HP50g simplifing a root
10-09-2020, 02:31 PM (This post was last modified: 10-12-2020 06:30 AM by Albert Chan.)
Post: #20
 Albert Chan Senior Member Posts: 2,682 Joined: Jul 2018
RE: HP50g simplifing a root
(10-09-2020 12:20 AM)Albert Chan Wrote:  Unfortunately, our divisors based cube root simplify routines were flawed !
Real rational root for a is a divisor of n, *but* possibly divided by 4 ...

Another problem is we had assumed there is only 1 real root for a.
Lets rearrange the cubic to match form x³ + 3px - 2q = 0

c³ = A² - R
4*a³ = 3*c*a + A
a³ + 3*(-c/4)*a - 2*(A/8) = 0

→ Cubic discriminant = p³ + q² = (-c/4)³ + (A/8)² = (-c³ + A²) / 64 = R/64
→ If R < 0, we got 3 real roots.

see https://proofwiki.org/wiki/Cardano%27s_Formula

Quote:XCas> find_cbrt(81,30,-3) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → $$\frac{9}{2} + \frac{i}{2}\cdot\sqrt{3}$$

XCas> find_all_a(A,R) := solve(surd(A*A-R,3) = a*a - (A/a-a*a)/3 , a)

XCas> find_all_a(81, 30^2 * -3) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → [-3, -3/2, 9/2]
XCas> simplify( [-3+2i*sqrt(3), -3/2-5i/2*sqrt(3), 9/2+i/2*sqrt(3)] .^ 3)

$$\qquad\quad [81 + 30i\sqrt3\;,\; 81 + 30i\sqrt3\;,\; 81 + 30i\sqrt3$$

If we consider integer as simplest, then $$\sqrt[3]{81 + 30i\sqrt3} = -3+2i\sqrt3$$

Comment: I was wrong on above example.

x³ = y does not imply ³√y = x. We need to consider branch-cut
In other words, simplified form should match numerical evaluated values.

>>> (81 + 30j * 3**0.5) ** (1/3)
(4.5+0.86602540378443837j)
 « Next Oldest | Next Newest »

 Messages In This Thread HP50g simplifing a root - peacecalc - 09-29-2020, 09:22 PM RE: HP50g simplifing a root - Albert Chan - 09-29-2020, 11:47 PM RE: HP50g simplifing a root - Albert Chan - 09-30-2020, 02:22 AM RE: HP50g simplifing a root - Albert Chan - 09-30-2020, 10:50 PM RE: HP50g simplifing a root - Albert Chan - 10-01-2020, 07:31 AM RE: HP50g simplifing a root - peacecalc - 09-30-2020, 05:33 AM RE: HP50g simplifing a root - peacecalc - 10-01-2020, 02:20 PM RE: HP50g simplifing a root - Albert Chan - 10-01-2020, 05:22 PM RE: HP50g simplifing a root - peacecalc - 10-04-2020, 06:05 PM RE: HP50g simplifing a root - Albert Chan - 10-04-2020, 11:48 PM RE: HP50g simplifing a root - peacecalc - 10-04-2020, 07:36 PM RE: HP50g simplifing a root - peacecalc - 10-05-2020, 11:36 AM RE: HP50g simplifing a root - Albert Chan - 10-05-2020, 05:01 PM RE: HP50g simplifing a root - peacecalc - 10-06-2020, 05:25 AM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 09:40 AM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 12:06 PM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 04:13 PM RE: HP50g simplifing a root - Albert Chan - 10-07-2020, 06:12 PM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 12:20 AM RE: HP50g simplifing a root - Albert Chan - 10-09-2020 02:31 PM RE: HP50g simplifing a root - Albert Chan - 10-11-2020, 06:28 PM RE: HP50g simplifing a root - Albert Chan - 10-12-2020, 03:17 AM RE: HP50g simplifing a root - Albert Chan - 10-24-2020, 02:19 PM RE: HP50g simplifing a root - Albert Chan - 10-12-2020, 10:54 PM RE: HP50g simplifing a root - CMarangon - 10-12-2020, 11:45 PM RE: HP50g simplifing a root - grsbanks - 10-13-2020, 06:46 AM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 05:21 PM RE: HP50g simplifing a root - Albert Chan - 10-10-2020, 03:58 PM RE: HP50g simplifing a root - Albert Chan - 10-10-2020, 04:49 PM RE: HP50g simplifing a root - peacecalc - 10-12-2020, 08:49 PM RE: HP50g simplifing a root - peacecalc - 10-13-2020, 06:30 AM RE: HP50g simplifing a root - peacecalc - 10-13-2020, 06:36 AM

User(s) browsing this thread: 1 Guest(s)