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ON NUMERICAL APPROXIMATION IN SYNTHESIS
FOR PRESCRIBED AMPLITUDE RESPONSE
by
WALTER JAMES HENRI HARRIS
B.A.Sc,, U.B.C., 1962
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
Master of A p p l i e d S c i e n c e
i n the Department of
E l e c t r i c a l E n g i n e e r i n g
We ac c e p t t h i s t h e s i s as conforming t o the
r e q u i r e d s t a n d a r d
Members of the Department
of E l e c t r i c a l E n g i n e e r i n g
THE UNIVERSITY OF BRITISH COLUMBIA
NOVEMBER, 1964
In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of
the requirements f o r an advanced degree at the U n i v e r s i t y of
B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y
a v a i l a b l e f o r reference and study* I f u r t h e r agree that per
m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y
purposes may be granted by the Head of my Department or by
h i s r e p r e s e n t a t i v e s . I t i s understood that, copying or p u b l i
c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed
without my w r i t t e n permission*
Department
The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8 5 Canada
ABSTRACT
T h i s t h e s i s d e s c r i b e s a new method of o b t a i n i n g s u i t a b l e
e q u a l - r i p p l e r a t i o n a l f u n c t i o n s f o r use i n the s y n t h e s i s of
networks. I t has two p r i n c i p a l advantages over o t h e r methods
used f o r t h i s purpose* F i r s t l y , i t i s capable of f i n d i n g the
b e s t a p p r o x i m a t i o n to a g i v e n magnitude f u n c t i o n , phase f u n c t i o n
or b o t h s i m u l t a n e o u s l y . The e r r o r of a p p r o x i m a t i o n may be
weighted as d e s i r e d a t any f r e q u e n c y i n the range of a p p r o x i
m a t i o n . Secondly* i t i s e a s i l y adapted f o r use on anautomatic
computer* T h i s e n a b l e s q u i c k comparison of the a p p r o x i m a t i o n s
produced by u s i n g r a t i o n a l f u n c t i o n s of d i f f e r e n t o r d e r s of
c o m p l e x i t y *
i v
ACKNOWLEDGEMENT
The author would l i k e t o thank Dr. A. D. Moore f o r h i s
s u p e r v i s i o n and guidance throughout t h i s r e s e a r c h p r o j e c t and
the s t a f f of the computing c e n t r e f o r t h e i r generous a s s i s t a n c e *
The p r i n c i p a l p a r t of t h i s r e s e a r c h was c a r r i e d out w i t h the
support of the N a t i o n a l R esearch C o u n c i l w i t h a d d i t i o n a l
a s s i s t a n c e from a B» C* Telephone Co. S c h o l a r s h i p .
i
TABLE OF CONTENTS
Page
L i s t of I l l u s t r a t i o n s ••»•»....»..... * ........ » i i i
Acknowledgement ..•••.••••••.........»...»**»•. i v
1. I n t r o d u c t i o n .»••••»....».» ..... 1
1-1. P r o p e r t i e s of Impedances ...... ...... 2
1- 2« Types of F u n c t i o n s t o be A p p r o x i
mated ...••••«•••..........*.... 5
2. M a t h e m a t i c a l Background 7
2- 1. N o t a t i o n 7
2-2. Statement of the A p p r o x i m a t i o n
Problem 9
2-3. E x i s t e n c e and Uniqueness ............ 9
3. Methods of R a t i o n a l A p p r o x i m a t i o n ......... 12
4. A New Method 19
4-1. D e s c r i p t i o n 19
4-2. S o l u t i o n of a S u b s i d i a r y Problem .... 21
4-3. The Computer' Program ................ 23
5. Examples 27
6. C o n c l u s i o n .....»..••••........ 33
7. Appendix ......... 34
References 48
i i
LIST OF ILLUSTRATIONS
F i g u r e Page
1. A p p r o x i m a t i o n to I l l u s t r a t e d Response .. 28
2. A p p r o x i m a t i o n to G a u s s i a n Response ..... 29
3. E r r o r i n A p p r o x i m a t i o n to G a u s s i a n Response ................. ........ 30
4. Comparison of Weighted E r r o r i n A p p r o x i m a t i o n to G a u s s i a n Response ..... 31
5. A p p r o x i m a t i o n to I l l u s t r a t e d Response .» 32
i i i
1. INTRODUCTION
H i s t o r i c a l l y , s u i t a b l e r a t i o n a l f u n c t i o n s f o r the
s y n t h e s i s of networks have been o b t a i n e d by two g e n e r a l
c l a s s e s of methods: one has been the use of an analogue of
the network which a l l o w e d e a s i e r m a n i p u l a t i o n of p o l e s and
zeroes than the a c t u a l network, and the o t h e r has been the
use of v a r i o u s a n a l y t i c f u n c t i o n s and t r u n c a t e d s e r i e s .
P r o b a b l y the b e s t example of the former i s the p o t e n t i a l
a n a l o g y . F o r the l a t t e r , the B u t t e r w o r t h and B e s s e l a p p r o x i
mations to the low-pass f i l t e r c h a r a c t e r i s t i c are among the
b e t t e r known. F o r many a p p l i c a t i o n s these methods produce
s u f f i c i e n t l y good r e s u l t s . However, when f i l t e r s are r e q u i r e d
to have p r e c i s e l y s p e c i f i e d magnitude and phase r e s p o n s e s , as
f o r example i n l o n g telephone networks, i t i s o f t e n d e s i r a b l e
to be ab l e t o s p e c i f y the maximum d e v i a t i o n from the d e s i r e d
f u n c t i o n and to make t h i s v a l u e as s m a l l as p o s s i b l e f o r a g i v e n
degree of network c o m p l e x i t y . To b e g i n the s y n t h e s i s , i t i s
f i r s t n e c e s s a r y t o o b t a i n e i t h e r the p o s i t i o n s of the p o l e s
and zeroes o r , e q u i v a l e n t l y , the c o e f f i c i e n t s of the r a t i o n a l
f u n c t i o n . A p p r o x i m a t i o n s o b t a i n e d by u s i n g the p o t e n t i a l
analogy i n the form of an e l e c t r o l y t i c tank g i v e the p o l e and
zero p o s i t i o n s d i r e c t l y w h i l e c l a s s i c a l a n a l y t i c approaches
u s u a l l y r e s u l t i n one of these s e t s of q u a n t i t i e s e x p r e s s e d
by an e x p l i c i t f o r m u l a i n terms of t a b u l a t e d f u n c t i o n s . However,
i n the case of a p p r o x i m a t i o n s o b t a i n e d by d i r e c t l y r e d u c i n g the
maximum e r r o r , the amount of c a l c u l a t i o n i s , g e n e r a l l y ,
c o n s i d e r a b l y g r e a t e r .
The p o t e n t i a l analogue has the advantage of p e r m i t t i n g
a b e t t e r i n t u i t i v e f e e l i n g f o r the s e n s i t i v i t y of the f i n a l
response t o s m a l l d e v i a t i o n s i n p o l e and zero p o s i t i o n s ,
but the di s a d v a n t a g e of b e i n g capable of o n l y l i m i t e d a c c u r a c y
A n a l y t i c a p p r o x i m a t i o n s may r e s u l t i n an e a s i l y c h a r a c t e r i z e d
r esponse; f o r example, w i t h the Ghebyshev a p p r o x i m a t i o n s to
a c o n s t a n t , the maximum passband r i p p l e i s d i r e c t l y a v a i l a b l e .
A d i s a d v a n t a g e i s t h a t these methods have o n l y been developed
f o r s imple forms of responses* The n u m e r i c a l methods to be
c o n s i d e r e d i n t h i s paper e x p l o i t the c a p a b i l i t i e s of modern
high-speed automatic computers? t h i s a l l o w s d i r e c t r e d u c t i o n
of the maximum e r r o r of response w i t h r e s p e c t t o p r a c t i c a l l y
any a r b i t r a r y f u n c t i o n , w i t h a c c u r a c y l i m i t e d o n l y by the
number of s i g n i f i c a n t f i g u r e s r e t a i n e d d u r i n g c a l c u l a t i o n s .
A l t h o u g h q u e s t i o n s of the e x i s t e n c e and uniqueness of a
" b e s t " r a t i o n a l f u n c t i o n a p p r o x i m a t i n g a d e s i r e d f u n c t i o n w i l l
be d e a l t w i t h l a t e r , i t i s n e c e s s a r y f i r s t t o c o n s i d e r the
l i m i t a t i o n s imposed f o r the sake of p h y s i c a l r e a l i z a b i l i t y .
1—1» P r o p e r t i e s of Impedances
The network f u n c t i o n s t o be approximated can be
r e p r e s e n t e d as e i t h e r t r a n s f e r impedances or d r i v i n g - p o i n t
impedances. The f o l l o w i n g r e q u i r e m e n t s must be met by a
p h y s i c a l l y - r e a l i z a b l e d r i v i n g - p o i n t impedance (or adm i t t a n c e )
produced by a network of lumped j. l i n e a r , p a s s i v e , t i m e -
i n v a r i a n t elements?
1. Z ( s ) i s a r a t i o n a l f u n c t i o n of the complex f r e q u e n c y
s, and can be w r i t t e n i n the two e q u i v a l e n t forms;
3
n n 7 a . s 1 ( s - s . ) /_ l | | V i 7
«<•) = - i # =K. ^ m : n+m
X v 1 (s - s.)
i=0 i = 1+n
2. P o l e s and zeroes of Z ( s ) are e i t h e r r e a l or occur i n
complex conjugate p a i r s o r , e q u i v a l e n t l y , a l l the
c o e f f i c i e n t s a^ and are r e a l .
3. The r e a l p a r t s of a l l p o l e s and zeroes of Z ( s ) must
be n o n - p o s i t i v e .
4. P o l e s and zeroes of Z ( s ) must be simple w i t h r e a l ,
p o s i t i v e r e s i d u e s i f the r e a l p a r t i s z e r o ,
5. The c o n s t a n t m u l t i p l i e r K must be r e a l and p o s i t i v e .
6. | m — n| must be e i t h e r 0 or 1.
The req u i r e m e n t s f o r an open c i r c u i t t r a n s f e r impedance are the
same except t h a t the r e a l p a r t s of the zeroes of Z ( s ) are not
r e s t r i c t e d and m — n ^ - 1.
U s u a l l y s p e c i f i c a t i o n s f o r response are g i v e n f o r r e a l
f r e q u e n c i e s , i . e . , f o r s'= jtt,. I n some c a s e s , the impedance i s
s e p a r a t e d i n t o r e a l and i m a g i n a r y components. I f the impedance
i s w r i t t e n i n the form
m, + n,
where the m's and n's denote r e s p e c t i v e l y the even and odd
p a r t s of the p o l y n o m i a l s , t h e n , f o r s = jw, the even p a r t s
are r e a l and the odd p a r t s are i m a g i n a r y , and
Re Z( s ) m l m 2 ~ n l n 2
(m 2) - ( n 2 ) '
Im Z(s ) m 2 n l ~ m l n 2
(m 2) - ( n 2 )
Assuming t h a t Z ( s ) i s a n a l y t i c i n the r i g h t h a l f - p l a n e f t h a t
i s , f o r a l l s e Re(s) > 0, then e i t h e r the r e a l p a r t or the
im a g i n a r y p a r t can be o b t a i n e d from the o t h e r , t o w i t h i n an
added r e s i s t a n c e or r e a c t a n c e r e s p e c t i v e l y .
F o r some purposes i t i s p r e f e r a b l e t o use p o l a r
c o - o r d i n a t e s and d e s c r i b e the impedance i n terms of i t s magni
tude and phase a n g l e . F o r t h e o r e t i c a l purposes, i t i s more
common to use the squared magnitude or the l o g a r i t h m of the
magnitude r a t h e r than the magnitude i t s e l f . The "squared-
magnitude" f u n c t i o n i s non-negative f o r a l l to, and may be
o b t a i n e d from
Z ( s ) . Z ( - s ) = ( m i ) 2 - ( n 2 ) 2
( m 2 ) 2 - ( n 2 ) 2
by s e t t i n g s = jw, so t h a t
Z(j«.)|2 = Z(j«) . Z(-j».)
