniquely by (12) with (16), suppose P,,, and P,,,+ 1 satisfy the orthogonal conditions (10) and suppose (14) holds, then we have k+, PROOF.
= A,+$‘,,,
.
From (14), we have
Gw
M. Okabe
The orthogonal
I Comput. Methods Appl. Mech. Engrg.
375
123 (1995) 371-383
conditions with respect to P,,, and P,,,+l give
L]x m-2iPm+J = L[x~-~~+‘P,,,+J
- A,+1L[xm-2iP,]
= 0,
i = 1, . . . , K, .
Hence 0 = L[x”P,+,] = L[xm+l
pm+11 - 4n+,wRI
which completes the proof.
9
0
5. Strong existence of the discrete orthogonal polynomials THEOREM, 5.1. The discrete orthogonal polynomials recurrence rule holds
P,,, are strongly existing so that the following
P1=x,
PO=l,
P m+l=xP,,,-A,P,_l,
(29)
mZ1,
where A,
= m2(2n + 1 - m)(2n + 1 + m)/4(2m
- 1)(2m + l)n2.
PROOF.
(30)
It suffices to prove that the polynomials uniquely determined orthogonal conditions (10) and (13) or (14). For m =0 and 1, we have
by (29) with (30) realize the
P0 = 2n + 1 , L[x2P,] 8, = A& L[x3P,]
(31)
= c1,2i)0 = (2n + l)(n + 1)/3n = J2 , = c~,~~)O, = c1,3i)l = (2n + l)(n + 1){3n(n
(32)
+ 1) - 1}/15n3 = J4.
Suppose here that P,,, by (29) with (30) satisfies the orthogonal evidently we have
conditions
and (13) upto m, then
L]x m+1-2jPm+l] = L[X”+2-2jP,] - A,L[Xm+‘-2jp,_,] =o, j=l,..., Km+l. Next, suppose that (14) holds upto n - 1. Then from Lemma 4.5, (14) holds for n. Noting (32) guarantees (14) for m = 0 and 1, (14) is thus proved. Cl Lemma 4.6 further guarantees (28), and hence strong existence of P,,, is proved. THEOREM
5.2.
For the discrete orthogonal polynomials P,,,, the following relations hold
i), = 2n + 1 ) p,,,=A,p,,_,
,
mE1.
(33)
6. Extended discrete orthogonal polynomials
Let N be the number of sampling points by N=2n+l.
(34)
376
M. Okabe / Comput. Methods Appl. Mech. Engrg.
123 (1995) 371-383
Substitution of (34) into Theorem 5.1 then gives the extended discrete orthogonal are also adequate for even N (Appendix B). THEOREM 6.1. The extended discrete orthogonal polynomials recurrence rule of the form
{P,},“::
polynomials
are strongly existing with the
P1=x,
P()=l,
Pm+l =xP,,, -A,P,,_,
,
which
(35)
mZ1,
where A,
= m2(N - m)(N + m)/(2m
THEOREM
6.2.
- 1)(2m + l)(N - 1)” .
For the extended discrete orthogonal polynomials P,,,, the following relations hold
P()=N, P,,,=A,i),_I, THEOREM Ci/m+l
=
REMARK
(37)
m21.
6.3. Cilm
(36)
Coefficients in (7) of the extended discrete orthogonal polynomials satisfy +
A,c~_~,,,_~
,
i = 1, . . . ,
K,
(38)
.
6.4.
Roots of (N - l)‘“P* with respect to N are N = 0 and N = *i (i = 1, . . . , m). That is, orthogonal polynomials are not in infinite series. For N points, only P,, to PN_I are
the discrete adequate. THEOREM
6.5.
Let
D=d/dx,
(39)
then (35) gives DP, = 1,
DP,=O,
DP,+,
=xDP,,,-A,DP,,_,+P,,,, D2P,=0,
D*P,,=O, D2P,+*
(40)
mZ1,
=xD*P,,,-A,D2P,,_1
+2DP,,,
,
mZ1.
(41)
7. Application examples and concluding remarks
We have thus developed significant applications. 7.1.
One-dimensional
the discrete orthogonal
polynomials in finite series. Here we refer to some
data smoothing (curve fitting)
Within (-1, I), let the trial function be expressed as f(x) = C,
a,P,(x)
.
(42)
Here a,,, denotes the coefficient to be determined, and M is the polynomial degree of our trial function. We then introduce a least-square functional for 2n + 1 sampling points by *=+,jt
where f:
i--n
{f(iln)-ft>*,
denotes the measured data at the sampling point of x = i/n.
