GSB A runs a self contained secant 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 to be solved. Note that some of the 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. 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.

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.

There are 3 solutions depending on the guesses.

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

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

4 ENTER 2 GSB A points to the correct answer of 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.

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 Secant solver with multiple roots try choosing both guesses to be on the same side of a root. This may allow a particular root to be found. Otherwise just bracket as usual.

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

A | Secant solve routine - needs two guesses in x and y | |

D | Recover root after program stop | |

E | Formula to Solve = 0 | |

1 | Iteration loop |

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

.0 | x0 | |

.1 | x1 | |

.2 | x0-x1 | |

.3 | f(x0) | |

.4 | f(x1) | |

.5 | x2 |

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

000 | 019 | 43 20 | g x=0 | 038 | 43 32 | g RTN | |||||

001 | 42,21,11 | f LBL A | 020 | 42 36 | f RAN # | 039 | 42,21,15 | f LBL E | |||

002 | 44 .0 | STO . 0 | 021 | 10 | ÷ | 040 | 36 | ENTER | |||

003 | 33 | R⬇ | 022 | 30 | − | 041 | 36 | ENTER | |||

004 | 44 .1 | STO . 1 | 023 | 44 .5 | STO . 5 | 042 | 36 | ENTER | |||

005 | 42 36 | f RAN # | 024 | 45 .1 | RCL . 1 | 043 | 6 | 6 | |||

006 | 44 .3 | STO . 3 | 025 | 44 .0 | STO . 0 | 044 | 30 | − | |||

007 | 42,21, 1 | f LBL 1 | 026 | 45 .4 | RCL . 4 | 045 | 20 | × | |||

008 | 45 .0 | RCL . 0 | 027 | 44 .3 | STO . 3 | 046 | 8 | 8 | |||

009 | 45,30, .1 | RCL − . 1 | 028 | 45 .5 | RCL . 5 | 047 | 40 | + | |||

010 | 44 .2 | STO . 2 | 029 | 44 .1 | STO . 1 | 048 | 20 | × | |||

011 | 45 .1 | RCL . 1 | 030 | 45 .4 | RCL . 4 | 049 | 4 | 4 | |||

012 | 32 15 | GSB E | 031 | 43 34 | g RND | 050 | 20 | × | |||

013 | 44 .4 | STO . 4 | 032 | 43,30, 0 | g TEST x≠0 | 051 | 7 | 7 | |||

014 | 45 .1 | RCL . 1 | 033 | 22 1 | GTO 1 | 052 | 48 | . | |||

015 | 45 .4 | RCL . 4 | 034 | 45 .1 | RCL . 1 | 053 | 5 | 5 | |||

016 | 45,20, .2 | RCL × . 2 | 035 | 43 32 | g RTN | 054 | 30 | − | |||

017 | 45 .3 | RCL . 3 | 036 | 42,21,14 | f LBL D | 055 | 43 32 | g RTN | |||

018 | 45,30, .4 | RCL − . 4 | 037 | 45 .1 | RCL . 1 |