The phase angle i s an odd f u n c t i o n h a v i n g the p r i n c i p a l
v a l u e ,
Q(<p) = tan' -1 m 2 n l m l n 2 j ( m 1 m 2 - n x n 2 )
s = J«
T h i s f u n c t i o n i s c o n t i n u o u s except a t f r e q u e n c i e s c o r r e s
ponding to p o l e s or zeroes l y i n g on the j<o a x i s , a t which
p o i n t s the phase changes d i s c o n t i n u o u s l y by an i n t e g e r
m u l t i p l e + 71. * A g a i n assuming no r i g h t - h a l f - p l a n e p o l e s or
z e r o e s are p r e s e n t , the magnitude or the phase may be o b t a i n e d
from each o t h e r . By f a c t o r i n g the numerator and denominator
of Z ( s ) . Z ( - s ) and r e t a i n i n g o n l y the l e f t - h a l f - p l a n e p o l e s
and z e r o e s , Z ( s ) can be o b t a i n e d . Thus, knowing any one of
Z ( s ) , Re [z(s)]|, Im [ z(s)], 0(«), Z ( s ) , Z ( - s ) , or Z(jft>)| 2 , airy one
of the o t h e r s c a n , i n t h e o r y , be o b t a i n e d , s u b j e c t to the s t a t e d
a s s umptions.
I n t h i s t h e s i s * o n l y a p p r o x i m a t i o n s to s p e c i f i e d
magnitude f u n c t i o n s w i l l be t r e a t e d i n d e t a i l , a l t h o u g h the
a l g o r i t h m to be developed can be r e a d i l y extended f o r a p p r o x i
mating o t h e r f u n c t i o n s d e r i v e d from Z ( s ) .
1-2. Types of F u n c t i o n s to be Approximated
Network f u n c t i o n s to be approximated may be s e p a r a t e d
b r o a d l y i n t o two c a t e g o r i e s : those a p p r o p r i a t e f o r d e s c r i b i n g
band f i l t e r s and those a p p r o p r i a t e f o r e q u a l i z e r s * O f t e n the
s p e c i f i c a t i o n s f o r t r a n s m i s s i o n through such networks are
g i v e n i n terms of the a t t e n u a t i o n . T h i s q u a n t i t y i s
p r o p o r t i o n a l to the l o g a r i t h m of the magnitude of the t r a n s f e r
impedance, i . e . *
A = -k • l o g | Z | , where k i s r e a l and p o s i t i v e *
Band f i l t e r s are c h a r a c t e r i z e d by d i s t i n c t passbands
( i n t e r v a l s of ft) over which the a t t e n u a t i o n i s s m a l l ) and s t o p -
bands / ( i n t e r v a l s of ft) over which the a t t e n u a t i o n i s l a r g e )
6 a l t e r n a t i n g a l o n g the w - a x i s . S p e c i f i c a t i o n s f o r t h i s type
of f i l t e r are o f t e n g i v e n i n terms o f :
( i ) p e r m i s s i b l e passband r i p p l e , i . e . , the r a t i o of
maximum to minimum t r a n s m i s s i o n w i t h i n a passband;
( i i ) minimum a t t e n u a t i o n i n the stop-bands;
and ( i i i ) w i d t h of t r a n s i t i o n - or guard-bands.
E q u a l i z e r s g e n e r a l l y do not r e q u i r e sharp t r a n s i t i o n s ,
but o f t e n r e q u i r e c a r e f u l c o n t r o l of e i t h e r magnitude or
phase, o r b o t h . Examples are networks used to shape the
fr e q u e n c y c h a r a c t e r i s t i c of the feedback l o o p of an a m p l i f i e r ,
t o e q u a l i z e the frequency-dependent l o s s i n a t r a n s m i s s i o n
l i n e , or t o s i m u l a t e t r a n s m i s s i o n - l i n e impedances.
F o l l o w i n g a d i s c u s s i o n of some methods which have been
used f o r f i n d i n g r a t i o n a l - f u n c t i o n a p p r o x i m a t i o n s , a new
method w i l l be shown which seems more s u i t a b l e f o r e q u a l i z e r
a p p l i c a t i o n s .
2. MATHEMATICAL BACKGROUND 7
2-1• N o t a t i o n
For convenience, the independent v a r i a b l e w i l l be
denoted by x i n p l a c e o f tt or . L e t f ( x ) be a s p e c i f i e d
squared-magnitude f u n c t i o n and l e t the r a t i o n a l f u n c t i o n
a p p r o x i m a t i o n t o f ( x ) on the i n t e r v a l x ^ b be g i v e n by n
Yl p i x
p ( x ) = E M = i^J2 . v 7 • • m ± Q(x)
m
0
m Z ^ - x
i = 0
The e r r o r of a p p r o x i m a t i o n w i l l be d e f i n e d by
e.(x) = f ( x ) - F ( x ) .
Si n c e the number of c o e f f i c i e n t s of the r a t i o n a l f u n c t i o n
o c c u r s r a t h e r o f t e n , l e t i t be r e p r e s e n t e d by
n = n + m + 2 . c
When i t i s c o n v e n i e n t t o r e f e r t o the f u l l s e t of numerator and denominator c o e f f i c i e n t s , t h i s p o i n t i n n c ~ d i m e n s i o n a l space w i l l be c a l l e d R»
I n o r d e r t o p r o v i d e an e x a c t mathematical statement of
the problem t o be s o l v e d , i t i s n e c e s s a r y t o d e f i n e a p r e c i s e
. 8 measure of the c l o s e n e s s of the a p p r o x i m a t i o n to the d e s i r e d
f u n c t i o n . T h i s w i l l be done i n terms of the norm of the
" e r r o r " f u n c t i o n . A norm i s a r e a l s c a l a r q u a n t i t y d e f i n e d
on a f u n c t i o n space. I t i s denoted by | and must s a t i s f y
the f o l l o w i n g t h r e e c o n d i t i o n s :
! • I gU) I ^ ° J A N D 1 gU) || = 0 i f and o n l y i f g(x) = 0
2, |j c g ( x ) || = |c | * |g(x) I f o r any r e a l c o n s t a n t c 3. I h ( x ) + g(x) I ^ ||g(x) I + || h ( x ) I
f o r a l l g (x) and h ( x ) i n the f u n c t i o n space. The two l i n e a r
spaces and t h e i r norms which w i l l be of i n t e r e s t a r e :
1. The Space L p (v C l)
The space lP c o n s i s t s of the t o t a l i t y of a l l
f u n c t i o n s measurable i n the i n t e r v a l (a,b) whose
a b s o l u t e v a l u e t o the p - t h power i s i n t e g r a b l e i n
the sense of Lebesgue. The- norm i n L P , c a l l e d the
l e a s t p ^ t h norm, i s d e f i n e d by
|g(t)|~ d t J b P > 1 !/P
2. The Space C
The space C c o n s i s t s of the t o t a l i t y of a l l con
t i n u o u s f u n c t i o n s of the p o i n t s P of a bounded,
c l o s e d s e t S i n E u c l i d e a n space of any d i m e n s i o n a l i t y ,
The norm i n C, c a l l e d the Chebyshev norm, i s d e f i n e d
by g || = max | g(P)
P € S
e (R*) || C ||e(R)
9 S e t t i n g g = e ( x ) = f ( x ) - F ( x ) , the norm of the e r r o r
f u n c t i o n , | e || , i s a s u i t a b l e measure of the c l o s e n e s s of
the a p p r o x i m a t i o n to the s p e c i f i e d f u n c t i o n .
2-2» Statement of the A p p r o x i m a t i o n Problem
G i v e n f ( x ) , a r e a l - v a l u e d c o n t i n u o u s f u n c t i o n d e f i n e d
on a s e t S, f i n d a r a t i o n a l f u n c t i o n F(R ,x) w i t h R EP» a
bounded s e t , such t h a t
M f o r a l l R e P
Such a r a t i o n a l f u n c t i o n w i l l be c a l l e d a b e s t a p p r o x i m a t i o n ,
2-3, E x i s t e n c e and Uniqueness
F o r the Chebyshev norm, e x i s t e n c e and uniqueness f o r
the case of a p p r o x i m a t i o n of a c o n t i n u o u s f u n c t i o n over a s e t
c o n t a i n i n g no i s o l a t e d p o i n t s are a s s u r e d by the f o l l o w i n g
theorem a t t r i b u t e d t o Chebyshev:
Theorem: Gi v e n a c l o s e d ( f i n i t e or i n f i n i t e ) i n t e r v a l [a , V ]
on the real-number a x i s and a r e a l , s i n g l e - v a l u e d
f u n c t i o n , f ( x ) , c o n t i n u o u s i n [ a , b ] , t h e r e t h e n e x i s t s
a t l e a s t one f u n c t i o n
P (x) F (x) = TTV-T n,mv Q m ( x )
such t h a t IUI = max | f ( x ) - F (x) |
xrc[a,bj '
f o r t h i s f u n c t i o n i s not g r e a t e r than f o r any o t h e r
r a t i o n a l f u n c t i o n of the same o r d e r . Moreover, t h i s
f u n c t i o n F n j m ( x ) = F n m ( x ) l s u n i q u e , i f we c o n s i d e r
two r a t i o n a l f u n c t i o n s as i d e n t i c a l when th e y ^
c o i n c i d e a f t e r b e i n g reduced t o t h e i r l o w e s t
terms. The f u n c t i o n e t a k e s on i t s maximum
a b s o l u t e v a l u e f o r a t l e a s t n c - min(jj,,D) p o i n t s
i n [ a f b j , where n-^ and m-a) are the h i g h e s t
powers of x w i t h non-zero c o e f f i c i e n t s o c c u r r i n g # #• i n P (x) and Q (x) r e s p e c t i v e l y a f t e r r e d u c t i o n n ** m r u
to l o w e s t terms. P r o o f s of the above may be found
i n books of A c h i e s e r [ l ] or R i c e \j-5^\ •
F o r the l e a s t p - t h norm, the p r o o f of e x i s t e n c e of
a b e s t r a t i o n a l f u n c t i o n a p p r o x i m a t i o n to a f u n c t i o n on an
i n t e r v a l can be e s t a b l i s h e d as f o l l o w s . I n [V],-pp, 1 0 - 1 1 ,
i t i s shown t h a t the l e a s t p - t h norm s a t i s f i e s the C o n d i t i o n
E of Young f o r a p o l y n o m i a l a p p r o x i m a t i n g f u n c t i o n . Thus
Theorem 1-4 of R i c e \l5~[ proves e x i s t e n c e f o r the p o l y n o m i a l
c a s e . Then from an argument p a r a l l e l to t h a t of Lemma 3 - 5
of R i c e [ 1 5 ] , the r e s u l t can be extended to r a t i o n a l
f u n c t i o n s . Moreover, A c h i e s e r £l^ proves t h a t the b e s t a p p r o x i
m a t i o n i s unique f o r any s t r i c t l y n o r m a l i z e d f u n c t i o n space
which e s t a b l i s h e s uniqueness f o r iP i f 1 <^ p <̂ oo*
F o r p o l y n o m i a l a p p r o x i m a t i n g f u n c t i o n s a well-known
theorem of P o l y a and J a c k s o n (see [ l 5 ~ ] , p. 8 ) shows t h a t the
sequence of b e s t l e a s t p - t h a p p r o x i m a t i o n s as p —*-00 con
t a i n s a convergent subsequence. F u r t h e r the l i m i t of t h i s
subsequence i s a b e s t a p p r o x i m a t i o n u s i n g the Chebyshev norm.
U n f o r t u n a t e l y , t h e r e i s l i t t l e p u b l i s h e d m a t e r i a l c o n c e r n i n g
r a t i o n a l f u n c t i o n a p p r o x i m a t i o n u s i n g the l e a s t p - t h norm,
p a r t i c u l a r l y the p o s s i b i l i t y of convergence f o r i n c r e a s i n g p.
