BisecnewtP3 - 4 Solvers for the hp15c with single entry of guesses - J E Patterson

J E Patterson - jepspectro.com - 20231119

Description

The four solvers in this program have their own particular strengths. GSB A Enters up to two guesses. GSB B runs a Bisection Solver. GSB C runs a Secant Solver. GSB D runs a Newton-Raphson Solver. GSB E runs SOLVE, the built-in solver. Guesses only need to be entered once to run alternative solvers.

Using the hp-15c Simulator by Torsten Manz, solutions to most root finding problems can be found. In this version, guesses are saved for use with alternative solvers.

Bisection Solver

GSB A saves guesses from the stack for use with any of the solvers in this program. Guesses don't need to be re-entered when trying another solver. They are recalled from memory instead.
GSB B runs a bisection solver which acts on f(x) = 0 under label 0.
This program was written to test the hp-15c Simulator by Torsten Manz.

Label 0 can hold any equation of the form f(x) = 0. Note that some of the hp-15C Owner's Handbook examples may require some extra ENTER statements at the beginning as the stack is expected to be filled with x. The open box example on page 189 requires three ENTER statements at the beginning.

A problem from the hp-15c Owner's Handbook - page 189:

A 4 decimetres by 8 decimetre metal sheet is available, i.e. 400 mm by 800 mm.
The box should hold a volume V of 7.5 cubic decimetres, i.e. 7.5 litres.
How should the metal be folded for the tallest box in decimetres.
We are using decimetres rather than mm because equation entry is simplified.
Let x be the height.
Volume V = (8 - 2x)(4-2x)x. There are two sides and two ends of height x.
Rearrange to f(x) = 4((x - 6)x + 8)x - V = 0 and solve for the height x.

Instructions:

There are 3 solutions depending on the guesses.
Initial guesses can be recovered with RCL 1 and RCL 2

0 ENTER 1 GSB A GSB B gives x = 0.2974 decimetres or 29.74 mm - a flat box.
1 ENTER 2 GSB A GSB B gives x = 1.5 decimetres or 150 mm - a reasonable height box.
2 ENTER 3 GSB A GSB B can't find a root as there are none in this interval. Press any key to stop.
3 ENTER 4 GSB A GSB B can't find a root as there are none in this interval. Press any key to stop.
4 ENTER 5 GSB A GSB B gives x = 4.206 decimetres or 420.26 mm - an impossible box height.

Bisection Solver Procedure:

Enter two guesses x1 and x0.
Do
Set flag 1
Calculate f(x0)
Let x2 = (x1 - x0)/2
Calculate f(x2)
If f(x2)*f(x0) < 0 let x1 = x2, clear flag 1
If flag 1 is set let x0 = x2
Loop until f(x2) = 0 within an error tolerance determined by the FIX, SCI or ENG significant digits setting.
Display root x2.

Notes:

Choose guesses which bracket the root of interest.
There 3 roots of which two are useful.
The practical range, given the available metal, is 0 to 8 decimetres.

Secant Solver

GSB A saves guesses from the stack for use with any of the solvers in this program. Guesses don't need to be re-entered when trying another solver. They are recalled from memory instead.
GSB C runs a secant solver which acts on f(x) = 0 under label 0.

RAN# is substituted for f(x0)-f(x1) if a divide by zero error would occur. If the equation to be solved has a square root term an error will occur for negative inputs. Use g TEST 2, 0 as the first two equation program statements. This is a test for a negative input and sets it to zero as the lowest valid input. Negative inputs may occur during the iteration.

Instructions:

There are 3 solutions depending on the guesses.
Initial guesses can be recovered with RCL 1 and RCL 2

0 ENTER 1 GSB A GSB C gives x = 0.2974 decimetres or 29.74 mm - a flat box.
1 ENTER 2 GSB A GSB C gives x = 1.5 decimetres or 150 mm - a reasonable height box.
2 ENTER 3 GSB A GSB C gives x = 1.5 decimetres or 150 mm - a reasonable height box.
3 ENTER 4 GSB A GSB C gives x = 4.2026 decimetres or 420.26 mm - outside the guess interval and an impossible box.
4 ENTER 2 GSB A GSB B points to a correct answer of x = 1.5 decimetres in spite of a potential divide by zero error.

