Methods for solving nonlinear algebraic equations in computer programs for nuclear magnetic resonance spectroscop

Adler, R. J.

Ku, S. Y.

English

NASA Technical Reports Server

A METHOD OF SOLVING SETS OF NONLINEAR 
ALGEBRAIC EQUATIONS 
Progress Report No.  5 
March 10, 1968 
R. J. Adler 
S. Y. Ku 
Case Western Reserve University 
University Circle 
Cleveland, Ohio 
Grant No. NGR-36-003-021 
Prepared for 
National Aeronautics and Space Administration 
Washington, D. C. 20546 
https://ntrs.nasa.gov/search.jsp?R=19680010725 2020-03-12T08:44:55+00:00Z
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A METHOD OF SOLVING SETS OF NONLINEAR 
ALGEBRAIC EQUATIONS 
Progress Report N o .  5 
Chemical Engineering Science Divis ion 
Research Report N o .  March-10-68 
by 
R. J.  Adlar 
S .  Y .  Ku 
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Chemical Engineering Science Divis ion 
School of Engineering 
Case Western Reserve University 
Cleveland, Ohio 44106 
FOREWARD 
The research reported here  i s  p a r t  of t h e  a c t i v i t y  of the Chemical 
Engineering Science Division, The Div is ion ' s  teaching and research 
e f f o r t s  are d i r ec t ed  toward an understanding and synthes is  of processes  
and materials which involve chemistry as w e l l  as engineering. These in- 
clude t h e  broad areas of reac t ions ,  separa t ions ,  and chemical and phys ica l  
s t ruc tu re .  A general  desc r ip t ion  of t h e  Divis ion 's  a c t i v i t i e s  i s  given by 
CHEMICAL ENGINEERING SCIENCE AT CASE WESTERN RESERVE UNIVERSITY, a r epor t  
i s sued  i n  September of each year.  
Research a c t i v i t i e s  a r e  sponsored by agencies of t h e  U. S. Government 
and i n d u s t r i a l  companies. Duplicate copies  of a l l  research r epor t s  are 
automatical ly  i ssued  t o  companies who p a r t i c i p a t e  i n  the  Chemical Engineer- 
i ng  Science I n d u s t r i a l  Sponsors Program. A t  t h e  present  t i m e ,  P i t t sburgh  
P l a t e  G l a s s  Company-Chemical Division has joined t h i s  newly organized 
program. 
Robert J. A d l e r  
Head 
Chemical Engineering Science Division 
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ABSTRACT 
The second generat ion computer program being developed as p a r t  of  
t h i s  research program has been applied t o  a problem from t h e  f i e l d  of 
nuc lear  magnetic resonance spectroscopy. 
supplementary techniques of  p a r t i t i o n i n g  and some l imi ted  forms of 
f ac to r ing  toge ther  with an improved algorithm for  e l imina t ing  polynomial 
va r i ab le s .  The problem solved consis ted of f i v e  mul t iva r i a t e  polynomial 
equat ions,  four  non l in t a r  and one l i n e a r ,  i n  f i v e  va r i ab le s .  Twelve 
real so lu t ions ,  which are permulations of t w o  " t rue  so lu t ions , "  were 
shown t o  e x i s t .  The problem i l l u s t r a t e s  t h e  c a p a b i l i t i e s  of t h e  second 
generat ion program and t h e  usefulness  of supplementary techniques.  
The program incorpora tes  t h e  
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TABLE OF CONTENTS 
Page 
1 INTRODUCTION o . a . e . . o 1 
2 PROBLEM DESCRIPTION 0 . . o . . 2 
3 SOLUTION BY ELIMINATION e . o o e . . . 4 
4 SOLUTION BY SUPPLEMENTARY TECHNIQUES COMBINED WITH ELIMINATION . . . 6 
5 CONCLUDINGREMARKS e 0 e a .  o . e 8 
6 PERSONNEL . . . .  9 
7 REFERENCES.. . . * .  . . . . . . . . . . . . . . . * .  . . 10 
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1 A. INTRODUCTION 
This r epor t  is t h e  f i f t h  semi-annual progress  report made by C a s e  
Western Reserve Universi ty  under NASA Research Grant NGR-36-003-021 
e n t i t l e d  "A Method of Solving Sets of Nonlinear Algebraic Equations." 
