OST - SOLVE alternative, numerical solution of a lenear equation using Ostrowski's method.

Input

"PSLV" : name of the equation to be solved 
"SOLVE" : name of the root variable to solve
REG x : first initial guess
IND "SOLVE" : second initial guess

Resources to be used

RSLV"  : registors
REG 00 : iteration counter
REG 01 : xn
REG 02 : xn+1
REG 03 : xn+2
REG 04 : f(xn)
REG 05 : f(xn+1)
REG 06 : f(xn+2)
REG 07 : t work memory
REG 08 : "PSLV"
REG 09 : "SOLVE"
REG 10 : epsilon
REG 11 : pivot counter
REG 12 : range expander

Output

REG t : registers
REG z : status
 0: a root is found
 1: |f(x)| < epsilon (2E-12) *Note1
 2: xn+1==xn
 3: f(xn+1)=f(xn)
 4: |f(xn+1)| > |f(xn)|
 5: Iteration limit (30 times)
 6: Diverged
 7: Bad guess or f(x) is constant
REG y : the value of the equation at the root
REG x : solution (root) for the unknown

Program list for the HP 42S

OST - Rev 2.93: Aug. 26 2011 - Copyright(c) 1998-2011 Takayuki HOSODA, All rights reserved.
00 { 492-Byte Prgm } 01 LBL "OST" 02 RCL "REGS" 03 STO "RSLV" 04 R↓ 05 STO 01 06 0.1 07 STO 00 08 1E-12 09 STO 10 10 0.03 11 STO 11 12 0 13 STO 12 14 RCL "SOLVE" 15 STO 08 16 RCL "PSLV" 17 STO 09 18 RCL 01 19 RCL IND 08 20 X≠Y? 21 GTO 00 22 1.01 23 × 24 X≠0? 25 GTO 00 26 0.01 27 LBL 00 28 STO 02 29 STO IND 08 30 XEQ IND 09 31 STO 05 32 X=0? 33 GTO 12 34 ABS 35 RCL- 10 36 X>0? 37 GTO 01 38 1 39 GTO 13 40 LBL 01 41 RCL 01 42 STO IND 08	43 XEQ IND 09 44 STO 04 45 X=0? 46 GTO 11 47 ABS 48 RCL- 10 49 X>0? 50 GTO 02 51 1 52 GTO 13 53 LBL 02 54 RCL 01 55 RCL+ 02 56 0.5 57 × 58 GTO 04 59 LBL 03 60 1 61 RAN 62 14 63 × 64 + 65 16 66 ÷ 67 RCL 02 68 RCL- 01 69 × 70 RCL+ 01 71 LBL 04 72 STO 03 73 STO IND 08 74 XEQ IND 09 75 STO 06 76 LBL 05 77 RCL 06 78 X=0? 79 GTO 13 80 ABS 81 STO 07 82 RCL- 10 83 X>0? 84 GTO 20 85 1	86 GTO 13 87 LBL 20 88 RCL 06 89 RCL- 05 90 RCL 06 91 RCL- 04 92 × 93 X≠0? 94 GTO 06 95 ISG 11 96 GTO 21 97 7 98 BEEP 99 GTO 13 100 LBL 21 101 RCL 02 102 RCL- 01 103 RCL 12 104 X=0? 105 X<>Y 106 STO 12 107 RCL 06 108 RCL+ 06 109 RCL- 05 110 RCL- 04 111 ABS 112 1E-8 113 X≤Y? 114 GTO 25 115 -1.5 116 RCL× 12 117 STO 12 118 RCL+ 01 119 STO 02 120 STO IND 08 121 XEQ IND 09 122 STO 05 123 GTO 03 124 LBL 25 125 0 126 STO 12 127 GTO 03 128 LBL 06	129 RCL 03 130 RCL- 01 131 X≠0? 132 GTO 07 133 ISG 11 134 GTO 03 135 2 136 GTO 13 137 LBL 07 138 RCL 06 139 RCL÷ 05 140 ABS 141 1 142 - 143 ABS 144 10 145 RCL× 10 146 - 147 X≥0? 148 GTO 08 149 ISG 11 150 GTO 03 151 3 152 GTO 13 153 LBL 08 154 RCL 06 155 ABS 156 RCL 05 157 ABS 158 X≥Y? 159 GTO 09 160 1E-8 161 X<Y? 162 GTO 09 163 3 164 RCL+ 11 165 RCL- 00 166 X≤0? 167 GTO 09 168 4 169 GTO 12 170 LBL 09 171 RCL 03	172 RCL- 02 173 RCL 06 174 RCL- 04 175 × 176 RCL× 05 177 RCL 03 178 RCL- 01 179 RCL 06 180 RCL- 05 181 × 182 RCL× 04 183 ÷ 184 STO 07 185 5 186 RCL- 00 187 X<0? 188 GTO 23 189 RCL 06 190 ABS 191 10 192 RCL× 05 193 ABS 194 X>Y? 195 GTO 23 196 ISG 11 197 GTO 22 198 3 199 GTO 13 200 LBL 22 201 RCL 04 202 ABS 203 X≤Y? 204 GTO 01 205 RCL 01 206 RCL+ 02 207 0.5 208 × 209 STO 01 210 STO IND 08 211 XEQ IND 09 212 STO 04 213 GTO 03 214 LBL 23	215 1 216 RCL- 07 217 X≠0? 218 GTO 10 219 6 220 BEEP 221 GTO 12 222 LBL 10 223 RCL 02 224 RCL 07 225 RCL× 01 226 - 227 X<>Y 228 ÷ 229 STO IND 08 230 X<> 03 231 X<> 02 232 STO 01 233 XEQ IND 09 234 X<> 06 235 X<> 05 236 STO 04 237 ISG 00 238 GTO 05 239 5 240 LBL 13 241 RCL 06 242 RCL 03 243 GTO 14 244 LBL 12 245 RCL 05 246 RCL 02 247 GTO 14 248 LBL 11 249 RCL 04 250 RCL 01 251 LBL 14 252 STO IND 08 253 RCL "RSLV" 254 X<> "REGS" 255 R↓ 256 RTN 257 END

*Note1: 'epsilon' is hard coded to 2E-12 for HP 42S and 1E-16 for Free42 at line # 08, however,
it can be replaced by a command to use external variable e.g. RCL "EPS".

Download

'OST' (ost-2.93.raw, 495 byte, raw program file for the Free42)

Example

Test function to be solved

f(x) = eX - 3X2	plot
00 { 19-Byte Prgm } 01 LBL "Test" 02 RCL X 03 E↑X 04 LASTX 05 X↑2 06 3 07 × 08 - 09 RTN 10 .END.

Operation

preparation	1st root	2nd root	3rd root
"Test" ASTO "PSLV" "X" ASTO "SOLVE"	0 STO "X" 1 XEQ "OST"	0 STO "X" -1 XEQ "OST"	3 STO "X" 4 XEQ "OST"
Results	0.91000757249	-0.458962267536	3.73307902863

SEE ALSO

Ostrowski's method - a root-finding method
Rational interpolation program for the HP 42S
Online calculator - Synthesize/Analize microstrip transmission line

EXTERNAL LINKS

556 : Karl Schneider, 2005, HP SOLVE/INTEG on all RPN-based models, Karl Schneider

190798 : question for lyuka and the Ostrowski method
191098 : Lyuka and the Ostrowski method's for Root Seeking
191462 : Results of new root-seeking methods