However, e x i s t e n c e cannot be a s s u r e d i n the case of
a r a t i o n a l - f u n c t i o n a p p r o x i m a t i o n to a f u n c t i o n d e f i n e d on a
d i s c r e t e s e t f o r e i t h e r the Chebyshev or the l e a s t p - t h norm.
Moreover, even i f the r a t i o n a l f u n c t i o n does e x i s t , i t may have
p o l e s w i t h i n the range of approximation,. Examples of these
c o n d i t i o n s may be found i n | 3I ,
3. METHODS OF RATIONAL APPROXIMATION
The statement of the problem i m m e d i a t e l y suggests one
type of approach to a s o l u t i o n ; namely, to c o n s i d e r || e |j as a
f u n c t i o n of i t s c o e f f i c i e n t s * R, and to attempt to f i n d a
minimum of t h i s f u n c t i o n * For the l e a s t p - t h norm, the space
d e f i n e d by
INI ^ k 5 e * < k < ~
i s s t r i c t l y convex, a f a c t w hich g r e a t l y s i m p l i f i e s the t a s k
o f f i n d i n g such a minimum* F o r the Chebyshev norm, the space
as d e f i n e d above i s a p o l y t o p e ; thus the methods of convex
programming can be used t o o b t a i n a b e s t a p p r o x i m a t i o n . M i n i
m i z a t i o n of a f u n c t i o n can be a c c o m p l i s h e d on a d i g i t a l
computer by t e s t i n g the v a l u e of t h i s f u n c t i o n on a d i s c r e t e
s e t of p o i n t s to p r o v i d e i n f o r m a t i o n about the l o c a t i o n of a
g l o b a l minimum or l o c a l minima. These procedures may be
c l a s s i f i e d by t h e i r method of c h o o s i n g t e s t p o i n t s as e i t h e r
s e q u e n t i a l or n o n s e q u e n t i a l *
Most s e q u e n t i a l methods f o r convex f u n c t i o n s use the
g r a d i e n t of the f u n c t i o n as an i n d i c a t i o n of the d i r e c t i o n t o te)
ward the minimum. L e t r be the v a l u e s of the c o e f f i c i e n t s (k)
e x c l u d i n g q Q , e v a l u a t e d a t the k - t h i t e r a t i o n , h be the s t e p (k) s i z e of the k - t h s t e p and D be an n -1 d i m e n s i o n a l d i r e c t i o n r c v e c t o r p r o v i d i n g the d i r e c t i o n of the change of R. The g r a d i e n t
methods t o be d e s c r i b e d a l l use the i t e r a t i v e e q u a t i o n
r ( k + l ) = r ( k ) + h ( k ) # D ( k )
13 but d i f f e r i n t h e i r c h o i s e of h and D.
( i ) U n i v a r i a t e or R e l a x a t i o n Method
As the name su g g e s t s , o n l y one c o e f f i c i e n t i s
changed d u r i n g each s t e p ; hence D
v e c t o r of the form
0 0
00 i s a column
D 00 • t h 3 row
0 1 0 *
0
The i n d e x j i d e n t i f y i n g the c o e f f i c i e n t to be 3 f
changed a t each s t e p i s the one f o r which —— or d f
f =
2 < a 2 f 2 ^ i s the l a r g e s t i n magnitude, where
The s t e p s i z e i s
OO r ( k ) <52f 5 r . 2
0
( i i ) S t e e p e s t Descent, Newton's and Optimum G r a d i e n t
Methods
These methods change a l l the independent (k)
v a r i a b l e s a t each i t e r a t i o n . D ' can be w r i t t e n
i n the form
•>(K) —LB-1! A 0 0] where B i s a p o s i t i v e - d e f i n i t e w e i g h t i n g m a t r i x and
14
00 <? ri .00 ,00
d f ' c3 rn -1 ,00
F o r the Method of S t e e p e s t Descent, B i s the u n i t
m a t r i x of o r d e r n - 1 . Newton's Method uses the c
H e s s i a n m a t r i x f o r B. I t i s d e f i n e d by
b. *2t
The s t e p s i z e may be chosen i n a number of ways.
One of these i s to s a t i s f y the i n e q u a l i t y
e || ( k + 1 ' < ||e||( k' f o r each s t e p . Another i s t o (k)
choose h so t h a t the new p o i n t i s a t the i n t e r - ? s e c t i o n of the c o - o r d i n a t e axes and the tangent
(k) hyperplane t o the f u n c t i o n a t R , From a n a l y t i c
(k) geometry, the v a l u e of h v ' t o a c h i e v e t h i s can be
c a l c u l a t e d from the f o r m u l a
,00 GO
( B - 1 A ^ ) t . ( B - 1 A< k>)
The s t e p s i z e f o r the optimum g r a d i e n t method i s
chosen t o mi n i m i z e || e || (k+l) a i o n g -th e l i n e B-^" A ^ k ^ ,
An a p p r o x i m a t i o n i s u s u a l l y used f o r t h i s s t e p s i z e
[5J s i n c e i t s c a l c u l a t i o n i s o f t e n l e n g t h y .
An example of a n o n s e q u e n t i a l method of m i n i m i z i n g a
f u n c t i o n i s the method of d i r e c t l y t e s t i n g - t h e v a l u e of the
f u n c t i o n a t ev e r y p o i n t o f an n - d i m e n s i o n a l g r i d . The l o c a t i o n
of the minimum i s t a k e n t o be the p o i n t f o r which the s m a l l e s t
15 v a l u e of the f u n c t i o n i s observed.
U s i n g the p u r e l y random method, a c e r t a i n number of t e s t
p o i n t s are chosen from a volume c o n t a i n i n g the minimum, w i t h
t h e i r l o c a t i o n s determined by an ( n c ~ l ) - d i m e n s i o n a l p r o b a b i l i t y -
d e n s i t y f u n c t i o n . The s m a l l e s t v a l u e a c h i e v e d i n t h i s sample
i s c o n s i d e r e d to be a t the l o c a t i o n of the minimum. I f S i s
the c o n f i d e n c e l e v e l t h a t a p o i n t of the random d i s t r i b u t i o n
f a l l s w i t h i n a hypercube of s i d e d^ and the s e a r c h volume i s
a hypercube of s i d e then d e f i n e
n -1 a = °
The number of p o i n t s , p, which s h o u l d be t e s t e d to ensure the
minimum found to be w i t h i n d^ of the p o s i t i o n of the t r u e
minimum w i t h c o n f i d e n c e l e v e l S, assuming a u n i f o r m d e n s i t y
f u n c t i o n , i s
_ log (1 - S) p - l o g tl - a)
More complex methods have been proposed which reduce the c a l c u l a t i o n
time by s t a r t i n g the s e a r c h on a coarse g r i d over a l a r g e
volume (or randomly w i t h i n t h i s volume), t h e n s y s t e m a t i c a l l y
reduce the volume of the s e a r c h . Such methods attempt to
combine the b e s t f e a t u r e s of s e q u e n t i a l and n o n s e q u e n t i a l
methods. The amount of c a l c u l a t i o n r e q u i r e d u s i n g n o n s e q u e n t i a l
methods i n c r e a s e s e x p o n e n t i a l l y w i t h the number of independent
v a r i a b l e s and the p r e c i s i o n of l o c a t i o n of the minimum* I n
r e t u r n f o r t h i s i n c r e a s e of c a l c u l a t i o n t i m e , the f u n c t i o n t o be
m i n i m i z e d by these methods i s not r e q u i r e d to be d i f f e r e n t i a b l e
or even c o n t i n u o u s throughout the volume.
16 F o r the Chebyshev norm, the above methods cannot be used
s i n c e the r e q u i r e d d e r i v a t i v e s are not a v a i l a b l e . I n s t e a d , one
of the a l g o r i t h m s based on convex programming may be used,
1, The f i r s t a l g o r i t h m t o be d e s c r i b e d was proposed
by Loeb, To s t a r t t h i s i t e r a t i v e p r o c e d u r e , choose
an a r b i t r a r y r a t i o n a l f u n c t i o n F ( R ^ } , x ) ; then a t
the k - t h s t e p s e l e c t c o e f f i c i e n t s R
mize
(k) which m i n i -
max x e [ a , b ]
Q TkZIT (x) f ( x ) Q ( k ) ( x ) - P ( k ) ( x )
The t r i v i a l s o l u t i o n , R = 0, i s a v o i d e d by f i x i n g
one c o e f f i c i e n t ,
2. A s i m i l a r a l g o r i t h m d e s c r i b e d by Cheney and Loeb £4]
d i f f e r s o n l y i n t h a t the f u n c t i o n t o be m i n i m i z e d i s
under the r e s t r i c t i o n t h a t R (k)| I 1 < 1 ^ V
3. The t h i r d method of t h i s type was proposed by Loeb
[133 and i n a s l i g h t l y m o d i f i e d form by G o l d s t e i n £7])
C o n s i d e r the system of i n e q u a l i t i e s
P ( x ) - ( f ( x ) + M)Q(x) < 0 a l l x € [a,b] - P ( x ) + ( f ( x ) - M)Q(x) < 0 > I'(M)
R = 1
T h i s system i s i n c o n s i s t e n t i f M ^ M , where M i s
the v a l u e of M a s s o c i a t e d w i t h the b e s t a p p r o x i m a t i o n .
Choose b (0) m so t h a t z : (0) x 1 4 0, x e [ a , b ] ;
= 0; H* 0) = f ( x ) ; and a,± = 0, i = 0, 1, ... f n.
At the k - t h s t e p
M ( k ) = L ( k - D + H ( k - 1 ) 2
I f the system I ' ( M ^ ) ) i s i n c o n s i s t e n t , t h e n = M^^, „(k) r r ( k - l ) , „(k) D ( k - l ) y A i i , . H v ' = H ' and R v ' - R v . I f the system i s
(k) c o n s i s t e n t * t h e n choose R t o s a t i s f y i t ;
I > > = L ( k - X > and H<k> =M ( k>.
Th i s a l g o r i t h m p r o v i d e s upper and lower bounds f o r
the e r r o r of the a p p r o x i m a t i o n and these converge
m o n o t o n i c a l l y t o the v a l u e of the b e s t a p p r o x i m a t i o n .
To conclude t h i s s e c t i o n , two d i r e c t approaches t o o b t a i n i n g
a b e s t r a t i o n a l f u n c t i o n a p p r o x i m a t i o n w i l l be p r e s e n t e d .
1. I t e r a t i o n u s i n g the z e r o e s of the e r r o r
T h i s method* developed by Maehly [ l 4 ^ , i s s t a r t e d by
cho o s i n g n c—1 p o i n t s , x^, such t h a t a ^ x i x 2 x , S b. Prom t h i s s e t of p o i n t s , the c o e f f i c i e n t s n —± ^ c of the r a t i o n a l f u n c t i o n which i s equal t o f ( x ) a t
the p o i n t s x^ are found by s o l v i n g the system of l i n e a r
e q u a t i o n s
f ( x A ) * Q(x.) - P ( X i ) = 0 1 < i ̂ n c - l
Next, modify the x^ so t h a t the maxima of | e ( x )
w i l l be more n e a r l y e q u a l . One method of a d j u s t i n g
these z e r o e s i s based upon the assumption t h a t the
maximum d e v i a t i o n between two a d j a c e n t zeroes i s
a p p r o x i m a t e l y p r o p o r t i o n a l t o the d i s t a n c e between
these z e r o e s . Hence, the zeroes s h o u l d be moved
c l o s e r t o the p o i n t s which have the l a r g e s t d e v i a t i o n .
I t e r a t i o n u s i n g the maxima of the e r r o r
Choose an a r b i t r a r y s e t of n c i n i t i a l p o i n t s f o r the
extrema of the e r r o r curve such t h a t a ^ x ^ <^ x 2 <̂ • •
x , <Tb* and a v a l u e of E ^ ^ f o r the d e v i a t i o n a t n -1 \ ' c these p o i n t s * An example of an a l g o r i t h m of t h i s t y p
i s Bemes Second A l g o r i t h m . A t the k - t h s t e p , attempt (k)
t o f i n d a r a t i o n a l f u n c t i o n which d e v i a t e s by E from f ( x ) w i t h a l t e r n a t i n g s i g n a t the p o i n t s x^.
