Golden ratio min/max

52ac791d-9b34-4f78-afc7-47ea6cdbdfd0 66 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Golden-Ratio search [min/max]

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / 71 line :

a b + 2 / iter 21 mat

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 sys

x f x 4^2 x 3^* - 7 x 2^* - 2 - eval :

ε 1015 -^:

By setting f(x):=eval(,)by default, it will cope with all "_not defined" As well, you may have to use local definition. Some functions will ONLY accept higher 'ε' ... 10^{-14} ... 10^{-13}

t0 1 time :

X f 1.5 - 1 - y1 y2 ≤ ε Extrema f 0.5 - 0.5 y1 y2 ≥ ε Extrema f 2.5 3 y1 y2 ≤ ε Extrema 31 mat :

rootsDichotomy local command line style x0 x1 TOL D a x0 : b x1 : 12 mat xn b a + (2 / : ε TOL : 12 mat 21 line n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 21 line 31 line :

1 time t0 - 438.4 s *

on failure , increase TOL TOL 1012 -^:

r 2.5 - 1.5 - TOL D 3.5 4 TOL D 21 mat : 1.922518473318 - 3.852153430325 21 mat

x f X 1 el X 1 el f + 20 red X 2 el X 2 el f + 20 red X 3 el X 3 el f + 20 red 3 5 mat r 1 el 0 . 14 black 1 5 mat r 2 el 0 . 14 black 1 5 mat 4 1 sys

X 1.2655644606697 - 2.81605475000001 108 -^* - 2.76556437660193 31 mat

r f vectorize 11013 -^* - 31013 -^* 21 mat

Hyper fast/accurate Golden search ...

x f x 4^2 x 3^* - 7 x 2^* - 2 - :

Assign ...
scalar Derivative x D x f x diff :

t0 1 time :

x f

Min ... X1 x D x 1 - roots : 1.26556443707463 -

Max ... X2 x D x 0.25 - roots : 0.000000000000000

Min ... X3 x D x 2 roots : 2.765564437074640

X1 X2 X3 31 mat f vectorize 6.5923 - 2 - 39.3452 - 31 mat

1 time t0 - 0.66 s *

The discrete Golden ratio method ...

start end mesh x f Xend decimals Algo

L H N # f range n Algo U 0 : dx H L - N 1 - / : 12 mat i 1 N 1 + (range U i el L dx - (dx i * + : for j 1 range range u j el U j el : y j el U j el f eval : 21 line for u n round vectorize y n round vectorize augment 41 line :

data 2.5 - 44000 x f 40009 Algo :

lookup(value,table,col)

value table col lookup k 1 : j k table rows range table j col el value ≡ index k el j : k k 1 + : 21 line continue if for k 1 ≡ index NO match : index if 31 line :

value table col lookup

data

ROW data 2 col Min data 2 lookup : 324111 mat

data ROW row 2.766 39.345 - 12 mat

Project NOTE ... It may be tempting to spline the data set then run the scalar Golden ratio. It will NOT be more accurate than the piecewise method ... simply deceptive.

sub data 5002000 range 12 range el :

sub

ROW sub 2 col Min sub 2 lookup : 26111 mat

sub ROW row 1.266 - 6.592 - 12 mat

ROW sub 2 col Max sub 2 lookup : 104011 mat

sub ROW row 0.0001 - 2 - 12 mat

X 1.266 - 2.816 108 -^* - 2.766 31 mat

Carbon12 ... original data set

Carbon12 0.175 317.303 0.35 195.232 0.525 124.387 0.7 43.733 0.875 40.463 1.05 55.722 1.225 90.599 1.4 115.668 1.575 184.332 1.75 140.736 1.925 100.409 2.1 57.902 2.275 42.643 2.45 58.992 2.625 98.229 2.8 206.131 2.975 316.212 3.15 345.64 3.325 284.605 3.5 235.559 3.675 90.599 3.85 35.014 4.025 36.104 4.2 11.035 242 mat :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Matlab data set pts Carbon12 + 12 red plot :

