Bisection Solver - J E Patterson

J E Patterson - jepspectro.com

Description

J E Patterson - jepspectro.com - 20231119

Bisection Solver

This solver performs like the built-in solver but it is somewhat slower.

GSB A runs a self contained 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.

To compare with the built-in solver press 1 ENTER 2 f SOLVE E.

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

A 4 decimetre 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.

0 ENTER 1 GSB A gives x = a flat box with a height of 0.2974 decimetres or 29.74 mm.
1 ENTER 2 GSB A gives x = a reasonable height of 1.5 decimetres or 150 mm.
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 returns the impossible box height x = 4.206 decimetres or 420.26 mm.

Solver:

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. Adjust for other roots. Guesses can be tested with GSB E. For the Bisection solver look for a sign change.

In SCI and ENG modes on the DM15 and a real hp-15C some roots are not always found. This is because RND is implimented 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 GSB D. Alternatively change to FIX mode - e.g. FIX 4 before solving.

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

Program Resources

Labels

Name	Description
A	Bisection solve routine - needs two guesses in x and y
D	Recover root after program stop
E	Formula to Solve = 0
1	Bisection loop updates x2 and either x0 to x1 or x1 to x0
2	Store x0 to x1 and x2 to x2
3	Store x1 to x0 and x2 to x1

Storage Registers

Name	Description
.0	x0
.1	x1
.2	x2 = (x0+x1)/2
.3	f(x0)
.4	f(x2)

Flags

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

Program

Line	Display	Key Sequence	Line	Display	Key Sequence	Line	Display	Key Sequence
000			019	32 3	GSB 3	038	45 .1	RCL . 1
001	42,21,11	f LBL A	020	43, 6, 1	g F? 1	039	43 32	g RTN
002	44 .0	STO . 0	021	32 2	GSB 2	040	42,21,15	f LBL E
003	33	R⬇	022	45 .4	RCL . 4	041	36	ENTER
004	44 .1	STO . 1	023	43 34	g RND	042	36	ENTER
005	42,21, 1	f LBL 1	024	43,30, 0	g TEST x≠0	043	36	ENTER
006	43, 4, 1	g SF 1	025	22 1	GTO 1	044	6	6
007	45 .0	RCL . 0	026	45 .2	RCL . 2	045	30	−
008	32 15	GSB E	027	43 32	g RTN	046	20	×
009	44 .3	STO . 3	028	42,21, 2	f LBL 2	047	8	8
010	45 .1	RCL . 1	029	45 .2	RCL . 2	048	40	+
011	45,40, .0	RCL + . 0	030	44 .0	STO . 0	049	20	×
012	2	2	031	43 32	g RTN	050	4	4
013	10	÷	032	42,21, 3	f LBL 3	051	20	×
014	44 .2	STO . 2	033	45 .2	RCL . 2	052	7	7
015	32 15	GSB E	034	44 .1	STO . 1	053	48	.
016	44 .4	STO . 4	035	43, 5, 1	g CF 1	054	5	5
017	45,20, .3	RCL × . 3	036	43 32	g RTN	055	30	−
018	43,30, 2	g TEST x<0	037	42,21,14	f LBL D	056	43 32	g RTN