The research seeks t o  develop and eva lua te  combined a lgeb ra i c  and numeri- 
c a l  procedures f o r  t h e  computer so lu t ion  of systems of mul t iva r i a t e  poly- 
nomial equations.  
obtained with a f irst  generat ion computer program and developments 
underlying a second generat ion computer program. This report descr ibes  
the  r e s u l t s  of applying the  second generat ion computer program now being 
completed t o  t h e  s o l u t i o n  of a problem i n  nuclear  magnetic resonance (NMR) 
spectroscopy. 
Previous Progress Reports' have discussed t h e  r e s u l t s  
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2 .  PROBLEM DESCRIPTION 
Through the a n a l y s i s  of NMR spectral d a t a  it is poss ib l e  t o  obta in  
the  values  of c e r t a i n  parameters which are of considerable  value t o  
s t r u c t u r a l  organic  and inorganic  chemists. These parameters,  called chemical 
s h i f t s  and spin-spin coupling constants ,  are in t imate ly  r e l a t e d  t o  t h e  elec- 
t r o n i c  and nuclear  conf igura t ions  of molecules. Pos tu la ted  s t r u c t u r e s  can 
o f t e n  be confirmed o r  excluded so le ly  from NMR spectra. 
The conventional technique f o r  a n a l y s i s  of NMR spectra is an i t e r a t i v e  
one, where t h e  sp in  parameters are i n i t i a l l y  es t imated and i t e r a t i v e l y  i m -  
proved u n t i l  t he  ca l cu la t ed  and experimental spectra are brought i n t o  agree- 
ment. Convergence and uniqueness a re  usua l ly  problems. When more than one 
or  t w o  parameters are involved, it is d i f f i c u l t  and c o s t l y  to  seek a l l  of t h e  
possible so lu t ions  and impossible t o  be s u r e  t h a t  t h e  correct one has been 
found . 
An a l t e r n a t i v e  procedure researched by Professor Whitman’ i n  the  Chemistry 
Department a t  Case Western Reserve Universi ty  c o n s i s t s  of  deducing from 
quantum-mechanical r u l e s  an idea l ized  spectrum which agrees  with t h e  main 
f e a t u r e s  of the  experimental  spectrum. The sp in  parameters are then determined 
so t h a t  t h e  ca l cu la t ed  and idea l ized  spectra agree p rec i se ly ;  i n  p r a c t i c e  only 
t h e  l i n e  p o s i t i o n s  are matched. 
are obtained for t h e  sp in  parameters i n  terms of t he  experimental  NMR s p e c t r a l  
l i n e  pos i t i ons .  
technique produces a system of nonl inear  equat ions i n  the  NMR sp in  parameters. 
I n  many i n t e r e s t i n g  cases  e x p l i c i t  equat ions 
However, i n  several  of t h e  m o s t  i n t e r e s t i n g  cases3 t h i s  
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I n  p a r t i c u l a r ,  t h e  general  three-proton system (called t h e  ABC case) 
y i e l d s  a system of f i v e  equations--four of which are nonlinear-in t h e  
five-spin parameters x i  x2, X J ,  y1 and y2 : 
For t h e  s p e c i f i c  ABC s y s G e m  vinyl  ch lo r ide ,  t h e  experimentally determined 
constants ,  which are funct ions of the l i n e  p o s i t i o n s ,  are: 
-20.432 C l  = 
c2 = 1161 711 
c3 = 1284 545 
c4 = -36507 222 
~5 =: 378712.368 
(The number of s i g n i f i c a n t  f i gu res  
shown are n o t  j u s t i f i e d  by experimental 
accuracy. However, t hese  constants  
were employed i n  o rde r  t o  fac i l i t a te  
t h e  comparison of Professor  Whitman's 
s o l u t i o n  and our  solut ions.)  
Using i t e r a t i v e  numerical techniques, Professor  Whitman found one solut ion.  
Because of d i f f i c u l t y  experienced i n  obtaining convergence, and because it 
w a s  suspected t h a t  more than one s o l u t i o n  might ex is t ,  t h e  problem w a s  
brought t o  our a t t e n t i o n .  
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3.  SOLUTION BY ELIMINATION 
The second g e n e r a t i o n  program under  development w a s  a p p l i e d  t o  s o l v e  
t h i s  problem i n  t w o  d i f f e r e n t  ways.  
