HOW TO SOLVE THE COMPLETE CUBIC BY THE METHOD OF 'BASES'

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP82-00039R000200120026-9
• Release Decision: RIPPUB
• Original Classification: R
• Document Page Count: 12
• Document Creation Date: December 22, 2016
• Document Release Date: April 25, 2012
• Sequence Number: 26
• Publication Date: August 29, 1952
• Content Type: REPORT

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STAT
-OW Ol_B he GOSH NL_ 8 ,e Cubic by the Mf ethod of $
:13. M. Shunnragskiy
Source; T'ablitsy dlya Resheniya Kubicheskikh Uravneniy
Metodom Osnov /ab1es for the Solution of Cubic
Equations by the Method of Bases70
Moscow/Leningrad: 1950?
STAT
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6-9
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-CTED
~AITIrINF011MAilUP~
HGJTO VE THE COMPLETE, CUB, G BY T1, ETHQD OF ' 1 A$ ES'
B. M. 5hunyagskiy
tionaame from a small hard-back book entitled
1'he fa~.low~.ng a.n~ orms, ~,o~,u..
~ote: r
heskikh Uravneniy Metodom UsnoV' /abler for the
I''abla.tsy dlya Resheria.ya l~uba.c
' thod of Bases7, by B. M. Shumyagskiy; State Press
~tion of Cubic 'I~a~ua~,ions by the Me .~
Literature, M0Q/Leningrad: 190.
of Tec~~n:~ca~.~Theor, eta.ca~. lanatian
The information consists of the book's (Contents', 'Foreword,, and I~p
of T abies' wa.th a sample of the tablea?
~
Contents
Foreword
i xplanation of the tables
Table I. Values of Z and bt for A
1
from ..0.000,000,008, 0 to 300
Table II. Values of Zl and a for A
from -0,000,000.008 to ~07
Table IIIa. Values of Zl for A
from -bp7~ to -300
Table Irub. Values of Z2 for A
from -67 to ..300
Foreword
nragellt tables are intended for the solution of complete cubic
mb,r~
pages
3
L. l2
13''30
3l~70
7l-l2L
l2-l3~
equations by
reduction to the trinomial equations of special form.
,ay their
The idea oz v~1J
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RESTRICT,
thad is contained in the author's article IIIschjsleniyQ Shayrash" +TCal.cul.us af'
me
ski ~boxn~.k ~athematic~1. Symposium Volume
+Sho rush+7, pulalisl~Qd f,n Mater~atial7e
y
e sh ' is an Ashkenazf,c Hebrew word or 'root' J
J~a' C~93a), 3. Note: Shayra
of these tables all computations will be
It is assumed that during employment
. but in cases not requiring great accuracy compute
conducted an the +rithmometer?,
Lions can be done on a slide rule. for remarks, instruction, and advice, which
The author Will be very grateful fo any
should be directed to the address: Moscow, 4rlikov 3, Gostekhizdat.
p1iOfl o f the Tables
Section 1.
In these tables the roots of the equation
(1)
z3+A.-A=C
are neater detail cancerning the properties of
liven for various values of A. For ~
,~ ? A = p, see the author's article enta.tled
trinomial equations of the type
+Shayrosli+++,'~7, published in Maematicheski,
+1lschisleniye Shoyrosh~~ "Calculus of - hlcenaz~ic
? me (1930, 3? Note; +Shoyrosh is As
Sbarnik Mathematical Sympas~.urn?], Volume
-r--" roots which we eill
r each value of A in the tables correspond three
for ? raot+~,~ ~o s bol
r bases ('roots') of A and designate by the Ym
call the th~.rdworde ?,,,,,,,,,,,,,, .
(2)
Section 2. The Contents of the Tables
Table (l ), corresponding to positive;
. I coma.ens the bases (roots) of equation
?oralues of A from 8.l0 to 300? To each value of A corresponds one ree].. basis
( toot) and bases froots).
.. m lex#con~ugate
twa co p
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 LI
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RESTRICTE,
In the table are given the values of the real basis (root), deeignated bye,
and the vaJ.uea of the quotient fromividing the absolute value of the coefficient
of i (the complex roots by the absolute value of the real root
~.mag~.nary' part) o~ the zl, this quotient is designated in the table by alpha ' p
of z is known, then the value z is expressed thus:
II' the value 1 2,3
:MM: - L;A
3 2 = 1 1
The preceding equation follows from the fact that the sum of all three roots of
equation (1) equal 0.
