POLYNOMIAL CURVE AND SURFACE FITTING/67531/metadc130895/... · Curve Fitting by Spiine Functions I...
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POLYNOMIAL CURVE AND SURFACE FITTING APPROVED: Major Professor ™ Minor Professor irector of the Department of Mathematics — can of the Graduate School
Transcript of POLYNOMIAL CURVE AND SURFACE FITTING/67531/metadc130895/... · Curve Fitting by Spiine Functions I...
POLYNOMIAL CURVE AND SURFACE FITTING
APPROVED:
Major Professor ™
Minor Professor
irector of the Department of Mathematics
—
can of the Graduate School
POLYNOMIAL CURVE AND SURFACE FITTING
THESIS
Presented to the Graduate CounciI of the
North Texas State University in Partial
Fulfillment of the Requirements
For the Degree of
MASTER OF SCIENCE
By
Ann Dowdy Capps, B. A.
Denton, Texas
January, 1968
PREFACE
The main p rob lems of n u m e r i c a l a n a l y s i s i n v o l v e p e r f o r m i n g
a n a l y t i c a l o p e r a t i o n s , such as i n t e g r a t i o n , d i f f e r e n t i a t i o n , f i n d i n g
z e r o e s , i n t e r p o l a t i o n , and so f o r t h , of a f u n c t i o n when a l l t h e d a t a
a v a i l a b l e a r e some samples o f t h e f u n c t i o n . T h e r e f o r e , t h e pu rpose
o f t h i s paper i s t o i n v e s t i g a t e t h e f o l l o w i n g p rob lem: g i v e n a s e t
o f da ta p o i n t s ( x . , y . ) w h i c h a r e samples of some f u n c t i o n , d e t e n r i i n e
an a p p r o x i m a t i n g f u n c t i o n . F u r t h e r , ex tend t h e p rob lem t o t h a t of
d e t e r m i n i n g an a p p r o x i m a t i n g f u n c t i o n f o r a s u r f a c e g i v e n some samples
( x . , y . , z . . ) of t h e s u r f a c e .
• V 'J
The c l a s s of a p p r o x i m a t i n g f u n c t i o n s t o be I n v e s t i g a t e d is t h e
s e t g e n e r a t e d by l i n e a r c o m b i n a t i o n s o f 1, x , x 2 , . . . , x n ; t h a t i s ,
t h e s e t o f p o l y n o m i a l s . T h i s s e t is i m p o r t a n t f o r two reasons : 1) i f
P ( x ) i s an e l emen t o f t h e a p p r o x i m a t i n g s e t , t h e n P (x+k ) is an e lement
o f t h e s e t ; and 2) i f P ( x ) is an e lemen t o f t h e s e t t h e n P ( k x ) is a l s o
i n t h e s e t . In ( 1 ) , o n l y t h e c o e f f i c i e n t s o f P(x-i-k) d i f f e r f r o m P ( x ) ,
and (1 ) i m p l i e s t h a t t h e l o c a t i o n of t h e o r i g i n does n o t a f f e c t t h e
p rob lem of f i t t i n g a p o l y n o m i a l t o t h e d a t a . S ta temen t (2 ) i n d i c a t e s
t h a t s c a l e changes do n o t a f f e c t t h e p r o b l e m .
In C h a p t e r I , some methods a r e d e r i v e d f o r f i t t i n g an n"^' degree
p o l y n o m i a l t o (n+1) d a t a p o i n t s ( x . , y . ) . I n c l u d e d i n t h e s e e r e t h e
methods of Lagrange and Newton. In a d d i t i o n , t h e p r o c e d u r e of s o l v i n g
a sysTern o f n+1 l i n e a r e q u a t i o n s f o r t h e c o e f f i c i e n t s o f t h e p o l y n o m i a l
i i i
IV
is d iscussed. Chapter I is concluded w i t h an i n v e s t i g a t i o n of curve
f i t t i n g by s p l i n e f u n c t i o n s . The d e f i n i t i o n f o r t h e se t of s p l i n e
f u n c t i o n s is g iven a long w i t h t h e proo fs of theorems which are necessary
f o r t h e u t i l i z a t i o n of t h e f u n c t i o n s .
In some cases where t h e data po in t s ( x . , y j ) a re known t o con ta i n
e r r o r f rom measurement, t h e method of l eas t squares curve f i t t i n g may be
u t i l i z e d . In Chapter I I , t h e method of l eas t squares f o r n ^ degree
po lynomia ls is de r i ved . In a d d i t i o n , procedures are cons idered where
r e s t r a i n t s a re placed on t he data p o i n t s such as we igh t i ng t h e da ta , o r
r e q u i r i n g t h a t t h e l eas t squares curve pass th rough s p e c i f i c p o i n t s .
F i n a I l y in Chapter I I I , t h e problem of f i t t i ng a su r face t o a se t
of data po in t s (x. , y . ,z . . ) is cons idered . The se t of cubic s p l i n e J ' J
f u n c t i o n s c f Chapter I is extended t o a se t of f u n c t i o n s of two v a r i -
ab les , and va r ious i heoreros o f t h e f i r s t chapter are extended in Chapter
I I I t o theorems concern ing b i c u b i c s p l i n e f u n c t i o n s . Chapter I I I is
concluded w i t h a mathematical procedure f o r computat ion of a b i c u b i c
spl i ne f u n c t i o n .
In each of t h e chap te r s , examples are inc luded f o r t h e t o p i c s
d iscussed. Since t h r e e of t h e t o p i c s i nvo l ve lengthy computa t ion , a
d i g i t a l computer was used as a means of c a l c u l a t i n g t h e examples. For
each of t hese , a computer program was w r i t t e n , and each has been
inc luded in t h e appendix s e c t i o n a long w i t h a d e s c r i p t i o n of how data
can be inpu t t o t h e program. The computat ion of a s p l i n e f u n c t i o n of
Chapter ! is shown in Appendix A. The procedures developed in Chapter
I I f o r l eas t squares curve f i t t i n g are programmed and are shown in
•Appendix B. F i n a l l y , the program f o r t h e mathematical procedure,
descr ibed in Chapter I I I , t o determine a b i c u b i c s p l i n e f u n c t i o n
is i l l u s t r a t e d in Appendix C.
TABLE OF CONTENTS
Chapter Page
I . METHODS OF POLYNOMIAL CURVE-FITTING 1
By Use of L inear Equations By t h e Formula of Lagrange By Newton's Formula Curve F i t t i n g by Sp i ine Funct ions
I I . METHOD OF LEAST SQUARES 24
Polynomials of Least Squares Least Squares Polynomial Approx imat ion w i t h
Restra i n ts
I I I . A METHOD OF SURFACE FITTING 37
B icub ic Sp l ine Funct ions Theorems t o Aid in Sp l ine Func t ion Computation Procedure f o r Computation of a Sp l ine Surface
APPENDIX A . . .
APPENDIX B 61
APPENDIX C
BIBLIOGRAPHY 77
v 1
CHAPTER
METHODS OF POLYNOMIAL CURVE FITTING
By Use of L inear Equat ions
Given a se t of (n+1) data po in t s ( x . , y ) , where i = 0, 1 , . . . , n;
n a polynomial of degree n, P (x) = £ a .x 1 may be f i t t e d th rouch
n i =0 1
these p o i n t s . This polynomial may be determined by s imply s u b s t i t u t i n g
eacn p o i n t ( x . , y . ) i n t o t o o b t a i n n+1 l i n e a r equal ions in t h e n+1 unknowns, a 0 , a1, . . . , a n From Cramer's Rule of s o l v i n g l i n e a r
equat ions (3 , p. 83 ) , i f t h e determinant
?.
(1 -1 ) |V| =
xo xo*
X1 X1 • • • x
0 n
l
1 xn x R2 . . . x n
is not ze ro , a s o l u t i o n f o r aQ , an e x i s t s .
I h e o r e n M J ^ I f x j 4 Xj f o r i 4 j , t hen t h e va lue of t h e d e t e r -
minant (1 -1) is not zero .
F i r s t i t must be shown t h a t | v | i n (1-1) is t h e product of
a l l d i f f e r e n c e s Xj - x - f o r which i > j ; id e s t ,
C - 2 ) | V | = n (x j —x:). 0<j< i <n
For ri = 1, j V | = xx - x Q , which obv ious l y s a t i s f i e s ( 1 - 2 ) . Assume
equa t ion (1-2) is t r u e f o r n = k; t h a t i s ,
( 1 - 3 )
Then fo r n = k+1
II (x. - x . ), 0 < j c i <_k 1 ^
|V| x f
X K + 1 • x0
k+1 • X1
X X2
' k+1 k + 1 " " k+1
The f o l l o w i n g is t r u e f o r determinants: i f t h e ma t r i x B is obta ined
from A by m u l t i p l y i n g a column of A by a constant and adding each
element of t h i s column t o another column of A then |B[ = |A[ (3, p. 75)
Thus,
( 1 - 4 ) |V| -
x0 0
x l ^ x i - x o )
x k+1 k+1 <x'
k+1 k+1 k+1 "X n )
where (1-4) is obtained by adding ( - x 0 ) t imes the (k+1) s
I
vk+2) column. Cont inu ing in t h i s manner,
1 0 0 . . . o
1 ( x i - x 0 ) X ^ X j - X q ) . . . x f ( x i - x 0 )
column t o the
1 " W ^ X M I X K + I X 0 '
This example may be expanded by co fac to rs t o y i e l d
( x r x 0 ) X ^ X J - X Q )
x k + i x k + T x » '
| v | (Xp-Xn) X p(X o —X n)
x i ( x r x o ' x2(x2~Xg)
( X k + l X ( ) ) X ' k+ i V f - ' x k + i x k+1
x n )
Another "theorem of determinants s ta tes i f a ma t r i x B is obta ined from
A by m u l t i p l y i n g a row of A by a constant c , t h e n | B | = C |A | . There-
f o r e ,
(1-5) ( x i - x o ) ( x 2 - x o ) - - - ( xk + T x o )
Xi xf k
x i
1 x 2 A 2
1 xk+1 xk+1 • k
' • xk+1
( x 1 - x 0 ) ( x 2 - x 0 ) . . . ( x k + T x 0 ) | V ' | .
Now from t h e assumption in ( 1 - 3 ) , s ince the determinant [V»| in (1-5)
has (k+1) terms,
I V ' | = [ ( X 2 - X 1 ) ( X 3 - X L ) . . ( x k + T x 1 ) ] . [ ( x 3 - x 2 ) ( x 4 - x 2 ) . . ( x k + T x 2 )] . . [ ( x k + T x k ) ]
F i n a l l y from equat ions (1-5) and the d e f i n i t i o n of [ V ' | , f o r each
p o s i t i v e in teger n, equat ion (1-3) is t r u e ; t hus ,
( 1 ~ 6 ) | v | = n ( x . - x . ) . 0<j < i <n
By hypothes is , x. ^ x . f o r i ^ j . There fo re , each f a c t o r in (1-6)
is nonzero and |V[ 4 0. V is known as t h e Vandermonde m a t r i x .
Example 1: Consider t he f o l l o w i n g po in ts ( 0 , . 9 9 9 ) , ( 30 , .5000 ) ,
(60 , .8660) , (90 ,1 .000) . A cubic may be f i t t e d t o these po in t s by
determin ing the unknowns aQ , a^, a 2 , a 3 in t he general cub ic .
y ~ a0 + a l x + 92 x 2 4 a jX^.
The four r e s u l t i n g l i n e a r equat ions are
a 0 = 0.000
a0 + 30aj + 900a2 + 27000a3 = 0.500
a0 + 50ej 3600a- + 216000a3 = 0.866
a0 4 90a } + 8100a2 + 729000a3 --- 1.000
= - 6859-101 ; a1 = - . 6 0 0 0 * 1 0 " 6
| VJ = [ ( 3 0 - 0 ) (60-0) (90 -0 ) ] . [ (60 -30) (90-30) ] - [ (90-60)1
= C162* 1 0 3 ) ( 1 8 - 1 0 2 ) ( 3 - 1 0 ) - 8748-106 .
By Cramer's Rule
1 0 0 0
1 .500 900 27000
1 .866 3600 216000
1 1.000 8100 729000
S i m i l a r l y , s o l v i n g f o r a 2 and a 3 y i e l d s
y = 0 + .1781 -10" 1 x - .1999-10~4x2 - .6000• 10~Gx3.
As a check, cons ider t h e p o i n t (30, .5000) .
y = 0+ ( . 1781 • 10-1 ) ( . 3 - 1 0 2 ) + (~. 1999- 10_ l f) ( . 9 * 103)-f ( - . 6 - 1 0 ~ 5 ) ( . 2 7 ' 105)
= ( .5343 > + ( - . 1799•1Q- 1 ) + ( - .1620•10~ 1 ) = .5000.
By t h e Formula of Lagrange
Other methods e x i s t f o r f i t t i n g an n"^ degree polynomial th rough
a s e t of (n+1) p o i n t s . One such method is t h a t of Lagrange.
Given (n+1) va lues of a f u n c t i o n ( x Q , y 0 ) , (x1,yL),...,(xn,y ) a
f u n c t i o n pass ing th rough these p o i n t s , of t h e form
P n ( x ) = A 0 ( x - x ! ) ( x - x 2 ) ( x ~ x 3 ) . . . ( x - x n )
(1 -7 ) + A X ( x - x 0 ) ( x - x 2 ) ( x - x 3 ) . . . ( x - x R )
+ A n ( x ~ x 0 ) ( x - x x ) ( x - x 2 ) . . . ( x - x j ) ,
may be determined. C l e a r l y each te rm of (1 -7 ) is a polynomial of
degree n; t hus P n ( x ) is a polynomial of degree n. The constants AQ,
A ^ . . . , An may be determined by r e q u i r i n g t h a t each of t h e (n+1)
po in t s s a t i s f y ( 1 - 7 ) .
L e t t i n g ( x Q , y 0 ) be a s o l u t i o n of P n ( x ) , yQ = A „ ( x - x ) ( x - x )
( x 0~ xn
) -> a n d
J/JL A0 - ( x 0 - x 1 ) ( x 0 - x 2 ) . . . ( x 0 - x )
S i m i l a r l y , l e t t i n g each of ( x ^ y ^ , ( x 2 , y 2 ) , . . . , ( xn > Y n ) be a s o l u t i o n
f o r ( 1 - 7 ) ,
_Yi_ A i = ( x
1 - x o > < x1 - x
2) - - - ( x r x
n) >
A2 = Cx2-x0 ) ( x g - X j ) . . . ( x 2 - x ) ,
An = ' V V ' V V - ' W l 1 '
Then f rom these equa t ions f o r AQ, A 1 , A , t h e fo rmula f o r Lagrange
is ^ 3 ) (x **xp ) . . , (X—Xp ) • yQ (X~-Xfl ) (x—x > ) . »« (x~ Xp ) * yi
pn
( x ) = T x 0 - x 7 H ^ - x 2 T ~ T ( ^ ~ > ^ ) + • ( T - x o ) ( x i ^TTTTTG^) (1 -8 )
(x-xp ) (x—Xi ) . . . (x-Xn-1 ) t yn
+ ( x n - x 0 ) ( ^ - x 1 ) . r . ( ^ - ^ l _ 1 ) .
ExampIe 2 . Lagrange's Method y i e l d s t h e f o l l o w i n g equa t i on f o r
( 0 , . 0 0 0 0 ) , ( 3 0 , . 5 0 0 0 ) , ( 6 0 , . 8 6 6 0 ) , ( 9 0 , 1 . 0 0 0 ) .
( x - 3 0 ) ( x - 6 0 ) ( x ~ 9 0 ) « 0 ( x - 0 ) ( x - 6 0 ) ( x - 9 0 ) . .5000 P 3 ( x ) = ( 0 - 3 0 ) ( 0 - 6 0 ) ( 0 - 9 0 ) + ( 3 0 - 0 ) ( 3 0 - 6 0 ) ( 3 0 - 9 0 )
( x - 0 ) ( x - 3 0 ) ( x - 9 0 ) - .8660 ( x - 0 ) ( x - 3 0 ) ( x - 6 0 ) . 1.000 + (60 -0 ) (60 -307 (60 -90 ) + ( 9 0 - 0 ) ( 9 0 - 3 0 ) ( 9 0 - 6 0 )
.5000 .8660 = 54000 [x (x -60) ( x -90 ) ] - 54000 [ (x) (x-30) (x-60)]
1 + 162000 [ (x ) ( x -30) ( x -60 ) ] .
As a check,
.5000 .5000'54000 P3 C30) - 54000 [30(30-60) (30-90) ] = " 54000 " " = .5000.
The of her t h ree po i n ts check in a s i m i l a r way.
By Newton's Formula
New+on's Formula is s t i l l ano ther way of w r i t i n g a p o l y n o m i a l .
I t i s use fu l i n t h a t t h e number o f p o i n t s can be increased w i t h o u t
r e p e a t i n g a l l c o m p u t a t i o n . Let ( x Q , y 0 ) , (x , y ) , . . . , ( * n > y n ) be
n+1 p o i n t s t h rough which t h e n+h degree po lynomia l Pn(><) is passed.
P (x ) can be expressed as
(1 -9 ) P ( x ) - y + ( x - x )P , ( x ) , n 'o o n-1
where P _-| (x ) i s a po lynomia l o f degree n - 1 . I t is obv ious f rom (1 -9 )
t h a t F3n(XQ) ~ yQ • Thus t h e prob lem reduces t o d e t e r m i n i n g t h e p o l y -
nomial P n _ i ( x ) . Now f rom ( 1 - 9 ) ,
Pn_! f x ) - Poi2S2zttL ( x - x 0 )
Thus t h e f o l l o w i n g must be t r u e f o r i = 1, 2, . . . , n:
^n 1 ^X i ^ ~ Pn x i )~Pn ^xn ) ~ Yi ~Yn > x i ~ x o x i ' x o
which means t h a t Pn__| (x ) passes t h rough t h e p o i n t s o f t h e fo rm
^ x j ' Vi~Vn ^• The v a l u e s y ; - y n a r e c a l l e d " d i v i d e d d i f f e r e n c e s " x, -xQ x. -xQ
and w i l l be denoted by [ x . , x Q ] .
S i m i l a r l y ,
(1 -10) pn - 1 ( x ) = [ x i ' x
0] + < X " V P n - 2 ( x ) '
where Pp_2 ^ x ) * s 3 po lynomia l of degree n - 2 .
Then P . (x ) = [x , x ] and f o r i = 2, 3 , . . . , n, P (x) is f rom n—• l l u n -2
( 1 - 1 0 )
f h - 1 ( x ) - [ x 1 , x 0 ] Pn_2 (x) = x - x x
and
Pn-1 (x; ) - [ x , , x n ] [xT , x n ] - [ x i , x n ]
( t - m P „ . 2 ( > < i , = V x i = V * i
t h e l a s t te rm in (1-11) is the d i v i d e d d i f f e r e n c e of t h e d i v i ded
d i f f e r e n c e and is denoted by [ x . , , x ] .
By express ing each succeeding polynomial in terms of a polynomial
whose degree is less by one, t h e f o l l o w i n g general express ion is
de r i ved :
d - 1 2 ) P . ( x ) = [ x - , x ,_ 1 , . . , x ] + ( x - x ; ) P ( x ) , J J J 1 0 J n - ( j + 1)
where 1 <_ j <_ n-1 and i - j , j + 1 , . . . , n.
By s u b s t i t u t i n g each of these equat ions i n t o ( 1 - 9 ) , Newton's
fo rmula is
(1-13) y ( x ) = y 0 + ( x - x 0 ) [ [ x 1 , x 0 ] + ( x - x 1 ) { [ x 2 , x 1 , x 0 ] + ( x - x 2 ) { . . . } } ] .
A t a b l e may be cons t ruc ted t o c a l c u l a t e t h e d i v i d e d d i f f e r e n c e s :
X V 0 0 r [ V x o ]
X I Y1 t X 2 ' X l ' X 0 ]
[ X 2 ' X 0 ] [ X 3 ' X 2 ' X ] ' X 0 ]
X2 , [ X 3 ' X l f x 0 ]
X y [ ^ • X » ,
3 3 • * •
• * •
ExampIe : By us ing t h e data of Example 1, determine a cub ic poly-
nomial passing th rough the data by Newton's Method.
0 .0000 .01067
30 .5000 .00012 .01444 .00000
60 .8660 .00000 . 0 1 1 1 1
90 1.0000
8
Thus,
y ( x ) = 0 + ( x - 0 ) { , 0 1 6 6 7 + ( x - 3 0 ) [ . 0 0 0 0 0 + ( x - 6 0 ) ( . 0 0 0 0 ) ] } .