(43)
M. Okabe
Differentiation Q/da,
I Comput. Methods Appl. Mech. Engrg.
123 (1995) 371-383
371
of (43) with respect to a, gives
= 5 a,L[P,P,] I=0 =a,i),-
-
i: P,(iln)f: i=-n
i P,(iln)fT i=-n
=o.
Immediately a, = [ i: P,(iln)fT] i=_n
(44)
Ii), .
The trial function (42) at the sampling point j of x = j/n takes the value of (45) If we express f( j/n)
in convolution
form by
f(jln) = i: wi.fT,
(46)
1=--n
then the weight wi can be written as P,(i/n)P,( m=O
w{ = 2
j/n)li),
.
(47)
At the origin of x = 0, (47) is written as wp = 5
m=o
Pm(i/n)Pm(0)/Pm
For non-negative P2k(O) = (-l)Lz, Px+1(0)
.
(48)
integer k, we have 7
= 0.
(49) (50)
It is obvious in (50) that (46) with (48) has the precision of order 2k + 1 even if our trial function (42) is of degree 2k. This feature is realized only at the origin. Savitzky and Ziegler [7] present explicit values of WYfor several odd points (N = 2n + 1) which have been efficiently used, for example, in spectroscopic data smoothing with some modification by Bromba and Ziegler [8]. At the other sampling points except for the origin, however, no formulas are proposed and consequently data smoothing has never been performed near ends of the spectroscopic range. It is emphasized that the proposed recurrent structure of the discrete orthogonal polynomials enable us in least-square data smoothing to adopt the so-called p-convergence procedures originally developed in finite element fields [9, lo]. Here the approximation degree M in our trial function (42) is automatically increased until sufficient smoothness is obtained with required accuracy. REMARK
7.1. In computations, recurrence rules (35) and (37) with (36) are needed without explicit formulations of Pm. For explicit form of P,,, upto 5, see Appendix C. First and second derivatives of f(x) can also be computed by using the recurrence relations (40) and (41), respectively. REMARK 7.2. For N points, our polynomial trial space is to be restricted This feature is clearly observed in Remark 6.4.
to degree N - 1 at most.
378
M. Okabe
REMARK
Legendre
I Comput. Methods Appl. Mech. Engrg.
123 (1995) 371-383
7.3. Our discrete orthogonal polynomials can thus be recognized as a discrete version of the polynomials.
7.2. Multi-dimensional data smoothing (surface fitting) Extension to multi dimensions is straightforward. can be written as fb,
Y) = 7
aijpi(x)pj(Y> .
7
Then we have in least-square aij
where
=
f
&
[C
ZZ
(51)
sense
pi(x,,P,(Yq)fzq]
P 4
In two dimensions, for example, our trial function
lpi’j
(52)
9
denotes the measured data at the sampling point (x,, y,).
7.3. Numerical integration Integration
of the discrete orthogonal
’ PZkdx = 2 i
I -1
(-l)i~i,2k/(2k
polynomials gives
- 2i + 1) ,
(53)
i=O
1
P 2k+l
dx
=
o
(54)
3
I -1
where k designates non-negative integer. For the F(x) function, we then have the following numerical integration formula in Newton type with N sampling points at equal intervals 1’ F(x)d_x=f -1
21”zG{i p=l
k=O
(-l)‘ci,,,/(2k-2i+l) i=O
1
.
{p2k
@2k>
1 F(x,)
)
(55)
where the abscissa is given by x,=(2p-N-l)/(N-I),
p=l,...,
N.
(56)
For N points, the formula due to the trial function of degree M = N - 1 is undoubtedly effective. REMARK
7.4.
the most
Noting (54), the integral formula utilizing upto P2k has the precision of order 2k + 1.
REMARK
7.5. For this integral use, the coefficient recurrence explicit form of those coefficients are given in Appendix C.
rule (38) is needed with (36). Some
7.4. Optimization example through the multiple regression analysis The problem we consider is to determine the calibration for peach brix (sugar content) t by the near infrared spectroscopic data from the fruit surface. Typical spectrum of the peach reflectance is shown in Fig. 2 which is composed of data at equal intervals by 2 nm from 650 nm to 1050 nm. It can be seen that the near infrared spectrum of fruits are so smooth that the polynomial interpolations could work well. The calibration model we adopt is t = a, + i
ai In ri _
(57)
i=l
Here, ri denotes the measured reflectance
at wave length Ai (A, = 850 nm, A, = 880 nm, A, = 910 nm,
M. Okabe
I Comput. Methods Appl. Mech. Engrg.