In Table II are given the bases ('roots') of negative values of A greater than
?6.7~ to the number -0.000,0,00,008 inclusively.
To these values of A also correspond one real negative basis (roots and two complex
Just as in Table I, Table IT also gives the real negative
conjugate bases (roots).
I
bas es ( root) z and value of (fit ( as above : 2.
1
s' negative values of A from -6.7 to .-300. To these values of
Table III contaan
A correspond three real bases (roots). one negative and two positive.
In the table the negative basis (root) is designated by zl, and the smaller
positive basis (root) is designated by z2.
the table consists of two parts: Table xila contains values of
Correspondingly,
z-1 (negative), and Table Txlb contains values of z2 (positive). The third root is
found from the formula z -
distribution of roots with respect to the tables is illustrated by the
The
diagram:
S'
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STi
Table k ~ Table
:ii: L
~F
s
r.r
A
A $ ? / 1 ?'Y ?I T l.
wA .? ww4 .rnl wr +----
The tab/es give accuracy to four decimal places for the real bases (roots) and
an
to three decimal places or complex bases (root).
~~rst case and the 1~th place for the second case
The lath and nth places for thef ~.
In most of the cases the yntexpolata.an p
are ermits
found by linear ~.nterpalatiox`Y? 1 l3,near
the reap. basis (root). The pages where
one to find ~ accurate places of
sis (root) are
interpolation gives only tour places (but not five) for the real ba
er corner off' the pages
d with an asterisk placed in the left upp
note
Section 3. Solution of the Complete Cubic ]quation
,
bic e nation and a~'tex~rards the
First we shall show how to solve the trinorUia1 cu c
complete cubic.
THE equation (~
y +py~ q"a
to an equation of the form (1)
can be reduced
ICT
~~rrlrry~~ /r 1
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to do th?s it is necessary to set in equation (Li)
y m .? q,!/p (S)
We then obtain
P'1
The equation obtained is an equation of form
In our symbols the equation Ls solved:
y
Let the following equation be given
x3+ax2*bx+c~0
(1),
The equation given will be solved in this order;
1. Eliminating the second term (ax2) ira this equation (7), we reduce it to
(b)
('7 )
the form (14)e
2. The equation of form (14) is reduced to an equation of the form (1) and solved
in accordance with formula (6).
Setting in equation (7)
x = y - a/3,
we obtain
y3 + (b
3
ab
3
this is an equation of form (Ii.), in which we have
3
2a3 ab
lEST
(g)
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in which A p3/q2.
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,rtoyo1 }l~u1)x
setting in equat on (8) y " q /p, we obtain
1) V ? M "'' I~~ .v may, .Liic':1 soiJ3 --
Wr'
,Mnn-h,n?,ria?ruyp.N~.~~.M1~Mi~. ,i
1 e L 'rh,
,.__t_,, C23_M
(9)
thus we have arrived at equation (1), in which
y
L2 ( 312
wMrr -n.-NW,NwWAnw,ArNwI,__wMn M1wmn+~M~^~" '
,_c1 " ' c
s~.;~ (root) in equation (9) be known to uss Lot US
Now let the value of the basis
find the value of the root of the original equation (7).
We have
y? qZ/A,
using our symbol, we can write,
A
3(b
Ww.WYI~AMMiJMN1.11`~yMLA+NI~M1V.1
:9i b Nv~1c)~
( ar
by s ettin
and
< raww
p,Wp"quen~4ri/+r~~MOMShr.An~hY,Yret.4e, n
~~~2 r
3 t
For b 0, we obtain
w.W~
For a 0, we obtain
1 I~,~J' ~1M' V 0 t)]9
a
_4_.%_,___rw_+.on.n. ,+wx?.~r.t.,rM+w.hn,^MYl.w,uiww
Mrt'w
f:&~91~ n+.r..9 Vl -F 21
y ~WM,IWxiYWYM41..MR,~'~
~~ /,w 1~ ~N1nwYWnw+1
I N?
wNw, y("/4 r
as f oll.ows
(10)
(13)
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 we get
T
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an equation of the form mx + may + k" o,
For L Y~
k2rri
Thus the s olution of a cubic equation of farm
an a qu ation of the f oxm
2
z
ba;~es (root a) of' (namely, za.' z2' ~) are
ii: A ~ ~ 6..7, then all three
CTED
(7
reduces to the aa~?ution of
whose roots we shall call the tkaix'd-a1^der bases (' roots ' ) of A.
