Halley' Method

Description

Halley' Method


References


Example


The programs B - E are from Valentin's article.
The step-size h is stored in register 0, while the name/number of the program is specified in the variable I.
So let's assume we want to solve program B with step-size h = 10-4 and starting guess 2:

RCL MATRIX B
STO I

EEX 4 CHS
STO 0

2 A

Intermediate values of |dz| are displayed:
((  running  ))
0.148589460
((  running  ))
0.002695411
((  running  ))
0.000000017
((  running  ))
0.000000000
((  running  ))
1.854105968

Should that annoy you just remove the PSE command in line 058

Definitions


These definitions are used:

Registers


Intermediate results are kept in these registers:
0: h
1: x
2: y
3: f-x
4: f-y
5: fx
6: fy
I: program

Program Resources

Labels

Name Description
 A Halley's Irrational Formula
 B Find a root of : x^x = pi
 C Find all roots of: ( 2 + 3i ) x³ - (1 + 2i ) x² - ( 3 + 4i ) x - ( 6 + 8i ) = 0
 D Attempt to find a complex root of: x³ - 6x - 2 = 0
 E Solve Leonardo di Pisa's equation: x³ + 2 x² + 10 x - 20 = 0
 0 calculate the next approximation
 1 calculate f(z + w) | w in {-h, 0, h}

Storage Registers

Name Description
 0 h
 1 x
 2 y
 3 f^{-}_x
 4 f^{-}_y
 5 f_x
 6 f_y

Program

Line Display Key Sequence Line Display Key Sequence Line Display Key Sequence
000 042 43 36 g LSTΧ 084 3 3
001 42,21,11 f LBL A 043 10 ÷ 085 42 25 f I
002 44 1 STO 1 044 34 x↔y 086 20 ×
003 42 30 f Re↔Im 045 20 × 087 1 1
004 44 2 STO 2 046 43 36 g LSTΧ 088 36 ENTER
005 42,21, 0 f LBL 0 047 33 R⬇ 089 2 2
006 45 0 RCL 0 048 30 090 42 25 f I
007 16 CHS 049 11 √x̅ 091 30
008 32 1 GSB 1 050 40 + 092 20 ×
009 44 3 STO 3 051 10 ÷ 093 3 3
010 42 30 f Re↔Im 052 45 0 RCL 0 094 36 ENTER
011 44 4 STO 4 053 20 × 095 4 4
012 0 0 054 44,30, 1 STO 1 096 42 25 f I
013 32 1 GSB 1 055 42 30 f Re↔Im 097 30
014 44 5 STO 5 056 44,30, 2 STO 2 098 20 ×
015 42 30 f Re↔Im 057 43 16 g ABS 099 6 6
016 44 6 STO 6 058 42 31 f PSE 100 36 ENTER
017 45 0 RCL 0 059 45 0 RCL 0 101 8 8
018 32 1 GSB 1 060 43 11 g 102 42 25 f I
019 36 ENTER 061 43,30, 8 g TEST x<y 103 30
020 36 ENTER 062 22 0 GTO 0 104 43 32 g RTN
021 45 3 RCL 3 063 45 1 RCL 1 105 42,21,14 f LBL D
022 45 4 RCL 4 064 45 2 RCL 2 106 43 11 g
023 42 25 f I 065 42 25 f I 107 6 6
024 40 + 066 43 32 g RTN 108 30
025 34 x↔y 067 42,21, 1 f LBL 1 109 20 ×
026 43 36 g LSTΧ 068 45 1 RCL 1 110 2 2
027 30 069 45 2 RCL 2 111 30
028 2 2 070 42 25 f I 112 43 32 g RTN
029 10 ÷ 071 40 + 113 42,21,15 f LBL E
030 34 x↔y 072 36 ENTER 114 2 2
031 45 5 RCL 5 073 36 ENTER 115 40 +
032 45 6 RCL 6 074 36 ENTER 116 20 ×
033 42 25 f I 075 22 25 GTO I 117 1 1
034 36 ENTER 076 42,21,12 f LBL B 118 0 0
035 40 + 077 14 119 40 +
036 30 078 43 26 g π 120 20 ×
037 1 1 079 30 121 2 2
038 43 36 g LSTΧ 080 43 32 g RTN 122 0 0
039 43 33 g R⬆ 081 42,21,13 f LBL C 123 30
040 10 ÷ 082 2 2 124 43 32 g RTN
041 43 33 g R⬆ 083 36 ENTER