379
123 (1995) 371-383
Fig. 2. Typical measured spectrum of the peach reflectance.
through the mulitple A, = 950nm, A, = 970 nm). The coefficients a, and ui are to be determined regression analysis. From the spectral curve of Fig. 2, for simplicity, we prescribe the approximation degree to be M = 3. Our optimization problem is now written as Find the half width n of sampling points yielding the maximum multiple correlation due to the calibration model (57) with approximation degree of M = 3.
factor
It is emphasized that our optimization is not dependent apparently on the smoothing accuracy but directly on the statistical result. Fig. 3 shows the obtained correlation factor y vs. II. Here, 72 peach data are used. By the best
.88
0
.86 OQ 0
Q
0
o”
0 0
0 0
0
.82 0
.80
L-__
0
--
--A_
10
20
-s
n 30
Fig. 3. Multiple correlation factor due to the polynomial smoothing of sampling points 2n + 1 with approximation
degree M = 3.
380
M. Okabe
I Comput. Methods Appl. Mech. Engrg.
123 (1995) 371-383
Fig. 4. Smoothed result of Fig. 2 by the polynomial smoothing according to the best solution of n = 24 and M = 3.
solution of n = 24 (i.e. sampling points of N = 49) with A4 = 3, the measured spectrum of Fig. 2 is smoothed as shown in Fig. 4. Our solution is thus adequate also in the smoothing accuracy sense. It is especially remarked that the smoothed curve is quite natural even near both ends of the spectral range. We further note in these practical applications that the optimization strategy could be constructed to find the best combination of hi, M and IZ. REMARK 7.6. Strictly speaking, the orthogonal polynomial system is always problem-oriented so long as the full or partial orthogonality is aimed. To the finite element method related to some typical differential equation in one dimension, for example, the so-called sparse polynomials are designed so that the partial orthogonality in the finite element matrix is realized [lo]. From the view-point of the least-square finite element method, our discrete orthogonal polynomials can be regarded as the specific system designed for only one element of Fig. 1 having several nodes at equal intervals. Here the full orthogonality is realized with the sequential use in finite difference manner. For spectra with sharp absorption, on the other hand, other discrete orthogonal polynomials should be designed with some partial orthogonality under the least-square finite element method with several finite elements, details of which will be reported in due course. Here only Co continuity is aimed at the connecting node between two adjacent finite elements which corresponds to the relevant absorption.
Acknowledgment
This work is financially supported by Maki Manufacturing
Appendix
A. Summation
Co., Ltd., to which the author is grateful.
rule
We present here some recurrence
rules related to
(A.1)
i=O
Notice r;’
that _ z;-”
THEOREM
A.1.
= nk
The following even recurrence
(A.21 relation holds
M. Okabe
I Comput. Methods Appl. Mech. Engrg.
2(2m + l)Z*, = (2n + l)nrn(n + 1)” - 2 i
381
123 (1995) 371-383
(mC2k + 2 mCZk+l)Z*m_*k- {1+ (-l)“]Z,
7 nZ?l.
k=l 64.3)
Here . C. denotes the combination and (Y = [(m - 1)/2], PROOF.
.
64.4)
In case of m = 1, (A.3) holds such that
I, = (2n + l)n(n + 1)/6.
(A-5)
Suppose (A.3) holds upto m - 1. Then we have 2(2m + l)[ZFi - Zri”] - 2 2
= (2n + l)nm(n + 1)” - (2n - l)(n - l)mnm
(mC2k + 2 mCZk+l)n2m-2k - (1 + (-l)“}n”
= n2m ,
k=l
which completes the proof. REMARK
AL’.
0
For example, we have explicitly
Z4= (2n + l)n’(n + l)*/lO -Z2/5, Z, = (2n + l)n’(n + 1)3/14 - 51,/7,
(A.61
Z8= (2n + l)n4(n + 1)4/18 - 141,/9 - Z4/9. THEOREM
A.3.
2mZ2,_,
The following odd recurrence
= nm(n + 1)” - 2 i
relation holds
mC2k+IZzm-2k_1 .
(A-7)
k=l
PROOF.
In case of m = 1, (A.7) holds such that
Z,=n(n+1)/2.
G4.8)
Suppose (A.7) holds upto m - 1. Then we have 2m[Z&_,
- ZriIi]
= nm(n + 1)” - (n - l)Yzm - 2 i:
mC2k+1n2m-2k-1
k=l
=n
which completes the proof. REMARK
A.4.
2m-1
7
0
For example, we have explicitly
Z3= n’(n + 1)2/4, Z,=n3(n+1)3/6-Z3/3,
(A-9)
Z,=n4(n+l)4/8-Z,.