Check: y(30) - 301.01667] = .5001.
From t h e d e r i v a t i o n o f t h e fo rmu la , t h e po in t s f o r t h e t a b l e do
not n e c e s s a r i l y need t o be in numerical o r d e r . Thus a p o i n t may be
added a t the bottom of t h e t a b l e and t h e next h igher o rder degree p o l y -
nomial may be computed f rom the a l t e r e d t a b l e .
ExampIe 4; Given t he po in t s ( 2 , 8 ) , ( 0 , 0 ) , (3 ,27) t h e f o l l o w i n g
t a b l e is c o n s t r u c t e d .
2 8 4
0 0 5 19
3 27
Thus t h e approx imat ing polynomial is
y ( x ) = 8 + ( x - 2 ) [ 4 + ( x - 0 ) 5 ] = 3x 2 - 6x .
Now (1 ,1 ) may be added t o t h e t a b l e and [Xg,Xg] , [ x 3 , x 1 , x 0 ] , and
[ x 3 , x 2 , X p x „ ] c a l c u l a t e d . Then t h e va lues f o r de te rmin ing a cubic
appear i n t he tab Ie:
2 8
0 0
3 27
1 1
4
19
7
There fo re ,
y ( x ) = 8+ (x -2 ) {4+x [5+(x -3)1 ] }
- 8+(x--2) (x2h 2x+4)
= v 3
Theorem 1 .2 . Given n+1 sample p o i n t s , t h e c o r r e s p o n d i n g n"^ degree
po l ynom ia l pass ing t h r o u g h t h e p o i n t s i s u n i q u e l y de te rm ined .
P r o o f : Assume t h e po l ynomia l y ( x ) is o b t a i n e d by one method of
a p p r o x i m a t i o n f o r t h e n+1 sample p o i n t s ^ x Q J Y Q ^ • • •> (x ,y ) ,
and t h e po l ynomia l zCx) i s o b t a i n e d by u s i n g t h e same p o i n t s in a n o t h e r
method. The d i f f e r e n c e y ( x ) - z C x ) is a f most a po l ynomia l o f degree n.
y ( x . ) - z ( x . ) = 0 f o r i = 0 , 1, 2 , . . . , n, s i n c e ( x . , y . ) i s a s o l u t i o n
f o r both y ( x ) and z ( x ) . Thus y ( x ) - z ( x ) is a po l ynomia l hav ing t h e
f o l l o w i n g s o l u t i o n s : ( x 0 , O ) s ( x ^ O ) , . . . , (x , 0 ) . T h e r e f o r e y ( x ) - z ( x )
is t h e po l ynom ia l 0 . x ° , o r y ( x ) - z ( x ) = 0 . Thus y ( x ) = z ( x ) .
Curve F i t t i n g by S p l i n e F u n c t i o n s
The f o r e g o i n g methods i n d i c a t e ways o f f i t t i n g an n"l*h degree p o l y -
nomia l t o n+1 data p o i n t s . As n inc reases t h e number n o f l i n e a r
e q u a t i o n s which must be s o l v e d i n v o l v e c o e f f i c i e n t s o f l a rge magn i tude ;
x n . Thus a usual method i s t o f i t t h e data by means o f a s e t o f
p o l y n o m i a l s o f degree men such t h a t each s e t o f (m+1) c o n s e c u t i v e
p o i n t s i s f i t w i t h t h e c o r r e s p o n d i n g m"^ degree p o l y n o m i a l ; t h e (m+1)£^
p o i n t of t h e p r e c e d i n g po lynomia l becomes t h e f i r s t p o i n t o f t h e nex t
p o l y n o m i a l . Except f o r t h e common p o i n t , t h e c o n s e c u t i v e p o l y n o m i a l s
a r e independent o f each o t h e r , and t h e y u s u a l l y i n t r o d u c e d i s c o n t i n -
u i t i e s i n t h e f i r s t d e r i v a t i v e a t t h e j u n c t i o n p o i n t s . S p l i n e f u n c t i o n s
a r e d e f i n e d so t h a t t h e s e d i s c o n t i n u i t i e s do no t o c c u r .
Def f n i t ion : Given a s e t o f p o i n t s ( x . , y . ) , i = 0 , 1, 2 , . . . , n
where x Q < x 1 < . , . < x , a r e a l f u n c t i o n d e f i n e d f o r a l l r e a l x in an i n t e r -
v a l [ x , x n ] i s c a l l e d a s p l i n e f u n c t i o n of d a y e e k f o r t h e poj n t s and
denoted by P(x) i f t h e f o l l o w i n g p r o p e r t i e s a re t r u e :
10
1) For each ( x . , y . ) , P ( x . ) = v . ; t ' i i i
2) P(x) is of c l ass Ck~1j P(x) has k-1 cont inuous d e r i -
v a t i v e s ;
3) P(x) is equal on each i n t e r v a l [ x . ^ , x . ] t o a polynomial of
degree k or less .
Def i n i t ion: Each x . in (1) above w i l l be c a l l e d a j o i n t .
Sp l i ne f u n c t i o n s of degree one r e s u l t i n f i t t i n g s t r a i g h t l i n e
segments between each p a i r of j o i n t s [ x . , x . ^ 1 3 > ' = 0 , U 2, n - 1 ,
wh i l e s p l i n e f u n c t i o n s of degree two have cont inuous f i r s t d e r i v a t i v e s .
Sp l i ne f u n c t i o n s of odd degree k, where k = 2m-1, are a l so impor tant
f o r t h e f o l l o w i n g reason: Of a I I t h e f u n c t i o n s Q(x) e C ^ " ' , such t h a t
Q(x. ) = y . , f o r i = 0, 1, 2 , . . . , n, t h e s p l i n e f u n c t i o n P ( x ) e C k-1 } • • . j I t f IMC- 3 p i I I I O l U l l ^ l i W M i \ / \ J t , ^
minimizes t h e i n t e g r a l
xn
J [ Q ' T x ) ] ~ d x .
This s ta tement is proven f o r cub ic s p l i n e f u n c t i o n s in Theorem 1.3.
Cubic s p l i n e f u n c t i o n s are examined in d e t a i l s ince they g i ve approx-
imate ly t he shape of a t h i n beam or " s p l i n e " which is fo rced t o go
th rough t h e p o i n t s ( X j , y . ) . This r e s u l t can be seen by c o n s i d e r i n g
t h e i n t e g r a I / [ Q " ( x ) ] 2 d x as an approx imat ion t o t h e s t r a i n energy o f a
t h i n beam, which is /Q" 2 / ( 1+Q'2 ) 5 / ' 2 d x . Thus t h e approx imat ion is a
good one i f t h e data po in t s represent a smooth near ly h o r i z o n t a l cu rve .
F u r t h e r , i f t h e cub ic s p l i n e approx imat ion i s t o c l o s e l y resemble
passing a s p l i n e th rough t h e g iven p o i n t s , then c o n d i t i o n s must be
s p e c i f i e d a t t he end p o i n t s . Namely, t h e s lopes sQ and sR s h a l l be
•given a t xQ and x n . Th is resembles the s p l i n e being clamped a t t he two
end p o i n t s .
11
Theorem 1.3 Given ( x - , y . ) , i = 0, 1, 2, n where x ( )<x1<. . . <x n ,
l e t P(x) be a s p i i n e f u n c t i o n of degree t h ree such t h a t P(x - ) - y j a t each
j o i n l x . . Let Q(x) be a f unc t i on having cont inuous f i r s t and second
d e r i v a t i v e s on O o > xR ] such t h a t Q(x . ) ~ P ( x . ) , Q ' ( x 0 ) = P ' ( X Q ) , and
Q ' ( x n ) - P ' ( x n ) . Then
rx n ^n / [Q" ( x ) ] 2 dx >_ J [ P " ( x ) ] 2 i
x0
Proof : Let 3(x) = Q(x) - P ( x ) .
(1-14) J [ Q " ( x ) ] dx > J r P " ( x ) l 2 d x . x0 " x0
f [ Q v ( x ) ] 2dx - f [ n p u ( x ) ] 2 dx
= ^ { [ Q " ( x ) ] 2 -2 [Q" (x ) ] [P , ! ( x ) ] f [ p» (x ) ]2 }dx x o xo
x 0 i^n / n + 2J [Q" ( x ) ] [ P1' ( x ) ] dx - 2 J [ P " ( x ) ] 2 d x
xo xo
= / [ & " ( * > ] 2 a * + 2 J [P" (x)] f p" (x)] dx *0 x 0
^ n 1 /»x i +1
= / f e " ( x ) ] 2dx + 2>„ / B" (x) • P" (x) dx x 0 i -0 x .
i n t e g r a t i n g each term of t he summation term by pa r t s y i e l d s
2dx - p [ P u U ) ] 2dx x Q x Q -Xp n -1
J [ e , f ( x ) ] 2dx + 2 ^ [ 0 ' ( x )P" (x ) - B ' ( x . ) P " ( x . ) ] x 0 i =0 i + 1 1 + 1
n-1 r x i + 1
-2 j ? J P " l ( x ) 8 ' ( x ) d x . i=0 xf
The f i r s t summation term s i m p l i i i e s t o 2 [ g1 (x ) P" ( x p ) - p' (:< 0 )P" (x 0 ) 3 ,
because £ ' l X j i _ i ) - 8 ' ( x j ) - 0 a t each of t he i n t e r i o r j o i n t s . The second
summation term in tegra tes simply s ince P I M ( x ) is a cons tan t . Thus
12
[ Q " ( x ) ] 2 d x - 2 d x = J ' ^ " ( x ) ] 2 d x + 2 [ e ' ( x n ) P » i x n ) - e « ( x 0 ) P « ( x 0 ) ]
n^,1
- 2 L P " ' ( i + h i n t e r v a l ) [ g (x ) - g ( x . ) ] . 1=0 1 + 1 1
In t h e second t e r m 0 ' ( x n ) = B - ( X ( ) ) = 0 s i n c e P ' ( x Q ) - Q ' ( x 0 ) and
P' ( x n ) - <?f C x n ) . The summat ion t e r m is 0 s i n c e S ( x ) = Q ( x ) - P ( x ) and
Q ( x . ) - p ( x . ) f o r 0 <_ i <_ n . F i n a l i y ,
£ ' [ Q n ( x ) ] 2 d x - £ r [ P " ( x ) ] 2 d x - ^ [ 3 " ( x ) ] 2d x > 0 ,
f [Qn ( x ) ] 2 d x J [ P " ( x ) ] 2 d x .
J h s o n ^ J ^ ^ The re i s e x a c t l y one c u b i c p o l y n o m i a l R ( x ) = £ a Cx-a) m
m=0
which assumes g iven va lues R(x> and R ' ( x ) a t end p o i n t s on an i n t e r v a l
[ a , b ] where a f b. T h i s p o l y n o n i a I i s
f , itrx . f 3 C R ( b ) - R ( a ) ) - R ' ( b ) - 2 R ' (a ) 1 ( 1 - 1 5 ) R(x ) - R (a )+R ' ( a M x - a H L ( b - a ) 2 ~ (b-cTF J * ( x - a ) 2
f — , R' C b) +R' (a ) 1 +*- ( b - a ) 3" T b - a ) 2 J ' ( x - a ) 3 .
P r o o f ^ Since R(x )= | ( x _ a , m a n d R 1 ( x ) = | ^ T h e
m-0 m - ]
s y s t e m o f l i n e a r e q u a t i o n s r e s u l t i n g f r om t h e f o u r g i v e n c o n d i t i o n s may
be w r i t t e n as m a t r i c e s R = Q. A, i . e .
a 0 R(a)
1
o o o
R('b) 1 (b--a) ( b - a ) 2 " ( b - a ) 3
R» (a ) 0 1 0 c
R ! ( b ) c 1 2 ( b - a ) 3 ( b - a I 2
X
0
U 1
a 2
a 3
The dererrr. i narvl of 1h<= c o p f f i r i o n - r ~i»+r-;v n • t o " r r k ' i e R r i r i x 0 is equa l t o t h e v a l u e
13
|Qj ( b - a ) 2 (b-a)3
2 ( b - a ) 3 ( b - a £
( b - a ) (b~a)2 (b -a )3
1 0 0
1 2 ( b - a ) 3 ( b - a f
- - 3 ( b - a P + 2 ( b - a ) 4 = - ( b - a ) 4 .
Thus |Q| / 0 s i n c e a / b, and t h e system o f l i n e a r e q u a t i o n s has a
un ique s o l u t i o n ; Le_. , t h e r e i s one c u b i c which is d e f i n e d by t h o p o i n t s
( a , R ( a ) ) , ( b , R ( b ) ) , and t h e d e r i v a t i v e a t each o f a and b. By t h o
a p p l i c a t i o n o f Cramer 's R u l e , A may be de te rm ined t o produce t h e
r e s u l t s in ( 1 - 1 5 ) .
iM i£LL tL9 ILL Given an i n t e r v a I [ xQ .x n ] w i t h j o i n t s x ^ x ^ . . .<x n ,
P (x ) w i l l be c a l l e d a p i e c e w i s e c u b i c po l ynom ia l over [ x x 1 i f o ' n
P ( X ) = R i - i ( x ) o n + h e i n t e r v a l where R._1 ( x ) is a c u b i c
p o l y n o m i a I .
C o r o l l a r y 1 .5 . I f y . and s l opes s. a r e g i v e n a t each j o i n t
x . , i - 0 , 1, n , ("here e x i s t s e x a c t l y one p i e c e w i s e c u b i c p o l y -
nomia l P(x)eC~ which s a t i s f i e s P ( x . ) = y . and P ' ( X j ) - s . .
From Theorem 1.4 t h e r e e x i s t s one po l ynomia l R (x ) J'~1
p a s s i n g t h r o u g h y ^ and y^ and hav ing t h e s l o p e a t each o f x . 1 and
Xj such t h a t = Sj_, and R - . = s . f o r j = , / 2 >
Thus P(x ) is d e f i n e d on t h e i n t e r v a l [ x , x ] as 0 n
( 1 - 1 6 ) P (x ) - R. . ( x ) f o r [ x , , , x . ] . ^ J J
Le t x 0 , x l } x 2 be such t h a t Ax o x i xQ -f 0 arid
Ax 2 - x 2 X l / 0 , b u t n o t n e c e s s a r i l y X q / X j , . L e + v ( x ) a n d w ( x ) b Q
c u b i c p o l y n o m i a l s such t h a t v < X j ) = w f X ] L ) = Y l , and v ' ( x j )
Then v " ^ ) - w " ^ ) i f and o n l y i f
w ' ( X j ) -• S 1
1 4
( i ""*** i / ^ ^ ^ "1 p ^
A x 1 v ' ( x 0 ) + 2 ( A x 1 4 A x 0 ) s 1 , ' - A x 0 w , ( x 2 ) = 31ax^ Lw(x? ) - y j + A ^ L Y l - v ( x Q ) J j ,
Assure v (Xj ) - w"'^x^). Let each of v ( x ) and w(x) be
de f ined in t h e form of R(x) in Theorem 1 .4 .
F~2(R(b)-R(a)) R ' ( b ) + R ' ( a ) l R " ( x ) = 6L (b -a )3 " + ' (b -a )2T J < x - a )
j"3(R(b)-R(a) ? R' (b) +2R' (a) ] + 2L (b -a )2 - ( b - a ) ~ ~ j .
F u r l h e r , l e t a - x^ , and b ~ x^ such t h a t
r 3 ( v ( x n ? -v(x , ) ) v ' f x p H P s , i V ' C x , ) ^ < - « „ ) ; • -
P i V ' X0 ) -y-i ) 1 = -AX0 L - i x 0 " - v» (x0 ) -2s r J .
S i m i l a r l y , l e t t i n g a = x^ and b - x9 ,
2 [" 3(w(x? ) - y i ) -j w " ( x 1 ) - AXj L AXj ' ~ w ' ( x 2 ) - 2 S l J .
l i then f o l l o w s t h a t (1-J7) is t r u e . Re t rac ing t h i s proof p ro -
v ides t h e proof of t h e converse; t h a t is g iven equat ion ( 1 - 1 7 ) , then
v " ( x 1 ) = w"(Xj ) .
' f A is a rea l (nxn) m a f r i x such t h a t
( , " 1 8 > l a i i l > % k l 1 = 1,2 k= 1, kf- i
then A is nons ingu la r .
Assume |A| = 0; i . e . , A is s i n g u l a r . Then t h e system of
equat ions
' a i l X > + a i 2 * 2 + • • • + a i nx
n = 0
a 2 1 X l + a22X2 + + a2 „ * „ = 0
a n l x l 1 a n 2 x 2 + + an i l
x „ = 0
15
has a n o n - t r i v i a l s o l u t i o n (X j , x 2 , . . . ,x ) . Let rn be t h e index such
t h a t |xm | = m a x { [ x j | , | x 2 [ , . . . , | x | } . Then t h e m~^ equat ion may be
wr i t t e n
i=n ammxm ~ " £ a m i x i
i = 1, i f m i =n
| | ^ a rn i x i f '> = 1, i M
lamml•f ml = lEamixil i EIami I"IxiI - EIam1 I"Ixmh
IsmmI — ^1am i ! »
which is a c o n t r a d i c t i o n t o t he hypothes is in (1 -18 ) . Thus j A j f 0,
or A is nons i rigu I a r .
Core I I a ry 1 »8. Le*f P(x) be a p iecewise cub ic polynomial of c lass
C1 w i t h j o i n t s x 0 , x1, . . . , x n . For g iven y? = P ( x . ) , i - 0 , 1, . . . , n
and s0 = P ' ( X q ) , sn = P ' ( x n ) , t h e r e e x i s t s e x a c t l y one se t of values
S j ~ ^ ~ n - ^ ' s u c h t h a t P(x)eC 2 .
P roo f : Since P t x k C 1 , t h e r e does e x i s t a se t of values s. = P ' ( x - ) , J J
j ~ K 2 , n - 1 . Let R j_ j (x) and R . ( x ) be cub ic po Iynomia I (s ) such
t h a t R j_ j (x) = P(x) on t h e i n t e r v a l [ x j _ . | , X j ] and Rj<x) = P(x) on t he
i n t e r v a l [ x j , X j + j ] , j = 1, 2, . . . , n - 1 . Since P(x) is a cub ic po lyno-
mial on each open i n t e r v a l ( x j . . i , X j ) , P " (x ) e x i s t s f o r each p o i n t in the
i n t e r v a l . Assume t h a t (x j ) = R"j ( x - ) a t each j o i n t x1 , x2 , . . . ,
*n-1 •
From Theorem 1 .6 , R , f j _ ^ ( x j ) = R J - ' x j ) i f and on ly i f
( 1 _ ] 9 )A X j R j - 1 ( X j - 1 ) f 2 ( A x j + A x j - t }* R j ( x j > + A x j - 1 R j ( x j + 1 }
f A X ) - 1 AXr ] " ' ^ A X j ( R j ( X j + 1 ) _ Y j 5 + A X j - 1 ( y j " " R j - 1 ( X j ~1
16
where Ax.«=? x . - x . , . Let s . ~ P ' ( x . ) , 1hen equa t ion (1 -19) may be vJ J J J
w r i t t e n as
f Ay. t 1 A x . s . ! + 2(Ax.+Ax. , ) s . + Ax. 1 s . , 1 = 3LAx. - 'Ax". + Ax? Ax. J .
J J -1 J J -1 J J -1 J+1 J - 1 J J J-1
The c o e f f i c i e n t m a t r i x Q of t h e unknowns ( s x , s 2 , . . . , s ^) is
2(Ax l-fAx^ ) A>^ 0 0 . . . 0 0
Ax> 2(AX2+AX1) AXj 0 . . . 0 0
0 AXg 2 (AXg HA x, ) A>c, . . . 0 0
0 0 0 0 . . . 2(AxR+2^ xn_3 ^ A x n~3
0 0 0 0 . . . Ax 1 2 (Ax +.Ax 0 ) n - l n - l n-l..
