298
B. MOND
very bad. The non-defective cases, with several vectors nearly parallel, were sometimes worse than the defective cases.
N = 6
s = -4
actual roots
0,
approximations (7070)
N = 6
s = -6
actual roots
approximations (7070)
N = 6
s = -6.5
actual roots
approximations (7070)
0
2,
.0228 -.0225
2
-1-4-10-18
1.995,2.004 -1
—6 0,0
-4
4,4
-10
-18
6, 6
-6
-.0056 ± .111* 3.969 ± .363* 6.036 ± .2215*
-3
-3
0
2.5
4.5
6
7, 7.75
.0066 2.436 5.103 5.151 7.401 ± .405*
The effect of higher precision is seen in the following example which was run on
both the 7070 (8 digits) and the 1604 (10 digits)
N = 10
s = -14
actual roots
approximations
0
12
22
30, 30
36, 36
40,40
42,42
-2 X 10"7
12.00004
21.995
29.638 ± 1.134*
34.993 ± 2.94*
40.024 ± 3.39*
43.347 ± 1.49*
University
Rochester,
(CDC 1604)
approximations
(IBM 7070)
.0026
11.898
20.195
23.843 ± 7.16*
33.926 ± 12.92*
36.977
46.387 ± 10.99*
52.594
of Rochester
New York
1. J. Brauner
& D. J. Wilson,
"Intramolecular
mechanism," J. Phys. Chem., v. 67, 1963,p. 1134-1138.
reactions
II: A weak energy transfer
2. P. J. Eberlein,
"A Jacobi-like method for the automatic computation of eigenvalues
and eigenvectors of an arbitrary matrix," J. Soc. Indust. Appl. Math., v. 10, 1962, p. 74-88.
Multivariate
Polynomial Approximation
Equidistant Data
for
By B. Mond
Abstract. The theory of polynomial approximation for evenly spaced points is
extended to multivariate polynomial approximation. It is also shown how available
tables prepared for univariate approximation can be used in the multivariate case.
1. Introduction. Assume f(x) is given for x = x, , x2, ■■■, xn and it is desired
to approximate f(x) by a polynomial of degree p, 1 5= p < n, i.e.
V
f(x)~Yl
aiX1.
Received November 12, 1963.
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MULTIVARIATE
POLYNOMIAL APPROXIMATION
FOR EQUIDISTANT
DATA
299
In order to determine the coefficients a,- so as to minimize
n
r~
p
X f(Xj) — X ffliZ/
¿=i L
»=°
one must solve a set of p + 1 normal equations. If the rr¿ are evenly spaced, the
problem can be greatly simplified by a change of variable and the use of orthogonal
polynomials [4]. This simplification will now be extended to approximation
by
multivariate polynomials.
2. Bivariate Polynomial Approximation. Assume that f(x, y) is defined on a
finite planar set of points {(xi , 2/¿)} (*' = 1, 2, • • • , n;j = 1, 2, • • • , m) and that it
is desired to approximate f(x, y) by a bivariate polynomial of the form
S
T
X X akhxkyh
h=0 4=0
1 i£ r < n and 1 5» s < m. In order to determine the coefficients au (k = 0,1, • • • ,
r; A = 0, 1, • • • , s) so as to minimize
XX
s
r
""12
f(xi, y¡) — X Jl akhX?y,h
h—0 4-0
J—l ¿=1
J
one must solve a system of (r + l)(s + 1) normal equations.
Assume, now, that the x< and y¡ are evenly spaced, i.e.
xk+\ — Xk = h
(k = 1, 2, • • • , n — 1)
2/i+i - 2/4 = Ä2
(fc = 1, 2, • • • , m — 1).
Thus, if we write
x — a + ih\
(1)
y = b + jh2,
where a = Xi — hi and 6 = yx — h2, (x, y) will represent the coordinates of the
given points when * = 1, • • • , n; j = 1, • • • , m. Let x and y be the midpoints of
the given ¡c<and */3{x = (xi + xn)/2;jj
= (t/i + ym)/2]. The horizontal and vertical distances from the midpoint divided by the uniform spacing [(x — x)/hi
and (j/ — *7)/Ä2] are then equal respectively, by virtue of equations (1), to
i - (n + l)/2 and j - (m + l)/2.
Let F« (A = 0,1, ■■■ ,r; k = 0,1, • • • , s) be a set of polynomials
degree r and s in * — (n + 1 )/2 and / — (to + 1 )/2 , i. e.
of exact
Pm = «to + «ÎS[*- (n + l)/2] + afi[j - (to + l)/2]
(2)
+ aîki[i -
(n+
l)/2][j
-
(to + 1 )/2] + • • •
+ o£S[*- (n + l)/2]"[/ - (to + l)/2]*.
Poo will always be taken equal to one.
If, now, we approximate f(x, y) by a polynomial
B
of the form
r
ft-0 4-(I
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300
B. MOXD
and solve for the constants bkh in the sense of least squares, we obtain (r + 1)
(s + 1) normal equations with the augmented matrix
~E2>o« EL*»«—EL** EEfi» •••SE fi.-EE ft.
ESftoft. EEfi'i-'-SEfi^fii---
SE ft. fin EE/fe,í,)ñ.
ZEftoPo., EEftift. —EEft1.—
.IZpMpr, EEftift.---EEft.ft.--where the notation
EZ/fe.tó"
EZft»ft» ZE/fc-.s^».
EX ft2* EE/C*,w)fi._
X X PhkPcd means
X X Pul« - (n + l)/2,i - (to + 1)/2]PC,[*- (n + l)/2, j - (to + l)/2].