Appendix
B. Extension to arbitrary
number
of sampling
points
In case of N = 2n, (4) is replaced by l(2n - 1)2k ,
(B-1)
382
M. Okabe I Comput. Methods Appl. Mech. Engrg.
which can explicitly be obtained through Theorems (2n - 1)2k+2{J&;2 -Jc’}
123 (1995) 371-383
A.1 and A.3. Then we have immediately
= (2n - 3)2k{(2n - 3)*./g;;’
- (2ti - l)*Jg-“}
.
(B.2)
Now (19) and (B.2) can unifiedly be expressed as (N - 1)2k+2{J~~+2- Jk)} = (N - 3)2k{(N - 3)25g7;’ - (N - l)‘J$_“}
)
03.3)
with notation Iz = [N/2],
.
(B-4)
Unified expressions related to (25) and (17) can further be written as (N - 1)2”{(N - l)zL’“‘[X”+2P,]
- {(N - 1)2 + (ZV- 3)*c’,“,,“}L’“‘[X”P,]}
= (N - 3)2”{(N - 3)2L(n-*)[Xm+2Pm] - (N - 1)2(1 + c’,“,jJL’“-‘qx”P,]} clim = m(m - 1){3(N* - 1) - (m - 2)(m + 1)}/6(2m
- l)(N - l)* .
)
(B.5) (B.6)
Theorem 6.1 can thus be proved. The discrete operator of (1) is now unifiedly expressed as
W~41 =jl f&J 7
(B.7)
with the abscissa of (56).
Appendix C. Explicit formulas We present here explicit formulas of our discrete orthogonal that coefficients in Table 1 are to be examined by (B.6) and
polynomials upto degree 5. It is noted
360(2m - 1)(2m - 3)(N - 1)4c2,, = m(m - l)(m - 2)(m - 3)[45(N2 - l)* - 30m(m - 1)(N2 - 1) + (m - 4)(m + 1)(5m2 - 7m + 6)] . Table 1 Explicit formulas of Pm upto degree 5. PO= 1 p0 = N P, =x P,=N(N+1)/3(N-1) P* =x2 - cl,* p, = 4N(N + l)(N’ - 4)/45(N - 1)3 cl12 = (N + 1)/3(N - 1) P3=x3-cI13x p3 = 4N(N + 1)(N2 - 4)(N* - 9)/175(N - 1)5 cl13 = (3N2- 7)/5(N - 1)2 P, =x4 - c,,4x2 + czi4 pd = 64N(N + l)(N* - 4)(NZ - 9)(N* - 16)/11025(N - 1)’ c,,~ = 2(3N* - 13)/7(N - 1)’ c~,~ = 3(N + l)(N’ - 9)/35(N - I)’ Ps =x5 - CIISX3+ c,,5x p, = 64N(N + l)(N’ - 4)(N2 - 9)(N* - 16)(N* - 25)/43659(N c,,~ = lO(N’ - 7)/9(N - 1)2 c*,~ = (15N4 - 230N’ + 407)/63(N - 1)4
- 1)’
cc.11
M. Okabe I Comput. Methods Appl. Mech. Engrg. 123 (199.5) 371-383
383
References [l] B.M. Irons, Engineering application of numerical integration in stiffness method, J.A.I.A.A. 14 (1966) 2035-2037. [2] M. Okabe and N. Kikuchu, Some general Lagrange interpolations over simplex finite elements with reference to derivative singularities, Comput. Methods Appl. Mech. Engrg. 28 (1981) l-25. [3] S.A. Coons, Surfaces for computer aided design of space form, MIT. Project MAC, MAC-TR-41, 1967. [4] A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977). [5] S. Haykin, ed., Nonlinear Methods of Spectral Analysis, Topics in Applied Physics, Vol. 34, 2nd edition (Springer-Verlag, Berlin, 1983). [6] R.N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965). [7] A. Savitzky and M.J.E. Golay, Smoothing and differentiation of data by simplified least squares procedures, Anal. Chem. 36 (1964) 1627-1639. [8] M.U.A. Bromba and H. Ziegler, Efficient computation of polynomial smoothing digital fulters, Anal. Chem. 51 (1979) 1760-1762. [9] I. Babuska, B.A. Szabo and IN. Katz, The p-version of the finite element method, SIAM J. Numer. Anal. 18 (1981) 515-545. [lo] M. Okabe, One-dimensional self-adaptive interpolations in the p-convergence procedure, Comput. Methods Appl. Mech. Engrg. 41 (1983) 69-89.