Re: Solve--Solutions
- To: mathgroup at yoda.physics.unc.edu
- Subject: Re: Solve--Solutions
- From: danl (Daniel Lichtblau)
- Date: Wed, 9 Mar 1994 13:08:42 -0600
>Begin forwarded message:
>Date: Wed, 9 Mar 94 08:16:02 PST
>From: fateman at peoplesparc.Berkeley.EDU (Richard J Fateman)
>To: danl at wri.com, mathgroup at yoda.physics.unc.edu
>Subject: Re: Solve--Solutions
>Actually, I'd suggest that WRI's version of the cubic formula
>is defective and provides unnecessary "I" values. This bug
>was corrected in Macsyma, as I recall, about 20 years ago.
----------------------------
Malarkey.
We start with Mathematica. I use the problem posted in the original
note in all examples. Mathematica gives an answer that contains 10
instnaces of Sqrt[-1].
In[1]:= sol = Solve[11*x^3 - 20*x^2 - 10*x + 22 == 0, x] // InputForm
Out[1]//InputForm=
{{x -> 20/33 + (730*2^(1/3))/(33*(-36074 + 66*I*58479^(1/2))^(1/3))
+
(-36074 + 66*I*58479^(1/2))^(1/3)/(33*2^(1/3))},
{x -> 20/33 - (365*2^(1/3)*(1 + I*3^(1/2)))/
(33*(-36074 + 66*I*58479^(1/2))^(1/3)) -
((1 - I*3^(1/2))*(-36074 +
66*I*58479^(1/2))^(1/3))/(66*2^(1/3))},
{x -> 20/33 - (365*2^(1/3)*(1 - I*3^(1/2)))/
(33*(-36074 + 66*I*58479^(1/2))^(1/3)) -
((1 + I*3^(1/2))*(-36074 +
66*I*58479^(1/2))^(1/3))/(66*2^(1/3))}}
In[2]:= N[sol]
Out[2]//InputForm=
{{x -> 1.61319327207777 + 4.440892098500627*10^-16*I},
{x -> -1.015665880292489 - 5.551115123125783*10^-17*I},
{x -> 1.220654426396537 - 4.440892098500627*10^-16*I}}
Here is what macsyma has to say. This also contains 10 instances of
Sqrt[-1]. Took me over an hour with the manual to get some rough
equivalent to the N[] function.
mosquito [2] danl> macsyma
Starting MACSYMA without a window system...
This is Macsyma 417.100 for SPARC Series Computers.
Copyright (c) 1982 Massachusetts Institute of Technology.
All Rights Reserved.
Enhancements (c) 1982, 1991 Symbolics, Inc. All Rights Reserved.
Type "DESCRIBE(TRADE_SECRET);" to see important legal notices.
Type "HELP();" for more information.
Checking password file:
/usr-local/macsyma_417/system/passwd-50383386-417.text
(C1) display2d : false;
(D1) FALSE
(C2) sol : solve(11*x^3 - 20*x^2 - 10*x + 22 = 0, x);
(D2) [X =
(-SQRT(3)*%I/2-1/2)*(SQRT(19493)*%I/(363*SQRT(3))-18037/35937)^(1/3)+
730*
(SQRT(3)*%I/2-1/2)/(1089*(SQRT(19493)*%I/(363*SQRT(3))-18037/35937)^(
1/3))
+20/33,X =
(SQRT(3)*%I/2-1/2)*(SQRT(19493)*%I/(363*SQRT(3))-18037/35937)^(1/3)
+730*(-SQRT(3)*%I/2-1/2)/(1089*(SQRT(19493)*%I/(363*SQRT(3))-18037/35
937)^(1/3
))+20/33,X =
(SQRT(19493)*%I/(363*SQRT(3))-18037/35937)^(1/3)+730/(1089*(SQRT
(19493)*%I/(363*SQRT(3))-18037/35937)^(1/3))+20/33]
(C3) nsol : dfloat(sol);
(D3) [X =
0.6703397612488522*(0.8660254037844386*%I-0.5)/(0.2220608965979688*%I
-0.5019061134763614)^(1/3)+(-0.8660254037844386*%I-0.5)*(0.2220608965
979688*%I
-0.5019061134763614)^(1/3)+0.6060606060606061,X =
(0.2220608965979688*%I
-0.5019061134763614)^(1/3)*(0.8660254037844386*%I-0.5)+0.670339761248
8522*(
-0.8660254037844386*%I-0.5)/(0.2220608965979688*%I-0.5019061134763614
)^(1/3)
+0.6060606060606061,X =
(0.2220608965979688*%I-0.5019061134763614)^(1/3)
+0.6703397612488522/(0.2220608965979688*%I-0.5019061134763614)^(1/3)
+0.6060606060606061]
(C4) really_truly_nsol : maplist(rectform, nsol);
(D4) [X = 1.220654426396538,X =
-2.775557561562891E-17*%I-1.01566588029249,X =
1.61319327207777]
(C5) quit();
Bye.
mosquito [3] danl>
You want a third opinion? I count 12 instances of I this time.
elvis [3] danl> maple
|\^/| Maple V Release 2 (Wolfram Research Inc.)
._|\| |/|_. Copyright (c) 1981-1992 by the University of Waterloo.
\ MAPLE / All rights reserved. MAPLE is a registered trademark of
<____ ____> Waterloo Maple Software.
| Type ? for help.
> sol := solve(11*x^3 - 20*x^2 - 10*x + 22 = 0, x):
> lprint(sol);
(-18037/35937+1/1089*I*19493^(1/2)*3^(1/2))^(1/3)+730/1089/(-18037/35
937+1/
1089*I*19493^(1/2)*3^(1/2))^(1/3)+20/33
-1/2*(-18037/35937+1/1089*I*19493^(1/
2)*3^(1/2))^(1/3)-365/1089/(-18037/35937+1/1089*I*19493^(1/2)*3^(1/2)
)^(1/3)+20
/33+1/2*I*3^(1/2)*((-18037/35937+1/1089*I*19493^(1/2)*3^(1/2))^(1/3)-
730/1089/(
-18037/35937+1/1089*I*19493^(1/2)*3^(1/2))^(1/3))
-1/2*(-18037/35937+1/1089*I
*19493^(1/2)*3^(1/2))^(1/3)-365/1089/(-18037/35937+1/1089*I*19493^(1/
2)*3^(1/2)
)^(1/3)+20/33-1/2*I*3^(1/2)*((-18037/35937+1/1089*I*19493^(1/2)*3^(1/
2))^(1/3)-
730/1089/(-18037/35937+1/1089*I*19493^(1/2)*3^(1/2))^(1/3))
> nsol := evalf(sol):
> lprint(nsol);
1.613193272+1.0E-10*I -1.015665881 1.220654427-2.0E-10*I
> quit;
bytes used=502468, alloc=393144, time=1.63
--Dan'L