data Carbon 12 α from superSmoothlet

C12 0.175 317.303 0.175 317.3009 0.175 317.2863 0.1751 317.2466 0.1752 317.1693 0.1754 317.0419 0.1757 316.8518 0.1761 316.5865 0.1767 316.2335 0.1774 315.7802 0.1783 315.2141 0.1794 314.5227 0.1807 313.6934 0.1822 312.7137 0.184 311.571 0.1861 310.2529 0.1884 308.7468 0.1911 307.0402 0.1941 305.1205 0.1975 302.9752 0.2013 300.5918 0.2054 297.9621 0.2099 295.096 0.2148 292.0076 0.22 288.7112 0.2255 285.2209 0.2314 281.551 0.2376 277.7157 0.244 273.7291 0.2508 269.6055 0.2578 265.3591 0.265 261.0041 0.2724 256.5547 0.2801 252.025 0.288 247.4294 0.296 242.782 0.3042 238.0971 0.3125 233.3887 0.321 228.6712 0.3296 223.9587 0.3383 219.2655 0.3471 214.6034 0.356 209.9754 0.3649 205.382 0.374 200.8238 0.383 196.3015 0.3922 191.8155 0.4014 187.3665 0.4106 182.955 0.4198 178.5817 0.4291 174.2471 0.4384 169.9519 0.4477 165.6966 0.4571 161.4818 0.4664 157.308 0.4757 153.176 0.4851 149.0862 0.4944 145.0393 0.5036 141.0358 0.5129 137.0763 0.5221 133.1614 0.5312 129.2921 0.5404 125.4708 0.5495 121.7001 0.5585 117.9828 0.5675 114.3217 0.5765 110.7194 0.5854 107.1787 0.5944 103.7024 0.6033 100.2931 0.6121 96.9537 0.621 93.6868 0.6298 90.4951 0.6386 87.3815 0.6474 84.3486 0.6562 81.3992 0.665 78.536 0.6737 75.7617 0.6825 73.0791 0.6912 70.491 0.7 67.9999 0.7088 65.6084 0.7175 63.3172 0.7262 61.1269 0.735 59.0379 0.7438 57.0508 0.7525 55.1661 0.7612 53.3841 0.77 51.7055 0.7788 50.1307 0.7875 48.6602 0.7962 47.2944 0.805 46.034 0.8138 44.8794 0.8225 43.831 0.8312 42.8894 0.84 42.055 0.8488 41.3284 0.8575 40.7101 0.8662 40.2004 0.875 39.8 0.8838 39.5089 0.8925 39.3256 0.9012 39.248 0.91 39.2742 0.9188 39.4023 0.9275 39.6303 0.9362 39.9561 0.945 40.3779 0.9538 40.8936 0.9625 41.5013 0.9712 42.1991 0.98 42.9849 0.9888 43.8568 0.9975 44.8128 1.0062 45.8509 1.015 46.9693 1.0238 48.1658 1.0325 49.4386 1.0412 50.7856 1.05 52.2049 1.0588 53.6945 1.0675 55.2521 1.0762 56.8753 1.085 58.5619 1.0938 60.3093 1.1025 62.1154 1.1112 63.9777 1.12 65.8939 1.1288 67.8617 1.1375 69.8787 1.1462 71.9426 1.155 74.051 1.1638 76.2015 1.1725 78.392 1.1812 80.6198 1.19 82.8829 1.1988 85.1787 1.2075 87.505 1.2162 89.8593 1.225 92.2395 1.2337 94.6427 1.2425 97.0655 1.2512 99.5037 1.26 101.9536 1.2688 104.411 1.2775 106.8722 1.2862 109.333 1.295 111.7897 1.3038 114.2382 1.3125 116.6746 1.3212 119.095 1.33 121.4954 1.3388 123.8718 1.3475 126.2204 1.3562 128.5371 1.365 130.8181 1.3738 133.0593 1.3825 135.2569 1.3913 137.4068 1.4 139.5052 1.4088 141.5483 1.4175 143.5325 1.4262 145.4548 1.435 