(k) The v a l u e o f EK ' f o r each s u c c e e d i n g s t e p i s found
by s o l v i n g an i t e r a t i v e e q u a t i o n . The e r r o r curve
i s t h e n examined f o r the l o c a t i o n of the l a r g e s t
d e v i a t i o n between a d j a c e n t p o i n t s x^. These are
then t a k e n as the s e t of p o i n t s f o r the next s t e p of
the i t e r a t i o n *
4. A NEW METHOD 19
4-1. D e s c r i p t i o n
The methods which have been d e s c r i b e d f o r o b t a i n i n g
b e s t r a t i o n a l f u n c t i o n a p p r o x i m a t i o n s have s e v e r a l l i m i t a t i o n s
f o r use i n f i l t e r s y n t h e s i s . F i r s t l y , they cannot be used to
approximate a g i v e n phase response s i n c e t h i s i s a t r a n s c e n
d e n t a l func-tion of the c o e f f i c i e n t s of F ( s ) . S e c o n d l y , they
cannot be used f o r the s i m u l t a n e o u s a p p r o x i m a t i o n of two or more
f u n c t i o n s , f o r example i n m i n i m i z i n g the e r r o r i n a p p r o x i m a t i o n
f o r g i v e n magnitude and phase r e s p o n s e s . F i n a l l y , t h e r e i s no
c o n v e n i e n t way t o i n t r o d u c e the r e q u i r e m e n t s of p h y s i c a l
r e a l i z a b i l i t y i n t o the a l g o r i t h m .
I n an attempt t o overcome these l i m i t a t i o n s , a new
a l g o r i t h m was developed which s u c c e s s f u l l y c i r c u m v e n t s the
f i r s t two o b j e c t i o n s but s t i l l does not handle the r e a l i z a b i l i t y
c o n s t r a i n t s . A s i m i l a r method d e v i s e d by L i n v i l l [ l l ] has the
advantage of e x p o s i n g r e a l i z a b i l i t y c o n d i t i o n s f o r simple c o n t r o l
but does not l e n d i t s e l f e a s i l y to a u t o m a t i c computation, a s e r i o u s
d i s a d v a n t a g e because the r e q u i r e d c a l c u l a t i o n can e a s i l y be
e x c e s s i v e i f done m a n u a l l y .
A t each s t e p of the procedure a s e t of " c o r r e c t i o n
f u n c t i o n s " , C(ft>), i s o b t a i n e d . The c o r r e c t i o n f u n c t i o n s used
are the l i n e a r terms of a T a y l o r s e r i e s e x p a n s i o n of the
f u n c t i o n of F ( s ) of i n t e r e s t w i t h r e s p e c t t o a l l the c o e f f i c i e n t s
of F ( s ) except the c o n s t a n t term i n the denominator. For
s i m p l i c i t y i n t e s t i n g t h i s on o n l y the squared-magnitude
f u n c t i o n , the d e r i v a t i v e s used are t a k e n w i t h r e s p e c t t o the
c o e f f i c i e n t s of |F(J»)|2, and P ( s ) i s l a t e r o b t a i n e d by-
f a c t o r i n g . That l i n e a r c o m b i n a t i o n of c o r r e c t i o n s i s found
which m i n i m i z e s the norm of the d i f f e r e n c e between the l i n e a r
c o m b i n a t i o n and the e r r o r f u n c t i o n . To a v o i d s i n g u l a r
e q u a t i o n s , the c o r r e c t i o n f u n c t i o n w i t h r e s p e c t t o the term
b i s not used. The l i n e a r c o m b i n a t i o n i s found most con-o v e n i e n t l y by c o n s i d e r i n g th$ g i v e n f u n c t i o n and the c o r r e c t i o n
f u n c t i o n s a t a d i s c r e t e s e t of f r e q u e n c i e s . I n t h i s case i t
i s n e c e s s a r y t o f i n d a " b e s t " s o l u t i o n t o the overdetermined s e t
of l i n e a r e q u a t i o n s
n - 1 c ^ C
d ( " i ) A j = f - F K ) + P±; l < i < m
= e((o i) + p i
where m i s the number o f d i s c r e t e f r e q u e n c i e s chosen, g i s the
v e c t o r of r e s i d u a l s and A i s the v e c t o r of c o r r e c t i o n s t o the
r a t i o n a l f u n c t i o n c o e f f i c i e n t s . I f the importance of e r r o r s of
a p p r o x i m a t i o n i s g r e a t e r a t some f r e q u e n c i e s than a t o t h e r s ,
a w e i g h t i n g f u n c t i o n , W(<o), may be i n t r o d u c e d so t h a t the system
of e q u a t i o n s becomes
C
V(<o.) \ Ci'oj.jA. = ¥(».).e (».) + 0±; 1 <. i ̂ m P r o o f of the e x i s t e n c e and uniqueness of a " b e s t " s o l u t i o n
of the s e t of e q u a t i o n s which m i n i m i z e s || P || f o r the Chebyshev
norm may be found i n \_8~\ w h i l e i t i s s t a t e d i n [ 9 ] t h a t the
s o l u t i o n a l s o e x i s t s f o r the l e a s t p - t h norm and t h a t the
s e r i e s of s o l u t i o n s f o r i n c r e a s i n g p have a convergent sub
sequence whose l i m i t i s the s o l u t i o n f o r the Chebyshev norm.
A f t e r d e t e r m i n i n g the c o r r e c t i o n s A ^, these are t h e n
added t o the o r i g i n a l c o e f f i c i e n t s R^. I f the c o r r e c t i o n s are
s m a l l enough the p r o c e s s i s c o n s i d e r e d to have converged;
o t h e r w i s e the m a t r i c e s [ c j and [E~\ are r e c a l c u l a t e d w i t h the
new c o e f f i c i e n t s , R^, and the p r o c e s s i s r e p e a t e d .
4-2. S o l u t i o n of a S u b s i d i a r y Problem
The method j u s t d e s c r i b e d r e q u i r e s t h a t a " b e s t "
s o l u t i o n t o a s e t of o v e r d e t e r m i n e d l i n e a r e q u a t i o n s be f o u n d ,
t h a t i s , a s e t of v a l u e s , A . , which w i l l m i n imize || P || .
I f the norm i s the l e a s t - s q u a r e s norm, the s o l u t i o n i s e a s i l y
c a l c u l a t e d by an e x p l i c i t f o r m u l a which may be d e r i v e d i n the m̂ ^
f o l l o w i n g way. I f S = >̂ (P^) i s a minimum, then the
p a r t i a l d e r i v a t i v e s a>s 1,2, . , n - 1 ' c
must be z e r o . E v a l u a t i n g these c o n d i t i o n s e x p l i c i t l y g i v e s
the s e t of l i n e a r e q u a t i o n s
m
E i = l
1,2, n - 1 c
w hich may be expanded to
m
T. i = l
n - 1 c
k = l C.(a.)A . 3 i J
0 0 — 1,2,
hence
m _ n - 1 c
i 4 L k = l
22
= 0 j = l , 2 , . . . f n c - l
m r n - 1 - i m
i = l L k=l J i = l
which may be r e w r i t t e n i n m a t r i x n o t a t i o n as
[ C * . C ] [ A ] - [ c * . . ]
The method which i s used f o r m i n i m i z i n g M = max pV l < i ^ m
i s an exhange p r o c e s s proposed by S t i e f e l [ l 7 J . I t i s based on
a theorem o r i g i n a l l y proved by de l a V a l l e e P o u s s i n which s t a t e s
t h a t the b e s t Chebyshev s o l u t i o n of a system of m l i n e a r
e q u a t i o n s i n n unknowns i s the b e s t s o l u t i o n of the subset of
n+1 of the m e q u a t i o n s which maximizes the d e v i a t i o n ||s|| , The
exchange p r o c e s s i s s t a r t e d by p i c k i n g an a r b i t r a r y subset of
n+1 e q u a t i o n s . A f t e r f i n d i n g the b e s t s o l u t i o n f o r t h i s subset
a s e a r c h i s made f o r a r e s i d u a l 8^ whose magnitude i s g r e a t e r
than the d e v i a t i o n of the s o l u t i o n f o r the s u b s e t . I f none i s
found, the s o l u t i o n to the subset must be the s o l u t i o n f o r the
e n t i r e s e t of m e q u a t i o n s . I f one i s found, the c o r r e s p o n d i n g
e q u a t i o n i s s y s t e m a t i c a l l y s u b s t i t u t e d f o r one of the n+1
e q u a t i o n s of the o r i g i n a l s u b s e t . The s u b s t i t u t i o n i s done so
t h a t a t e v e r y s t e p the d e v i a t i o n of the b e s t s o l u t i o n i n c r e a s e s .
The p r o c e s s t h e n s t a r t s a g a i n w i t h the new s u b s e t . S i n c e the
maximum d e v i a t i o n i s bounded above, and the number of com
b i n a t i o n s of e q u a t i o n s i s l i m i t e d , the a l g o r i t h m must converge
i n a f i n i t e number of s t e p s t o the b e s t Chebyshev s o l u t i o n .
23 4-3. The Computer Program
For ease of w r i t i n g , t h i s program has been w r i t t e n i n
a number of independent segments. The m a i n l i n e program, c a l l e d
MASTER, was w r i t t e n to p r o v i d e proper l o g i c a l f l o w and
communication between t h r e e s u b r o u t i n e s which perform f o u r
d i s t i n c t t a s k s . The f i r s t two, p r o v i d i n g an i n i t i a l a p p r o x i
m a tion by matching the r a t i o n a l f u n c t i o n to the d e s i r e d f u n c t i o n
a t a r b i t r a r y p o i n t s , and e v a l u a t i n g the response of a g i v e n
r a t i o n a l f u n c t i o n on a p r e d e t e r m i n e d s e t of p o i n t s , are performed
by the s u b r o u t i n e PMEV, S u b r o u t i n e MAIN a d j u s t s the c o e f f i c i e n t s
of the i n i t i a l r a t i o n a l f u n c t i o n so as to reduce the norm ( e i t h e r
l e a s t - s q u a r e s or Chebyshev) of the d e v i a t i o n from the g i v e n
f u n c t i o n w i t h the d e s i r e d w e i g h t i n g . F i n a l l y s u b r o u t i n e POLES
f i n d s the p o s i t i o n s of the p o l e s and zeroes of the r e s u l t i n g
r a t i o n a l f u n c t i o n and d e c i d e s upon i t s p h y s i c a l r e a l i z a b i l i t y .
F i n a l l y the m a i n l i n e program c o n t r o l s some p r i n t i n g so t h a t
the output w i l l appear more r e a d a b l e and i n l o g i c a l o r d e r .
i ) PMEV
The p o i n t matching s e c t i o n of t h i s s u b r o u t i n e i s
d e veloped as f o l l o w s . C a l l i n g the o r d e r of the numerator of the
r a t i o n a l f u n c t i o n NO and the o r d e r of the denominator N l , the
number of f r e e c o e f f i c i e n t s i s N0+N1+1 -- N8 s i n c e one
c o e f f i c i e n t can be assumed to be s p e c i f i e d i n advance f o r s c a l i n g .
T h e r e f o r e i f the r a t i o n a l f u n c t i o n i s equated to the d e s i r e d
f u n c t i o n a t t h i s number of p o i n t s the c o e f f i c i e n t s may be
u n i q u e l y d e t e r m i n e d . The e q u a t i o n s are d e r i v e d as f o l l o w s s
24 n
2^ _ i^o. T a.<o.2J 3 i m E b . (0.
