BisecnewtP2 - 4 Solvers for the hp-15c - J E Patterson

Description

20191013

The four solvers in this program have their own particular strengths. SOLVE, the built-in solver, is now accessed with GSB D. Using the hp-15c Simulator by Torsten Manz, solutions to most root finding problems can be found.

Bisection Solver

GSB A runs a bisection solver which acts on f(x) = 0 under label E. This program was written to test the hp-15c Simulator by Torsten Manz.

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

Bisection Solver Procedure:

Choose 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.
This solver can find multiple roots in the example f(x) = 0, coded under label E.
0 ENTER 1 GSB A, Answer is 0.2974.
2 GSB A, Answer is 1.5.
3 GSB A, Answer is 1.5.
4 GSB A, answer is 1.5.
5 GSB A, answer is 4.2026.
There 3 roots of which two are useful.
The previous root is the new first guess which gets pushed onto the stack by the second guess.
If not enough roots are found choose smaller intervals or widen the search.
The practical range, given the available metal, is 0 to 8 decimetres.
Secant solvers will not behave so neatly.

Secant Solver

GSB B runs a secant solver which acts on f(x) = 0 under label E.

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

Secant Solver Procedure:

Choose two starting points 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 C runs a Newton-Raphson solver which acts on f(x) = 0 under label E. Only one guess is required.

Instructions

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

0 GSB C gives x = 0.2974 decimetres or 29.74 mm - a flat box
1 GSB C gives x = 1.5 decimetres or 150 mm - a reasonable height box.
2 GSB C gives x = 1.5 decimetres or 150 mm - a reasonable height box.
3 GSB C 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 C gives x = 4.2026 decimetres or 420.26 mm - an impossible box.

Newton-Raphson Solver Prcedure:

Choose a starting point x1
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

GSB D runs the internal hp-15c solver which acts on f(x) = 0 under label E. Two guesses are required. A modified Secant method is used.

Instructions

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

0 ENTER 1 GSB D gives x = 0.2974 decimetres or 29.74 mm - a flat box.
1 ENTER 2 GSB D gives x = 1.5 decimetres or 150 mm - a reasonable height box.
2 ENTER 3 GSB D gives x = 1.5 decimetres or 150 mm - a reasonable height box.
3 ENTER 4 GSB D gives x = 4.2026 decimetres or 420.26 mm - outside the guess interval and an impossible box.
4 ENTER 2 GSB B gives a correct answer of x = 1.5 decimetres which is also outside the guess interval.

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 E. 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 E. For example f(x) = 3x² - 5x + 1 = 0 is entered using GTO E, 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	Bisection solve routine - two guesses in x and y	1	Bisection loop updates x2 and either x0 to x1 or x1 to x0
B	Secant solve routine - two guesses in x and y	2	Bisection - Store x0 to x1 and x2 to x2
C	Newton-Raphson Solve routine - one guess in x	3	Bisection - Store x1 to x0 and x2 to x1
D	SOLVE - internal solver - two guesses in x and y	4	Secant - Iteration loop
E	Formula to be Solved, f(x) = 0	5	Newton - iteration loop
0	Enter two guesses	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	32 15	GSB E	088	45 .4	RCL . 4
001	42,21,11	f LBL A	045	44 .4	STO . 4	089	43 34	g RND
002	32 0	GSB 0	046	45 .1	RCL . 1	090	43,30, 0	g TEST x≠0
003	42,21, 1	f LBL 1	047	34	x↔y	091	22 5	GTO 5
004	43, 4, 1	g SF 1	048	45,20, .2	RCL × . 2	092	45 .1	RCL . 1
005	45 .0	RCL . 0	049	45 .3	RCL . 3	093	43 32	g RTN
006	32 15	GSB E	050	45,30, .4	RCL − . 4	094	42,21,14	f LBL D
007	44 .3	STO . 3	051	43 20	g x=0	095	32 0	GSB 0
008	45 .1	RCL . 1	052	42 36	f RAN #	096	42,10,15	f SOLVE E
009	45,40, .0	RCL + . 0	053	10	÷	097	43 32	g RTN
010	2	2	054	30	−	098	42,21, 0	f LBL 0
011	10	÷	055	45 .1	RCL . 1	099	44 .0	STO . 0
012	44 .2	STO . 2	056	44 .0	STO . 0	100	44 2	STO 2
013	32 15	GSB E	057	34	x↔y	101	34	x↔y
014	44 .4	STO . 4	058	44 .1	STO . 1	102	43,30, 5	g TEST x=y
015	45,20, .3	RCL × . 3	059	45 .4	RCL . 4	103	32 6	GSB 6
016	43,30, 2	g TEST x<0	060	44 .3	STO . 3	104	44 .1	STO . 1
017	32 3	GSB 3	061	43 34	g RND	105	44 1	STO 1
018	43, 6, 1	g F? 1	062	43,30, 0	g TEST x≠0	106	43 32	g RTN
019	32 2	GSB 2	063	22 4	GTO 4	107	42,21, 6	f LBL 6
020	45 .4	RCL . 4	064	45 .1	RCL . 1	108	26	EEX
021	43 34	g RND	065	43 32	g RTN	109	16	CHS
022	43,30, 0	g TEST x≠0	066	42,21,13	f LBL C	110	7	7
023	22 1	GTO 1	067	44 .1	STO . 1	111	40	+
024	45 .2	RCL . 2	068	44 1	STO 1	112	43 32	g RTN
025	43 32	g RTN	069	42,21, 5	f LBL 5	113	42,21,15	f LBL E
026	42,21, 2	f LBL 2	070	45 .1	RCL . 1	114	36	ENTER
027	45 .2	RCL . 2	071	32 15	GSB E	115	36	ENTER
028	44 .0	STO . 0	072	44 .4	STO . 4	116	36	ENTER
029	43 32	g RTN	073	45 .1	RCL . 1	117	6	6
030	42,21, 3	f LBL 3	074	43 20	g x=0	118	30	−
031	45 .2	RCL . 2	075	12	eˣ	119	20	×
032	44 .1	STO . 1	076	26	EEX	120	8	8
033	43, 5, 1	g CF 1	077	4	4	121	40	+
034	43 32	g RTN	078	10	÷	122	20	×
035	42,21,12	f LBL B	079	44 .2	STO . 2	123	4	4
036	32 0	GSB 0	080	45,40, .1	RCL + . 1	124	20	×
037	42 36	f RAN #	081	32 15	GSB E	125	7	7
038	44 .3	STO . 3	082	45,30, .4	RCL − . 4	126	48	.
039	42,21, 4	f LBL 4	083	45,10, .2	RCL ÷ . 2	127	5	5
040	45 .0	RCL . 0	084	45 .4	RCL . 4	128	30	−
041	45,30, .1	RCL − . 1	085	34	x↔y	129	43 32	g RTN
042	44 .2	STO . 2	086	10	÷	130	43 32	g RTN
043	45 .1	RCL . 1	087	44,30, .1	STO − . 1