147.3118 1.4438 149.1004 1.4525 150.8172 1.4612 152.459 1.47 154.0226 1.4788 155.5047 1.4875 156.9021 1.4962 158.2115 1.505 159.4297 1.5138 160.5534 1.5225 161.5794 1.5312 162.5044 1.54 163.3253 1.5488 164.0386 1.5575 164.6413 1.5662 165.13 1.575 165.5015 1.5838 165.7534 1.5925 165.8868 1.6012 165.9036 1.61 165.8058 1.6188 165.5953 1.6275 165.274 1.6362 164.8438 1.645 164.3068 1.6538 163.6648 1.6625 162.9198 1.6712 162.0737 1.68 161.1284 1.6888 160.0859 1.6975 158.9481 1.7062 157.717 1.715 156.3944 1.7238 154.9824 1.7325 153.4829 1.7412 151.8977 1.75 150.2288 1.7588 148.4787 1.7675 146.6511 1.7762 144.7506 1.785 142.7814 1.7938 140.748 1.8025 138.6547 1.8112 136.5058 1.82 134.3058 1.8288 132.059 1.8375 129.7698 1.8462 127.4425 1.855 125.0815 1.8638 122.6913 1.8725 120.276 1.8812 117.8402 1.89 115.3882 1.8988 112.9243 1.9075 110.453 1.9162 107.9785 1.925 105.5054 1.9338 103.0375 1.9425 100.5783 1.9512 98.1305 1.96 95.6971 1.9688 93.281 1.9775 90.8852 1.9862 88.5124 1.995 86.1658 2.0038 83.8481 2.0125 81.5624 2.0213 79.3115 2.03 77.0983 2.0388 74.9258 2.0475 72.7968 2.0562 70.7144 2.065 68.6814 2.0738 66.7008 2.0825 64.7754 2.0912 62.9082 2.1 61.1021 2.1088 59.3598 2.1175 57.6833 2.1263 56.0743 2.135 54.5347 2.1438 53.0661 2.1525 51.6702 2.1612 50.349 2.17 49.104 2.1788 47.9372 2.1875 46.8501 2.1962 45.8446 2.205 44.9225 2.2138 44.0854 2.2225 43.3352 2.2312 42.6736 2.24 42.1024 2.2487 41.6232 2.2575 41.238 2.2662 40.9483 2.275 40.756 2.2838 40.6628 2.2925 40.6695 2.3012 40.7772 2.31 40.9867 2.3188 41.2989 2.3275 41.7146 2.3362 42.2349 2.345 42.8606 2.3537 43.5925 2.3625 44.4316 2.3712 45.3788 2.38 46.4349 2.3888 47.6008 2.3975 48.8775 2.4062 50.2658 2.415 51.7667 2.4238 53.381 2.4325 55.1095 2.4412 56.9533 2.45 58.9132 2.4588 60.9896 2.4675 63.1817 2.4762 65.4878 2.485 67.9067 2.4938 70.4369 2.5025 73.0769 2.5112 75.8254 2.52 78.6809 2.5288 81.6421 2.5375 84.7074 2.5462 87.8754 2.555 91.1448 2.5638 94.5141 2.5725 97.9818 2.5812 101.5467 2.59 105.2071 2.5988 108.9618 2.6075 112.8093 2.6162 116.7481 2.625 120.7769 2.6338 124.8936 2.6425 129.0934 2.6512 133.3712 2.66 137.7215 2.6688 142.139 2.6775 146.6185 2.6862 151.1546 2.695 155.7419 2.7038 160.3752 2.7125 165.0492 2.7212 169.7585 2.73 174.4977 2.7388 179.2617 2.7475 184.0451 2.7562 188.8424 2.765 193.6485 2.7738 198.4581 2.7825 203.2656 2.7912 208.066 2.8 212.8538 2.8088 217.6236 2.8175 222.3694 2.8262 227.085 2.835 231.7642 2.8438 236.401 2.8525 240.9892 2.8612 245.5225 2.87 249.995 2.8788 254.4003 2.8875 258.7324 2.8963 262.9851 