2j i — X p 2 y « « » ^ . N 8
Now assume b Q = 1 and m u l t i p l y by the denominator of the
r a t i o n a l " f u n c t i o n , w h i c h g i v e s
n m V a.«. 2 j = f ( « . 2 ) . Y b.«. 2 j; i = 1, 2, .. . , p 3=0 3=0
T h i s may be w r i t t e n i n m a t r i x form as
A =
6) (I)
[>]•
2n - f ( u 1
2n - f ( « 2
2n - f ( J p
•
*
2 ) « 2 . P
ao f(<V' a l •
f (* 22)
a
b i *
b m
f(« 2 ) p
r, / 2 \ 2m . . - f ( « J 1 ) « > 1
2<* 2m p p
The p a r t s o f the program which perform output e d i t i n g
and e v a l u a t i o n of a g i v e n r a t i o n a l f u n c t i o n are s t r a i g h t f o r w a r d
and s h o u l d r e q u i r e no f u r t h e r e x p l a n a t i o n .
( i i ) MAIN
T h i s s u b r o u t i n e i s used to min i m i z e the norm of the
d e v i a t i o n of the r a t i o n a l f u n c t i o n from the d e s i r e d f u n c t i o n .
The method used has been d e s c r i b e d p r e v i o u s l y . The f i r s t step
i s t o c a l c u l a t e the p a r t i a l d e r i v a t i v e s
F(j«) 2
2
j = 0,1, NO
j = 1 , 2 , N l \ i = 1,2, ,,,, NOPB
where NOPB i s the number of p o i n t s of the d i s c r e t e s e t of
f r e q u e n c i e s . The l i n e a r e q u a t i o n s are formed by e q u a t i n g the
t o t a l d i f f e r e n t i a l s t o the d e v i a t i o n s . B o t h s i d e s of the
e q u a t i o n are the n m u l t i p l i e d by the w e i g h t i n g f a c t o r (a f u n c t i o n
of f r e q u e n c y ) . I n the program the a r r a y of r a t i o n a l f u n c t i o n
c o e f f i c i e n t s i s c a l l e d X and the a r r a y of the c o r r e c t i o n s t o
these c o e f f i c i e n t s i s c a l l e d DX.
The t a s k of f i n d i n g the b e s t s o l u t i o n to the above s e t
of e q u a t i o n s o c c u p i e s most of the remainder of s u b r o u t i n e MAIN.
When the s o l u t i o n has been found, the ' c o r r e c t i o n s are added to
the r a t i o n a l - f u n c t i o n c o e f f i c i e n t s . The new r a t i o n a l f u n c t i o n i s
used t o c a l c u l a t e a new s e t of e q u a t i o n s and the p r o c e s s i s
r e p e a t e d u n t i l a l l the c o r r e c t i o n s become s u f f i c i e n t l y s m a l l .
O p t i o n a l l y , i f a r e c o r d of the r a t e of convergence i s d e s i r e d ,
the r a t i o n a l - f u n c t i o n c o e f f i c i e n t s may be p r i n t e d out at, each
s t e p b e f o r e the new p a r t i a l d e r i v a t i v e s are c a l c u l a t e d .
( i i i ) POLES 26
S u b r o u t i n e POLES c a l c u l a t e s the l o c a t i o n of the p o l e s and
zeroes of a r a t i o n a l f u n c t i o n and s t o r e s these i n the common
ar e a f o r l a t e r p r i n t i n g . The work i s d i v i d e d among t h r e e 2
s u b r o u t i n e s . The f i r s t * c a l l e d ROOT, f i n d s the zeroes i n w of f i r s t the denominator and then the numerator of the r a t i o n a l
2
f u n c t i o n , b o t h of which are p o l y n o m i a l s i n (0 . The second
s u b r o u t i n e , c a l l e d ACCEPT, d e c i d e s i f the zeroes found c o r r e s p o n d
to a p h y s i c a l l y r e a l i z a b l e impedance. T h i s i n f o r m a t i o n i s a l s o
s t o r e d i n the common s t o r a g e a r e a f o r l o g i c a l r o u t i n g i n the
m a i n l i n e program. F i n a l l y , t o o b t a i n the a c t u a l p o l e and zero
l o c a t i o n s , i t i s n e c e s s a r y t o take the square r o o t of the
complex numbers f o u n d . T h i s i s done by s u b r o u t i n e SQROOT* ( i v ) A u x i l i a r y Programs
APPROX c a l c u l a t e s the squared magnitude f u n c t i o n
WEIGHT c a l c u l a t e s the w e i g h t i n g f u n c t i o n
DSOLTN s o l v e s a s e t of l i n e a r e q u a t i o n s i n double p r e c i s i o n
5» EXAMPLES 27
To t e s t the a l g o r i t h m d e v e l o p e d , i t has been programmed
i n F o r t r a n IV f o r an IBM 7040 computer. The e s s e n t i a l p a r t s
of t h i s program are i n c l u d e d i n the Appendix f o r r e f e r e n c e .
The r e s u l t s of a p p l y i n g t h i s program to t h r e e d i f f e r e n t magni
tude f u n c t i o n s and f o r v a r y i n g o r d e r s of r a t i o n a l f u n c t i o n
c o m p l e x i t y are shown i n the accompanying diagrams. With the
e x c e p t i o n of the one i l l u s t r a t i n g l e a s t - s q u a r e s f i t , the e r r o r
c u r v e s show the a l t e r n a t i o n of equal p o s i t i v e and n e g a t i v e
d e v i a t i o n s c h a r a c t e r i s t i c of the Chebyshev f i t .
Two f e a t u r e s of the r e s u l t i n g a p p r o x i m a t i o n s u s i n g the
Chebyshev norm are apparent when comparing a p p r o x i m a t i o n s t o
ampl i t u d e f u n c t i o n s w i t h sharp breaks i n response w i t h a p p r o x i
mations to smooth f u n c t i o n s . F i r s t l y , f o r the same degree o f
c o m p l e x i t y , the maximum e r r o r of a p p r o x i m a t i o n i s u s u a l l y
g r e a t e r f o r those f u n c t i o n s w i t h abrupt changes of s l o p e t h a n f o r
smooth amp l i t u d e f u n c t i o n s . S e c o n d l y , the r e d u c t i o n of the
e r r o r of a p p r o x i m a t i o n w i t h i n c r e a s i n g o r d e r of c o m p l e x i t y of the
r a t i o n a l f u n c t i o n i s much slowe r i f the f u n c t i o n has abrupt
changes of sl o p e t h a n o t h e r w i s e .
f(« 2) jF(j«>)| ! 2
R e l a t i v e w e i g h t i n g o f e r r o r
0.0 ^ t o £ 1.0 1
0.8
0.6
0,4
0.2
0.0
f u n c t i o n t o be approximated
| F ( » | 2 = 1.1176470
1.0 + 0. 5010014856s'
|F(j«o)r =
|F(j6>)l
0.97921836 - 1-.3413799© + 0.50357208a 4
1.0 - 1.3669154tt 2+ 0.43 263364654
+ 0.078848972a
0.85236581 1.0 - 1.2856817«2+ 1.8589182co 4- 0.63662153co 6
+ 0.0649470776)'
0.0 0.5 1.0 1.5 2.0
F i g u r e 1. A p p r o x i m a t i o n to I l l u s t r a t e d Response t o CO
F i g u r e 2» A p p r o x i m a t i o n to G a u s s i a n Response
.-4
E r r o r i n A p p r o x i m a t i o n to Gaussian Response w i t h minimum percentage e r r o r f o r 0 ^ 0) ̂ 2
F i g u r e 3.
(|F(j«)! 2 - f(l» 2)} :¥T(« 2) f(« 2) = exp (-o 2)
f VTtto 2)= exp (to 2)
F i g u r e 4. Comparison of Weighted E r r o r i n : A p p r o x i m a t i o n to Gaussian Response
F i g u r e 5. A p p r o x i m a t i o n t o I l l u s t r a t e d Response
t o
6. CONCLUSION
The new a l g o r i t h m f o r f i n d i n g " b e s t " r a t i o n a l f u n c t i o n
a p p r o x i m a t i o n s has been t e s t e d on s e v e r a l d i f f e r e n t magnitude
f u n c t i o n s and found t o be e n t i r e l y p r a c t i c a l f o r use on a
computer such as the IBM 7040, t a k i n g about h a l f a minute t o
f i n d a b e s t r a t i o n a l f u n c t i o n of g i v e n degree of c o m p l e x i t y and
t e s t t h i s f o r p h y s i c a l r e a l i z a b i l i t y . Because of d i f f i c u l t y of
programming, the a b i l i t y of the a l g o r i t h m to f i n d b e s t
s i m u l t a n e o u s a p p r o x i m a t i o n s to g i v e n magnitude and phase
responses has not y e t been t e s t e d .
S i n c e t h i s a l g o r i t h m produces a b e s t f i t to a g i v e n
f u n c t i o n on a g i v e n s e t of f r e q u e n c i e s , i t f r e q u e n t l y happens
t h a t the r e s u l t a n t i s not p h y s i c a l l y r e a l i z a b l e . P r o b a b l y the
most i n t e r e s t i n g e x t e n s i o n of t h i s work would be to r e f o r m u l a t e
the r e a l i z a b i l i t y c o n d i t i o n s i n such a form t h a t they c o u l d be
i n t r o d u c e d as c o n s t r a i n t s , a l l o w i n g c a l c u l a t i o n of the b e s t
r e a l i z a b l e a p p r o x i m a t i o n w i t h a r a t i o n a l f u n c t i o n of s p e c i f i e d
degree.
34
APPENDIX
COPY OP FORTRAN IV COMPUTER PROGRAM FORTRAN SOURCE LIST MASTER
SOURCE STATEMENT C LOAD SUBROUTINES PMEV,MAIN,POLES,ACCEPT,ROOT,SQROOT,APPROX, C WEIGHT.OSOLTN
DOUBLE PRECISION X I lh) ,RR I 30 ) ,R I (30) .RTR (1 SI ,RTIt 1 *>) , 1 WR1,WR2,WA1,WA2,WA3,WA4,WA5 LOGICAL I SHI,ISW2tYES,ERROR,SW2,SW5«SWNUM,SWDEN COMMON N0,N1,UR1,WR2,WA1,WA2,WA3,WA4,WA5,NOPR,ISW1,ISW2,YES,X,RR, 1 RI,RTR,RTI,ERR0R,N3,N4,N5,N6,N7,N8,N9,SWNUM,SWDEN
C NO IS THE ORDER OF THE NUMERATOR, Nl IS THE ORDER OF THE DENOM-C INATOR. WR1 AND WR2 ARE THE ENDS OF THE R ANfiF DF APPROXIMATION, C WA1 AND WA2 ARE THE ENDS OF THE RANGE OF INITIAL POINT MATCHING. C WA3 AND WA4 ARE THE ENDS OF THE RANGE OF EVALUATION OF THE RESULT C AND WA5 IS THE INTERVAL OF THIS EVALUATION. NOPR IS THE NUMBER OF C POINTS FOR BEST FIT. IF ISW1=.TRUE. THEN LEAST SQUARES FIT, ELSE C MINMAX FIT. ISW2=.FALSE. FOR SUPPRESSION OF INTERMEDIATE PRINT. £ I F = . T B t l F . 1 M T T 1 A I PH I S F U A I I I A T F n . IF SWS = . TRUF. THEN C POLES AND FINAL EVALUATION PRINTED. IF SW5 = .FALSE. THESE ARE C PRINTED ONLY FOR A PHYSICALLY REALIZABLE FUNCTION. C EVALUATE FINAL RATIONAL FUNCTION IF SW5=.TRUE. OR YES=.TRUE.
100 START = CLOCMO.) READ!5,1) N0,NI,WR1,WR2,WA1,WA2,WA3,WA4,WA5,N0PR,ISW1,ISW2,SW2,SW5 C.AH SKIP TO I I ) '. : WRITE(6,3)N0,N1,WR1,WR2,WA1,WA2,WA3,WA4,WA5,N0PR,ISW1,ISW2,SW2,SW5 YES=.FALSE.
