 (12C Platinum) Cubic Equation - Printable Version +- HP Forums (https://www.hpmuseum.org/forum) +-- Forum: HP Software Libraries (/forum-10.html) +--- Forum: General Software Library (/forum-13.html) +--- Thread: (12C Platinum) Cubic Equation (/thread-12164.html) (12C Platinum) Cubic Equation - Gamo - 01-12-2019 09:27 AM ALG mode program solution of a Cubic Equation by Newton's Method. f(x) = aX^3 + bX^2 + cX + d = 0 Successive approximations to a root are found by Xi+1 = 2aXi^3 + bXi^2 -d / 3aXi^2 + 2bXi + c Guess X0 ------------------------------------------------------ Remark: This program is use to solve for "REAL ROOT" ------------------------------------------------------- Procedure: f PRGM // Each new program or GTO 000 a [R/S] b [R/S] c [R/S] d [R/S] X0 [R/S] Display shown each successive approximation until root is found. If more than one Real Solutions enter another guess and [R/S] Maximum of 3 Real Root. ------------------------------------------------------- Example: x^3 - 4x^2 + 6x - 24 = 0 f [PRGM] or [GTO] 000 1 [R/S] 4 [CHS] [R/S] 6 [R/S] 24 [CHS] [R/S] 20 [R/S] // My starting guess Display successive approximation search and stop when root is found. Answer Display 4 X=4 --------------------------------------------- -2x^3 + 3x^2 + 4x - 5 = 0 f [PRGM] or [GTO] 000 2 [CHS] [R/S] 3 [R/S] 4 [R/S] 5 [CHS] [R/S] 10 [R/S] ...............display 1.8508 0 [R/S] .................display 1 5 [CHS] [R/S] ..........display -1.3508 ----------------------------------------------- Program: ALG Mode Code:  STO 0 R/S STO 1 R/S STO 2 R/S STO 3 R/S STO 4 x 2 x RCL 0 + RCL 1 x RCL 4 X^2 - RCL 3 ÷ (RCL 4 x 3 x RCL 0 + (RCL 1 x 2) x RCL 4 + RCL 2) =  STO 5 - RCL 4 = X=0 GTO 049 RCL 5 PSE GTO 009 RCL 5 GTO 008 Gamo RE: (12C Platinum) Cubic Equation - Albert Chan - 01-12-2019 01:39 PM Hi Gamo It might be better if iteration formula is *not* simplified: X(i+1)= Xi - (aXi^3 + bXi^2 + cXi + d) / (3aXi^2 + 2bXi + c) Simplified form may introduce subtraction cancellation error on the *last* iteration. BTW, Professor Kahan had a systemetic way to get a good guess X0: https://apps.dtic.mil/dtic/tr/fulltext/u2/a206859.pdf, page 5 Using your examples: f(x) = x^3 - 4x^2 + 6x - 24 f(4/3) = -2.7475³, f'(4/3) = 2/3 ≥ 0 guess = 4/3 - (-1)(2.7475) = 4.0809 x = 4.0809 → 4.0023 → 4.0000 f(x) = -2x^3 + 3x^2 + 4x - 5 f(1/2)/-2 = 1.0772³, f'(1/2)/-2 = -1.6583² < 0 guess = 1/2 - 1.324718 (1) max(1.0772, 1.6583) = -1.6968 x = -1.6968 → -1.4145 → -1.3536 → -1.3508 Update: if we need the other roots, Kahan's algorithm suggested this: Deflate cubic to quadratic: a X² + e X + f = 0 IF |x³| > |d/a| THEN (f=-d/x, e=(f-c)/x) ELSE (e=ax+b, f=ex+c) RE: (12C Platinum) Cubic Equation - Albert Chan - 02-04-2019 04:10 PM Another way to get good guess is "look" at shape of curve Example 1: x^3 - 4x^2 + 6x - 24 = 0 x((x-2)² + 2) = 24 ~ 2.88³ Any guess > 2 work. Say, guess = 3 X = 3 -> 4.6667 -> 4.1220 -> 4.0051 -> 4.0000 Example 2: -2x^3 + 3x^2 + 4x - 5 = 0 x^3 - 1.5x^2 - 2x + 2.5 = 0 x((x-0.75)² - 