2.905 267.1522 2.9138 271.2277 2.9225 275.2053 2.9312 279.0789 2.94 282.8424 2.9488 286.4896 2.9575 290.0143 2.9662 293.4104 2.975 296.6717 2.9838 299.7928 2.9925 302.771 3.0013 305.6043 3.01 308.2906 3.0188 310.828 3.0275 313.2144 3.0362 315.4479 3.045 317.5264 3.0538 319.448 3.0625 321.2105 3.0712 322.8121 3.08 324.2507 3.0888 325.5243 3.0975 326.6308 3.1062 327.5684 3.115 328.3349 3.1237 328.9284 3.1325 329.3469 3.1412 329.5883 3.15 329.6506 3.1588 329.5326 3.1675 329.2359 3.1762 328.7626 3.185 328.1152 3.1938 327.2959 3.2025 326.307 3.2112 325.1508 3.22 323.8296 3.2287 322.3457 3.2375 320.7014 3.2462 318.8989 3.255 316.9406 3.2638 314.8288 3.2725 312.5658 3.2812 310.1537 3.29 307.5951 3.2988 304.8921 3.3075 302.047 3.3162 299.0621 3.325 295.9398 3.3338 292.6828 3.3425 289.2958 3.3512 285.7843 3.36 282.1533 3.3688 278.4082 3.3775 274.5542 3.3862 270.5966 3.395 266.5407 3.4038 262.3918 3.4125 258.155 3.4212 253.8357 3.43 249.4391 3.4388 244.9705 3.4475 240.4352 3.4562 235.8384 3.465 231.1854 3.4738 226.4814 3.4825 221.7317 3.4912 216.9416 3.5 212.1164 3.5088 207.2613 3.5175 202.3821 3.5262 197.4847 3.535 192.5747 3.5438 187.6579 3.5525 182.7403 3.5612 177.8274 3.57 172.9252 3.5788 168.0395 3.5875 163.176 3.5962 158.3404 3.605 153.5387 3.6138 148.7766 3.6225 144.0599 3.6312 139.3943 3.64 134.7857 3.6488 130.2398 3.6575 125.7625 3.6662 121.3596 3.675 117.0368 3.6838 112.799 3.6925 108.6481 3.7012 104.5848 3.71 100.6101 3.7188 96.7248 3.7275 92.9298 3.7362 89.2261 3.745 85.6143 3.7538 82.0956 3.7625 78.6706 3.7712 75.3404 3.78 72.1057 3.7888 68.9675 3.7975 65.9267 3.8062 62.984 3.815 60.1405 3.8238 57.3969 3.8325 54.7542 3.8412 52.2132 3.85 49.7748 3.8588 47.4393 3.8675 45.2044 3.8762 43.0674 3.885 41.0254 3.8938 39.0755 3.9025 37.215 3.9112 35.4411 3.92 33.7508 3.9288 32.1413 3.9375 30.6099 3.9462 29.1537 3.955 27.7699 3.9638 26.4556 3.9725 25.208 3.9812 24.0243 3.99 22.9017 3.9988 21.8373 4.0075 20.8283 4.0162 19.8719 4.025 18.9652 4.0337 18.1057 4.0425 17.2918 4.0512 16.5222 4.0598 15.7956 4.0683 15.1107 4.0767 14.4661 4.085 13.8605 4.0931 13.2927 4.1011 12.7613 4.1089 12.265 4.1164 11.8025 4.1237 11.3725 4.1307 10.9737 4.1375 10.6047 4.1439 10.2643 4.1501 9.9511 4.1558 9.6638 4.1612 9.4012 4.1662 9.1618 4.1708 8.9445 4.175 8.7479 4.1787 8.5709 4.1821 8.4125 4.1851 8.2717 4.1877 8.1475 4.19 8.0388 4.192 7.9445 4.1937 7.8637 4.1951 7.7953 4.1964 7.7382 4.1973 7.6915 4.1981 7.6541 4.1987 7.625 4.1992 7.6031 4.1995 7.5874 4.1998 7.5769 4.1999 7.5706 4.2 7.5673 4.2 7.5661 5002 mat :