Since 2 | A X - + A X . i j >_ |Ax . | + | A X . , | , by Lemma 1.7 | q[ > 0 . Thus t h e U s j \ J u
system of equa t ions de r i ved f rom (1-19) is l i n e a r l y independent and
s ; , j = 1, 2, . . . , n - 1 , may be determined u n i q u e l y . Let 3 . , j : : 1, 2, J J
n - 1 , where s . = R ' ( x . ) , be t h e se t such t h a t (1-19) is t r u e . Then J J
f rom Theorem 1.6 R " . _ ^ ( x . ) = R " . ( x . ) . Thus P(x )eC 2 . j j j j
N o t a t i o n : Let S ( x ; x Q , x 1 , . , . , x n ) , where x 0 < x 1 < . . . < x n , denote t h e
se t of a I I cub i c s p l i n e f u n c t i o n s P(x) on t h e i n t e r v a l [ x 0 , x n ] a n d w i t h
j o i n ts XQ , x^ , . . . , x^ ,
Theorem 1 .9 . For each s e t { y 0 , y 1 } . . . , y n , s Q , s } of va lues t h e r e
e x i s t s one P(x) e S ( x ; x Q , x 1 , . . . , x ^ ) such t h a t
p ( x ; ) = y { , i = 0 , 1 , 2 , . . . , n ; P ' ( x 0 ) = s 0 ;
P roo f : From Coro! Iary 1 .8 , g i ven y j = P ( x - ) , s 0 = P ' ( x Q ) ,
sn = P' (Xj.t) t h e r e e x i s t s one s e t Sj , j = 1, 2, n -1 such t h a t
P(x) e C 2 . F u r t h e r f rom C o r o l l a r y 1 .5 , t h e r e e x i s t s one p iecewise
po lynomia l of c l a s s C1 such t h a t
17
P(x. )--=y. and P' (x, )=s, .
Thus f o r g iven (Xj>Yj ~ so > an (^ = ^ c r e e x i £ " t ' s o n e
P(x) e S ( x ; x Q , x i , . . ) .
C o r o l l a r y 1.5 and Theorem 1.4 may be used in a computat ional
• scheme f o r t h e e v a l u a t i o n of P(x) f o r g iven ( x j , y j ) , s Q , and s n .
Each S | , i = 1, 2, . n - 1 , is computed f rom equat ion (1 -17 ) . Then
s ince P(x) equals a cub ic polynomial R j _ i ( x ) in each i n t e r v a l
[ x , _ 1 , x , ] and each of ( X j _ j , y j _ j ) , ( x - , y . ) , sj_-j and Sj is known,
equa t ion (1-15) may be u t i l i z e d t o compute each R j _ j ( x ) , 1 = 1 , 2 ,
. . n .
Example 5: Given t h e data p o i n t s ( 0 , - 2 . ) , ( 1 , 3 . ) , ( 2 , 2 . ) , ( 3 , - 1 . )
arid (4, .5) w i t h s .5 and s = . 5 , determine t he r e s u l t i n g cub ic
s p l i ne f u n c t i o n .
F i r s t equat ion (1-17) is used t o determine Sj , s^
s o l u t i o n m a t r i x of t h e f o l l o w i n g must be determined:
; t h a t is, t h e
—1
O
o «
S1 11 .5
1. 4. 1. X C
"2 r -12 .0
0 . 1. 4. -S3- - 5 . 0
.This y i e l d s s = 2 .875 , s2 = - 3 . 6 3 3 , s3 = - . 3 4 2 . F i n a l l y , by equa t ion
(1 -15 ) , t h e s p l i n e f u n c t i o n is
P(x) - -2 .000 4 0.50Gx + 11 . 1 2 5 X 2 6.625x , where 0<x<1;
P(x) = 3 . 0 0 0 + 2 . 8 7 5 ( x - 1 ) - 5 . 1 1 7 ( x - 1 ) 2 + 1 . 2 4 2 ( x - 1 ) 3 , where 1<x<2;
P(x) !.000 - 3.633(x--2) - i . 3 9 2 ( x - 2 ) 2 + 2 . 0 2 5 ( x - 2 ) 3 , where 2<x<3;
P(x) - -1 .000 - 0.542(x~3) + 4 .683(x -3 ) - 2.842{x-~3) , where 3<x<4,
18
A computer program f o r t h i s scheme is shown in Appendix A; g iven t h e
i n i t i a l p o i n t s and end s lopes the program computes t h e r e s u l t i n g P (x ) .
Theorem 1.10. The se t S { x ; x o > x , . . . , x n > of s p l i n e f unc t i ons w i t h
j o i n t s x , x^ , x^ is a v e c t o r space over t h e f i e l d o f rea l numbers.
Proo f : Let each of P ( x ; x Q , x 1 , . . . , x ) , Q ( x ; x 0 , X j , . . . , x ) and
3 R (x ; x , x , . . . , x ) be elements o f S. F u r t h e r , l e t P. = E p. . x^
o 1 n ' i - 1 j t o i - ^ j
denote P on t he i n t e r v a l [ x . e a c ^ ° f £. a n d b_ be rea l
numbers. Let "+" i n d i c a t e t h e normal a d d i t i o n of f u n c t i o n s , namely
s p l i ne f u n c t i o n s .
i . ) (P+Q) + R = p + (Q+R), s ince in each i n t e r v a l [ x . t h e
s p l i n e f u n c t i o n is a cub ic po lynomia l , and t h e a s s o c i a t i v e p rope r t y
holds f o r cub ic po l ynomia l s .
i i . ) By t h e same reasoning as in ( i ) , (F-fQ) = (Q+P).
i I i . ) T h e se t S has a zero Z. where Z = 0 on each i n t e r v a l t x . , , x . ] , i - 1 ' i '
and P + Z = P.
i v . ) Each element of S has an inverse s ince
P + ( -P) = Z;
i . e . , P. + ( - P . ) = £ p . , .x^ -! 5 f ( - p . . . )x^ - 0 = Z
' - 1 ' - 1 j=0 > J j = 0 ! " 1 ' J
f o r each i n t e r v a l [x . ^ x j ] -
v . ) In each i n t e r v a l [ x . ^ > X - L
3 . 3 a (P. .+Q. .) = a ( ^ D . . .xJ + X q . . . x J )
• - 1 ' - 1 j=o' ' ~ 1 ' J j =0 , _ 1 ' J
J -J- ^ Y. rt ^ - a l o . , ,xJ + a £ q . .
aP . . -i- aQ . ,. i - 1 i - 1
19
T h e r e f o r e , a(P+Q) - aP + aQ.
v i )
3 (a+b)P. . = ( a + ' b ) Z p . , .x^
' - 1 j = 0 1~ 1 * J
3 . 3 . = a l p , ] ;XJ + b Z P ; 1
j = o ! l , J j = 0 1 ' ' J
aP. . + bP. , i - 1 i - l
I h e r e f o r e ,
(a+b)P = aP + bP.
* v i i ) a (bP. , ) = a Z b p . 1 ,xJ* = abP j _ 0 i - ' > J
a (bP) = (ab) P.
v i i i ) The re i s u n i t y e lement 1 i n t h e s e t o f r ea l numbers such t h a t
1 »P = P.
The above e i g h t c o n d i t i o n s g i v e t h e d e f i n i t i o n o f a v e c t o r space ove r
a f i e l d ( 3 ) . Thus t h e s e t S { x ; x 0 , x 1 , . . . , x n } i s a v e c t o r space ove r
t h e f i e l d of r ea l numbers.
Theorem 1 .11 . The s e t o f <(>. ( x ) e S{ x ; x 0 , x 1 , . . . , xn } , i = 0 , 1 . . . . ,
n+2, d e f i n e d by f o l l o w i n g t h e c o n d i t i o n s , i s a b a s i s f o r t h e v e c t o r
space } j
f 0 \ f ] <t>, ( x . ) = <5. j = \ 1 i = j , <J)' . (x 0 )=<J>'.(X )=0 f o r i , j = 0 , 1 , . . . , n ;
J ' J J J n ( 1 - 2 0 ) <J>n+1 (Xj ) = $ n + 2 ( x | > = ° . f o r i = 0 , 1 , . . . , n ;
< t ' n+1 ( x 0 ) = < f ' n+2 ( x n ) = ^ ' n + l ^ i * = $ 'n+2 (x0 ) = ° *
F u r t h e r , t h e space 5 has d imens ion ( n + 3 ) .
P r o o f : The s e t o f (n+3) v e c t o r s ,
(1-21) (1,0,...,0 ), (0,1,0,...,0 ), . . . , (0,0,...,0,1 > n+3 n+3 n+3
20
forms a bas i s f o r t h e v e c t o r space of d imension (n+3) over t h e f i e l d
of r ea l numbers ( 3 ) . From Theorem 1.9 , t h e r e is a o n e - t o - o n e cor respon-
dence between each v e c t o r Cy0 ^ , . . . , S q , s n ) and each element P(x)
i n S{x;xQ ,Xj , . . . , x n } , F u r t h e r , each element f . ( x ) i n (1 -20) maps t o
t h e i"*1"1 v e c t o r i n t h e s e t o f u n i t v e c t o r s i n (1 -21) Let (v v s Y Q > Y >
Yn> sov s r , ^ rnaP P(x) e S. Then
( y 0 » y i ^ • • - , y n , s 0 , s n ) - y 0d ,o , . . . ,o ) + . . . +yp(o,o,.,.,o,i,o,o)
+ s Q ( 0 , . . . , 0 , 1 , 0 ) + s ( o , o , . . . , o , 1 ) ;
and
P<x) = Vo ( i ) o(x> + y i ' i ' i ( x ) + . . . + y n ( ! ) n ( x ) + s 0 < ( , n + 1 ( x ) + S n < f ) n + 2 ( x ) .
t h a t i s , t h e s e t o f ^ ( x ) , k = 0 , 1 n +2 spans t h e space S.
F u r t h e r , t h e s e t <f>k(x) is l i n e a r l y independent s i n c e t h e v e c t o r
( 0 , 0 , . . . , 0 ) maps t o t h e ze ro s p l i n e f u n c t i o n Z (x ) e S, and
2 ( x ) = 0'<j>o(x)+0 •<{>i(x)+ . . . +0*<f.n+^x).
I h u s , t h e s e t o f f u n c t i o n s $ R ( x ) d e f i n e d in (1 -20) forms a bas i s f o r
vec ( o r space S { x , x 0 , x 2 , . . . , } . Since <f>k(x) spans t h e space and
i s l i n e a r l y independent , t h e d imension of s e t S is (n+3) .
— m p l e 6 : R e P r e s e n t t h e s p l i n e f u n c t i o n in Example 5 by a l i n e a r
comb ina t i on o f bas i s e lements .
The bas is e lements <f>, (x ) of t h e v e c t o r space S { x ; 0 , 1 , 2 , 3 , 4 } have
been computed us ing t h e computer program i n Appendix A. These elements
a re :
4>0(x) 1.000 - 2 . 2 5 0 x 2 +1.250X3 on t h e i n t e r v a l [ 0 , 1 ]
- 0 . 7 5 0 ( x - 1 ) + 1 .300 (x -1 )2--0.550Cx— 1)3 2]
+ 0 . 2 0 0 ( x - 2 ) - 0 . 3 5 0 ( x ~ 2 ) 2 + 0 . 1 5 0 ( x - 2 ) 3 [ 2 3 ]
-C .050 ( x -3 )-l-Q. 1 0 0 ( x - 3 ) 2 - 0 . 0 5 0 ( x - 3 ) 3 [ 3 4 ] -
• 2<x)
•3Cx)
$ (x) = +3.000x2
= 1.000 -2.200(x-
-0 .800tx-2)+1.400(x-
0.200(x-3) -0 .400(x-
-0.750x?
+0.750(x-1)+1.500(x-
1.000 -2.250 (x-
-0 .750(x -3)+1 . 500(x-
0.000
-0.800(x-
0.800(x-2)+1.600 (x-
1.000-0.200(x-3)-2.600(x-
0.000
+0.200(x-
-0 .200(x-2) -0 .400(x-
0.800(x-3 > + 1.400(x-
I.OOOx -1.750x 2
-0.250(x-1)+0.433(x-
0.067(x-2)-0.117(x-
-0.017(x-3)+0,033(X'
0.000
-0.067(x-
0.067 ( x - 2 ) - 0 . 133(x-
-0.267 (x-3 ) -0 .467(x-
On each in te rva l [ x . _ | , x . ] , P(x)
-2.000x3 on the In te rva l
1 ) 2 +1 .200 (x - l ) 3
+ 5 ( x } =
Vx)
2)2-0.600(x-2
3)2+0.200(x-3
+0% 750 x3
1)2-1.250(x~1
2)2+1.250(x-2
3 ) 2 -0 .750(x -3
•1)2+0.800(X-1
•2)2-1.400 (x-2
•3)2+1.800(x-3
-1) 2 -0 .200(x-1
-2)2+0.600(x-2
-3)2-1.200(x~3
+0.750x3
-1 )2_o.183CX-1
•2)2+0.050(x-2
-3)2-0.017(x-3
-1)2+0.067(x-1
-2)2-0.200,(X-2
on the i n tervaI
on the in te rva l
on the i n te rva l
on the i n tervaI
on the i n tervaI
•3) +0.733(x-3
in Example 5 may be represented
21
0,1]
1,2]
2.3]
3 . 4 ] ;
1]
1,2]
2.3]
3 . 4 ] ;
0,13
,2]
2.3]
3.4] ;
0, 1]
,2)
2.3]
3 . 4 ] ;
0,1]
1,2]
2.3 ]
3 . 4 ] ;
0,1]
1,2]
2 ,3 ]
[ 3 , 4 ] .
by
22
P(x) = ~2'<f>0 (x)+3*<j>i (x)+2*<J>2 (x) — 1 • 413 (x)+0„5*<ftf (xH0.5*<|!g(x)+0.5'<f>^x).
Thus P(x) on t h e i n t e r v a l [ 0 , 1 ] is
P(x) - -2 .000+0 .500x-11 .125x 2 -6 .625x 3 .
S i m i l a r l y P(x) may be computed on each of t h e o the r t h r e e i n t e r v a l s .
CHAPTER BIBLIOGRAPHY
1. B i r k h o f f , G a r r e t t and de Boor, C a r l , "P iecewise Polynomial Surface F i t t i rig, " Proceed i ngs, Genera I Motors Research Labora to r i es Syrnpos ium: Approx imat ion of Func t ions , e d i t e d by H. L. Garabedian, E l s e v i e r , Amsterdam, E l sev i e r P u b l i s h i n g Company, 1965.
2. N ie l sen , Kaj L . , Methods of Numerical A n a l y s i s , New York , The MacmiI Ian Company, 1956.
3 . Per l i s , Sam, Theory of Ma t r i ces , Reading, Massachusetts, Addison-Wesley P u b l i s h i n g Company, 1958.
23
CHAPTER I I
METHOD Or LEAST SQUARES
Polynomials of Least Squares
There are cases when i t is no t d e s i r a b l e t o compute an a p p r o x i -
mat ion by f i t t i n g a polynomial e x a c t l y th rough n+1 data p o i n t s . Of ten
t h e data p o i n t s are ob ta ined by some method which is not exac t . This
is t r u e when t h e data is taken by some phys ica l measurement'. Thus
i t may be necessary t o determine an approx imat ing f u n c t i o n 1 hat w i l l
come as c lose as poss i b l e t o t h e po in t s and s t i l l r e t a i n y p r e d e t e r -
mined c h a r a c t e r i s t i c . Th is w i l l show t h e general na ture of t h e data
but w i l l no t n e c e s s a r i l y pass e x a c t l y th rough the g iven p o i n t s .
The method o f l eas t squares is one techn ique t h a t is used in
curve f i t t i n g t o determine a f u n c t i o n t o represent data t h a t c o n t a i n
e r r o r . The p r i n c i p l e of l eas t squares s t a tes t h a t t h e best represen-
t a t i o n of t h e data is t h a t which makes t h e sum of t h e squares of "the
residua.Is a minimum, where t h e res idua l Rj is de f ined as
Rj = y . - f ( x . ) , f o r i = 0 , 1 , 2 , . . . , n ;
and f is t he approx imat ing f u n c t i o n .
Th is p r i n c i p l e in one dimension is e q u i v a l e n t t o t h e assumption
"that t h e a r i t h m e t i c mean is t h e best es t ima te of a se t of measurements,
Suppose t h a t some q u a n t i t y is measured n+1 t imes .
x . — x+ | j i - 0 , 1 , 2 , . . . , n,
24
25
where e. is t h e e r r o r of t h e measurement and x is t h e t r u e va tue . i
The p r i n c i p l e of l eas t squares f o r one dirrension imp l ies t h a t t h e best
es t ima te x of x is t h a t number which minimizes t h e sum o f t he squares
o f t h e d e v i a t i o n s of t h e data from t h e i r es t ima te . Thus t h e f u n c t i o n
f (x) = S ( x . - x ) 2
i =0
must be minimi zed.
Theorem 2 . 1 . The p r i n c i p l e of l eas t squares f o r one dimension
impl ies t h a t t h e average
1 n x = n+1 Z x .
1=0 '
i s t h e best es t ima te ; i .e . , x = x .
n Proof : Since f ( x ) - £ ( x . - x ) , f rom t h e Calcu lus
i=0
df n dx = - 2 £ ( x . - x ) - 0
i=0
fU £U = E x . - Z x = o
1=0- i =0 n
= E x . - (n+1)x = 0 . 1=0
„ J L f r x = n+1 L x . = x .
i=0 ' a
Thus x g is a c r i t i c a l p o i n t f o r f ( x ) . Now
d 2 f df n dx = - 2 dx X ( x . ~ x)
1=0 1
= 2n+2>0.
There fore x = x is a minimum f o r f ( x ) , and t h e method of l eas t squares
imp l ies t h e average is t h e best es t ima te .
26
Examp Ie 1: The l e a s t square e s t i m a t e o f t h e numbers 2, 3,2,1, 2 , 3
may be found both ways.
x - (2+3+2+142+3)/6 = 13/6. 0
f ( x ) - ( 2 - x ) 2 + ( 3 - x ) 2 + (2~x) 2 + ( 1 - x ) 2 + ( 2 - X ) 2 + ( 3 - x ) 2
= 3 ( 2 - x ) 2 + 2 ( 3 - x ) 2 + ( 1 - x ) 2 .
df
cfx = - 6 ( 2 - x ) - 4 ( 3 - x ) - 2 (1 - x )
0 = -12 + 6x - 12 + 4x - 2 + 2x
26 = 12x
x = 13/6 , and x = x .
The p r i n c i p l e o f l e a s t squares is o f t e n used t o f i t a po lynomia l
of degree m th rough n+1 p o i n t s ( X j , y . ) , where m < n+1, and t h e n+1
data p o i n t s a re known t o c o n t a i n e r r o r . Th i s method s t a t e s t h a t a good
e s t i m a t e of t h e data p o i n t s is t h e po lynomia l which min imizes t h e sum
n+ 1 ( 2 " n Sm = Z [ y r P m ( x j ) ] 2 ,
i = 1
where Pm = amxm + am_^xm~ 1+ . . . +aQ . Thus,
n +*1 n+1 £ [ y , - ( a m x ' » + a m _ 1 x » ' - ' + . . . + a 0 ) ] = TR.2 = f ( a m , V , a , ) .
For t h e po lynomia l o f degree m, t h e prob lem w i l l be t o de termine t h e
c o o r d i n a t e s of a s i n g l e p o i n t ( a ^ ' , a [ n ^ 1 , . . . ,a ' ) i n an m+1 d imens iona l
Euc l idean space which min im ize f . A necessary c o n d i t i o n t h a t f ( a .a m m - T
. . . , a ) has a minimum is t h a t t h e r e e x i s t s a va lue (a ' a ' . . . , a ' ) o m m-1 ' ' 0
such t h a t t h e p a r t i a l d e r i v a t i v e s
27
9f 9am = 0 a t a, m m
( 2 - 2 ) 3f 9am_r = 0 a t am_ m-1 ~ m-1 '
9f
9 a 0 = 0 a t a0 = aQ» .
S e t t i n g t h e p a r t i a l d e r i v a t i v e of f t o zero y i e l d s
9f r 9Ri 3R2 8 Rn+1 (2 -3 ) Saj = 2LRiaaJ.
i + 1 1 + R29aj + . . . + Rn+^ 9aj J = 0 , j = 0 , 1 , . . . , m.