C N3 IS THE NUMBER OF NUMERATOR COEFFICIENTS JOF RATIONAL FUNCTION. N3=N0+l
C N5 IS THE POSITION OF THE FIRST DENOMINATOR COEFFICIENT N5=N3-n ;
C N6 IS THE POSITION OF THE SECOND DENOMINATOR COEFFICIENT N6=N5+1
C N7 IS THE NUMBER OF DENOMINATOR COEFFICIENTS OF RATIONAL FUNCTION. N7=N1*1 N8=N0+N1 N9=N8»1 ,
C NA IS THE TOTAL NUMBER OF COEFFICIENTS IN THE RATIONAL FUNCTION. N4=N9+1 SWNUM=NO.EQ.O SWDEN=N1.EQ.0
C INITIAL APPROXIMATION BY POINT MATCHING. : CALL PMFV 1.TRIIE..SW?)
IF (ERROR) GO TO 104 C REDUCE NORM OF ERROR.
CALL MAIN IF (ERROR) GO TO 104 IF(X(N3)»XU).LT.O.) GO TO 101 I F i » i M t ) . i T.n. \ r.n Tn m i
C FIND THE LOCATION OF POLES AND ZEROS AND CHECK FOR PHYSICAL C REALIZABILITY. IF THE RATIONAL FUNCTION IS REALIZABLE, YES*.TRUE.
103 CALL POLES GO TO 102
101 IF (SW5) GO TO 103 i n ? «;w?=<;MS.nR.VFS
C EVALUATE FINAL RATIONAL FUNCTION IF SW5=.TRUE. OR YES-.TRUE. CALL PMEV (.FALSE.,SW2)
C OUTPUT REALIZABILITY. DECISION. IF (.NOT.YES) WRITE 16,2)
35
FORTRAN SOURCE LIST MASTER SOURCE STATEMENT IF (YES) WRITE (6,11) IF (.NOT.SW2) GO TO 104 LsO :
C BEGIN PRINTING POLES AND ZEROS. IF (SWNUM) GO TO 105 WRITE (6,9) WRITE (6,7) L=2«N0 WRITE (6,8) ( R I ( I l . R R I I ) . 1 = 1 ,1 )
105 IF (SWDEN) GO TO 104 WRITE (6,6) WRITE (6,7) M=L+2»N1 L = L*1 WRITE (6,8) (RIII).RR(H,I=I ,H)
104 TOTAL=CLOCK(STARTJ/60. WRITE (6,10) TOTAL
C CHECK FOR MORE INPUT DATA. GO TO 100
1 FORMAT (2I3,7F5.2,14,4L3) 7 F O R M A T ( 4 4 H A T H P T M P F n A N T . F I S NfIT P H V S I f . A I I Y R F A I I 7 A R L F . / / 1 3 FORMAT (80H NO Nl WR1 WR2 WA1 WA2 WA3 WA4 WAS NOP 1R ISW1 ISW2 SW3 SW5//1X,13,I4,7F6.3,I5.4L6//)
6 FORMAT (/39X,5HP0LES) 7 FORMAT ( 18X,5HSIGMA,3*>X,5H0MEGA/) 8 FORMAT (D33.16,040.16) Q F f l P M A T I/*<>ytsn7FRn<;)
10 FORMAT (17H0ELAPSEO TIME WAS,F7.2,8H SECONDS) 11 FORMAT (40H0THE IMPEDANCE IS PHYSICALLY REALIZABLE.//)
END
36
FORTRAN SOURCE LIST PMEV SOURCE STATEMENT SUBROUTINE PMEV (SW1,SW2) DOUBLE PRECISION X<16),RR<30),RI<30),KTR(15),RT1(15),WR11WR21WAI,
1 WA2.Wft3.WA4.MA5,A(IS,IS),HI151,WSQt 15 1.YI 15).WT(1 ),FIINC(11 . 2 XN, NUM,DENtF,DEViWTDEV LOGICAL I SWl,ISri2tVEStERR0R,SWNUM,SWDEN, SW1,SW2»INDEX COMMON N0,N1,WK1, WR2, WA1, WA2, WA3, WA4, WA5,IMOPR,ISW1,ISW2,YES,X,RR,
1 Rl,RTR,RTI.ERROR,N3,N4,N5,N6,N7,N8,N9,SWNUM,SWDEN C IF SW1=.TRUE. PROGRAM DOES POINT MATCHING, OTHERWISE BYPASSES THIS •C PART OF THE PROGRAM. IF SW2=.TRUE. PRIJGRAH OflFS EVALUATION flF C RATIONAL FUNCTION.
IF (.NOT.SWl) GO TO 200 C THIS SECTION FINDS A RATIONAL FUNCTION WHICH IS EQUAL TO A GIVEN C FUNCTION (APPROX) AT (N0*N1*1> POINTS. EQUATING THE TWO AT THIS C NUMBER OF POINTS AND ASSUMING A VALUE (1.) FOR THE CONSTANT TERM X l.M THE DFMnMT N ATfltt Pftri\/ir»FS A SFT f lF I T NF Aft EQUATIONS WHflSF C SOLUTION GIVES THE DESIRED RATIONAL FUNCTION COEFFICIENTS.
XN = N8 XN=(WA2-WA1)/XN W(l)=WAl WSQ(1)=WA1»*2 DO ISO T = ?,N9 :
W(I)=W(I-1) «• XN 150 WSO(I)=WI1)»»2
CALL APPROX (N9,W,X) WRITE (6,3) (W(I),1=1,N9) DO 152 1=1,N9 A ( i , i) = i . no : IF (SWNUM) GO TO 101 DO 153 J=2,N3
153 A I I . J ) = A(I,J-l)»WSQ(I) 101 IF (SWDEN) GO TO 152
A(I,N5) = -XI I)»WSQ(I) IF (Nl .FO.l I GO TD 15? : : DO 154 J=N6,N9
154 A ( I , J ) = All,J-l)«WSQ(I) 152 CONTINUE
CALL DSOLTN (A,X,N9,15,XN) IF (XN .ECO. ) GO TO 300
X i f TUP n R n p g nr THE nFMOMjMATnR i s r .Rr-ATFa THAN 7 F x n f S H I F T THF C COEFFICIENTS FOUND AND INSERT tTHE ASSUMED VALUE OF I. FOR X(N5).
IF (SWDEN) GO TO 103 I=N4
155 X(I)=X(1-1) 1 = 1-1 I F n . r . F . N M an TO IS5
103 X(N5)=1.D0 C BEGIN EDITING FOR OUTPUT.
200 IF (N0.GT.N1) GO TO 201 LESS=N3 1NDEX=.TRUE.
: GO TO ?0? 201 LESS=N7
INDEX=.FALSE. 202 DO 250 1=1,N7
J=I+N3
37
FORTRAN SOURCE LIST PMEV SOURCE STATEMENT
250 Y ( I ) = X ( J ) WRITE (6,4) <X< I ) , Y ( I ) , 1 = 1,LESS) LESS=XESS+1 IF (.NOT.INDEX) WRITE (6,6) (X(I),I=LESS,N3) IF (NO.NE.N1.AND.INDEX) WRITE (6,5) (Y(I),I=LESS,N7) IF (.N0T.SW2) RETURN
C BEGIN EVALUATION OF RATIONAL FUNCTION. NOP=(WA4-WA3)/WA5 + 1.D0 WRITE (6.8) , DO 251 1=1,NOP XN=I-1 W(1)=WA3+XN»WA5
C APPROX GIVES DESIRED VALUE IN ORDER TO CALCULATE DEVIATION. CALL APPROX (i.W.FUNCJ CALL WEIGHT ll.W.WT) : XN=W(1)*»2 NUM=X(N3) IF (SWNUM) GO TO 203 DO 252 J=2,N3 K=N3*l-J
252 NUM=NUM»XN+X(K) 203 DEN=Y(N7)
IF (SWDEN) GO TO 204 DO 253 J=2,N7 K=N7*1-J
253 DEN=DEN»XN+Y(K) 204 F^NUM/DFN '. :
DEV=F-FUNC(1) WTDEV=DEV»WT(I)
251 WRITE (6,9) W( I).F,DEV,WTDEV RETURN
300 WRITE (6,10) ERRQR=. TRI.IF. RETURN
3 FORMAT (18X.43HP0INT MATCHING AT THE FOLLOWING FREQUENCIES/ I (D53.16))
4 FORMAT (//14X,22HNUMERAT0R COEFFICIENTS,12X,24HDEN0MINAT0R COEFFIC 1IENTS/(2036.16))
<i F O R M A T - < n 7 7 . 1 * ) : 6 FORMAT (D36.16) 8 FORMAT (/5X,5H0MEGA,14X,8HRESP0NSE,18X,9HDEVI AT I ON,9X, 1 18HWEIGHTED DEVIATION/)
9 FORMAT (IX,F9.3,3D27.161 10 FORMAT (14H0ERR0R IN PMEV)
END :
FORTRAN SOURCE L I ST MAIN SOURCE STATEMENT
SUBROUTINE MAIN REAL S IGMA(16) DOUBLE PRECIS ION X (16 ) , RR (30 F, R I f 3 0 ) , R T R ( 1 5 ) , R T I ( 1 5 i ) i W R I , W R 2 , WAI,
1 K A 2 , W A 3 , W A 4 , W A 5 , A ( 2 0 1 , 1 5 ) , B ( 2 0 1 ) , O X ( 1 6 ) , W ( 2 4 0 ) , F U N C ( 2 4 0 ) , 2 W T ( 2 4 0 ) , R ( 1 6 , 1 6 ) , W R 3 , N U M , D E N , 0 , B I G , S I G , F I , F 2 , S M , B I , D E T , E ( 2 0 1 ) 3 » L A M B D A ( 1 6 ) , M U ( 1 6 )
DOUBLE PRECIS ION DABS INTEGER IA (16 ) LOGICAL ISW1, ISW2,YES,ERROR, JY,SWNUM,SWDEN COMMON N0 ,N1 ,WR1 ,WR2 ,WA1 ,wA2 ,WA3 ,WA4 ,WA5 .N0PR , I SW l , I SW2 ,YES ,X ,RR ,
1 R I , R T R . R T I . E R R O R , N 3 , N 4 , N 5 , N 6 , N 7 , N 8 , N 9 , S W N U M , S W D E N E *ROR=.FALSE .
C IRUN1 COUNTS THE NUMBER OF LINEAR EQUATION ADJUSTMENTS IRUN1=0 NNP=N4 WR3=N0PR-l WR3=(WA2-WR1)/WR3 N1=N1-1 N4=N4-1 W(1)=WR1 PU 450 I=2,N0PR
450 W( I )=W( I-D+WR3 C GENERATE SQUARED MAGNITUDE FUNCTION AND WEIGHTING FUNCTION
CALL APPRDX (NOPR,W,FUNC) CALL WEIGHT (iMOPR , W , W T )
600 IRUN1=IRUNI+1 Z 1RUN2 COUNTS THE! NUMBER -OF EXCHANGES REQUIRED TO FIND CHEBYSHEV C SULUTION TO OVEROETERMINEO EQUATIONS.
1RUN2=0 NUP=0
C BEGIN ITERATIVE PROCEDURE. DO LOOP 650 CALCULATES THE EQUIVALENT C OVERJETERMI Nb0 LINEAR I ZED EQUATIONS
DO 650 1=1,NOPR C IF WtlGHTING IS ZERO DO NOT FORM THE ASSOCIATED TR IV IAL EQUATION.
IF (WT( I ) .EQ.O.) GO TO 650 C RESTURt N l AND N4 TO THEIR ORIGINAL VALUES FOR THIS SECT ION .