2.5625) = -2.5 ~ -1.36³ If RHS is 0, we have roots -0.8508, 0, 2.3508, with LHS interval signs = - + - + Any guess < -0.8508 work. Say, guess = -1.5 X = -1.5 -> -1.3649 -> -1.3509 -> -1.3508 Since LHS interval (0, 2.3508) also is negative, eqn might have 3 real roots. Indeed it does, 3 roots = -1.3508, 1.0000, 1.8508 RE: (12C Platinum) Cubic Equation - Thomas Klemm - 02-04-2019 08:13 PM Or then use the built-in polynomial solver with this program: Code: 01-       1    1 02-      34    x<>y 03-      24    ∆% 04-45,43 31    RCL PSE 05-      34    x<>y 06-      33    R↓ 07-      25    % 08-      40    + Examples: $$x^3 - 4x^2 + 6x - 24 = 0$$ Enter coefficients: 1 CF0 -4 CFj 6 CFj -24 CFj Enter guess: 20 R/S 4.0000 $$-2x^3 + 3x^2 + 4x - 5 = 0$$ Enter coefficients: -2 CF0 3 CFj 4 CFj -5 CFj Enter guess: 5 R/S 1.8508 Enter guess: 0.5 R/S 1.0000 The advantage is that the same program can be used for higher order polynomials. Cheers Thomas PS: Cf. HP-12C’s Serendipitous Solver RE: (12C Platinum) Cubic Equation - Gamo - 02-05-2019 04:24 AM Thanks Thomas Klemm The build in Polynomial Solver work very good. On Original 12C work well but On 12C Platinum need to adjust guesses differently. Your second example with two real roots. First guess 5 R/S display 1 Second guess 0.5 R/S display 1 Remark: second guess needed to be 3 to get answer For 1.8508 Can the guess be negative integer? I try -1 and Error 5 shown. Gamo RE: (12C Platinum) Cubic Equation - Thomas Klemm - 02-05-2019 06:09 AM (02-05-2019 04:24 AM)Gamo Wrote:  Can the guess be negative integer? It appears that you can't solve for negative roots. But you can just do a transformation: $$x \rightarrow -x$$ This transforms the equation to: $$2x^3 + 3x^2 - 4x - 5 = 0$$ Enter coefficients: 2 CF0 3 CFj -4 CFj -5 CFj Enter guess: 1 R/S 1.3508 And then transform the result back again to get the solution of the original equation: -1.3508 Cheers Thomas RE: (12C Platinum) Cubic Equation - Thomas Klemm - 02-05-2019 06:20 AM (02-05-2019 04:24 AM)Gamo Wrote:  Remark: second guess needed to be 3 to get answer For 1.8508 That's interesting. Overshooting this solution happens only with initial guess 10. So it seems that the algorithm is implemented differently. RE: (12C Platinum) Cubic Equation - Albert Chan - 02-05-2019 10:24 PM (02-04-2019 08:13 PM)Thomas Klemm Wrote:  $$x^3 - 4x^2 + 6x - 24 = 0$$ Enter coefficients: 1 CF0 -4 CFj 6 CFj -24 CFj Enter guess: 20 R/S 4.0000 Above entered coefficients order are good for root finding, and to evaluate polynomial. Example, with above entered polynomial, calculate f(X = 12.34) 1 Enter 12.34 [Δ%] [i] ; showed rate of 1134%, X = 1 + i NPV 0 PMT FV CHS ; showed 1320.018504 <-- f(X) Confirm with Horners rule: 12.34 Enter Enter Enter RCL 0 * RCL 1 + * RCL 2 + * RCL 3 + ; showed 1320.018504 Doing the same with reversed ordered entry is not as accurate: 24 CHS CF0 6 CFj 4 CHS CFj 1 CFj 1 Enter 12.34 [1/x] [Δ%] [i] ; showed rate of -91.89627229%, X = 1/(1 + i) NPV ; showed 1320.018507, +3 ULP error