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the modif ied van d e r  Waerden's e l i m i n a t i o n  algorithm was used. 
gram e l i m i n a t e d  the v a r i a b l e s  xi, x2, x3, and y i , i n  tha t  o r d e r ,  t o  y i e l d  
a u n i v a r i a t e  polynomial  of 30th  degree i n  y2. 
t h i s  f i n a l  polynomial  f a i l e d  because of overflow. 
by s u b s t i t u t i n g  y2 = 1Oy; reso lved  t h i s  d i f f i c u l t y .  
I n  t h e  f irst  method of solut ion,  only 
The pro- 
A first attempt t o  solve 
R e s c a l i n g  the problem 
Twelve real  solutions 
were found t o  ex i s t  as l i s t e d  i n  T a b l e  I. 
S e t  
1. 
2. 
3. 
4. 
5. 
6. 
-
* 
7. 
8. 
9. 
10. 
11 ' 
1 2  . 
3 
-9'28115 
1.47651 
-12.6274 
-9.28115 
-12.6274 
1.47651 
1.45397 
-7011808 
-14* 7679 
1.45397 
-14.7679 
-7.11808 
3 
1047651 
-12 e6274 
-9 -28115 
-12 06274 
1.47651 
-9028115 
-7-11808 
-14- 7679 
1.45397 
-14- 7679 
-7011808 
1- 45397 
3 
-12 -6274 
-9 -28115 
1.47651 
1.47651 
-9 28115 
-12-6274 
-1407679 
1.45397 
-7- 11808 
-7011808 
1- 45397 
-1407679 
YL 
-47.9895 
4.23412 
43- 7554 
47.9895 
-43.7554 
-4.23412 
-2- 53189 
46- 9489 
-44.4170 
2- 53189 
44.4170 
-4609489 
Table  I S o l u t i o n s  to  NMR Spectroscopy Problem 
yz 
-4 * 2 3412 
-43.7554 
47.9895 
43'7554 
4.23412 
-47.9895 
-46.9489 
44- 4170 
2.53189 
-44.4170 
46.9489 
-2053189 
* 
S o l u t i o n  found by P r o f e s s o r  Whitman u s i n g  i t e r a t i v e  numer ica l  methods. 
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These twelve solutions are permutations of two different sets of spin 
parameters. The first six solutions correspond to permutations of one 
set of spin parameters, while the last six are permutations of the second 
set. 
served spectral line positions. However, the two sets yield different 
intensities. In this particular problem the second set, which was found 
by Professor Whitman, more closely matches the experimental intensities. 
Using double-precision arithmetic, the total computing time was 535 
seconds. The maximum n b e r  of list cells employed simultaneously during 
the computation was 1064. Together with other auxiliary data structures, 
the peak number of memory words employed for data storage was about 8,000 
words. 
Both sets of spin parameters are in precise agreement w i t h  the ob- 
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4. SOLUTION BY SUPPLEMENTARY TECHNIQUES COMBINED WITH ELIMINATTON 
A second more efficient method of solution made use of the 
supplementary techniques of grouping and partitioning, The steps of the 
second solution are now described in detail. 
the equations become: 
After eliminating x1 and x2, 
The symmetry of Yi dnd y2 in the subsystem (2~1, (2d) and (2e) suggested the 
following groups: 
* 
Introducing P and Q into (2c), 
( 3 )  to (2) yields the following partitionable system. 
(2d) and (2e) by hand and appending System 
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-x32(Q2 - 3P) --[2cl(Q2 9 - 3P) + c ~ Q I  
2 
3 2  + 
+ 6(Q2  - 4P) (Q2 - 3P) 
+ c ~ c ~ Q  - Cl2(2$ - 9 ? ) ]  
9X3P + c1(2Q2 - 9P) - C2Q 
2Q(9P - 2Q2) I = Y1Y2 
Q =: Y i  + Y 2  
= 9c3(Q2 - 4P) 
= c 4  
= c5 
(4b 1 
(4c) 
The subsystem (4b) can be solved by e l imina t ing  f i r s t  x3 and then P t o  
obtain a 12th degree polynomial i n  Q. 
so lu t ions ,  each of which g ives  rise t o  a (x3, P, Q) so lu t ion  t o  (4b) .  