N1=N1+1 N4=N4+ 1 • uOP=NOP+l WSO=W( I ) « *2 NUK, = X(N3) IF (SWNUM) GO TO 601 DO 651 J=1 ,N0 K=N3-J
— 6 5 1 NUMNUM»wSQ+X(K) : 601 DEN=X(N4)
IF (SWDEN) GO TO 602 DO 652 J = l . N l K=N4-J
652 OEN=OEN»WSQ+X(K) ' Z THE MATRIX A ( I , J ) CONSISTS OF THE PARTIAL DERIVATIVES OF THE C RATIONAL FUNCTION AT FREQUENCY W(I) WITH RESPECT TO THE C COEFF IC I ENTS X t J ) ALL MULTIPL IED BY WEIGHT WT(I) EXCEPT THAT C OER IVAT IVcS WITH RESPECT TO X(N5) ARE OMITTED
602 A (NOP ,1 )=WT( I ) /DEN
SOURCE STATEMENT FORTRAN SOURCE LIST MAIN
N l = N l - l N4=N4-l IF I SWNUM) GO TO 603 '. DO 653 J=2,N3
653 A(NOP,J)=A(NOP,J-1)»WSQ 603 IT (SWDEN) GO TO 604
A(N0P,N5)=-WTII)»WSQ*NUM/DEN»»2 IF (Nl.EQ.O) GO TO 604 00 654 J=N6,N4
654 A(NOP,J)=A(NOP,J-l)«WSQ C THE MATRIX B ( I ) CONSISTS OF THE DIFFERENCE BETWEEN THE DESIRED C RESPONSE AND THE RESPONSE OF THE CURRENT RATIONAL FUNCTION C AT OMEGA = W(I) ALL MULTIPLIED BY THE WEIGHTING FUNCTION.
604 B(NOP)=WT(I)»(FUNC(I)-NUM/DEN) 650 CONTINUE '.
C ON THE FIRST PASS, FIND LEAST SQUARES F I T TO EQUATIONS AND C CORRESPONDING LARGEST RESIDUALS. ON SUBSEQUENT PASSES, USE THE C VALUES OF IA( ) FROM THE PREVIOUS PASS.
IF ( IRUN1.NE. L A N D . .NOT. ISW1) GO TO 200 IF (N0P.GT.N4) GO TO 606 nn ASS t = l f N 4
D X ( I ) = B ( I ) DO 655 J=1,N4
655 R ( I , J ) = A ( I ,J) CALL OSOLTN (R,DX,N4,16.0ET) IF (DET .EQ. 0.) GO TO 820 Gfl TO BOO
606 DO 656 I-1,N4 DO 656 J-1.N4 IF ( I . G T . J ) GO TO 607 R ( I , J ) = 0 . 00 657 K=1,N0P H I I T .11 =R I I T .1 I • A I K f I 1 « A I K f .11
GO TO 656 607 R ( I , J ) = R ( J , I ) 656 CONTINUE
DO 658 1=1,N4 D X ( I ) = 0 . 00 fttfl .1*1 ,N(1P
658 OX(I) 3DX(I)+A(J,I)»B(J) CALL DSOLTN (R,OX,N4,16,DET) IF (DET.EQ.O.) GO TO 820 IF (.N0T.ISW1) GO TO 608 IRUN2=-1 GO Tf) BOO
608 DO 659 1*1,NOP E ( I )«—B( I ) 00 660 J=1,N4
660 E( I)=E( I)«-A(I.J)»DX(J) IF (I.GT.NNP) GO TO 610 IF ( i - r . T . i > r.n m f,ciQ : I A ( 1 ) = 1 GO TO 659
609 K=I-1 GO TO 611
40
FORTRAN SOURCE L I ST MAIN SOURCE STATEMENT
610 K=NNP 611 00 661 L=1,K
LD=IA (L ) IF ( 0 A B S ( E ( L D ) ) . L E . D A B S ( E ( I ) ) ) GO TO 612
661 CONTINUE IF ( I . L E . N N P ) IA ( I )= I
659 CONTINUE GO TO 200
612 M=K+1 IF I I . G T . N N P ) M=M-1
613 IA(M)=IA(M-1) M=M-1 IF ( M . G T . L ) GO TO 613 IA ( L )= I GO TO 659
C SECTION CALCULATES bEST F I T TO NNP GIVEN EQUATIONS. LOCATION OF C EQUATIONS IS GIVEN BY THE VECTOR IA( ) , SIGN OF DEVIATIONS GIVEN C BY VECTOR SIGMA( ) , THE COEFF IC I ENTS OF THE LINEAR COMBINATION OF C ROWS ARE LAMBDA( ) . THIS SECTION FOLLOWS R I C E , PG . 1 7 4 . C F IRST CALCULATE REFERENCE DEV IAT ION , D.
200 ID=IA(NNP) DO 250 J=1 ,N4 JD=IA ( J ) L A M B D A ! J ) = - A ( I D . J ) DO 250 1=1,N4
250 R( It J )=A( JO , I ) CALL OSOLTN ( R , L A M B D A , N 4 , 1 6 , D £ T ) :
IF ( D E T . E Q . O . ) GO TO 820 LAMBDA(NNP)=1.
201 IRUN2=IRUN2*l NUM=0. PEN=0. DO 251 1=1,NNP S I G M A ( I ) = 1 . IF ( L A M B D A ! I ) . L T . O . ) S IGMA( I )=-1 . ID=IA ( I ) NUM=NUM-B(ID)*LAMBDA(I)
251 DEN=OEN»DABS( LAMBDA! I ) ) , D=NUM/DEN
C USING THIS VALUE FOR THE REFERENCE DEV IAT ION , CALCULATE THE C CORRESPONDING SOLUTION, OX( ) , GIVING THE BEST CHEBYSHEV F I T TO C THE NNP EQUATIONS OF IA( ) .
00 252 1=1,N4 ID=IA ( I )
—Dxin=B( iD»*s tGMA ( n»D : — DO 252 J=1 ,N4
252 R ( I , J ) = A ( I D , J ) CALL DSOLTN ( R , D X , N 4 , 1 6 , D E T ) IF ( D E T . E Q . O . » GO TO 820
C CALCULATE VECTOR E = A . DX - B AND FIND VALUE AND POSIT ION OF ~£ ELEMENT OF LARGEST MAGNITUDE.
B IG=0. DO 253 1=1,NOP E U ) = - 8 ( I ) DO 254 J=1 ,N4
FORTRAN SOURCE LIST MAIN SOURCE STATEMENT
254 E U ) = E ( I ) * A ( I , J ) « D X ( J ) IF (DABS(EU) ).LE.BIG) GO TO 253 B1G=DABS( E( I > ) MAX = I
253 CONTINUE C CHECK IF THIS LARGEST ELEMENT IS CONTAINED IN THE SET IA( ).
DO 255 1=1,NNP IF (MAX.EQ.IAII)) GO TO 800
— 2 5 5 CONTINUE C IF THE LARGEST ELEMENT IS NOT IN THE SUBSET IA( ), THIS MAY HAVE C OCCURRED BECAUSE OF ROUNDING ERRORS. THEREFORE, IF DEVIATION IS C CLOSE TO D, THEN ACCEPT THIS AS A SOLUTION.
IF (BIG.LE.1.000001D0«D) GO TO 800 C IF NOT A SOLUTION, IT WILL BE NECESSARY TO TRY A NEW SET OF X- F Q I I A T y n N S , T A I 1 . T U T S I S W I N F B Y F X f . H A N f c I hlCl O N F IIP T H P O R I I N A l
C SET BY THE LARGEST CURRENT OEVIATION. FOLLOWING RICE, EQUATION C (6-8.12) OF PG. 174. FIND COEFFICIENTS MU( ).
SIG=1. IF (E(MAX).LT.O.) SIG=-1. ID-.I A( NNP) DO 256 .J=),N4 J D - I A ( J ) MU<J)=-SIG»A(MAX,J)-A( ID,J) DO 256 1=1,N4
256 R( I , J ) = A ( J D , I ) CALL OSOLTN (R,MU,N4,16,DET) IF (DET.EQ.O. ) GO. TD 820 MUINNP)=1. F1=-SIG»B(MAX) F2=0. DO 257 1=1,NNP ID=IA(I) F1=F1-MU(I)»RIin) :
257 F2=F2+LAMBDA!I)*B(ID) C NOW EVALUATE THE ABSOLUTE VALUE OF THE DEVIATIONS THAT WOULD C RESULT FOR THE NEXT STEP WITH EACH POSSIBLE EXCHANGE AND SELECT C THE LARGEST OF THESE.
BIG=0. ; DO 258 J=I.NNP :
SM=MU(J)/LAMBDA!J) DEN=1. DO 259 1=1,NNP IF (I.NE.J) DEN=DEN+DABS!MU!I)-LAMBDA11 I*SM)
259 CONTINUE R I = n A R S M F l ^ F 7 « S M ) / 0 F N ) IF (BI.LE.BIG) GO TO 258 BIG=BI MIN=J
258 CONTINUE IA!MIN)=MAX
C T N OBTATM I A M R r > A < > F O R nTHFR T H A N T H F FIRST STFP IT IS NOT C NECESSARY TO SOLVE A SET OF LINEAR EQUATIONS, BUT INSTEAD USE C EQUATION !6-8.15> WITH J = MIN.
NUM=MU(MIN)/LAMBDA(MIN) DO 260 1=1,NNP
42
FORTRAN SOURCE LIST MAIN SOURCE STATEMENT
IF(I .NE.MIN) LAMBDA( I )=LAMBDA( I )»LAMBDA(MIN) • (MU( I ) /LAMBDA! I ) -NUM) 260 CONTINUE
LAMBDA ("MIN)=SIG«LAMBDA( MIN) C NORMALIZE COEFFICIENTS LAMBDA! ) TO PREVENT FLOATING POINT TRAPS.
BIG=0. DO 261 1=1,NNP Bl=0A8S(LAMBDA( I) )
261 IF (BI .GT.BIG) BIG = BJ DO 262 1=1,NNP
262 LAMBDA(I)=LAMBDA(I)/BIG GO TO 201
C OUTPUT SECTION. 800 J=:<I4+1 850 OX!J)=DX(J-1)
J = J _ 1
IF (J.GE..M6) GO TO 850 N4 = '>I4+1 DX(N5)=0. JY=.FAL S t .
C CHECK IF THE ABSOLUTE VALUES OF THE CORRECTIONS TO THE RATIONAL X" FUNCTION COEFFICIENTS ARE CONVERGING. IF NOT, REVISE THE SET OF C LINEAR EQUATIONS.
DO 851 1=1,N4 IF (DABS(DX( I ) ) ,GT. I .D-6*DABS(X( I ) ) ) JY=.TRUE.
851 X ( I ) - X(I)+DX(I) . IF ( .NOT. JY) GO TO 802
IF (.NOT.ISW2) GO TO 801 WRITE (6,1) IRUN2 WRITE (6,5) <X(I),I=1,N3) WRITE (6,2) WRITE (6,5) ( X ( I ) , I=N5 »N4)
801 N4=N4-1- GO TO 600'
302 N1=N1+1 D = 1 0 0 . » 0 WRITE (to,3) D.IRUN1 IF <ISW2) WRITE <7,4) NO,Nl , (X( I ) , I=1 ,N4) RETURN . •
820 ERROR=.TRUE. RETURN
1 FORMAT (23H-NUMERATOR COEFFICIENTS,8X,8HIRUN2 = » I 4 / ) 2 FORMAT (25H-OENOMINATOR COEFFICIENTS/) 3 FORMAT (22H0WEIGHTED DEVIATION IS ,F8 .3 / / / 20X ,14HFINAL SOLUTION,
1 16X,8HIRUiMl = ,14) 4 FORMAT (2137(6012) > 5 FORMAT (D24.lt>)
FND
43
FORTRAN SOURCE L IST POLES
SOURCE STATEMENT
SUBROUTINE POLES DOUBLE PRECIS ION X ( 1 6 ) , R R ( 3 0 ) , R l ( 3 0 ) , R T R ( 1 5 ) , R T I ( 1 5 ) . W R 1 , W R 2 , M A I ,
1 HA2 .Wft3.HAA. MAS, A( 1 6 ) , B ( 1 6 ) . S M 6 ) LOGICAL ISWl, ISW2,YES,ERROR,SWNUM,SWOEN COMMON N0,N1,WR1,WR2,WA1,WA2 * W A 3 * W A 4 , W A 5 » N O P R , I S W 1 , I S W 2 , Y E S , X , R R ,
1 R I , R T R , R T I , E R R O R , N 3 , N 4 , N 5 , N 6 , N 7 , N 8 , N 9 , S W N U M , S W D E N IDD=2«N1
0 PLACE COEFF IC I ENTS IN TEMPORARY LOCAT ION. oo 150 1=1.NA :
150 S ( I ) = X ( I ) C PLACE NUMERATOR COEFF IC I ENTS IN REVERSE ORDER IN A( ) .
DO 151 1=1,N3 J=N5-I
151 A ( J ) * S ( I ) X PLACE DENOMINATOR P.nFFFIC IFNTS TN RFVFRSF DROFR IN RI ) .
DO 152 1=1,N7 J=N7+l-I K=I+N3
152 B ( J )=S (K ) IF (SWDEN) GO TO 100
_C SUBROUTINE ROOT FINOS ZEROS OF OMFGA SQUARED OF DENOMINATOR C POLYNOMIAL.
