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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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) = 3x

ENTER ENTER g x

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 +.

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. |

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) |

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 | 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 |