Let us consider the two cases separately.
1. A be7
A Mb .7
In
according to one of the formulas (12)(15) (de
;r.
from the tables. Further, we find a
'on) the throe roots of the original equation (7).
pending; uponthe form of the eauatl
the bases ('roots') zis real, but the other two
If 6a7~, then one of ~,
we first find the real rant xl of equation
Vases are camplex~conjugate. In this case
in he same order as in the first case.
(7)
In order to find the comple roots of equation (7) we use farina (3)
x
read.. formula 10); `then we find ~l,
se we first find A according to
th s c a
z = ~l iz1DC
?,3 2
Hence, according to fornn La () we have:
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(1)
(16)
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TRICT
wan -
r,r1.IP
?aMl.
M'
,wrlwW Mlrr.ww",.'I^'Wi"", ywMn.
-A-()
formulas:
IN~AyMyl/MYMMIn11MiYWl,wMNwMJr~
2-?
AMP9C::)
t +
Finally,
z
I 6.7, to Table x (a~f A ~ a) or Table II (~.~
then we arnd, according .
- 6.7~ 0) the values of and ad r a c ?
< ~ ~ ~ find the values
of equation (7). Further, we
From Z we find the first root (xl)
1 ~
x2, ccording to foxmula (16) or (1r ). Thus;
of y2~3 a 3
For
10
for equation y3 * py + q 0,
we have y2a3 ., yl/2 . iyl cx
for equation x3 + ax2 * bx * c 0,
x:a : :t (x 4 i)(
we have x2 3 ,
~ 2
., roximate
00 or /A / '< 8 10 we can use the ollotidng app
/A/,~ 3
+L
A +L
0
If A > 300, then we haves
20 if A < in the
imaginary' part of the complex roots. We find three places immediately from the
The remaining places are found by linear interpolation:
table.
oe 210L.1
0, 202
Qob9
o7 8O0
69
o47` 69
1.2706
i5. o.69 ....,..L
1x2916
v@ue,of,the roots, the complex being found according to
Now we findthe.... .
formula (16); , . , :
\~}('~1~ ~WNrIlM7
P yyil/NMy
V~ ~tNWNI 1
C0'
r
0
(with an accuracy
to fifth place);
1 00998 8 1.0.998 $ t<
2
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200120026-9
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~sTwcr~u
0.49929 :1: i' 0?99$58? 1.2916
0QLL9929 ? i.1?2897
/Note: Three other examples are given in the original; namely:
716 x3 - S12x + !~67 = 0;
~.
x3-2x2 +x
5x+ 3=0,7
/~Exarnp1e of a page from Table Ij
Table I.
For A from 0.11682 to 0.23068 (p. 20)
A
d
z
(
A
d z 0<
1
1
0.11682
o.L.io
1.2020
0.16568
0.L5o 1.222
5
11787
10;
1
11787
Lj.11
2032
107
1189L
.
1412
20L
107
12001
L~13
2056
/ote:
Table II is similarto
108
table I in format.7
12109
141)4
2069
0.12218
0-L~15
1.2081
1109
of;.a.S
/etc; there are 6 x 8 L8 lines
like this per half side of page.7
Example of a page from Table IIIa.7
0w
T
C
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Wxlt~,1 4i( ." c
Table lila. For A from w 10.391.08 to ' 10.9Q6SL (p 80)
A
z1 A d1
100 39Lt08 s 3.t .0
633
10o61i.872 3.680
61
=
~t~ 3 there are b x 8 M !.8 etc; there are 6 x 8 L8
similar linese7 similar lin.es.7
Cote: Table, 11Th is like Table l la in 'ormat.7
-END
lCM~ ~
7
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