F u r t h e r ,
9R 9 Ri ?JiL 30
. . . , 9a = 1, i = 1 , 2 , . . . ,n+1. (2-4) 9am - 9am_1
Thus f rom t h e equat ions in (2 -3) and ( 2 - 4 ) , t h e f o l l o w i n g rrt+1 equat ions
a re obta i ned:
(2 -5)
0 1
a n I x? 0 i
+ ajEx- + a2£x 2 + . , .. + arrr
+ a Ex . 2
1 1 + a Ex
2 3 + • . . . + a ;
m
4" a Ex 3
1 1 + a Ex
2 4 + . , * . + a :
m £>^+2;
Ex. y. 1 1
E x.2y. 1 1
a Ex.m + a Ex . m + 1 + a Ex . m + 2 + . . . + a E X . 2 m - E x.my. 0 1 I ' 2 1 m 1 1 '1
The summation s igns in (2 -5) range from i, = 1 t o i = n+1 . This system
of equat ions is c a l l e d t h e se t of normal equat ions and i t s s o l u t i o n
y i e l d s a mini mum f o r f (am,am_^ , . . . ,aQ ) .
Theorem 2 . 2 . Prove t ha f i f x . / x . f o r i ^ j , then a unique s o l u t i o n , j
e x i s t s f o r t h e system of equat ions in ( 2 - 5 ) .
P roo f : The (m+1) x (m+1) c o e f f i c i e n t m a t r i x C may be w r i t t e n as
t he product of mat r i ces D and t h e t ranspose D' of D as shown in
equat ion ( 2 - 6 ) . D is an (n+1) x m m a t r i x .
28
(2-6) C =
n + 1 Ex. Ex 2 v m
. . Ex,
Ex . i
Ex. 2 | Ex 3 m+1
. . Ex.
w EXj 3 E x r , rrtf-2
. . E x j
v"m _ m+1 m+2 iXj Exj Zxj . . EX:
*2r
1 1 1 . . . 1
X 1 X 2 X 3
X,- X0 X;
V i
^ + 1 x
1 X,
1 X .
1 X =
X 2 2
. . X,
• • Xq
X • m Xn+l
= D' • D.
m., m_, rn m . . . x n + | 1 \ i +1 "n+1
Rows 1 through m+1 in D correspond t o the mat r i x V of Theorem 1.1.
From Theorem 1.1, the Vandermonde determinant | v j 4 0, s ince x, 4 x j ,
•for i 4 j . Fur ther , m+1 is the largest integer such t h a t D has an
(m+1) x (rn+1) submatrix w i th determinant not zero. From matr ix theory
(2, p. 58), the rank of D is m+1 and the rank of D' is m+1. F i n a l l y ,
C = D*D' has rank m+1 and is an (m+1) x (m+1) ma t r i x ; thus C is non-
s ingu la r and the system of equations in (2-6) has a unique s o l u t i o n
(2, p. 60) .
Example 2: F i t a least squares polynomial of degree 2 t o the data
po in ts ( x , y ) shown in Table I .
Table I
Xj .. Yi Ri -2 .0 18.1 8.66-10"^
0.0 3.9 -4.29-10-2 1.0 3.0 8.14.10-2 3.0 12.9 -7.79-10-2 4.0 24. 1 3.85-10-2
29
The computer program in Appendix B was used t o per form t h e c a l c u -
l a t i o n of t h i s problem. The f o l l o w i n g terms were computed f o r t h e
normal equa t ions :
Ex = 6 .0 ; Ex2 = 30 .0 ; Ex3 = 84.0 ; l x k = 354.0;
Ey =62.0; Exy =101.9; Ex2y=577.1.
The s o l u t i o n of t h e normal equat ions of (2-5) p rov ided t h e c o e f f i c i e n t s
f o r t h e leas t squares q u a d r a t i c equa t ion ,
P(x) = 2.018x2 - 2 .042x + 3 .943.
The r e s i d u a l , R. = y. - P ( x j ) , was computed a t each p o i n t ( X | , y j ) and
is shown in column 3 of Table I .
Least Squares Polynomial Approx imat ion w i t h R e s t r a i n t s
Weighted Residuals
For some data i t may be des i red t o g i ve more weight t o some of t he
obse rva t i ons than t o o the rs in f i t t i n g a polynomial t o t h e p o i n t s ; thus
t h e sum of t h e squares o f t h e res idua l s of (x { , y j ) may be weighted by
w. , where Wj i- 0 . Then equat ion (2-1) becomes
n+1 <2-7) Sm - Z - , | y , - V l * l + + V 1 2
1 = 1
n+1 Z w j R j - f (am ,am_i , . . . , a ) .
i = 1
F u r t h e r , t h e p a r t i a l d e r i v a t i v e s 8jr_ , j = 0 , 1, . . . , m are 3 a j
i L _ J E l k 3Rn+i 1 (2-8) 33j = ' 2 L w 1 R 1 3 a j + w ^ a a j + + wn+1 3a j J;
and
3R; 8 Rj 8R; (2-9) 3a m = 3a m_, = ... ; 3a = w,.
30
Us ing (2 -8 ) and (2 -9 ) and e q u a t i n g t h e p a r t i a l d e r i v a t i v e s t o ze ro
y i e l d s
3f 8a0 - 2 ( W l R : + W2R2 + . . . + w n + 1 R n + 1 ) = 0;
(2 -10) 3f 9a1 = 2 (w 1 x 1 R 1 + W2X2R2 + . . . + w n + 1 x n + 1 F^+ 1 > - 0;
3f 3Q
m 2 ( " 1 ^ R 1 + « 2 ^ R 2 + • • • + » n + 1 >£+, V , ) = 0.
Making t h e s u b s t i t u l i o n s f o r Rj produces t h e s e t o f normal equa t ions
f o r we ighted r e s i d u a l l e a s t squares .
I w . a 0 + S ( w . x . ) a 1 + . . . + £ (w. xrP )am = Z ( w , y . )
(2 -11) £ < " i X i ' a 0 + E < W i X i > a i + + K w , ^ j ' + 1 )am = E C w j ^ y , )
E ( w 1 x T ) a 0 + E ( " ix 7 + , ) a , + • • • + £ ( w , x ' . 2 m ) a - ) .
i i u i i i i i rn i i 7 \ The summation s i g n ranges f rom i = 1 t o i = n+i i n each t e r m of (2 -11)
Theorem : 2 . 3 . I f X. 1 * x j
for ? ¥ j , then a unique s o l u t i o n e x i s t s
for t h e system of l i n e a r equa t i on s i n ( 2 - 1 1 ) .
P r o o f : Denot ing t h e c o e f f i c i e n t m a t r l v I n (2 -11) by F, i t may be
w r i t t e n as t h e p roduc t o f two 1 matrices D f and E; t h a t i s ,
1 1 1 . . . 1 w 1
w X 1 1
w x 2
1 1
x x 1 2 X3 • ' * Xn4-1
w 2
w X 2 2
2 W 2 X 2
m , w x
2 2 (2 -12) F = 2 2
X X 1 2
x 2 . . 3 • tf+1 X W
3 W X 3 3
W X 2
3 3 m
w x 3 3
' m m m *m m wn+1 xrH-L
x X _ 1 2
x . . 3 *
_ Wn-H wn+1xn+1 2 wn+1xn+1 • • *
m wn+1 xrH-L
= D ' *E .
The m a t r i x D< was d e f i n e d In Theorem (2 -2 ) and i t was shown t h a t t h e
rank of D' was ( m f l ) . By m u l t i p l y i n g each row i o f E by 1, t h e m a t r i x
w.
31
E is equivalent to matrix D of Theorem 2.2. From matrix theory, two
equivalent matrices have the same rank (2, pp. 60-61). Since from
Theorem 2.2 the rank of D is (m+1), it follows that (m+1) is the rank
of E. Finally, F is the product of two matrices D' and E, each of which
has rank m+1. Again from matrix theory (2, p. 60), the rank of F is
m+1, and since F is an (m+1) x (m+1) matrix, F is nonsingular. Thus a
unique solution exists for the system of equations in (2-11).
Example 3: Fit a cubic polynomial to the data points (xj»yj) shown
in Table II. Assign the weight w., which appears in column 3, to each
data point (x.,y.)
Table II
Xi Yi w; Ri Ki
0.0 . 127462. 1.0 —.4143•10~2 -.8388-10~2
0.2 .133271 1 .0 -. 1681-10""2 -. 5335•10~2
0.4 .139413 1 .0 -.3802.10"^ -.3099-10~2
0.6 . 145905 1 .0 .9253-10~3 -.1546-10"2
0.8 .152753 1.0 .1337•10~2 -.5543*10 3
1.0 .159964 10.0 .1326-10"2 0 1.2 .167528 1 .0 . 1005-10~2 .2229• 1 0~3
1.4 .175459 1.0 .5102-10~3 .2449•10-3
1.6 .183731 1 .0 -.6282* 10-lf . 1562-10~3
1.8 .192319 1 .0 -.6165.10~3 .4810.10~4
2.0 - .201186 10.0 -.1066-10"2 0 2.2 .210262 1.0 -.1358-10~2 .5819-1O-^ 2.4 .219482 1 .0 -.1437-10"2 .2740•10~3
2.6 .228741 1.0 -. 1285 -10 — 2 .6587.10"3
2.8 .237937 1.0 -.8820-10"3 . 1227 - TO"2
3.0 .246848 10.0 -.3274*10~3 .1872 -10~2
3.2 .255362 1 .0 .3887•10~3 . 2600•10~2
3.4 .263251 1 .0 . 1160-10"2 .3298-10-2
3.6 .270285 1.0 .1880•10~2 .3852«10"2
3.8 .276221 1 .0 . 2427 • 10~2 .4136 * 10 ~ 2
4.0 .280822 1 .0 .2686.10~2 .4029•10-2
4.2 .283815 1 .0 .2506•10~2 .3375•10"2
4.4 .285005 1.0 .1815-10-2 .2095•10"2
4.6 .284214 1 .0 .5563•10~3 . 1267-10-3
4.8 .281286 1.0 -. 1303 • 1 0~ 2 -.2569•10~2
5.0 .276144 1 .0 -.3719'10"2 -.5953 • 10~2
32
The computer program in Appendix B was used again in t h e s o l u t i o n of
t h i s example. The terms making up t h e c o e f f i c i e n t m a t r i x f o r t h e normal
equat ions were computed:
Zw = .53000 '10 2 , Ewx = .11900Q-103 , Ewx2 - .347000-10 3 ,
Iwx3 = .116900' Id 1 , Zvtxh = .432783-10^, Ewx5 - . 171194- 105 ,
£wxe = .710744'105 ,£wy - .111506-102 , Ewxy = .280006-102 ,
Ewx2y - .869789-102 , Ewx3y = .304675-103 .
The s o l u t i o n o f t h e normal equat ions y i e l d e d t h e c o e f f i c i e n t s f o r t h e
requ i red cub ic po lynomia l ,
P(x) = ( - . 254534-10~ 2 ) x 3 + ( . 159268-10" 1 ) x 2 + ( .136511•10 _ 1 ) x + .131605.
The data in t h e f o u r t h column in Table I I show t h e res idua l va lue Rj
f o r each data p o i n t ( x . , y . ) .
Other Restra i n ts
Given n+1 data po in t s ( x . , y . ) w i t h weights w . , suppose i t is re-
qu i red t o min imize t h e sum
n+1 (2-13) Sm = I w | [ y , - VmCxr ) 3 2 ,
where Vm(x) is a polynomial of degree m w i t h k r e s t r a i n t s on V (x) a t
x = u . ; wh ere j = 1, 2, . . . , J . The rest ra i nts shaI I be def i ned as Kj
f o l l o w s :
(2-14) v m < r ) ( u ? ) = b r i ' r = 0 , 1 , 2 , . . . , R . , , O 1 J O
where a l l t h e b . are s p e c i f i e d f o r u . . That i s , t h e curve V (x) r J J rn
s h a l l pass t h rough -u . and s h a l l have R. s p e c i f i e d d e r i v a t i v e s where J J
Rj 0. Fu r the r i f any u. = x. } then Xj may be el iminated from t h e n+1
data p o i n t s , s i nce t h i s p o i n t w i l l have no e f f e c t on m in im iz i ng Sm in
33
(2 -13 ) . This is t r u e because t h e res idue ! y . - vm ^ u j ^ = Yj ~ b
0 j ' s a
cons tan t .
An example of a se t of r e s t r a i n i n g c o n d i t i o n s is t o r equ i r e t h a t
t h e curve V (x ) pass th rough t h e po in t s ( 0 , 0 ) and ( a , b ) , and have s lope
of 0 a t t h e o r i g i n ; then t h e b r - are de f ined as
boi = Vm(0) = 0
b02 = vm
( a ) = b
b u = v ; t o ) = o.
K l o p f e n s t e i n (1) de f ines a t r a n s f o r m a t i o n of t h e p o i n t s ( X j , y . )
such t h a t t he t rans fo rmed problem is o f s tandard l eas t squares t ype
w i t h o u t r e s t r a i n t s . Let W^_^(x) be t h e polynomial of degree ( k -1 ) which
s a t i s f i e s t h e k r e s t r a i n i n g c o n d i t i o n s . Then
(2-15) Vm (x) = W R i (x) + n k ( x ) V m _ k ( x ) ,
where,
J i _j_p. (2-16) n k ( x ) = n ( x - u . ) J .
j = i J
Although t h e ex i s tence of a unique Wk_-| (x) w i l l not be proven
g e n e r a l l y , Theorem 1.2 prov ides a proof where t h e k r e s t r a i n i n g con-
d i t i o n s r equ i r e (x) t o pass th rough k g iven p o i n t s . F u r t h e r ,
Theorem 1.4 , shows t h a t t he re is a unique cub ic polynomial which passes
th rough two g iven po in t s w i t h s lopes g iven a t each of these p o i n t s . The
f a c t t h a t Vm(k) in equat ion (2-15) s a t i s f i e s t h e k_cond i t i ons is ev i den t
s ince t h e second te'rm is zero a t each o f t h e r e s t r a i n i n g p o i n t s .
From ( 2 - 1 3 ) , i t f o l iows t h a t
n+1 (2-17) ^ = - [Wk_fX|> + n k ( x , ) V ,(X|>]}2
j=1
34
is a l g e b r a i c a l l y e q u i v a l e n t t o
n+1 (2-18) S = £ * , { [ 7 . - W ( x . ) ] - J I k ( x . )V (x , ) } 2
m j = i 1 1 k-1 ' K 1 m-k 1
n U ] 2 f y j - Z ^ k - l ^ i i \ 2
= L w j n k ( x t ) \ n k ( x ; ) - v (x . )j . j = l m-k 1
Thus, w i t h t h e t r a n s f o r m a t i o n
y! = n. (x . ) , w.' = w.nf (x. ) , i k i i i k i '
t h e equat ion in (2-17) reduces t o
n+1
( 2 - ' 9 ) Sm = . ^ w i ' ( y i ' - V m - h ) ) 2 -
Equat ion (2-19) shows t h a t t h e problem is now a s tandard l eas t squares
equat ion of degree (m-k ) .
In p r a c t i c e , K l o p f e n s t e i n ' s t r a n s f o r m a t i o n prov ides a general method
f o r f i t t i n g a l eas t square polynomial w i t h r e s t r a i n t s . However, severa l
c o n d i t i o n s must be cons idered in i t s use. F i r s t , t h e number o f data
p o i n t s n+1 > m, t h e degree of t h e leas t squares po l ynomia l . Second,
t h e number of r e s t r a i n t s k should s a t i s f y the r e l a t i o n k < m. F i n a l l y ,
in t h e f i r s t paragraph i t was s t a t e d t h a t an x. data p o i n t should be
e l i m i n a t e d i f x . = u . . I f a f t e r these d e l e t i o n s , t h e number of data
poinhs remain ing is less than o r equal t o m-k, then t h e method f a i l s .
Example 4: F i t a l eas t squares polynomial of degree 3 t o t h e
data p o i n t s in Table I I . Assign a weight of 1 to each data p o i n t and
r e q u i r e t h a t t h e polynomial pass through t h e po in t s (1 .0 , .159964) and
(2.0,.201186).
The computer program in Appendix B was u t i l i z e d a l a s t t ime t o
35
per form t h e c a l c u l a t i o n of t h i s example. In summary, t h e polynomial
Wj(x) passing through t h e two requ i red p o i n t s was determined:
Wj (x ) = C.4122-10" 1 )x + .1187.
The remain ing data p o i n t s were t rans fo rmed by K l o p f e n s t e i n ' s method, and
t h e l eas t squares l i n e through these p o i n t s was computed:
V j ( x ) - - ( . 242338 -10~ 2 ) x + .855398.
F i n a l l y , t h e requ i red polynomial V3(x) was determined:
V 3 ( x ) - Wi(x) + ( x - 1 ) ( x - 2 ) V i ( x )
- - ( . 2 4 2 3 - 1 0 " 2 ) x 3 + (.8580* 10~2)x2 - .9016x + . 1 8 2 9 M 0 " 1 .
The res idua l va lue Kj = y j - V3(X j ) f o r each data p o i n t ( x j , y j ) is
shown in the l a s t column of Table I I .
CHAPTER BIBLIOGRAPHY
1. K l o p f e n s t e i n , K. L . , "Cond i t i ona l Least Squares Polynomial Approx i -ma t i on , " Mathematics of Computat ion, XV I I I (October, 1964), 659-662.
2. P e r l i s , Sam, Theory of Ma t r i ces , Reading, Massachusetts, Addison-Wesley P u b l i s h i n g Company, 1958.
36
CHAPTER I I I
A METHOD OF SURFACE FITTING
B icub ic Sp l i ne Funct ion
In Chapter I , curve f i t t i n g by s p l i n e f u n c t i o n s was discussed
f o r f u n c t i o n s of one v a r i a b l e . This method may be extended t o f u n c t i o n s
of two v a r i a b l e s , where a b i c u b i c s p l i n e f u n c t i o n is f i t t e d t o a se t of
p o i n t s ( x . , y . , z . . ) t o y i e l d a smooth sur face over t h e reg ion c o n t a i n i n g J o
t h e data po i n t s .
Def i ni t i o n : A b i c u b i c polynomial s h a l l be de f ined as a rea l
f u n c t i on
3 C ( x , y ) - £ ^ n x n y m ,
m, n=0'
where a is a rea l number. mn
Def i n i t i o n : Given a se t of p o i n t s (x. , y . , z . . ) , i = 0 , 1, . . . , n, ^ J ^ J
j = 0, 1, . . m , in a r ec tangu la r g r i d ; a rea l f u n c t i o n de f ined f o r a l l
rea I ( x , y ) on a reg ion R, XQ X _< x^ and yQ y _< y^ , is ca I led a
b i c u b i c s p l i n e f u n c t i o n f o r t h e po in t s and denoted by P (x , y ) i f t h e
f o l l o w i n g p r o p e r t i e s are t r u e :
1) For each Cx. ,y . ,z . . ) , P(y. .y. ) - z. • • J 1J i J i J
2) P (x , y ) has cont inuous f i r s t and second d e r i v a t i v e s ; i . e . ,
FCx^y) is of c l ass C 2;
3) P ( x , y ) is equal on each rec tang le Rj . , x. ^ < x x. , y. ^ < y _< y. t o a b i cub i c po I ynomia ! .
37
38
D e f i n i t i o n : Each ( x . . v . ) i = 0 1 n. ; - n 1 — — — i ' y j ^» ' u> '» • ••» n ; J - u, 1, . . m w i I I
be ca I i a id po j n t .
g e f i n i + i ° n : G , V e n a r e 9 ' o n R, x0 <_ x <_ V yo < y ym w l t h g r i d
po in ts ( X j j y j ) i 0, 1, n; j = 0, 1, . n ; P ( X j y ) w i l l be ca l l ed
a P.j.ecewise b icub ic po lynomia I over R_ i f
P<x,y) - Cj_i ( x ,y )
rectangle K j j x ,_ , <. x <_ X | ; y . _ , s y <. y J ; where C, . , < x , y ) , s
a b icubic polynomial .
In Theorem 1.11 a set of sp l i ne funct ions in one va r iab le $ k ( x ) ,
k = 0, 1, n+2 was shown t o be a basis f o r the (n+3) dimensional
vec tor space S(x; x Q , X j , . . . ^ ) where the set S was def ined as the set
of a l l cubic sp l i ne funct ions on the i n te rva l [ x Q , x n ] w i th j o i n t s xQ ,
V x n . S i m i l a r l y , l e t 0 | ( y ) , I - 0, 1, . . . , m+2, denote the set
of basis elements f o r the (m+3) dimensional space S(y- y v v ) ' '
yo ' y i ' * ' ' > "m Now l e t T denote the set of a l l funct ions of the form
m-K? n+2 ( 3 _ 1 ) P(x ,y) = L S k | * k ( x ) 0 . ( y ) ,
1-0 k=0
where is a real number.