Secant Solver Procedure:

Enter two guesses x0 and x1.
If x0 = x1 add 1E-7 to x1.
Set f(x0) = RAN# a non-zero, non-integer starting value to avoid possible divide by zero. f(x0) is updated by f(x1) after one iteration.
Do
Calculate f(x1)
Let x2 = x1 - f(x1)*(x0 - x1)/(f(x0) - f(x1))
if f(x0)-f(x1) = 0 then substitute RAN#
If f(x1) = 0 let root = x1, display x1 and exit
else
x0 = x1
f(x0) = f(x1)
x1 = x2
Loop until f(x1) = 0 within an error tolerance determined by the FIX, SCI or ENG significant digits setting.
Display root x1.

Newton Raphson solver

GSB A saves guesses from the stack for use with any of the solvers in this program. Guesses don't need to be re-entered when trying another solver. They are recalled from memory instead.
GSB D runs a Newton-Raphson solver which acts on f(x) = 0 under label E. Only one guess is required. If two guesses are entered only the last one is used.

Instructions

There are 3 solutions depending on the guesses.
The initial guess can be recovered with RCL 1

0 GSB A GSB D gives x = 0.2974 decimetres or 29.74 mm - a flat box
1 GSB A GSB D gives x = 1.5 decimetres or 150 mm - a reasonable height box.
2 GSB A GSB D gives x = 1.5 decimetres or 150 mm - a reasonable height box.
3 GSB A GSB D gives x = 0.2974 decimetres or 29.74 mm - a flat box. Here the secant line now intersects the x axis below the smallest root.
4 GSB A GSB D gives x = 4.2026 decimetres or 420.26 mm - an impossible box.

Newton-Raphson Solver Procedure:

Enter a guess x1
If two guesses are entered only the last one is used.
Do
Calculate f(x1)
If x1 = 0 use 1 instead to avoid a divide by zero from the derivative f'(x1)
define h = 1/10000
calculate f'(x1) ≈ (f(x1+h) - f(x1))/h
calculate f(x1)/f'(x1)
subtract from x1
Loop until f(x1) = 0 within an error tolerance determined by the FIX, SCI or ENG significant digits setting.
Display root x1.

SOLVE - the hp-15c solver

Instructions

There are 3 solutions depending on the guesses.
Initial guesses can be recovered with RCL 1 and RCL 2

0 ENTER 2 GSB A GSB E gives x = 0.2974 decimetres or 29.74 mm - a flat box.
1 ENTER 2 GSB A GSB E gives x = 1.5 decimetres or 150 mm - a reasonable height box.
2 ENTER 3 GSB A GSB E gives x = 1.5 decimetres or 150 mm - a reasonable height box.
3 ENTER 4 GSB A GSB E gives x = 4.2026 decimetres or 420.26 mm - outside the guess interval and an impossible box.

Notes:

A TEST=0 e^x statement in the Newton-Raphson solver program is a way of entering 1 without it attaching to the following EEX statement. TEST=0 10^x or TEST=0 COS could also be used. Also the obvious TEST=0 1 ENTER would also do.

There are useful discussions on Newton's solver for the hp-12C here and here.

SOLVEkey.pdf has a good explanation of the additional tricks used to solve difficult equations using the built-in Solver.

In SCI and ENG modes on the DM15 and a real hp-15C some roots are not always found. This is because RND is implemented slightly differently in the hp-15C Simulator. Just stop the iteration by pressing any key and examine the register where x is held. This can be done with RCL . 1. Alternatively change to FIX mode - e.g. FIX 4 before solving.

Guesses can be tested with GSB 0. For the Bisection solver look for a sign change. For the Secant solver with multiple roots try choosing both guesses to be on the same side of a root. This can alter the direction of the iteration. For the Newton-Raphson solver try moving the guess closer to the expected root, or to the other side.

Equations are entered under label 0. For example f(x) = 3x² - 5x + 1 = 0 is entered using GTO 0, changing to program mode with g P/R and deleting any entered function. X is placed on the stack twice as it is needed in two terms.

ENTER ENTER g x² 3 X X<>Y 5 X - 1 +

The roots are 0.2324 and 1.4343, using guesses 0,1 and 1,2.

An alternative form of this equation is: f(x) = x(3x-5)+1 = 0

The program is shorter: ENTER ENTER 3 X 5 - X 1 +.

Program Resources

Labels

Name	Description	Name	Description
A	Enter two guesses GSB A	1	Bisection loop updates x2 and either x0 to x1 or x1 to x0
B	Bisection solve routine	2	Bisection - Store x0 to x1 and x2 to x2
C	Secant solve routine	3	Bisection - Store x1 to x0 and x2 to x1
D	Newton-Raphson Solve routine	4	Secant - Iteration loop
E	SOLVE - internal solver	5	Newton - iteration loop
0	Formula to be Solved, f(x) = 0	6	If two guesses are the same, add 1E-7 to R1.