Each of t hese  sets when subs t i t u t ed  i n t o  (4c) y i e l d s  a p a i r  of(x3,  y l ,  
y2,P,  Q) so lu t ions  to  (4b,c).  Each of these  (x3, y l ,  y2, P, Q) so lu t ions  
when s u b s t i t u t e d  i n t o  (4a) y i e lds  a ( X I ,  ~ 2 ,  x3, y l ,  y2,  P, Q )  solut ion 
f o r  t h e  e n t i r e  problem ( 4 ) .  A l l  twelve of t hese  so lu t ions  s a t i s f y  (1) and 
are i n  agreement with t h e  so lu t ions  of  T a b l e  I t o  a t  least seven s i g n i f i -  
cant f i g u r e s  . 
This polynomial i n  Q has  s i x  r e a l  
This method of so lu t ion ,  employing t h e  supplementary techniques of 
grouping and p a r t i t i o n i n g  combined with e l imina t ion ,  required only 161 
seconds of computing t i m e  and a peak number of l ist  cel ls  of 214. 
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CONCLUDING REMARKS 
From t h e  s tandpoin t  of NMR spectroscopy, this c a p a b i l i t y  of  f ind ing  
a l l  so lu t ions ,  i n  this case a "double" so lu t ion ,  is of considerable  i m -  
portance, s ince  many workers have neglected t h e  p o s s i b i l i t y  of  mul t ip le  
so lu t ions .  
among o t h e r  th ings ,  be of s i g n i f i c a n t  practical  value to  NMA spectrosco- 
p i s t s ,  s ince  it w i l l  permit t h e  ana lys i s  of NMR s p e c t r a  t o  be performed 
unambiguously f o r  ABC, ABCD, e t c . ,  systems. I t  should be emphasized tha t  
the method provides  e s s e n t i a l l y  the  only means of determining t h e  unique- 
ness ,  and thus  the  r e l i a b i l i t y ,  of spectral analyses  f o r  such systems. 
The method of so lu t ion  being developed i n  t h i s  research w i l l ,  
The example presented i s  r ep resen ta t ive  of t h e  capabilities of t h e  
second generat ion computer program as it presen t ly  e x i s t s ;  approximately 
fou r  nonl inear  va r i ab le s  can be handled i n  t y p i c a l  cases by using elimina- 
t i o n  only. This example a l s o  serves  t o  i l l u s t r a t e  t he  usefu lness  of 
supplementary techniques,  which can increase  the  size of problems t h a t  can 
be handled. The mos t  cr i t ical  computing factor is  memory space. Propa- 
ga t ion  of round-off errors and computing time do no t  appear t o  be as cri t i-  
cal. 
of l a r g e r  memories w i l l  permit somewhat l a r g e r  problems t o  be treated i n  
t h e  fu ture .  
Fur ther  development o f  supplementary techniques and t h e  a v a i l a b i l i t y  
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PERSONNEL 
Personnel associated with the research during the last  six-month 
period include Professor R. J. Adler and graduate students S. Y .  Ku, 
C. G. Sperling and W. R. Brown. M r .  Ku is  a Ph.D. candidate and 
MeSSrS. Sperling and Brown are M.S. candidates. All three of these men 
w i l l  be writ ing theses during 1968. 
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REFERENCES 
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1. Progress Reports on A METHOD OF SOLVING SETS OF NONLINEAR ALGEBRAIC 
EQUATIONS (Grant No. NGR-36-003-021). 
No. 1 by R. J. Adler and G. E. Haynam (September, 1965) 
No. 2 by R. J. Adler and E. J. Emmer (April, 1966) 
No. 3 by R. J. Adler and S. Y. Ku (November, 1966) 
No. 4 by R. J. Adler and S. Y. Ku (July, 1967) 
2. Whitman, D. R., "Computer Assignment Technique for Analysis of High- 
Resolution NMR Spectra," J. Chem. Phys., 36, (No. 8, 19621, 2085- 
2093. 
3. Whitman, D. R., "Direct Analysis of NMR Spectra, 'I J. Mol. Spect., 10, 
(April, 1963), 250-262. 
4. Renewal Proposal for Research on A METHOD OF SOLVING SETS OF NON- 
LINEAR ALGEBRAIC EQUATIONS (Case Institute of Technology No. E 032566, 
Principal Investigator, R. J. Adler) (December, 1965). 
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A method of solving sets of nonlinear algebraic equations Progress report

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