Lemma 3 . K The set T is a vector space over the f i e l d of real
numbers.
m+2 n+2 Proc lL Let each of P ( x , y ) = £ Z <xk[ * k ( x j e , ( y ) , QCx.y)
m+2 n+2 I £ c
I =0 k=0
z z Y k | * k ( x > 0 | <y) and R(x,y) = E E ? k [ $ k ( x ) 0 , ( y ) be elements of T. Let each
of a and b be real numbers and l e t «+» denote the normal add i t i on of
funct ions,.
39
i ) T is a s s o c i a t i v e w i t h respect t o a d d i t i o n .
(P+Q) + R
= [ZZa k |4. k (x )e 1 (y) + EEYkl<!»k(x)e i ( y ) ] + E Z C k | ^ ( x ) 0 | (y)
= EEa(<1(j)1(x)0j (y) + [EEyk j <}>k (x )0 j (y) + EE^k] <j>k (x)e ( ( y ) ]
= P + (Q+R).
i i ) S i m i l a r l y , P+Q = Q+P.
i i i ) The se t T has a ze ro , namely Z = EEO^(X)0 (y) such t h a t
P+Z = P.
i v ) Each element of T has an inverse s ince
P + <-P) = Z;
SSo tkI ^ x^ 01 ^ + 2 ^ ~ a k l t y j = z -
v) The d i s t r i b u t i v e law ho lds :
a(P+Q) - alESo f (x)0 (y) + EZy d. <x)6 ( y ) ] Kl K I KI K [
= a E E ak | ^ k
( x ) 0 | (y> + a E r Y k i V x ) 0 i ( y )
= aP + aQ.
v i ) (a+b) P = (a+b)EEt*k| <(>k(x)0 (y) - aP + bP.
v i i ) a (bP) = a[bEEak |<f)k(x)0 | ( y ) ] = (ab)EEak j <}>k (x)6 ( (y) = (ab)P.
v i i i ) There is a u n i t y element 1 such t h a t 1«P = P.
Thus, t h e se t T is a vec to r space over t he f i e l d o f rea l numbers.
Theorem 3 . 2 . Let t h e r e be g iven a se t o f g r i d poi nts ( x . , y .,z. .), ^ <J J
i = 1, . . . , n; j = 0, 1, . . . , m over a reg ion R, x < x < x . v < v 0 — — n ' 'o — y
~ Ym" A l s ° ' e + + h e r e b e 9 ' v e n +he p a r t i a l d e r i v a t i v e w i t h respect t o x
a t each (x0 ,y^ . ) and ( x ^ y ) ; t h e p a r t i a l d e r i v a t i v e w i t h respect t o y
a t each ( x ^ y ^ ) and ( x . , y m ) ; t he p a r t i a l d e r i v a t i v e w i t h respect t o x
and y a t each corner s r i d p o i n t ( x „ , y 0 ) , <x n , y 0 > , ( x , , , ^ ) , <x n ,ym>.
Then the re e x i s t s a unique P(x ,y )eT which s a t i s f i e s these c o n d i t i o n s .
40
Proo f : From t h e hypo thes is , t h e f o l l o w i n g is g i ven :
z . j - P (x . , y .) i - 0 , 1 , . . . , n; j - 0 , 1 , . . . , rnj
U i i = P v ( x i ' y i ) i = 0 ' n ' J = 0 • • -»m ; (3 -2) 'J x 1 J
v i j = Py
( x { ' Vj } j = 0 , ^ ;
w . . = P ( x . , y . ) i - 0 , n ; j=0 ,m. i j xy i j
Recal l f rom Theorem 1.11 t h e d e f i n i t i o n of t h e basis elements of
< M x ) , k = 0, 1, . . . , n+2, f o r S (x ; x ,x , . . . , x ) . ^ u i n
ro i / - k (3 -3) <l>k(x j) = 6 . r = \ 1 i=k ; ^ ( x q ) = ^ ( x n ) = 0 I , k=0,1 n;
( 3 - 4 ) * n + 1 ( x i 5 - c f 'n+2 ( x i5 = 0
( 3 ~ 5 ) ' W V = • i + 2( x n ) = 1 ' ^n+1 ( x
n5 = W V = ° -
The 9 | ( y ) are s i m i l a r l y de f i ned . From ( 3 - 1 ) ,
m+2 n+2
z u = p < x i ' y j ' = | 0 J ^ k ' V V V -
m+2
z i j V V + B i i V x i ) + " - + W , i W x i " e i ( V j ) '
and m+2
(3 -6 ) = Z eM
0 | ( y . ) , i = 0 , 1 , . . . , n; j =0, 1, . . . m; 1=0 1 J
s ince f rom ( 3 - 3 ) , <^ (x . ) = 0 f o r I i- k and 1 f o r j = k. The same is
t r u e f o r 0 j ( y . ) ; t h e r e f o r e ,
z i j = P j j> i r r°> • > • • • >n; j = 0 , I , . . . , m .
n+2 m+2 u i j = P x < x i ' V j ' = | o | I ^ i ^ < X | ' 9 | < y J ) i =0 ,n ; j = 0 , 1 m.
n t ? f 1 i J * =0 ( 3 - 8 ) U i J = J o V ; < x , l »
I h i s is t r u e from ( 3 - 3 ) , ( 3 - 4 ) , and t h e cor responding equat ions f o r
<S|!yj>.
41
rn+2 n+2 V i i = P v < X i , Y i ) ^ ^ k<x i } 9 ! ( y i 5 f = 0 ' 1 » * " » n ; J"°>m. 'J y i J ( = 0 k=.0 k i k i I j
„ m+2 31 ,m-M / J=0 v . j . = | l 6 | | e ; ( y j ) e i i m + 2 t j = m .
.Equat ion (3 -9 ) f o l l o w s f rom ( 3 - 3 ) , ' (3 -4 ) , and ( 3 - 5 ) .
m+2 n+2 w i j = p x y ( > : i , y j ' = | 0 J j y k ' v i ' v i = o - ^ j = ° ' m -
For i=0,
, , m i 2 3n+1,m+1 r i =0 , j=0 ( 3 ~ 1 0 ) w j j =
( ^ )3
n + 1 , I 0 ! ( Y j ) = 3n+2,m+2 { i=0, j=m.
For i - n ,
Thus t h e r e e x i s t s a unique set of 6 . . , i = 0, 1, n+2; j - 0 ,
1, rn+2, f o r t he g iven c o n d i t i o n s . The se t of 0. . is de f ined by ^ J
equat ions (3 -7) Ih rough (3—11). F i n a l l y , t h e unique se t B. . de f ines a ^ J
unique P(x ,y )eT s a t i s f y i n g t h e g iven c o n d i t i o n s in ( 3 - 2 ) .
Coro1 I a ry 3 . 3 . The s e t T is t h e se t of a l l b i c u b i c spl ine f u n c t i o n s
over t h e reg ion R w i t h g r i d po in t s ( x . , y . ) , i = 0 , 1, n; j = 0 , 1, ^ *J
. . . , m.
P roo f : Let Q(x ,y ) be a b i c u b i c s p l i n e f u n c t i o n over t he reg ion R
w i t h g r i d p o i n t s ( x . , y . ) . There e x i s t values of Q(x ,y ) f o r each con-
d i t i o n in ( 3 - 2 ) . The re fo re , f rom Theorem 3 . 2 , Q(x ,y )eT.
Let P ( x , y ) e T ; j _ . e . ,
m+2 n+2 P (x , y ) = £ ^ 3 [ < | ^ | < ( x ) 0 | ( y ) .
1-0 k=0
<j>k(x) and 0 ( ( y ) each have j o i n t s x Q , x l 5 . . . , xn and y 0 , Y l , y^
r e s p e c t i v e l y . Fu r the r they are each p iecewise cub ic and of c lass C^.
42
Thus, any I i near combi nat ion of 4> k ( x ) a n d 0 j ( y ) is of c iass C2 and is
p iecewise b i c u b i c over t h e reg ion R. The re fo re , P (x , y ) is a b i c u b i c
s p l i ne f u n c t i o n .
Coroi I ary 5 . 4 . The se t of b i c u b i c s p l i n e f u n c t i o n s V = {<f> ^ (x) *
0 . ( y ) , k = 0, 1, . . . , n+2; I - 0 , 1, m+2; w i t h j o i n t s x. f o r <jv (x) I K
and w i t h j o i nts y^ f o r 0 j ( y ) } is a bas is f o r t h e b i cub i c s p l i n e f unc t i ons
over R w i t h g r i d po in t s ( x . , y . ) .
P roo f : From C o r o l l a r y 3 .3 , i f Q(x ,y ) is a b i c u b i c s p l i n e f u n c t i o n
over R, t hen Q(x ,y ) can be represented as a l i n e a r combinat ion of t h e
<f>k(x) • 0 | ( y ) .
F u r t h e r , t h e se t V is l i n e a r l y independent; because i f t h e b i cub i c
s p l i n e f u n c t i o n Q(x ,y ) = 0 over R, then
m+2 n+2 m+2 n+2 = 1 I e k i , M x ) e i < y ) = £ To-«j> ( X ) e . ( y ) .
1=0 k=0 |=o k-0 1
Theorems t o Aid i n Sp l i ne Funct ion Computation
The above theorem and c o r o l l a r i e s show t h a t t he re is a unique
b i c u b i c s p l i n e f u n c t i o n over R of t h e form
m+2 n+2 P<x,y) = I £ P k , < M x ) 0 . ( y ) ,
1=0 k=0
which s a t i s f i e s t h e c o n d i t i o n s g iven in ( 3 - 2 ) . The procedures developed
in Chapter I may be used t o compute the <f>k(x), k = 0, 1, . . . , n+2 and '
0 | ( y ) , I = 0 , 1, . . . , m+2 and thus determine P (x , y ) f o r t h e g iven con-
d i t i o n s . However, i t is p o s s i b l e by use of t h e f o l l o w i n g two theorems
t o u t i l i z e a more d i r e c t means of computing 1he b i cub i c s p l i n e f u n c t i o n
f o r t h e c o n d i t i o n s g iven in ( 3 - 2 ) . This is done by ex tend ing Theorems
l . 4 and 1.6 t o theorems i n v o l v i n g two v a r i a b l e s .
43
By d e f i n i t i o n , t h e b i c u b i c s p l i n e f u n c t i o n P (x , y ) equals a b i c u b i c
polynomial i n R j j ; x j _ i <_ x <_ Xj ; y j _ i <_ y <_ y j . This b i c u b i c polynomial
may be w r i t t e n in t h e form
<3-12) c <x,y) = | £ ^ J < x - x > k < y -y ) ' ,
'J I =0k-0 J
where t h e s u p e r s c r i p t s on i n d i c a t e t h e p a r t i c u l a r R. . in R.
The f o l l o w i n g Theorem 3 .5 determines a b i c u b i c s p l i n e f u n c t i o n f o r
t h e spec ia l case where n = m = 1. Al though one method f o r t h i s d e t e r -
m ina t ion was shown in Theorem 3 . 2 , a d i f f e r e n t method w i l l be presented
s ince i t is use fu l in t h e computat ion of o the r s p l i n e f u n c t i o n s .
Theorem 3 . 5 . There is e x a c t l y one polynomial c . . ( x , y ) which ^ J
assumes g iven values f o r e . . ( x , y ) , _9_ c. . ( x , y ) , _9c. . ( x , y ) , and J 9x J 3 y J
JL c j j ( x , y ) a t each of t h e f ou r corners o f t h e rec tang le R . . . 8x9y
Proo f : Let each of t h e f o l l o w i n g be g iven a t rn = i - 1 , i ; n = j - 1 , j :
zmn = c i j ( V V ' u = 3 c. .
r ~z i i \ 7i— (3.-13) 3x
mn — " i j ^ ' ^ n ^ '
vmn = - i c | j ( V Y n > : 9y J
wm n
=
8 x9y
S u b s t i t u t i n g each o f t h e f o u r ( xm > y n > z
r f l n ) i n t o equat ion (3-12) and t h e
o the rs ( V V V i ^ and ( x m , y n , w m n ) i n f o t h e a p p r o p r i a t e
d e r i v a t i v e y i e l d s s i x t e e n l i n e a r equat ions in s i x t e e n unknowns a k | ,
where k , I = 0, 1, 2, 3. I h i s system may bo expressed in m a t r i x n o t a t i o n
44
(3-14)
1 0 0 0
0 1 0 0
1 Ax • Ax2 Ax3
0 1 2Ax 3AX2
a0 0 a01 a02 ^
a10 a l l a l 2 ^3
a 2 0 a 2 1 a 2 2
_a30 a31 a32 ^
1 0 1 0
0 1 Ay 1
0 0 Ay2 2Ay
0 0 Ay3 3A'/
e . . i j
_
where Ax X -X ; Ay = y.-y, ; and 1 i - 1 j j - 1
mn
z v mn mn
u w mn rnn
Equation (3-14) may be r e w r i t t e n in t he form
(3-15) X* A* Y = B.
Each of t he matr ices X and Y is nons ingu lar and has an inverse; t h a t i s ,
1 0 0 0
(3-16) X < • - 1 =
0 1 0 0
-3 3 - l Ax2 Ax A>? Ax
_2 -2 _1_ Ax3" AX2 AX3" AX2
and Y"1^
0 -3 Ay3
0 1 - 2 1 Ay Ay2
0 0 3 - 2 Ay2 Ay3
0 0 -1 1 Ay Ay^
The next equat ions f o l l o w from (3-15) and t h e p rope r t i es of matr ices in
t he s o l u t i o n of l i n e a r systems.
X" 1«X« A« Y - X-^B
l -A-Y-Y" 1 = X~]*B»r1
(3-17) A = X ^ B - r 1 .
There fo re , from equat ion (3 -17 ) , t he re does e x i s t a unique ma t r i x A
45
d e s c r i b i n g a unique se t ; k, 1 = 0 , . . . , 3; and i t f o l l o w s t h e r e is
one polynomiaI . of t h e form of (3-12) s a t i s f y i n g t h e c o n d i t i o n s in
( 3 -13 ) .
Theorem 3 .6 . I f t h e values in (3 -2 ) are g i ven , t hen , f o r P (x , y )
of t h e form ( 3 - 1 ) , t h e values
u i j " P x ( x i ' y j ) J = 0 , 1 , . . . , m ) ,
(3-18) v . j = py
( x i ' Y j 5 ( i = 0 , 1 , . . . , n ; j = 1 , 2 , . . . , m - 1 ) ,
wi j ^ px y ( x j , y j > ( i - 1 , 2 , . . , n -1 , f o r j - 0 , m ) ( i = 0 , 1 , . . , n; j = 1 , 2 , . . , m-1) ,
are un ique ly determined by t h e f o l l o w i n g 2n + m + 5 l i n e a r systems of
a l l t oge the r 3mn + n + m - 5 equa t ions :
f o r j = 0, 1, . . . , m,
(3-19) Ax ._ 1 u i + I ) j + 2 ( A X . _ 1 + A x i ) u . j + A x i u j _ 1 > j =
ferJ- . A * i 1 31 Ax. ( z i + 1 ) J - z i j } + 4 x i _ 1 ( z . . - z . . 1 j j ) J )
i —1,2,...,n—1•
f o r j=0 ,m,
(3-20) AX; i W ; , , ;+2(AX; < +Ax- )w : ;+AX. W- , . = i - l i + l , j i - i i IJ i i - l , j
fA x ; i Ax; 1 3 1 4 k , < v , + 1 j J - v | J ) « X | . l ( v I J - v | . 1 > J ) J ,
i — 1 , 2 , . . . , n— 1
f o r i = 0 , 1 , . . . , n ,
(3-21) A y j _ t v i ) j + l+2(Ay j ._1+Ay j . J V j j + A y j V j ^ . , , =
3[ Ay. ( z i ^ j+1 ~Z i j ) + ^ y j - 1 Czi j ~ z i , J - 1 ' J j
j—1,2 , . . . ,m—1j
f o r i = 0 , 1 , . . . , n ,
46
(3 -22) Ay. .w. . , +2 (Ay . . -fAy. )w. .H-Ay. w. . . = yJ"1 i , J + 1 XJ-1 XJ 'J yJ i , J - 1
r A y j - i A x u i 31 Ay. (u. . , -u . . )-tAy. , (u. . - u . . , )| , L yJ I f J + 1 IJ \ H 'J I , J " 1 J '
J - 1 , 2 , . • . , m - 1
P r o o f : The v a l u e s in (3 -2 ) a re g i ven ; t hese are i l l u s t r a t e d in
F i g u r e 1, where a r e g i o n R is shown w i t h (n+1)x(m+1) g r i d p o i n t s a long
w i t h t h e g i ven va lues a t each g r i d p o i n t .
z u z z . . . z u j=m * * * * J v w v v v w
j = m - 1
j = 1
J=0
z u z z z u * * * *
z u z z z u * * * *
z u z z z u * * * *
V w V V V w
o ii 1=1
CM
II i=n
F i g u r e 1
A long each g r i d I i n e y = y . , j = 0 , 1, . . . , m,
P ( x , y ) - p j . ( x ) e S ( x ; x 0 , x 1 , . . . , x n ) and P x ( x , y .) = P j ( x ) -
By C o r o l l a r y 1.8 of Chapter I , t h e numbers p ' j ( X j ) = P ( x r , y j ) = U t j , J vj U
i = 1, 2, . . . , n - 1 , a re u n i q u e l y de termined i f p j ( x j ) , 1 = 0 , 1, . . . ,
n, and p l ( x 0 ) , P j ( xn ) a r e known. Since P j ( x j ) = Z j j and p j ( x Q ) = u 0 j ,
^ j ^ X n ' ~ U n j a r e 9 ' v e n ^ o r J = • • * > m> '"I" f o l l o w s f rom C o r o l l a r y
1.8 t h a t u . ., ( i = 1 , 2 , . . . , n - 1 ; j = 0 , 1 , . . . - , m ) is u n i q u e l y determined
by t h e m+1 s e t s o f n -1 equa t ions in ( 3 - 1 9 ) .
A long each g r i d l i n e x = x . , i - 0, 1, . . . , n,
P(x»y) = s , ( y ) e S ( y ; y 0 , y i , . . . , y m ) , and P y ( x j , y ) -- s» j (y ) .
47
By s i m i l a r reasoning as in t he preceding paragraph, t he equat ions in
(3-21) determine v . . ( i - 0 , 1 , . . . , n ; J - 1 , 2 , . . . , m ~ 1 ) un ique ly , given
the va lues of ( 3 - 2 ) . F igure 2 i nd i ca les the parameters which have now
been determined.
j=m z u *
V w
z u * V v*U •
z u * V w
j=m— 1 z u * V
z u * V
z u * V
z u * V
•
J=1 z u V*
z u V*
#
•
z u V*
J=o z u v*w
z u V * v* U •
z u * V w
1=0 i - 1 1=2 i =n
F igure 2
Along each g r i d I i ne y = y . , j = 0, m, vi
P y l x . V j ) = I * k C x > t E B k | e | < y j ) ] = q j ( x )£S Cx;x0 , Xj x n ) ;
an<^ ^ x y ' x ' ' ~ ^x^ * ^ ' n c e 1 j ' x j ' = w j j » ' ~ n> a 'id q j ( X j ) = v . . , ^ J ^ j
i - 0, 1, . . . , n, are given f o r j = 0, m; equat ion (3-20) is of the form
of C o r o l l a r y 1.8 and t h e r e f o r e un ique ly determines W j j , ( i = 1,2,... n-1-
j = m), g iven the values in ( 3 - 2 ) .
F i n a l l y , f o r each i = 0, 1, . . . , n; P x ( x , , y ) = ze, ( y ) t z e k l ^ ( x f ) ]
= r f ( y ) e S ( y ; y o , y i , . . . , y m ) a n d P x y ( X | , y ) - r ( ( y > .
For each i = 0, 1, r | < y j ) = P x ( x „ y j ) . j = 0, 1, m, , s
e i t h e r g iven or i t can be un ique ly determined from (3 -19 ) ; arid r ' ( y - ) , ^ J
j - 0, m is e i t h e r g iven or can be determined from (3 -20 ) . Co ro l l a r y
1.8 is used a las t t ime t o conclude t h a t w- - ( i — 0 1 n- i - 1 ? IJ V l J - i j Z ,
48
. . . , r n - l ) , is un ique ly determined by equat ion (3 -22 ) , w i t h (3-19) and
( 3 - 2 0 ) , g iven ( 3 - 2 ) .
The re fo re , t h e g iven values in (3 -2) and equat ions (3-19) th rough
(3-22) determine un ique ly t h e Z j j , u - j , v - j and w j j a t each g r i d p o i n t ,
F i gu re 3 i l l u s t r a t e s t he conc lus ions of t h e theorem.
