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.

The box should hold a volume V of 7.5 cubic decimeters, i.e. 7.5 litres.

How should the metal be folded for the tallest box in decimeters.

We are using decimeters 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.

0 ENTER 1 GSB A gives x = 0.2974 decimeters or 29.74 mm - a flat box.

1 ENTER 2 GSB A gives x = 1.5 decimeters 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 decimeters or 420.26 mm - an impossible box height.

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.

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.

There are 3 roots depending on the guesses.

0 ENTER 1 GSB B gives x = 0.2974 decimeters or 29.74 mm - a flat box.

1 ENTER 2 GSB B gives x = 1.5 decimeters or 150 mm - a reasonable height box.

2 ENTER 3 GSB B gives x = 1.5 decimeters or 150 mm - a reasonable height box.

3 ENTER 4 GSB B gives x = 4.2026 decimeters or 420.26 mm - an impossible box.

4 ENTER 2 GSB B points to the correct answer of x = 1.5 decimeters in spite of a potential divide by zero error.

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.

The root x1 is found if f(x1) = 0 within an error tolerance determined by the FIX, SCI or ENG significant digits settng.

0 GSB C gives x = 0.2974 decimeters or 29.74 mm - a flat box

1 GSB C gives x = 1.5 decimeters or 150 mm - a reasonable height box.

2 GSB C gives x = 1.5 decimeters or 150 mm - a reasonable height box.

3 GSB C gives x = 0.2974 decimeters 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 decimeters or 420.26 mm - an impossible box.

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 - exit if f(x1) = 0 within an error tolerance determined by the FIX, SCI or ENG significant digits settng.

Display the root x1

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 GSB D. 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 root or to the other side.

Name | Description | Name | Description | ||
---|---|---|---|---|---|

A | Bisection solve routine - needs two guesses in x and y | 1 | Bisection loop updates x2 and either x0 to x1 or x1 to x0 | ||

B | Secant solve routine - needs two guesses in x and y | 2 | Bisection - Store x0 to x1 and x2 to x2 | ||

C | Newton - Raphson Solve routine - needs one guess in x | 3 | Bisection - Store x1 to x0 and x2 to x1 | ||

D | Recover root after program stop | 4 | Secant - Iteration loop | ||

E | Formula to be Solved, f(x) = 0 | 5 | Newton - iteration loop |

Name | Description | |
---|---|---|

.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) | |

.5 | Secant x2 |

Number | Description | |
---|---|---|

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

Line | Display | Key Sequence | Line | Display | Key Sequence | Line | Display | Key Sequence | |||
---|---|---|---|---|---|---|---|---|---|---|---|