strip from duplicated [2 start ... 1 end] C12 C12 3 C12 rows 1 - (range 12 range el :

C12 rows 497

Golden ratio

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 71 line :

X C12 1 col : Y C12 2 col : 12 mat x f X Y x cinterp : TOL 106 -^: New X Y augment : 41 line

New pts 2 1 sys

from superSmoothlet
original data [Matlab] C12 pts 21 sys

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 line

X f 0.75 1.2 y1 y2 ≤ TOL Extrema 1 el f 1.25 2 y1 y2 ≥ TOL Extrema 1 el f 2 2.5 y1 y2 ≤ TOL Extrema 1 el f 3 3.25 y1 y2 ≥ TOL Extrema 1 el 41 mat :

iter f 0.75 1.2 y1 y2 ≤ TOL Extrema 2 el f 1.25 2 y1 y2 ≥ TOL Extrema 2 el f 2 2.5 y1 y2 ≤ TOL Extrema 2 el f 3 3.25 y1 y2 ≥ TOL Extrema 2 el 41 mat :

cinterp X 0.903 1.598 2.288 3.149 41 mat

ainterp 0.903 1.598 2.288 3.149 41 mat

Golden ratio iterates iter 2829282641 mat

20221212

Davide Carpi

golden section search algorithm. # F a b ε maxIter target GoldenSearch φ 15 sqrt + 2 / : a b > x.lw b : x.up a : 21 line x.lw a : x.up b : 21 line if i 0 : d 2 φ - (x.up x.lw - (* : x.1 x.lw d + : x.2 x.up d - : f.x1 x.1 F : f.x2 x.2 F : i maxIter < x.up x.lw - abs ε ≤ break x.1 F x.2 F > (target min ≡ (& (x.1 F x.2 F < (target max ≡ (& | x.lw x.1 : x.1 x.2 : f.x1 f.x2 : x.2 x.up 2 φ - (x.up x.lw - (* - : f.x2 x.2 F : 51 line x.up x.2 : x.2 x.1 : f.x2 f.x1 : x.1 x.lw 2 φ - (x.up x.lw - (* + : f.x1 x.1 F : 51 line if i i 1 + : 21 line if 11 line while x.up x.lw + 2 / i 11 mat 21 mat 101 line :

t0 1 time :

x f x 4^2 x 3^* - 7 x 2^* - 2 - eval :

x f

ε 1015 -^:

iter 200 :

X x f 1.5 - 1 - ε iter min GoldenSearch x f 1 - 1 ε iter max GoldenSearch x f 23 ε iter min GoldenSearch 31 mat :

X 1.26556443438215 - 7411 mat 21 mat 2.78848466495801 108 -^* - 7411 mat 21 mat 2.76556439576251 11511 mat 21 mat 31 mat

X 1 el (1 el X 2 el (1 el X 3 el (1 el 31 mat 1.26556443438215 - 0.000000027884847 - 2.76556439576251 31 mat

1 time t0 - 4.2 s *

Nuances between the two Golden ratio. ε 10^{-12}... GoldenSearch ... 2.3s ε 10^{-12}... Extrema ........ 0.5s ε 10^{-15}... Extrema ........ 7.2s ε 10^{-15}... GoldenSearch ... 3.2s Matter of preference wrt desired 'ε' ... only 7 decimals agree ? However ... reconstruct ± last 15th D

Davide 1.26556443438215 - 0.000000027884847 - 2.76556439576254 31 mat

Jean 1.26556446066969 - 0.000000028160548 - 2.7655643766019 31 mat

1.26556443438215 - 0.000000027884847 - 2.76556439576254 31 mat 1.26556446066969 - 0.000000028160548 - 2.7655643766019 31 mat - 0.00000002628754 0.000000000275701 0.00000001916064 31 mat

1.26556443438215 - 0.000000027884847 - 2.76556439576254 31 mat f vectorize 6.592288948768570 - 2.000000000000000 - 39.345211051231400 - 31 mat

1.26556446066969 - 0.000000028160548 - 2.7655643766019 31 mat f vectorize 6.592288948768570 - 2.000000000000010 - 39.345211051231300 - 31 mat