J=m z u z u z u z u
X * X * V w V w V w V w
z u z u z u z u * X V w V w V w V w
z u z u z u . . . z u J* ~ 1 SL
1 V W V W V W v w
z u z u z u . . . z u v*w V*W V*W V*W
i=0 i=1 i=2 . . . . i=n
F i gu re 3
Procedure f o r Computation of a Sp l i ne Surface
Given the equat ions in ( 3 - 2 ) , t h e systems of l i n e a r equat ions in
(3-19) th rough (3-22) may be u t i l i z e d t o determine t h e u , V j j » a n c '
w, . in ( 3 - 1 8 ) . Each of t h e systems of equat ions i n (3-19) th rough
(3-22) can be expressed as a m a t r i x equa t ion where t h e c o e f f i c i e n t
m a t r i x is t r i d i a g o n a l and of t h e form shown in C o r o l l a r y 1.8 of
Chapter I .
A method w i l l now be descr ibed f o r t h e s o l u t i o n of a system of
l i n e a r equat ions whose c o e f f i c i e n t m a t r i x is t r i d i a g o n a l . The Crout
Reduct ion Method of s o l v i n g n_ l i n e a r equat ions is we l l known in numer-
i c a l a n a l y s i s (2, pp. 429-434) . Crout computes a der i ved m a t r i x
49
^ = I I a j j I I f r o m +he coef f ic i en t m a t r i x A = | | a. . [ | and a der ived ^ J
m a t r i x C1 = | | c | | f r o m t h e cons tan ts C = | J c , | | . These are then
employed in t he c a l c u l a t i o n of t h e s o l u t i o n se t x. . The elements of
t h e de r i ved mat r ices and t h e s o l u t i o n se t are computed as f o l l o w s :
'J 'J k=1 ik k j ' ( 3 ~ 2 3 ) i>J ;
( 3 ~ 2 4 ) a ! • = f a - • • Z a . 1 a ' "J L i J ^ i k kj-1 i i '
i < j ;
i - i ( 3 ~ 2 5 ) C« = f c . - y a> c ' T / a t •
1 ^ 1 [£'1 1 k k ' '
(3-26) x! - ci - i 'ikv k= i + 1
where _i_ and j_ range f rom 1 t o n. In ( 3 - 2 3 ) , t h e summation is zero when
j = 1. S i m i l a r l y t h e summation te rm of t h e o the r t h r e e equat ions is
ze ro when t h e upper l i m i t of t h e summation te rm is less than t he lower
l i m i t . The order of computat ion in C r o u t ' s Method is t o compute each
element in t h e f i r s t column of t h e de r i ved m a t r i x A' by ( 3 - 2 3 ) ; then t o
compute elements 2 ih rough n of t h e f i r s t row by (3—24). Next compute
t h e remain ing n - 1 elements of column 2 by (3-23) f o l l owed by t h e n -2
elements of row 2 by ( 3 - 2 4 ) . The procedure cont inues u n t i l a l l elements
of A' have been c a l c u l a t e d .
Since t h e elements on each row _i_ of a t r i d i a g o n a l m a t r i x are zero
except t h e elements a . ^ . ; and a. f + 1 ; t h e equat ions in (3-23)
th rough (3-26) have zero terms. There fo re , t h e equat ions reduce t o t he
f o I lowing:
50
(3-2.7) a.'. = a. . , f o r i > j ; 'J "J
(3-28) ^ = a x i ' a,' j = a i j - a i ' - , , j - a i ' , j - , f o r
(3-29) a.'. = a. . / a . ' . , f o r i < j ; I J I J I I
(3-30) c- = a / a ^ ; c y < c , " a , f , _ , •=,'_, >/=/ , , f o r i > l ;
( 3 - 3 , ) xn = c^ ; x. = c . ' - a . ' _ . + 1 . x H | f o r K n .
F u r t h e r , each c f and a.! te rm in (3-31) has an a.*, denominator in I ( y I "T* I j j
i t s e v a l u a t i o n in (3-30) and (3-29) r e s p e c t i v e l y . The re fo re , l e t
(3-32) d, = c , ; 0( = C j ~ s | J , - j ' dj ^ , ' ' o r i =2 ,3 , t h e n ,
< 3 - 3 3 ) x n = d n / a ^ n ; x, = " r a i > i + 1 - x i + 1 ) / a ; . , f o r i = n - l , n - 2 t .
Thus, by t h e use of (3-28) and (3 -32 ) , t h e s o l u t i o n of a system of
l i n e a r equat ions whose c o e f f i c i e n t m a t r i x is t r i d i a g o n a l is g iven by
( 3 - 3 3 ) .
The above method may be used t o so lve t h e systems of I i n e a r
equat ions (3-19) th rough (3-22) in Theorem 3 . 6 , which a id in computing
1 he s p I i ne f u net ion. Si nee on ly two d i s t i net coef f i c i en t matr i ces
appear in t h e systems of equat ions in (3-19) th rough ( 3 - 2 2 ) , equat ions
of t h e form of (3-28) need t o be computed on l y t w i c e . Equations (3-32)
and (3-33) are computed f o r the s o l u t i o n s of each of t h e 2m+n+5
systems. Th is computat ion y i e l d s t h e z . . , u. . , v . . , and w. . a t each ^ J ^ J J ^ J
of t h e g r i d po in t s ( x . , y . ) of t h e su r f ace , • ^ J
F i n a l l y , t h e s i x teen coef f i c i ent s of eac h b i cu b i c poIy nom i a I
C | j ^ X ' Y ^ ° n ^ r ' c ' s c l u a r e R j j a r e computed by equat ion ( 3 - 1 7 ) . Once
t h e c o e f f i c i e n t s a ^ of (3-12) are computed f o r each R. . , t h e e v a l u a t i o n J
o f t h e s p l i n e f u n c t i o n a t a p o i n t ( x ' , y » ) reduces t o f i n d i n g t h e ( i , j )
such t h a t ( x f , y ' ) e R . j and e v a l u a t i n g t h e cor respond ing b i c u b i c f u n c t i o n
51
a i (x* , y ' ) ; j jd e s t ,
c . j ( x ' , y ' ) = z».
- x a r " p l e 1 : F , t a b i c u M c s P l i n e f u n c t i o n t o t he data p o i n t s shown
I - F igu re 4 . Let x „ - 4000; 4x. = . ,000; y 0 . 6000; and 4 y . .
y j ~ y j - 1 1 0 0 0 ' w h e r e i and j each range f rom 1 t o 3 .
* * * z 1452 1368 1274 u - . 023 V - . 1 2 7 - . 1 1 8 - 185 " 1 q/I w - 259*10 ~5 ~ - . 194
^ I U - . 3 5 6 - 1 0 " 5
*
1199 -.088
* * *
z 1564 1515 1 4 4 4 u -.012 v
*
1368 - . 075
z 1647 1618 u - . 0 1 1 v
* * * * *
157-5 1530 - .048
* * * 1680 1650
*
z ^731 1718 U ~ - 0 2 0
v - . 0 6 2 - . 051 _ 054 ~ n t H w .900*10~5 ' - . 065
121 • 10""^
F igure 4
A program which appears In Appendix C was used f o r the computat ion
t h i s example. A t o t a l of n ine b i cub i c equat ions of t h e form
'3 3
E Eakl
(x-x - j ' C y - y . . ) k 1=0 k=0 k l 1 - 1 J -1
were determined.
c o e f f i c i e n t s a k j o f each is shown in m a t r i x form a long w i th
t h e reg ion R | J over which t he b i cub i c equat ion p e r t a i n s .
52
R 11
R 12
R 13
R 21
[4000<x<5000; 6000<y<7000]
Power of Y 0 X 0 1.731-103
1 2 3 Power of Y 0 X 0 1.731-103 -6.200 •10" .2 -4 .146 ' 10" •5 1.946* 10" 8
1 -2.000*10-2 9.000 •10" -6 2.773* 10" 9 . -2.773* 10" 12
2 2.421 • 10-5 9.004 • io- -9 -1.183- 10" 10 6.031- 10" 14
3 -1.721-10-8 -7.004 •10" -12 6.562- 10" 14 3.464- 10" 17
[ 5000<x<6000; 6000<y<7000]
Power of Y 0 X 0 1.718-103
1 2 3 Power of Y 0 X 0 1.718-103 -5.100 -10" -2 -9.135- 10"" 5 4.235* 10" 8
1 -2.321-10-2 5.996 •10" -6 -3.694- 10" 8 1.391* 10" 11
2 -2 .741-10" 5 -1.201 •10" •8 7.857- 10" 11 -4.363* 10" 14
3 1.262 *10"8 3.013 •10" -12 -3.077- 10" 14 1.686- 10" 17
[6000<x<7000; 6000iyl7000 ]
Power of Y 0 X 0 1.680-103
1 2 3 Power of Y 0 X 0 1.680-103 -5.400 •10" •2 QO
O
00 I 10" 5 2.948* 10" 8
1 -4.018 ' 1 0 ~2 -8.983 •10-•6 2.790* 10"" 8 -2 .279 ' 10" 11
2 1.044-10"5 -2.970 •10-•9 -1.374* 10" 11 6.937* 10" 1 5
3 -2 .686MO" 1 0 9.523 •10-•13 -8.666* 10" 1 5 6.358- 10" 18
[4000<x<_5000; 7000<y<8000]
Power of Y 0 X 0 1 .647 *10 3
1 2 3 Power of Y 0 X 0 1 .647 *10 3 -8.654 •10" 2 1.691 - 10" 5 -1.337- 10" 8
1 -1 .100 -10" 2 6.227 •10" 6 -5.546- 10" 9 -1.681 ' 10" 12
2 -2.476-10 ~5 -4.663 •10" 8 6.265* 10" 11 -4.588- 10" 14
3 6.761*1 0 ~9 2.029 -10" 11 -3.832 * 10" 14 2.889* 10" 1 7
53
R : [5000<X<6000; 7000<y<8000]
R2 3
31
Power of Y X 0 1.618 -103 -1 .067 * 1 0" 1 3 .570 -10 " 5 -3 .205-•10-.8
1 - 4 . 0 2 4 * 1 0 " 2 -2 .615 •10" 5 4.789-10™3 -6 .782- 10--1?
2 -4 .477•10~ 6 1.425' -10" 8 - 5 . 2 3 1 - 1 0 " 1 1 4.078- 10" -14
3 1.716*10~ 9 -7 .963 ' • 10~12 1 .980 ' 10_ l i f -1 .440- 10' -17
[6000<x<7000; 7000<y<8000]
Power of Y X 0
0 1 2 3 Power of Y X 0 1.575*103 -1 .265* i o " 1 7 .965-10 - 6 -1 .245- 10" -8
1 -4 .405-10~ 2 -2 .154* 10" 5 - 4 . 0 4 6 - 1 0 " 8 3.158* 10" •] 1
2 6.704-10~7 -9 .640* 10~9 7.071•10~ 1 2 -2 .416* 10" •15
3 -1 .625- 10""9 2.693* 10" 1 2 1 .040.10-1L f • -8 .694* 10" •18
[4000<x<5000; 8000<y<9000]
Power of Y X 0
0 1 2 3 Power of Y X 0 1.564-103 -9 .283* 10"2 -2 .320 .10~ 5
4.032- 10" 9
1 -1 .200-10~ 2 -9 .908* 10~6 - 1 . 0 5 9 . 1 0 " 8 9.498- 10" 12
2 -5 .462-10~ 5 -5 .896* 1 o~8 - 7 . 4 9 9 . 1 0 " 1 1 9.088- 10" 14
3 1.761*10~8 3.031* 1 0 " 1 1 4.834.10" l l + --5.957* 10" 17
R 3 2 : [5000<X<6000; 8000<y<9000]
Power o f Y o 1 2 X 0 1.515" 10 -1.314* 10"1 - 6 . 0 4 4 . 10~5 4.484 *10~8
1 -6.838* 10"2 -3 .692* 10~5 -1 .556- 10~8 1 .256 • i o - 1 1
2 - 1 .767* 10"6 3.195- 10~8 . 7 .002. 10" 1 1 -9 .782
J* •—1 1 O
•
3 -8 .491* i o " 1 0 -1 .157- 10" 1 1 -2 .340- 10"1'4 3.774 *10" 1 7
54
R 3 3 : [6000<X<7000; 8000<y<9000]
Power of Y 0 1 2 3 X
0 1.444-103 -1.479*10~1 -2.938*10~5 7.310*10~9
1 -7.447*10~2 -7 .715*10" 6 5 .429*10"8 - 4 . 9 8 9 * 1 0 - 1 1
2 -4.315* 10"5 - 2 . 7 4 7 ' I C f 9 -1 .780* 10~13 2.539* 10" l l f
3 2.780* 10"9 -2.577*10" 1 2 -1.568* 1 0 " l l t 1 .001*10" 1 9
In a d d i t i o n the f o l l o w i n g po in t s on the sur face were computed:
(4500, 6500, 1685); (5500, 7500, 1542); (6500, 8500, 1327).
CHAPTER BIBLIOGRAPHY-
1. De Boor, C a r l , "B i cub i c Sp l ine I n t e r p o l a t i o n , " Journal of Mathematics Phys i c s , XLI (September, 1962), 212-218.
2 . H i ldebrand, F. B . , I n t r o d u c t i o n t o Numerical A n a l y s i s , New York, McGraw-Hil l Book Company, 1956.
55
APPENDIX A
PROGRAM TO COMPUTE SPLINE FUNCTIONS
Method of Computation
The program, SPLINE, is w r i t t e n in t h e Fo r t ran language f o r a
CDC 6600 computer. I t u t i l i z e s c e r t a i n theorems which were proven in
Chapter I . The system of l i n e a r equat ions in (1-17) of C o r o l l a r y 1.8
is cons t ruc ted f rom t h e inpu t data : (x . , y . ) , i = 1, 2, . . . , n; s ^ t h e
d e r i v a t i v e a t x : and s , t h e d e r i v a t i v e a t x . The s o l u t i o n of t h e i n n
system of equat ions is determined by use of C r o u t ' s Method f o r s o l v i n g
l i n e a r equa t ions ; t h i s computat ion prov ides t h e s lopes s. a t each of
t h e i n t e r i o r p o i n t s i = 2, 3, . . . , n - 1 . F i n a l l y , equat ion (1-15) of
Theorem (1 -4 ) is u t i l i z e d t o determine t h e cub ic polynomial on each
i n t e r v a l (x. , , x . ) , where i = 2, 3, . . . . n. i - 1 i ' ' '
Each i n t e r v a l (x. , , x . ) and t h e c o e f f i c i e n t s of t h e r e l a t e d cub ic i - l i
po lynomia l are p r i n t e d on an ou tpu t l i s t i n g .
P repara t i on of Input Data
Data is input f o r t h e computer v i a punched cards . A d e s c r i p t i o n
f o r cod ing these cards f o l l o w s . I t inc ludes the parameter, i n c a p i t a l
l e t t e r s , t o be coded; t h e card columns in which t h e parameter is t o
appear; and t h e F o r t r a n fo rmat . In a d d i t i o n a d e f i n i t i o n of t h e para-
meter is g i ven .
Card Type J_
N_ Columns 1-10 Format 110 The number of data po in t s t o be read.
56
57
S1 Columns 11-20 Format F10.0 The d e r i v a t i v e a t t h e f i r s t p o i n t .
S n Columns 21-30 Format F10.0 The d e r i v a t i v e a t t h e f i n a l data p o i n t .
Card Type I I
X ( I ) , Y d ) , I=1,N Columns 1 - 1 0 , 1 1 - 2 0 , . . . , 7 1 - 8 0 Format 8F10.0 Data p o i n t s . Each x.. and y j is coded in p a i r s w i t h f ou r p a i r s per card . These are coded on successive cards u n t i l a l l of t h e N dafa p o i n t s have been coded. Fu r t he r t h e X ( I ) ' s must be in numerical o r d e r .
Tra i l e r Card Columns 1-10 Format 110 A -999 is coded in columns 7 th rough 10 t o t e rm ina te t h e program.
A l l parameters should be coded r i g h t j u s t i f i e d in t h e columns i n -
d i c a t e d . Any of t h e parameters S | , s , X ( l ) , o r Y(1) may inc lude a
decimal p o i n t i f necessary t o represent i t a c c u r a t e l y .
The program can compute any number of s p l i n e f u n c t i o n s in a g iven
computer j o b ; t h i s is done by coding Card types I and I I f o r each de-
s i r e d spl ine f u n c t i o n . Fo l l ow ing t h e data f o r t h e l a s t spl ine f u n c t i o n
a t r a i l e r card must be coded.
58
L I S ' i l N o OF FGKTRA'M PROGRAM
P R O G R A M S P U l f v t ( i N P U T , C U I F U r , I A P F 2 0 ) .
O l M f i i M S ! 0 ^ A ( 5 U » b O ) •> C ( 5 0 ) » X ( 5 0 ) > V ( 5 0 ) , t )E|_T X ( 5 0 ) » S ( S O )
5 REAO .10 » n » S ( 1 ) • S ( " i ) » ( X ( l ) , Y ( I ) i l = I , M>
1 0 F O R M A T ( I ]. 0 » 7 F 1 0 . 0 » / • ( 8 F 1 0 . 0 ) )
I T ( r O 2 0 0 1 2 0 0 > 1 5
1 5 0 0 2 0 I » 2 , N
2 o » E L r x ( I - I I = X ( D - U I - I I
DO 2 5 1 - 1 , 2 5 0 0
2 5 M i ) = 0
N K 2 = W - 2
1)0 1 0 0 I = 1 » N M 2
C ( I ) = 3 . * (OF L T X ( I ) (Y ( 1 + 2 ) - Y ( I + 1 ) } / b E L T X ( I + 1 ) +
i O E ' L T X ( I + ) ) » ( Y ( I + 1 ) " Y ( I ) ) / D t l T X ( J ) )
A ( i , i > - 2 , * ( d f l t x ( t + i ) H ) t v r x ( n )
I f ( I » E C ? « 1 ) 5 5 • t>5
5 5 C ( I ) ~ C ( U - S f 1 ) * O E L T X ( 1 + 1 )
6 0 TO 1 0 0
b 5 A ( I , 1 - 1 J 1 X ( 1 + 1 )
I F { I , ! : _0. tvt- '2) 7 5 » 8 0
7 5 C < I } = C ( t ) - S ( N ) # n £ L T X ( 1 )
o o r o l o o
f 'O A ( I i I + 1 ) - D E L TX ( I )
1 0 0 COtV T I t ' t J E
PR I XT 1 0 1
101 F C m - ' A r (JHJ » « I M r i AL HA I'R 1 CES#) DC J 2 0 I " l , r . > M 2
P H t N ' T 1 1 5 i ( A ( l » j ) i J = 1 » N M 2 ) » C ( I )
3 J 5 FCpr-'Aj (10F 12, A) l'*0 c o n t i n u e
C A L L C R O t ' T ( A , c , s ( 2 ) , N M 2 )
P R I N T 1 3 0 , ( S ( 1 ) s I - - 1 i N )
1 3 0 F O ^ N ' A T ( i H O i * S C L U T l H N ^ A T R I X * / ( 1 0 F 1 2 . 4 ) )
P R I N T 1 4 0 N i V i = N « 1
DC 1 7 5 I - 1 1 M M 1
O t * L T Y " Y ( I + l ) - Y ( I >
r 3 = ( 3 . 1 ' 0 E L T Y / D E L T X - S ( l + l ) ~ 2 . 1 : - S ( l ) ) / 0 E L T X { i )
T ' i - ( " 2 « ^ u c l - T Y / OF L ' f X { I ) + s ( I + 1 ) + s ( I ) ) / O E L T X ( I ) # « 2
P R I N T 1 7 0 , X { J ) j Y ( 1 ) » Y ( { ) ) S ( 1 ) ) T 3 > 1 4 , ' X ( I )
1 / 0 F O R M A T U HO » ft {<*» F b " « 0 » <*,<!•) F S . O * •&)•& i 5 X » 4 F 1 0 , 3 » 5 X »
1 H F! F r. T - X - & j F 1 0 * 3 )
1 / 5 C O N T I N U E "
1 4 0 F C R V A T < 1 H 0 » « S P L I N E F U N C T I ON , S (X ) . # / # CM I N T E R V A L ( A , B ) «
1 » * » S ( x ) --F { r ) ~ ^ • 3 6 \ 1 I T + ! ' T $ 2 + , WHERE f ~X - *A •
2 « / , 5 x , # ( A , B ) '•> J ?,')A 1 » 1 0 A >451 L t t » 1 OX ; # i 'H* t 1 OX , )
P R I N T 1H-.J» 5 ( 1 ) » 5 ( M 5 » ( X ( I ) , Y ( I ) » I = 1 * N ) 1 bO FCRiVA I { I H 0 1 / 1 * i \ ' R U f # * / ( 2 F 1 0 , 2 ) )
(30 f C 5
59
200 CONTINUE END SLffjfJC'JT IN£ CRCUr (ft»C»X»N,LlMl ) Oll^ENS 1OM A(U1M1»LIM1 >»C(1)»X(1) IH s ^ DO 560 1 = 1 i Ih IF{ A{ I » t ) ) 5 6 0 * 5 0 0 , 5 6 0
500 DO 51 'J K = I»M IF {A <K » 1 ) ) S ? 5 » 5 l O > b 2 S
510 CCM1nU£ PHI NT 518
51H FCR^aT ( 1H1 i#£>f<CK-2£fcO UIAGCNmL El EMFNT IfcPuT TG»» 1# CRCUT#)
C/U.L EXIT 5£5 00 540 J ~ 1s N
5 - A ( I t J ) A ( T » J ) = A ( K i J )
5^0 A(K * J ) = S s - c c I > C ( I ) - C (K) C ( K ) = S
560 CONTINUE DO 120 J=1 .N DC l i b 1=1»N I F ( J ~ 1 ) 1 1 5 » 1 1 5 » 4 5
45 S = 0 I F { t - J ) 0 0 » 7 5 » ? 5
60 L U ' s I - l DI v = A {I »1) GO TO bS
' 5 LIM=J-1 0 1 V - 1 •
65 IF ( l . I h - 1 ) 95»86»06 bb DC 90 K=ltLIM 90 S = S*A ( I , K) -"-A (K »J) 95 A {I » j ) ~ (A (I , J ) ~5) /1>I V
U 5 CONTINUE i<io c o n t i
00 200 1 = 1 «N S = 0 IF { I - I ) l a s » 1 8 5 1165
165 L l M = I " l • DC 180 K = 1»LIM
IbO S=S+A<I»K)*C(K) \t>b C ( ) ; ) » ( C ( J ) ~ S ) / ^ ( I > 1 ) 200 c o n t i n u e
X (M) -C (N) IF ( \ - i ) ^ ^ 5 j Z ' 8 b » ^ 3 S
235 J~N- ,1 Z<*0 S"-0
L T M = J +• 1
60
UC 26'"- I=L1N»N S = S + M J » ) ) # X ( I I X ( J ) = C ( J ) - S J J ~ 1 I F ( J ) c'tib »<?fs5 s £V)
? b 5 RETURN e n d
APPENDIX B
PROGRAM TO COMPUTE A LEAST SQUARES POLYNOMIAL
Method of Computation
The program KLOPLS can compute an n+h degree l eas t squares p o l y -
nomial f rom data po in t s ( x j , y j ) by t h e method der i ved in Chapter I I .