Storage Registers

Name	Description
1	Initial guess 1
2	Initial guess 2
.0	x0
.1	x1
.2	Bisection - x2 = (x0+x1)/2, Secant x0-x1, Newton - h
.3	Bisection - f(x0), Secant - f(x0)
.4	Bisection f(x2), Secant - f(x1), Newton - f(x1)

Flags

Number	Description
1	Bisection - If flag 1 is set x0 = x2 else x1 = x2

Program

Line	Display	Key Sequence	Line	Display	Key Sequence	Line	Display	Key Sequence
000			044	43 32	g RTN	088	10	÷
001	42,21,11	f LBL A	045	42,21,13	f LBL C	089	44 .2	STO . 2
002	44 .0	STO . 0	046	45 2	RCL 2	090	45,40, .1	RCL + . 1
003	44 2	STO 2	047	45 1	RCL 1	091	32 0	GSB 0
004	34	x↔y	048	42 36	f RAN #	092	45,30, .4	RCL − . 4
005	43,30, 5	g TEST x=y	049	44 .3	STO . 3	093	45,10, .2	RCL ÷ . 2
006	32 6	GSB 6	050	42,21, 4	f LBL 4	094	45 .4	RCL . 4
007	44 .1	STO . 1	051	45 .0	RCL . 0	095	34	x↔y
008	44 1	STO 1	052	45,30, .1	RCL − . 1	096	10	÷
009	43 32	g RTN	053	44 .2	STO . 2	097	44,30, .1	STO − . 1
010	42,21,12	f LBL B	054	45 .1	RCL . 1	098	45 .4	RCL . 4
011	45 2	RCL 2	055	32 0	GSB 0	099	43 34	g RND
012	45 1	RCL 1	056	44 .4	STO . 4	100	43,30, 0	g TEST x≠0
013	42,21, 1	f LBL 1	057	45 .1	RCL . 1	101	22 5	GTO 5
014	43, 4, 1	g SF 1	058	34	x↔y	102	45 .1	RCL . 1
015	45 .0	RCL . 0	059	45,20, .2	RCL × . 2	103	43 32	g RTN
016	32 0	GSB 0	060	45 .3	RCL . 3	104	42,21,15	f LBL E
017	44 .3	STO . 3	061	45,30, .4	RCL − . 4	105	45 2	RCL 2
018	45 .1	RCL . 1	062	43 20	g x=0	106	45 1	RCL 1
019	45,40, .0	RCL + . 0	063	42 36	f RAN #	107	42,10, 0	f SOLVE 0
020	2	2	064	10	÷	108	43 32	g RTN
021	10	÷	065	30	−	109	42,21, 6	f LBL 6
022	44 .2	STO . 2	066	45 .1	RCL . 1	110	26	EEX
023	32 0	GSB 0	067	44 .0	STO . 0	111	16	CHS
024	44 .4	STO . 4	068	34	x↔y	112	7	7
025	45,20, .3	RCL × . 3	069	44 .1	STO . 1	113	40	+
026	43,30, 2	g TEST x<0	070	45 .4	RCL . 4	114	43 32	g RTN
027	32 3	GSB 3	071	44 .3	STO . 3	115	42,21, 0	f LBL 0
028	43, 6, 1	g F? 1	072	43 34	g RND	116	36	ENTER
029	32 2	GSB 2	073	43,30, 0	g TEST x≠0	117	36	ENTER
030	45 .4	RCL . 4	074	22 4	GTO 4	118	36	ENTER
031	43 34	g RND	075	45 .1	RCL . 1	119	6	6
032	43,30, 0	g TEST x≠0	076	43 32	g RTN	120	30	−
033	22 1	GTO 1	077	42,21,14	f LBL D	121	20	×
034	45 .2	RCL . 2	078	45 2	RCL 2	122	8	8
035	43 32	g RTN	079	42,21, 5	f LBL 5	123	40	+
036	42,21, 2	f LBL 2	080	45 .1	RCL . 1	124	20	×
037	45 .2	RCL . 2	081	32 0	GSB 0	125	4	4
038	44 .0	STO . 0	082	44 .4	STO . 4	126	20	×
039	43 32	g RTN	083	45 .1	RCL . 1	127	7	7
040	42,21, 3	f LBL 3	084	43 20	g x=0	128	48	.
041	45 .2	RCL . 2	085	12	eˣ	129	5	5
042	44 .1	STO . 1	086	26	EEX	130	30	−
043	43, 5, 1	g CF 1	087	4	4	131	43 32	g RTN