000 | 040 | 44 .1 | STO . 1 | 080 | 12 | eˣ | |||||

001 | 42,21,11 | f LBL A | 041 | 42 36 | f RAN # | 081 | 26 | EEX | |||

002 | 44 .0 | STO . 0 | 042 | 44 .3 | STO . 3 | 082 | 4 | 4 | |||

003 | 33 | R⬇ | 043 | 42,21, 4 | f LBL 4 | 083 | 10 | ÷ | |||

004 | 44 .1 | STO . 1 | 044 | 45 .0 | RCL . 0 | 084 | 44 .2 | STO . 2 | |||

005 | 42,21, 1 | f LBL 1 | 045 | 45,30, .1 | RCL − . 1 | 085 | 45,40, .1 | RCL + . 1 | |||

006 | 43, 4, 1 | g SF 1 | 046 | 44 .2 | STO . 2 | 086 | 32 15 | GSB E | |||

007 | 45 .0 | RCL . 0 | 047 | 45 .1 | RCL . 1 | 087 | 45,30, .4 | RCL − . 4 | |||

008 | 32 15 | GSB E | 048 | 32 15 | GSB E | 088 | 45,10, .2 | RCL ÷ . 2 | |||

009 | 44 .3 | STO . 3 | 049 | 44 .4 | STO . 4 | 089 | 45 .4 | RCL . 4 | |||

010 | 45 .1 | RCL . 1 | 050 | 45 .1 | RCL . 1 | 090 | 34 | x↔y | |||

011 | 45,40, .0 | RCL + . 0 | 051 | 45 .4 | RCL . 4 | 091 | 10 | ÷ | |||

012 | 2 | 2 | 052 | 45,20, .2 | RCL × . 2 | 092 | 44,30, .1 | STO − . 1 | |||

013 | 10 | ÷ | 053 | 45 .3 | RCL . 3 | 093 | 45 .4 | RCL . 4 | |||

014 | 44 .2 | STO . 2 | 054 | 45,30, .4 | RCL − . 4 | 094 | 43 34 | g RND | |||

015 | 32 15 | GSB E | 055 | 43 20 | g x=0 | 095 | 43,30, 0 | g TEST x≠0 | |||

016 | 44 .4 | STO . 4 | 056 | 42 36 | f RAN # | 096 | 22 5 | GTO 5 | |||

017 | 45,20, .3 | RCL × . 3 | 057 | 10 | ÷ | 097 | 45 .1 | RCL . 1 | |||

018 | 43,30, 2 | g TEST x<0 | 058 | 30 | − | 098 | 43 32 | g RTN | |||

019 | 32 3 | GSB 3 | 059 | 44 .5 | STO . 5 | 099 | 42,21,14 | f LBL D | |||

020 | 43, 6, 1 | g F? 1 | 060 | 45 .1 | RCL . 1 | 100 | 45 .1 | RCL . 1 | |||

021 | 32 2 | GSB 2 | 061 | 44 .0 | STO . 0 | 101 | 43 32 | g RTN | |||

022 | 45 .4 | RCL . 4 | 062 | 45 .4 | RCL . 4 | 102 | 42,21,15 | f LBL E | |||

023 | 43 34 | g RND | 063 | 44 .3 | STO . 3 | 103 | 36 | ENTER | |||

024 | 43,30, 0 | g TEST x≠0 | 064 | 45 .5 | RCL . 5 | 104 | 36 | ENTER | |||

025 | 22 1 | GTO 1 | 065 | 44 .1 | STO . 1 | 105 | 36 | ENTER | |||

026 | 45 .2 | RCL . 2 | 066 | 45 .4 | RCL . 4 | 106 | 6 | 6 | |||

027 | 43 32 | g RTN | 067 | 43 34 | g RND | 107 | 30 | − | |||

028 | 42,21, 2 | f LBL 2 | 068 | 43,30, 0 | g TEST x≠0 | 108 | 20 | × | |||

029 | 45 .2 | RCL . 2 | 069 | 22 4 | GTO 4 | 109 | 8 | 8 | |||

030 | 44 .0 | STO . 0 | 070 | 45 .1 | RCL . 1 | 110 | 40 | + | |||

031 | 43 32 | g RTN | 071 | 43 32 | g RTN | 111 | 20 | × | |||

032 | 42,21, 3 | f LBL 3 | 072 | 42,21,13 | f LBL C | 112 | 4 | 4 | |||

033 | 45 .2 | RCL . 2 | 073 | 44 .1 | STO . 1 | 113 | 20 | × | |||

034 | 44 .1 | STO . 1 | 074 | 42,21, 5 | f LBL 5 | 114 | 7 | 7 | |||

035 | 43, 5, 1 | g CF 1 | 075 | 45 .1 | RCL . 1 | 115 | 48 | . | |||

036 | 43 32 | g RTN | 076 | 32 15 | GSB E | 116 | 5 | 5 | |||

037 | 42,21,12 | f LBL B | 077 | 44 .4 | STO . 4 | 117 | 30 | − | |||

038 | 44 .0 | STO . 0 | 078 | 45 .1 | RCL . 1 | 118 | 43 32 | g RTN | |||

039 | 33 | R⬇ | 079 | 43 20 | g x=0 |