F u r t h e r , a weight may be assigned t o each data p o i n t and r e s t r a i n t s
may be placed on t h e leas t squares curve as descr ibed in equal ion (2-14)
of Chapter I I . I f r e s t r a i n t s are g iven , t h e ou tpu t of t h e program con-
s i s t s of t h e c o e f f i c i e n t s of po lynomia ls W and V in equat ion ( 2 - 1 5 ) .
I f no r e s t r a i n t s are s p e c i f i e d , t h e c o e f f i c i e n t s of t h e s tandard leas t
squares polynomial a re l i s t e d . In a d d i t i o n , t h e res idua l a t x ' f o r a
p o i n t ( x ' , y ' ) may be computed and o u t p u t .
KLOPLS is w r i t t e n in t h e Fo r t ran computer language f o r a CDC 6600.
I t s length f o r c o m p i l a t i o n arid execu t ion is 60,000 o c t a l l o c a t i o n s .
P repara t ion of Input Data
Contro l Card 1
I TOT Columns 1-5 Format 15 To ta l number of r e s t r a i n t s t o be p laced on t h e l eas t squares p o l y -nomia l . 0 <_ ITOT <_ m, where m is t h e degree o f t h e des i red leas t squares po lynomia l . F u r t h e r , t h e sum of t h e KK values on t h e res -t r a i n t cards should be equal t o ITOT.
Res t ra j nt Card(s) •
Th is t ype of card is coded i f ITOT > 0 . I t s p e c i f i e s t h e r e s t r a i n t s o f equa t ion ( 2 - 1 4 ) .
61
62
KK . Columns 1-5 To ta l number of r e s t r a i n t s a t a p a r t i c u l a r XX.
Columns 6-15
Format I 5
Format F10.0 XX The X coo rd ina te va lue a t which the KK r e s t r a i n t s app ly .
(YY(L) ,L=1, KK) Columns 1 6 - 2 5 , 2 6 - 3 5 , . . . , 6 6 - 7 5 Format 6F10.0 YY(1) - t he va lue of t he curve a t XX.
YY(2) - t h e va lue of t h e f i r s t d e r i v a t i v e of t h e curve a t XX.
YY(KK)- t h e va lue of t h e (KK-1)S^ d e r i v a t i v e of t h e curve a t XX.
Con t ro l Card 2
Columns 1-5 LSD The degree of t h e leas t squares po l ynomia l .
IWS Columns 6-10
Format 15
Format 15 Swi tch i n d i c a t i n g whether t h e data p o i n t s w i l l have v a r y i n g we igh ts . IWS = 1 means a l l data p o i n t s w i l l have equal we igh t . IWS 4 1 means t h e t h e weight f o r each data p o i n t w i l l be read from each data p o i n t ca rd .
IRES Columns 11-15 Format 15 Switch i n d i c a t i n g whether r e s i d u a l s are t o be c a l c u l a t e d . IRES = 1 means no r e s i d u a l s w i l l be c a l c u l a t e d . IRES = 2 means data cards c o n t a i n i n g ( x ? , y ! ) w i l l be read and t h e r e s i d u a l w i l l be c a l c u l a t e d f o r ( x ' , y ' ) .
Data Poi n t Cards
One data p o i n t card is coded f o r each inpu t p o i n t .
X Columns 4-13 The absc issa f o r t h e inpu t data p o i n t .
X Columns 14-23 The o r d i n a t e f o r t h e input data p o i n t .
W Columns 24-33 The weight assigned t o (X ,Y) . W is coded on ly i f IWS i- 1 on Cont ro l Card 2,
Data Po in t T r a i l e r Card
Format F10.0
Format F10.0
Format F10.0
Fo l l ow ing t h e l a s t data p o i n t ca rd , a t r a i l e r card is coded which con ta i ns 999 in columns 1 th rough 3.
Residual Cards
These cards are coded i f IRES = 2. on Cont ro l Card 2 .
63
X' Columns 4-13 Format F10.0 The abscissa f o r t h e p o i n t where t h e res idua l is t o be c a l c u l a t e d
Y1 Columns 14-23 Format F10.0 T h e ' o r d i n a t e f o r t h e p o i n t where t he res idua l is t o be c a l c u l a t e d
Residual T r a i l e r Card
F o l l o w i n g t h e l a s t res idua l card , a t r a i l e r card is coded which con ta i ns 999 i n columns 1 th rough 3 .
End of Job Card
F o l l o w i n g t h e l a s t se t of data f o r a j o b , a card is coded which con ta i ns -99 in columns 1 th rough 3.
AM parameters should be coded r i g h t j u s t i f i e d in t h e columns
s p e c i f i e d . The parameters XX, YY(L), X, Y, W, X ' , and Y' may inc lude
a decimal p o i n t is necessary f o r t h e i r r e p r e s e n t a t i o n .
The program can compute any number of l eas t square curves in a
computer j o b . The End of Job Card is coded a f t e r t h e l a s t data t o
t e r m i n a t e t h e computer run.
64
LI ST IMG OF FOK fh'A> J P K O G H A M -
PHG';^A1-i K L C P L S ( I N P U T , O U T P U T )
D I M E N S I O N A ( 2 0 * 2 0 ) » C < 2 0 ) , B < 2 0 ) , X ( ? 0 ) , Y ( 2 0 ) > K ( 2 0 ) t
1 XL (1OUO j , YL (1UOO) t tili (<-J0) j VjL ( i 0 0 0 )
1 ISU-v = 0
P K ) N T 2
2 F C H ^ A T ( iHl s AST S M U a H E S POLYNOMIAL, C U P V F F I T T I N G * )
IX- 1
PfcAl) 5 » I T 0 r
5 F O R M A T ( [5,7F 1 0 . 0 )
if ( i r o n s c o , u o » i 2
12 IF{I 1 0 T - I ) 15, 11,lb
U K d A O 5 ) K K » X ( i ; » Y ( 1 )
K (I X ) - 1
B < 1 ) " Y ( U
i o e 6 = o
GO io n o
15 1 = 1
I 0 E G S I T 0 T - 1
c 5 H t A O 5 » K K f X ( I X ) . (Y (t) » L - 1 » K K )
K ( I X ) - K K
L ~ 1
A (I » I T C T ) s i ,
DC Vi K - 1 , I D L G
J D E G - M + 1
'*5 a (I »m,4)=A(I , M h + l >*X ( IX) bO C ( I ) ~ Y ( L )
L~| +1 1 = 1 + 1 IF ( L ~ K K ) 7 0 » 7 0 j 1 0 0
70 FlOEG-lO'cG
0 0 fib M = 2 »ITCT
A ( J ,H-I ) ---h (I - L j I) - F l O t G
fib FlOfc]G-FlLU-:G"l . A ( I , I T C T ) s O ,
GO TC bO 1 0 0 I S U ^ - I S U M + K K
I F ( I S U M - I T C T ) 1 U H » 1 0 5 « 1 0 5
1 02 iXr. X x + 1
GO TO 2 5 '
105 C A t L C K C U T ( A , C , H , H O T , 2 0 )
110 PKJK'T 115» I D L b , (I # b (1) » 1~1, I TOT )
115 FCpf-'AT (1 H'i) i'PCL t'NOH j .-\L C O E F F I C I E N T S F O R W { X ) w H F R F # »
I D F G H t E OF w IS*' 13/ ( 1 6 » K \ 2 , 3) ) PKJ NT 1 2 5 , (1 ,x {J) ,K (I) » I=s I, IX)
1 2 5 FC W A T (I tiO »5X j i-<LSTK41MTS*»3X > N U M B E R # / {1 ? F 10 .3 j 18) )
}3<J pt:A0 1 35 i L S D » J wS •» 1 H t S
135 F 0 R ; V A ' M 3 I S )
DO 171 t ~1»I 0 0 0
65
1^5 i-iP-AO ISO* I EM) ,XL ( I > » YL <1) >V-»L (1 > 150 F'Cfit/Ar ( U , 3 H 0 . U )
IF ( IEND-999) 160 •) 1 «0* 160 160 I F { I To r ) 5 0 0 9 1 7 1 , 1 6 1 161 CO 170 KK~) ,1X
I F ( X L ( I ) ~ X (KK>)1 JO 9145»170 170 CONTINUE 171 c o n t i n u e
L L - 1 0 0 0 GO TO hi2
180 L L " I •* 1 1^2 I F ( I W S - 1 ) 1 9 5 »1 OS »195 165 00 190 1 = 1 H I . 190 wt. { I ) - 1 » 195 IF ( T TOT)500 * 2 5 5 * 2 0 0 200 00 25u L=1»LL
V A t . B (1 ) IF t I T 0 T ~ 2 > 2 2 1 ? 2 ) 5 , 2 l 5
215 DO 220 i -^2>JT0T 220 V :AL~VAL*XL (L) +'<S (M) Z21 P l s l .
00 2 ^ 0 K ~ l » I X KK-K (>')
240 P i = pl<> (XL <L) - A (>••> > YL ( U = < Y L ( U - V A L . ) / P I
250 UL (L) «wL (L ) i ; 'PI<: -PI PP INT 252
252 FORMAT (1H0 5 22X^TRANSFORMED DATA POINTS# /3QX , <>X^ ? 1 OX » l»Ytf , 1UX »#¥<#)
255 USUfi = L S i ; - n O T PH1NT 2 5 6 , ( X L ( I I ) ?YL. ( H ) »Wl , (U> » U ~ 1 » L L )
256 FOw.vAT (20x , 3 F U . 3 ) CALL. POuYLS (LSOP >> XL J YL 1 V<L j LL »A 120 > C , OB) N0--LSDR+1 PRINT 2 /0»LSDR 00 275 1=1»NO f l r .LSOR-1 + 1 P R T M 272» I I » t i H U )
272 FORMAT(* A ( # » I 2 . « ) # » E 1 6 . B ) 2 / 5 CONTINUE 2 (0 FORMAT(1H0»*C0FFF. FOR LS POLYNOMIAL,DEGREE#,13)
GO fO ( 1 » 3 0 0 ) » I H E S 300 p i n NT 305 305 FOmwat ( 1H1 ,#«ESlOUAt. D IFF EtfENCf.S* / 1 OX ? *X« • 10X> »
18A-»«PES<*) 310 REAO J 50» IEMO? XX t YY
I F ( IENO«999)3?.0»1»320 320 VAL=B(1>
IF ( TTOT-1) 322 ' 3 ' i5 > 335 322 VAL"0»
P I s: 1 ,
66
GO "JO 3 o 5 3 3 5 DO 3 V J 11=2»I TOT 3'+0 V A V A L * X X + B (M) 3 ^ 5 P I ~ 1
0 0 3 6 0 M=1»JX KN=K(M)
3 6 0 P I - p 1 ^ ( X X - X ( M ) ) # * K K 3 5 5 V A L 2 - 0 8 ( 1 )
LL~LSUR+ I DO 3 8 0 Ms2» l . L
3 b 0 V a L c = V A L 2 » X X + l3B(f1) fiES"YY~ (VAL + P l ^ V A L i J ) PPTNT 3 9 b » X X » Y Y * K t f S
3 9 5 F O j ^ A T f 3fc 11 • 3 ) GO TO 3 1 0
5 0 0 CONTINUE F N f ) SUpROUT .(ME POLYLS (LSOR»XL» Y|.» vjU , L L » / \ 1 L I H I f C » B )
C « « « FORK NOrtMAL E u U ^ r i C M S AS MATRIX PRODUCT A#B = C C*-»* CP OUT TO SOLVE FOR C O E F F I C I E N T S B ? Wh'EPE THE LS C POLYNOMIAL I S riU)*A*<M + b ( 2 ) # A « < M M - l > * . . » + [ 3 ( M H ) .
i j l ! . : F ^ S l O r l XL (1J »YL ( 1 ) »WL ( 1 J »« ( L I M 1 » X M 1 ) , n U ) >C Q > NOrrLSuR+I DO 15 K « l » N O
15 A ( K 1 1 > ~ 0 * C» t f * FOpv F I !<ST CCLUf-N OF A
0 0 5 0 1=1 »LL V A L ~ ',v L ( t ) # X L ( I ) # » L 5 0 P t )045 K ™ 1 9NO J = f ; C " K + l A < J , 1) =A U » 1 ) • V A L
45 V A L s V A L « X L ( I ) 50 c o n t i n u e
FORV UPPEP TPlArvGULMP M/ iTUIX DO 7 5 t . - l U . S D H L I M " N O ~ L +1 DO 7 0 J » 2 « L I M
70 A ( j - 1 i l . + l } =A ( J » I J V5 CONTINUE
C # # * FORV LAST c o l u m n a n d c o n s t a n t c c l u h m 00 q o J = 1 , M 0 C < j ) = 0 .
9 0 A t ^ j N O J ' - O . 0 0 ]HQ I «1? LL V A I, = W l_ ( I ) V A L 2 - w L ( I ) "• YL ( £ ) 00 1 3 5 K = 1 » N 0 J = N C ~ K + I A ( J ,i\ 'G) - A < J»NC) + VAL
67
J 25 C ( J ) --C ( J ) +VmL2 V A L = V A L # X L ( I )
135 VA|.2 = VAL2»XL u ) 1'VO CO\iT I r-Uli"
LC\\ER TRiAMoULAR H A l K l X no 170J : 'K*LSDH VAL~A(J>NO) '^"LSOR™ ,J +1 DO 16 b K s 1 , M
165 A ( J + K?NC--K) =VAL 170 CONTINUE
PRINT 175 1 f b FORMAT t 1 r i-0 3 ttCGLif- F1C LENT MATRIX OF i\'ORfiA!. EQUATIONS'®*)
DO <>00 i^= l iNC P R I M ^Ob i (A {K1 J) i J~ 1 s NO) » C (K)
c 05 FGRt-'»iT ( 1 H * BE i 6 . d) 200 00NT I <nlit:
CALL CKOur ( A , C » ' H » w C t L I H l ) RETURN EM) SUnRQUTlNE CROUf (A i C j X t N I t, IM1 ) DIMENSION A ( L l M l , L l M i ) , C U ) * X ( 1 ) }H.zt\/2
DO 5 6 0 I ] t X h i l~ ( A ( I » I } ) (5fo0 »5uO f 5 6 0
5 00 1}0 S1 u K " I 9 N I F { A ( * U ) )525»5 iO»13 i ;5
5H) CCNTir-tHE PRl f \ ;T '518
S i b FORf/AT (1H 1 ? vEHRO^"2E»<0 DIAGONAL ELEHENT JfiPUT T0#» i # CUCUT*)
CAI.L EXIT 5 ^ 5 00 5 ^ 0 J = 1 , N
S ~ A (1 »J) A (T , J ) ~t\ {K »J)
5-40 A ( K » J ) = S S-C < I ) C ( I ) = C ( K ) C (K) =;S
5 6 0 CONTINUE 00 1^0 J - 1 s N DO l \ h I s 1*N I F ( J ~ l ) U 5 > U 5 i 4 5
45 5 = 0 I F { I - J ) 6 0 , 7 5 i 7 5
60 L l M = I - l 0 I V ~ A { I ) I ) 60 TO 65
75 L l M s J - 1 0 I V « 1 .
0 5 J f" (L I r i •" 1 } 9 5 i H 6 » H 6
68
66 00 <50 K - 1 » L J M 90 S=S + A( L »K) (K».J) 95 A ( 1 , J ) - ( A ( I , J ) - S ) / D i V
11S CON f I - U t I'dO CCNTIiJUb'
00 2 0 0 I - 1 » N s=o I F ( 1 - 1 ) 1 »185 > 16S
165 L -l M™ J. ~ 1 DO 180 K " 1 , L I M
180 S = S + A ( I i K ) '::"C (ft ) l b 5 C ( I ) " ( C ( I ) - S ) / A f I , I ) 200 C«MTlwUE
X ( ^ ) " C ( W ) IF ( f \ - 1 ) 2 & 5 i 2 8 b i 2 3 ^
235 J = N - 1 240 S~0
L I M ~ J + 1 DC 2 6 0 I=L1M»N
260 S™S + A ( J « I ) X ( I ) X ( J ) - C ( J ) - S J - - J - 1 JF < J ) 2 « 5 » 2 h t j f 2 ^ 0
205 ntTUK-' l ENO
APPEND IX C
PROGRAM TO COMPUTE BICUBIC SPLINE FUNCTIONS
The method descr ibed in Chapter M l f o r f i t t i n g a b i c u b i c s p l i n e
f u n c t i o n t o data is programmed in t h e F o r t r a n program SPLINES. The
inpu t c o n s i s t s of t h e data f rom g r i d p o i n t s t h a t are shown in F igu re 1
of Chapter I I I . The se ts of l i n e a r equat ions (3-19) th rough (3-22) in
Theorem 3 .6 are so lved t o y i e l d t he requ i red u . j , v j j > and w j j . Next,
t h e c o e f f i c i e n t s of t h e b i c u b i c polynomial f o r each rec tang le Rj . a re
determined by equat ion (3 -17 ) . These c o e f f i c i e n t s may be l i s t e d as
o u t p u t . F u r t h e r , t h e s p l i n e f u n c t i o n may be eva luated a t any ( x ' , y ' ) in
t h e reg ion R. The ( x ' , y ' ) is read by t h e program which then determines
t h e R. . such t h a t ( x ' y 1 ) e R . . . The b icub ic poIynomiaI f o r R s t is i j U 'J
eva lua ted a t ( x ' , y ' ) and t h e r e s u l t i n g p o i n t ( x ' , y ! , z ' ) is p r i n t e d .