¿-i ¿-i
If the a'tj in (2) are chosen so that the polynomials Phk are biorthogonal, i.e.
m
n
X X PuPcd = 0
j=i ¿=i
if h 9¿ c or k 9¿ d,
the coefficient matrix reduces to a diagonal matrix. The b« can then be written as
(3)
bhk= X X /(*i, yj)Pkt/T, X P» •
J
»
í
i
As in the univariate case [2], testing the appropriatness of the representation is
also facilitated by the use of orthogonal polynomials. Should a polynomial of
higher degree be desired, the coefficients bhkneed not be recalculated.
3. Constructing Biorthogonal Polynomial Tables. In approximating with orthogonal functions, many of the calculations necessary are independent of the data
[for example, the denominator in (3)] and need not be recalculated each time a
different set of data is to be fitted. Extensive tables are available for univariate
orthogonal polynomials [1], [2]. Their use makes the actual calculation of the coefficients quite easy.
By appropriately choosing the bivariate biorthogonal polynomials, univariate
tables that are already available can easily be modified for use in the bivariate case.
Theorem. Let Ph (h = 0, 1, • • • , r) and P* (fc = 0,1, • • • , s) be sets of orthogonal
polynomials in (* — (n + l)/2) and (j — (to + l)/2) respectively with P0 = 1.
Let Phk = PhPk • Phk (h = 0, 1, • • • , r; k = 0, 1, • • • , s) is then a set of bivariate
biorthogonal polynomials in (*' — (n + l)/2) and (j — (to + l)/2).
Proof. It follows from the definition that Phk will be a bivariate polynomial of
degree h and k in (* — (n + l)/2) and (j — (to + l)/2) respectively.
Biorthogonality of the bivariate polynomials follows from the orthogonality
of the univariate polynomials and the fact that
/ . 2^ PhkPcd — 2-j X PhPkPcPd — ¿-I PkPc X PhPd ■
3
i
3
i
i
i
Taking Phk = P/,Pk, it is possible to utilize, in bivariate polynomial approximation, tables that were constructed for use in the univariate case. In general, if one
regards the rectangular array of values for a given n in Fisher and Yates statistical
tables [2] as a matrix, then the corresponding matrix for bivariate biorthogonal
polynomials would be a Kronecker product [5] of corresponding matrices.
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MULTIVARIATE
POLYNOMIAL APPROXIMATION
FOR EQUIDISTANT
DATA
301
For example, from the entries for n = 3 and n = 4 in the Fisher and Yates
tables [2], one gets as the entry in the bivariate table for n = 3, to = 4
3
-3
1
2
1
1
1
1
2
2
2
2
3
4
1
2
3
4
3
.'!
3
3
-1
+3
+ 1
-1
-1
+3 + 1
-3
+ 1
-1
-1
+ 1 -1
+3 + 1
-3
+1
-1
-1
+ 1
+3 +1
-1
+ 1
1
2
3
4
60
-3
+ 1
-1
+3
-3
+ 1
-1
+3
■3
+1
12
60
-1
-1
-1
-1
0
0
0
0
+3
+1
-1
-1
+1
+1
-3
0
0
0
-1
0
0
0
0
0
+1
+1
+1
+1
-3
-1
+1
+3
8
40
+1
-1
+1
+1
-3
+3
■1
0
0
0
0
1
+3
-3
+1
P20
P
Pz
+1
+1
+1
+1
-2
-2
-2
-2
+1
+1
+1
+1
-3
+1
-1
-1
+1 -1
+3 +1
+6
2
+ 2 +2
-2
+2
-6
-2
-3
+1
-1
1
+1
1
+3 +1
-1
+3
-3
+1
+2
-6
+6
-2
+3
-3
+1
120
120
40
24
1
24
where the last line represents the sums of squares of all elements in the particular
column. All entries in the bivariate table are products of corresponding entries in
the univariate table.
Bivariate tables constructed as outlined here could be used in exactly the same
manner as recommended for univariate tables [see 2].
4. Extension to n Variables. The extension
The n dimension analogue to equation (3) is
to n variables
is straightforward.
bh,...K- E ••• £/(*£,*%, ■■■,sirVv..»./ E ••• n=i
E p*V
I * - - *»
where the Phl---hn are orthogonal polynomials
the coefficients in the approximation
f(xw, •••
X" ) ~
in n variables
¿~i ■' ' /-ibhi--.hnPhx...hn
¿_
and the bhl...h„ are
' • ' ¿-iPhi-.-hn
»1—1
so as to minimize
X ••• il-1E k*8\
••• ,*lB))- *n=0
E ••• »1=0
E &V..1JV.»
L
*n=l
As in the bivariate case, one can make use of univariate tables in multivariate
polynomial approximation by taking for the multivariate polynomials the products
of corresponding univariate orthogonal polynomials.
Aerospace Research Laboratories
Wright-Patterson
Air Force Base, Ohio
1. R. L. Anderson
& E. E. Houseman,
"Tables of orthogonal
to n = 104," Research Bulletin 292, Ames, Iowa, 1942.
2. R. A. Fisher
& F. Yates,
Statistical
Tables for Biological,
Research, Hafner Publishing Company, New York, 1957.
3. F. A. Graybill,
An Introduction to Linear Statistical
Company Inc., New York, 1961.
4. F. B. Hildebrand,
New York, 1956.
5. C. C. MacDuffee,
Introduction
polynomial
to Numerical
Agricultural
and Medical
Models, Vol. 1, McGraw
Analysis,
Theory of Matrices, Chelsea Publishing
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
values extended
McGraw
Hill
Company
Hill
Inc.,
Company, New York, 1936.