CALL ROOT (N1 ,B ) C CHECK PHYSICAL R E A L I Z A B I L I T Y .
CALL ACCEPT ( N l ) C FIND ZEROS OF OMEGA BY TAKING COMPLEX SQUARE ROOT OF ZEROS ABOVE.
CALL SQROOT ( N l ) C CHECK IF NECESSARY TO FIND ZEROS OF RATIONAL FUNCTION.
100 IF (SWNUM) GO TO 101 C IF SO , SHIFT POLE POSITIONS TO ALLOW ROOM FOR ZERO POS IT IONS .
J = I D D + 2 » N 0 I = IDD
102 RR1 J )=RR ( I ) R I ( J ) = R I ( I ) J = J-1 1 = 1-1 IF ( I . G T . O ) GO TO 102
C BEGIN ZERO LOCATION AND REAL I ZAB I L I TY TEST OF NUMERATOR. CALL ROOT (NO, A) '. CALL ACCEPT (NO) CALL SQROOT (NO)
101 RETURN END
44
FORTRAN SOURCt L I ST ROOT SOURCE STATEMENT
SUBROUTINE ROOT (M,B) DOUBLE PRECIS ION D (16 ) , RR t 3 0 ) , R I ( 3 0 ) , R T R ( 1 5 ) , R T I 1 1 5 ) ,WR1,WR2,WA1,
1 WA2,WA3,WA4,WA5, A( 16 , 2 > , B ( 1 6 ) , CI 16 ) , 0 (16) , TOL ( 2) , R . S , R 1 , S 1 , T , 2 X , Z , D R , D S
DOUBLE PRECIS ION DABS ,DSQRT LOGICAL ISWl , ISW2,YES,ERROR,SWNUM,SWDEN, IM, I NO COMMON N0 ,N l ,WR l ,WR2 iWA I ,WA2 .WA31WA4 ,WA5 ,NOPR , I SW l , I SW2 ,YES ,D ,RR ,
, 1 RI ,R T R , R T I , E R R O R , N 3 , N 4 , N 5 , N 6 , N 7 , N 8 , N ' 9 , SWNUM, SWDEN C M IS THE ORDER OF THE POLYNOMIAL, N IS THE NUMBER OF TERMS IN C THE REDUCED POLYNOMIAL.
N = M+1 DO 150 I = 1,N A I 1,1)=B( I )
150 At I ,2 )=b ( I) C T O L ( l ) IS THE TOLERANCE USED WITH THE REDUCED POLYNOMIAL AND C TOL ( ? ) FOR FULL POLYNOMIAL.
T 0 L ( l ) = l . D - 6 T 0 L ( 2 ) = 1 . D - 1 2 I =0
C TLi FIND ROOTS APPROXIMATELY IN ORDER OF INCREASING MOOULUS, START C WITH IN IT IAL APPROXIMATION FOR F IRST ROOT OF ZERO.
R l = 0 . S1=0. IM= .FALSE .
C THE VALUE OF IN DETERMINES IF THE ROOT IS TO BE EXTRACTED FROM C THE ORIGINAL POLYNOMIAL OR THE REDUCED POLYNOMIAL, IN = 1 FOR
Z REDUCED POLYNOMIAL. IM = . T R U E . WHEN THE LAST ROOTS ARE BEING C FOUND.
10U IF ( N . L T . 4 ) GU TO 112 C BEGIN 8AIRST0W*S METHOD WHICH FINDS QUADRATIC FACTORS, X » » 2 + R»X C + S, OF A POLYNOMIAL.
IN=1 115 KM)
I 'MO=IN.EO.l C B tG IN SYNTHETIC DIV IS ION OF TRIAL FACTOR INTO POLYNOMIAL. C BEGIN SYNTHETIC DIV IS ION OF TRIAL FACTOR INTO POLYNOMIAL.
R = R1 S=S1 Q ( i ) = A I l , I N ) C (1 )=0<1)
101 Q ( 2 J = A ( 2 , I N ) - R » Q ( 1 ) C ( 2 ) = Q ( 2 ) - R « C ( I )
C RftNGfc OF LOOP DEPENDS UPON VALUE OF IN . II- UND> NQ=N
— IF ( . N O T . I NO) N0=M+1 DC 151 1=3,NQ Q( I ) = A ( I » I N ) - R * U ( I - 1 ) - S » Q ( 1 - 2 )
151 C ( I ) = Q ( I ) - R » C ( 1 - 1 ) - S * C ( 1 - 2 ) C BEGIN CALCULATION OF CORRECTIONS TO ROOT POS IT ION.
X = R » C ( N Q - 2 ) + S*C(NQ-3) Z = C ( N 0 - 2 ) « « 2 + X * C ( N Q - 3 )
C CHECK FOR ZERO DENOMINATOR. IF I Z . E Q . O . ) GO TO 107 D R = ( C ( N Q - 2 ) » Q ( N Q - 1 ) - C ( N Q - 3 ) » Q ( N Q ) ) / Z D S = ( X » Q ( N Q - l ) + C ( N Q - 2 ) » Q ( N Q ) ) / Z
SOURCE STATEMENT FORTRAN SOURCE LIST ROOT
IF (R . N E . O . ) GO TO 102 IF ( S . E Q . O . ) GO TO 103
C CHECK FOR WILD CORRECTION. 102 IF(((R+0R)«»2+(S+0S)»«2I.GT.9.»(R»»2+S»»2I) GO TO 106 103 R=R+OR
S=S+DS 104 K = K+1
C TEST FOR EXCESS IVE ITERAT IONS . IF (K .GE.50) GO TO 105
C TEST FOR CONVERGENCE. IF ( PABS IDR ) .GT .DABS tR *TOLI IN ) ) ) GO TO 101 IF .<DABS(DS).GT.DABS<S*TULUN>> ) GO TO 101
C USE THE QUADRATIC FACTOR JUST FOUND AS THE IN IT IAL APPROXIMATION C FOR THE NEXT QUADRAT IC FACTOR.
105 R1=R S1 = S IF ( .NOT . IND ) GO.TO 109 IN = 2 GO TO 115
106 R=3.D0»R S=3.00»S GO TO 104
107 R=l.D-5 S=l.D-5 GO TO 104
C NOW THAT A QUADRATIC FACTOR HAS BEEN REMOVED FROM THE POLYNOMIAL X ITS ROOTS ARE FOUND AND ADDED TO THE OUTPUT ARRAY.
109 R=-.5U0*R S=R**2»S IF (S .GT.O.) GO TO 110 S=DSQRT(-S) DO 152 1 = 1,2 L = L + 1 RTRIL1=R RT I(L)=S
152 S=-S GO TO 111
110 S = 0S(JRT(S) L = L + 1 RTR(L)=R+S RTKL).=0. L=L + 1 RTRU)=R-S RT1(L)=0.
I l l IF <IM) RETURN C RtOUCE N AND SET UP NEW REDUCED POLYNOMIAL.
N = N - 2 DO 153 1 = 1.N
153 A «I,1 ) = Q (I) GO TO 100
C APPROACHING END OF JOB. POLYNOMIAL HAS BEEN REDUCED TO DEGREE C TWO OR L E S S .
112 IF IN .LT.2) RETURN IF- IN.GT.2) GO TO 113
C LINEAR FACTOR TERMINATION.
46
FORTRAN SOURCE L I ST ROOT
SOURCE STATEMENT
L = L + l R T R I L ) = - A ( 2 , 1 ) / A < 1 , 1 ) R T I ( L ) = Q . RETURN
C QUADRATIC FACTOR TERMINATION. 113 IM=.TRUE .
R = A ( 2 , 1 ) / A ( l , l ) S = A ( 3 , 1 ) / A ( 1 , I ) IF I M . L T . 3 ) GO TO 109 IN=2 GO TO 101 END
47
FORTRAN SOURCt L IST ACCEPT SOURCE STATEMENT
SUBROUTINE ACCEPT (Ml ) C SUBROUTINE ACCEPT TESTS IF THE POLES AND ZEROS OF THE BEST £ RATIONAL FUNCTION CORRESPOND TO. A P H V i i r A ^ I v R F A I I / A R I F F l l N i r T T H M .
DOUBLE PRECIS ION X 1 1 6 ) , R R ( 3 0 ) , R I ( 3 0 ) , R T R ( 1 5 ) , R T I ( 1 5 > , W R 1 , W R 2 , W A 1 , 1 WA2 ,WA3,WA4,WA5,F I LE l15 )
DOUBLE PRECIS ION DABS LOGICAL ISWl, ISW2,YES,ERROR,SWNUM,SWDEN COMMON N0,N1,WR1,WR2,WA1,WA2,WA3,WA4,WA5,N0PR, I SW1, I SW2,YES .X .RR ,
1 R I , RTR, RT I , ERROR,N3 ,N4 ,N5 ,N6 ,N7 ,NB ,N9 ,SWNUM,SWDEN INTEGER FILMRK FILMRK=0 Y E S = . T R U E . DO 150 1=1,Ml
C IF THE IMAGINARY PART OF THE ROOT OF OMEGA SQUARED IS NOT ZERO X (A TOLFRANf.F IIF I . F- l f i IS ftl I flWFD FOR RflUNO O F F ) , THFN THFRF WILL C BE A SET OF FOUR ROOTS OF OMEGA IN QUADRANTAL SYMMETRY. HENCE C SUCH A ROOT IS REAL IZABLE WITHOUT FURTHER EXAMINATION.
IF ( D A B S ( R T I ( I ) J . G E . l . D - 1 0 ) GO TO 150 C IF THE REAL PART IS LESS THAN ZERO, THERE WILL BE A PAIR OF ROOTS C ON THE SIGMA AX I S , HENCE NO FURTHER EXAMINATION NEEDED.
IF (RTR ( I ) . L . T . O . ) GO TO 150 : IF ( F I L M R K . E Q . O ) GO TO 103
C ALL OTHER ROOTS WILL BE IN PAIRS ON THE OMEGA AX I S . THESE WILL BE C REAL IZABLE ONLY IF THEY ARE OF EVEN ORDER. F I L E l ) CONTAINS A C STACK OF SUCH UNPAIRED ROOTS. FILMRK COUNTS THE NUMBER CURRENTLY C STORED IN THE STACK.
X F X A M T N F T H F F I I F T P S F F TF T H E M F H RDrtT I S SUFF1C. I F iMTI V f.l D S F C TO A PREVIOUS MEMBER.
DO 151 J= l , F I LMRK IF ( D A B S ( R T R ( I ) - F I L E ( J ) ) . L E . 1 . D - 5 * D A B S ( F I L E ( J ) ) ) CO TO 104
151 CONTINUE GO TO 103
X T H F NFV J pnnT MATr.HFn a P R F v r n n s M F M R F H . THF PRFVItlUS MEMBER IS C REMOVED FROM THE STACK AND THE STACK COUNTER DECREMENTED.
104 FILMRK=FILMRK-1 102 IF ( J . EQ . F I LMRK+1 ) GO TO 150
F I L E l J ) = F I L E ( J + l ) J = J*1 GO TO 102
103 FILMRK=FILMRK+1 F I L E ( F I LMRK )=RTR ( I )
150 CONTINUE C IF THE F I L E CONTENT IS NOT ZERO, THE RATIONAL FUNCTION IS NOT C PHYS ICALLY R E A L I Z A B L E .
f F I F r i M R K . i M F . n l V F S = . F A I S F . RETURN END
REFERENCES 48
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