P repara t i on of Input Data
Input t o t he program is from cards . B a s i c a l l y t h e r e are f o u r
t ypes : c o n t r o l ca rd ; data p o i n t ca rds ; e v a l u a t i o n p o i n t ca rds ; and
t r a i l e r c a r d .
Cont ro l Card
N_ . Columns 1-5 Format 15 The number of columns in t he input da ta .
M. . Columns 6-10 Format 15 The number of rows in t he input da ta .
KSW Columns 13-14 Format 12 Switch i n d i c a t i n g types of ou tpu t des i red . KSW = 10 means ou tpu t on ly t h e c o e f f i c i e n t s of t h e b i c u b i c p o l y -
nomia1s.
69
70
KSW = 01 means t o eva lua te t h e sur face a t ( x ' , y T ) data p o i n t s and p r i n t ( x ' , y ' , z T ) .
KSW = 11 means t o p r i n t both the above t ypes .
LAST Columns 16-20 Format 15 Switch i n d i c a t i n g l a s t problem. LAST >_ 0 i nd i ca tes t h a t inpu t f o r another spl ine f u n c t i o n f o l l o w s
t h e c u r r e n t s e t . LAST < 0 i n d i c a t e s t he l a s t se t of data f o r t he computer j o b .
I MESS(I) ,1=1,6 Columns 21-80 Format 6A10 Alphanumeric i n f o rma t i on which is p r i n t e d on t he ou tpu t l i s t t o i d e n t i f y the problem.
Data Po in t Cards
(XCI) ,1=1,N) Columns 1 - 1 0 , 1 1 - 2 0 , . . . , 7 1 - 8 0 Format 8F10.0 The X coo rd ina tes of t h e input g r i d p o i n t s . These must be coded in ascending numerical o r d e r . I f N > 8, t he X va lues are coded on success ive ca rds .
(Y(J ) ,J=1,M) Columns 1 - 1 0 , 1 1 - 2 0 , . . . , 7 1 - 8 0 Format 8F10.0 The Y coord ina tes of t h e input g r i d p o i n t s . These a l so must be coded in ascending o rder and on successive cards i f M > 8 .
( ( Z ( J , I ) , l = 1 , N ) , J = 1 , M ) Columns 1 - 1 0 , 1 1 - 2 0 , . . . , 7 1 - 8 0 Format 8F10.0 The Z va lues a t each g r i d p o i n t ( X j , y . ) .
(U(J ,1 ) ,J=1 ,M) Columns 1 - 1 0 , 1 1 - 2 0 , . . . , 7 1 - 8 0 Format 8F10.0 The p a r t i a l d e r i v a t i v e w i t h respect t o X f o r each g r i d p o i n t of column 1. I f M > 8, these parameters are cont inued on consecu t i ve ca rds .
(U(J ,N) ,J=1,M) Columns 1 - 1 0 , 1 1 - 2 0 , . . . , 7 1 - 8 0 Format 8F10.0 The p a r t i a l d e r i v a t i v e w i t h respect t o X f o r each g r i d p o i n t of column N. I f M > 8, these parameters are con t inued on successive ca rds .
( V ( 1 , I ) , 1 = 1 , N ) Columns 1 - 1 0 , 1 1 - 2 0 , . . . , 7 1 - 8 0 Format 8F10.0 The p a r t i a l d e r i v a t i v e w i t h respect t o Y f o r each g r i d p o i n t of row 1. These are coded on successive cards i f N > 8.
(V (M, I ) ,1=1 ,N) Columns 1 - 1 0 , 1 1 - 2 0 , . . . , 7 1 - 8 0 Format 8F10.0 The p a r t i a l , d e r i v a t i v e w i t h respect t o Y f o r each g r i d p o i n t of row M. I f N > 8, t h e V (M, I ) are con t inued on successive cards .
W ( J , I ) , I = 1 , N ; J = 1 , M Columns 1 -10 ,11-20 ,21-30 .31-40 Format 8F10.0 The p a r t i a l d e r i v a t i v e w i t h respect t o X and Y f o r each corner g r i d ' p o i n t : W(1,1) , W(1,N), W(M,1), W(M,N).
71
Eva lua t ion Po in t Cards
These cards are coded i f KSW f 10.
X' Columns 4-13 Format F10.0 The X coo rd i na te of a p o i n t f o r which t h e s p l i n e su r face is t o be eva lua ted .
Y' Columns 14-23 Format F10.0 The Y coo rd i na te of a p o i n t f o r which t h e s p l i n e sur face is t o be eva lua ted .
Tra i l e r Card
Th is card is coded i f KSW i- 10. I f f o l l o w s the l a s t e v a l u a t i o n p o i n t card and has a 999 coded in columns 1 -3 .
Each parameter is coded r i g h t j u s t i f i e d in the f i e l d s i n d i c a t e d .
The data p o i n t cards and e v a l u a t i o n p o i n t cards may inc lude decimal
numbers. The Z va lues on t he data p o i n t cards are coded by rows w i t h
e i g h t va lues per c a r d . A l I of t h e va lues on row one are coded consec-
u t i v e I y, f o l lowed by those f o r row two w i t h th6 i n i t i a l va lue of row two
coded in t h e next f i e l d a f t e r t h e n^^ va lue of row one. Th is procedure
con t inues th rough a i l m rows.
72
L I b r I M'.3 OF Fcr<TK«!M PROGRAM
PKCGHaM SPLlNtSfl -:PUT J OUTPUT) CCMyC-i Z(50,50)» 0(50*50)» V(50?50)» W(50,50) CGMPCU X(50}, Y(50)
COMMON !J i (50) ? U2(50)» C (50) » ANS(50)> A(4»4), 8(4)4) 01 Mf US I ON K S W (3) » I fif-I S S (o)
5 PEAO 10, N,M, (KSW(K) ,K = 1»3) »LAST, ( 1 M E S S U ) » I=l»6> 10 FORMAT (2I5,2X*3I1« I5»6A1U) 20 FGpvAT (iFlO.0)
PfcAO 20,. (X (1) 9 1 = 1 »N) PF-AD 20, (Y (J) , jssl fM) HEaO 20, ( (2 (J, I) , f~l,N) Rf.AO 20, (U (J) 1 ) , J=1 ,M) PEAO 20, (U (0 ? i•)) j «J™ 1 * H) RtAO 20, (V(l»l),1-1,M) RtAO 20, <V(M»I)»I~1»N> H L" A 0 2 0 , W ilil)* w ( 1 , N) , W (' < i 1 ) , w (M , N) PRINT 50, (IMtSS(l)»1=1,6)
50 FCp^AT U h l ,6A10) CALL SPGRID(N,H) P R I M 200, ( (u<J»I) » 1 = 1 » 1 0) »J=.l j iO) PRINT 200, ((V(J,I) , 1 = 1,10) »Jr: 1 »1 0) PPINT 200* ((W(J »I)» J ~ 1 i 10)> J"1•10)
200 f- 0 R U A 1 ( IH1 1 (10 F 13•> ) ) CALL SPOUT(KSW IF (LAST) 100,5>5
] o o continue end subroutine SPO^Jo{N»M) CCmn'GN <?(b0,50), U (50 > 50) t V(50»50), V.<(50,50) CCm^GM x (50),Y(50)
C^MN'GN U1 (5{;) t U2(5U) 9 C(b0)> AftS(50)« A (4,4), 0(454) C « S C L V t * StTS OF LINEAR EOUATIO^S IN (3"l9)
KM 1 -- N-l NM? ~ N-2 MSw = 1 DG 25 1«2>N
25 U1 (I-l) = X (I)» X(1-1)
DO 35 I~2,N
35 U2(I~1) =. 2 . M U 1 I J.) +Ui (I-l) ) OG 80 J=1,M DG 50 I~2)NM1
SO C(I-I) a 3.#(Ul(l"l)^{2(j tI + l ) - Z ( j , I ) ) / u l - t U
i + Ul <I)MZ(Jj I)~Z(J»i-l) )/ljl (I«»l) ) c(1) = C(1)-ul(2)*U(J,1) C(K'V2) " C(N'M2) - U1 (NMl) *U (vj, N) C^LL 1.1NS0L (U1 »'J2 * C • AUS , i\'I '2 j MSiv ) DG 75 I~Z» MMl
75 U(J,I) s AMs(I~i)
73
HO C C N T l f j U E
S O L V E S i iTS C F L I N E A R E Q U A T I O N S I N < 3 - ? l ) Kf-'l ~ M~ 1
- f/j—2 MSw = 1 0 G \ 2 5 v J — 2 s M
1 2 5 U K J - l l - Y ( J ) ~ Y ( J - 1 ) UC 1 3 5 J = 2 , M
} 3 5 U 2 ( J - l ) •- 2 , # ( U l ( J ) t U J ( J - U }
DO I d 0 ,1 1 »N 0 0 I S O
T j O C ( J - 1 ) = 3 . * ( U 1 ( J - l ' ) » ( 2 ( J + l , { ) » Z ( J , I ) ) / i j l { J ) I + U l ( J ) » ( Z { J , I ) » / ( j « l , | ) ) / u l ( j - i ) )
C ( ] ) C ( 1 ) - U l ( 2 ) » V < 1 j I >
C ( M f * ' 2 ) ~ C ( m M 2 } - U l (MM1 ) « V (M* I )
C L L L I w S 01. ( U 1 ? i J 2 •> C j A N S »i-'i f-i 2 j M S •'>) UC 1 7 : 3 J = 2 i M M 1
1 7 5 V ( J « 1 ) « A N s ( J " l ) 1 H 0 C O M I r-JUf£
c * * * s c ; i \ / e s e t s o f l i h e . a r e q u a t i o n s i n ( 3 - 2 0 ) k s w = 1 0 0 2 2 5 I ~ 2 » h
2 2 5 U l ( 1 - 1 ) ~ x ( I ) - / ( I ~ l )
DO 2 3 5 I ~ 2 » N
2 3 b t j 2 ( I - 1 ) = 2 , * t U ! ( U + U 1 ( 1 - 1 ) )
0 0 2 & 0 J - , I » M j H M l
0 0 2 h i ) I = 2 , N M 1 2 b O C < X - 1 > = 3 . *• (tJ 1 { [ ™ 1 ) & ( V ( J 1 I + 1 } ~ v ( J 1 I ) ) / t 1 1 ( 1 )
1 + U l < 1 ) « ( V ( J , J ) « v ( J * 1 - 1 ) ) / u l { I ~ 1 ) )
c ( i ) = c d ) » u i { 2 ) - « ' w ( j , n C ( N ^ 2 ) = C ( M M 2 ) '• U l ( N M l ) { J , M )
C A L L L I *nS01- ( U 1 1 U 2 1 C 1 A N 5 »sN! 12 > MSW)
DO 2 7 5 l a 3 , N M l
2 7 5 W ( J , 1 ) = A N i S ( I - l )
2BO C O N T I N U E C # # # S O L V E - S E T S C F L I N E A R E Q U A T I O N S I M ( 3 - 2 2 )
KSW = 1 0 0 3 2 5 J = 2 , M
3 2 5 U l ( J - l ) n Y ( J ) - f ( J * • 1 )
DC 3 3 5 3 3 5 i j 2 ( j ~ l ) = 2 , » ( u l ( J ) 4-IJl ( J ~ 1 ) )
0 0 3 8 0 I - 1 9 N DC 3 5 0
3 5 0 C ( J - 1 ) - 3 . # ( U l { J « l ) f t ( | j ( j t l , I ) . U { j , I ) ) / l i l ' { J )
I • U l < J ) # < U < J » I > - u U ~ l » I > ) / l j l ( J - l ) )
C ( 1 ) = C < 1 > -• U.i ( 2 ) # w ( l , I )
C {Mr.'2 ) -- c (f IMS ) - U 1 (MM 1 ) " M» I J C A | J , 1.1 ''iSOl. ( U I » U 2 * C ) ANS J •'•Hi2 « > ISW) 0 0 3 7 5 J = 2 » M M 1
3 7 5 "i ( J , 1 ) s Ai'.'S ( J " 1 ) 3 8 0 C O N T I N U E
74
i a o
ANS ( b 0 ) » > > (A ? A ) i B (!\ i 4 )
20
35
50
60 6 IS
RETURN END SUp R 0UT1 N S P C G £ F < I * J > CCmiyGiJ ^ ( 5 0 » b 0 ) » I j ( ! r > 0 f 5 0 ) » V ( 5 0 , S 0 ) f W ( U U , 5 0 ) CCM"C.-; A (SO) , Y ( 5 0 ) CGN.WO.-J U l { 5 0 ) » U 2 ( 5 G ) » - C ( 5 0 ) » U I M E I j S I O N Y I C M M DAT/H i r i j / 1 / DX ~ A ( l + l ) ~ X d ) DY S Y ( J + l ) - Y ( J ) 6 0 TC ( 2 0 » b 5 ) » I N I IN 1=2 DC 6 0 K = l f 2 0 0 6 0 L = l » 4 I K ( K - L ) 3 5 » 5 0 » 3 5 X I CK * L > - 0 Y 1 (L » K ) - 0 GO TO 60 X1 (:< i K ) - 1 , Y1 (K i K ) = 1 . cc r . 'TP- 'ne x i (4 , ?:) X I (3 » i ) X l ( 3 » 3 ) X 1 ( 3 « ^ l X I < 3 j 2 ) K 1 ( 4 » i ) X 1 Cm 3) X I (A »'t) Y 1 ( 4 * 4 ) Y1 ( 2 , / 0 Y 1 ( 1 , 3 ) Y I < 3 * 3 ) Y 1 ( * }»3) Y l ( ? » 3 ) Y1 ( 1 * 4 ) Y l (3« 4) 0 0 1 8 0 < = 0 0 1 B O L "
~ 1 * / (0 X "'** 0 X)
= - 3» # x 1 ( :m ?.) ~ - X I ( 3 , 1 ) - " 1 « / 0 X = ? , # X U 3 » / 0 = 2 , / U a * » 3 = - X l C u l ) = x i r * , 2 ) - l . / ( O Y « O Y ) --•Yl ( A , 4 J = - 3 , » Y 1 ( ^ s ^ ) - - Y 1 ( I » 3) = - 1 , / O Y = 2 » ^ Y I ( 4 > 3 ) = 2 . / 0 f * » 3 - - Y l < 1 »'+)
1 » 3 1 2 1 > 3 f 2
M = I + K / 2 N s J + L / 2 A (K • L ) " Z (N»M) A < K j L + 1 ) = V (N > m ) A ( K + 1 > U ) a U ( N , M ) A (K * 1»1- + 1) W (i •! 1 r-1 ) CALL f i A r i - U . I L ( X l » A » H » 4 ) CALL I-IATMUL (B j Y1 , A f 4 ) RETUh'fi ENO SU»HCUT INE MATMt »L <A. <3 , C i N) D IMENSION A ( N * > ) ) » Li (r j» N) , C (N»N)
75
DO 25 ~ 1 »N DO 25 L=1»N SUm=0, 00 20 f-1" X 9,N!
ZO SUM •" S'.Ji i A (K ? M) ":*tM M »L) 25 C ( K f L ) - SUM
PFTUNN FNf) s u f j n c u i ih[-: n u s n i . ( u i »ua»D»x,NC»Msw) DIMENSION U1 ( i ) »UJJ(J ) » C ( 1 ) » X ( 1 ) , 0 ( 1 ) GO TO (10 j25) i l - iSW
10 DO } 5 1=2,NO 15 U 2 ( l ) = U 2 ( I ) - U H l - ) ) ^ U l ( I + l ) / U < ? ( I - l )
MSW = 2 25 DO 35 I::2»NC 35 D ( T) =•• i J ( I ) ~ U1 ( 1 + 1 ) * l ) ( i ~ 1 ) / U 2 ( 1 - 1 )
X (N0} - 0 ( i iO ) /U2 CMC) t;OK\ M0~ 1 DO hO l = l»NiOMl J - NO •• I
6 0 x ( J > - < L > ( J ) - U 1 ( J ) # a I J + 1 ) ) / U 2 ( j )
END SUG^OUT;NI-: SPOUT (KSW,h»M» I MESS)
NSlON KSW (3) , I r^tSS f b) C0M.''0f-J ^ ( 5 0 , 5 0 ) j 0 ( 5 0 * 5 0 ) » V 150 5 50 ) , w ( 5 0 , 5 0 ) CCMVC^I x { 0 > , Y ( 50) COMMON HI (50) » 1J2(50)» C (50) » ANS(SO), A ( 4 » * ) > B ( 4 » 4 ) DO 1000 MM™1{3 . IF (KSw(^f i ) ) J 0 0 0 , 1000 , 15
15 GO TO ( 1 0 0 , 2 0 0 , 4 0 0 ) » M M 100 tvM 1 ~ M - 1
NKi 1 = W - l DC ] 45 J= I • l-'i'l 1 00 145 1 = 1? I'mjM 1 PRINT 115 , I ? J , ;< ( I ) j Y ( J)
115 FORI* AT (1 HO •# 1 = * . 13 »3X , * J = & »13 »5X » • E 1 2 . 5 1 5X » # Y - # » 1 P ; 1 2 . 5 / 1 5 X , * P C / F k CF Y*» 11X , *0<* , 1IX j 1»» U X ,«?.»» 11X t 2 * 3 * / 2 4 X , »X*)
CALL SPCOKF( I i J ) DO 145 L = l » 4 IX = L - l PRINT HO? I X t ( A ( V , K K ) » K K = 1 , 4 )
1^0 F O R M A T ( t 2 5 , l / » 4 F l 2 « 5 ) 1^5 CONTINUE
GO TO 1000 O t f * EVALDftTF S P L I iM F FUNCTION A I DATA POINTS
200 PRINT P.i) 1 201 FOHVAT ( lHO»^EVAt.U/<riON CF OATA^ / l 2X , * X * 1 ]. 1 X « #Y* ,
1 l l X » » Z « ) 204 RFAO 2 0 5 , I ALL , X A * YY
76
H'>5 F C f ^ A n i ; j , ? p 1 0 , 0 ) IF ( I/\LL'"*>99 ) 2 1 5 » 1 000 ,215
215 jf- (XX-X () ) )23bi£?.Q*d?.Q ?.£0 I r (XX-X <>!) ) 22b,22b»23b ?.cri IF (YY-Y ( 1) ) 235 i 230»230 230 fF {YY-Y (M) ) 2-V5?2^>235 235 PHI NT ^ 0 * XX? YY 2 * 0 FCRfAT (J. X , 2r 12«5, 4X t ^OUTSIDE RAM^F*) 2'»S 00 25*3 1=*»N
If* ( XX-X ( I ) ) 260?26b?25b 2'o5 CCNTI-mUF 260 1 = 1 - 1 / 2"5 on 2 / b J « 2 *>M '
IF {YY-Y (J) ) ? m } 28b i c l b 275 CCNTlilUr: 2tin j a j . i 2H5 CA|.L SPCOFF(I»J)
l i s 0 . 00 320 K ~ i ? 4 00 320 L = l * ' t KKsK-l L L s L - l L 'L - * A (K ? L) J-M ( A a-A ( I ) ) * »KK) ^ ( (YY"Y(J) ) L)
320 CCMTINUC PH1M 330? XX > YY» LL
330 F G r v a T ( 1 X , 3E12 .5 ) GO TO 204
4 00 CONTINUE 1000 CCNTIWUe
HtTUHW end
BIBLIOGRAPHY
Books
Hi Idebrand, F. B . , I n t r o d u c t i o n t o Numerical A n a l y s i s , New York, McGraw-H i I I Book Company, 1956.
N ie l sen , Kaj L.., Methods o f Numerical A n a l y s i s , New York, The Macrnil lan Company, 1956.
Pe ' r l i s , Sam, Theory o f Ma t r i ces , Reading, Massachusetts, Add ison-Wes I ey P u b l i s h i n g Company, 1958,
A r t i c l e s
B i r k h o f f , G a r r e t t and de Boor, C a r l , "P iecewise Polynomial Sur face F i t t i ng , " Proceedi ngs, Genera I Motors Research Labora to r i es Sym~ posiurn: Approx imat ion of Func t ions , e d i t e d by H. L. Garabedian, E l s e v i e r , Amsterdam, E l s e v i e r P u b l i s h i n g Company, 1965.
De Boor, C a r l , "B i cub i c Sp l i ne I n t e r p o l a t i o n , " Journal of Mathematics and Phys ics , XLI (September, 1962), 212-218.
K l o p f e n s t e i n , K. L . , " C o n d i t i o n a l Least Squares Polynomial Approx i -mat ion , "Mathematics of Computat ion, X V I I I (October, 1964), 659-662.
77
