/usr/share/gnudatalanguage/mpfit/mpfit.pro is in gdl-mpfit 1.85+2017.01.03-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 2054 2055 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085 2086 2087 2088 2089 2090 2091 2092 2093 2094 2095 2096 2097 2098 2099 2100 2101 2102 2103 2104 2105 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130 2131 2132 2133 2134 2135 2136 2137 2138 2139 2140 2141 2142 2143 2144 2145 2146 2147 2148 2149 2150 2151 2152 2153 2154 2155 2156 2157 2158 2159 2160 2161 2162 2163 2164 2165 2166 2167 2168 2169 2170 2171 2172 2173 2174 2175 2176 2177 2178 2179 2180 2181 2182 2183 2184 2185 2186 2187 2188 2189 2190 2191 2192 2193 2194 2195 2196 2197 2198 2199 2200 2201 2202 2203 2204 2205 2206 2207 2208 2209 2210 2211 2212 2213 2214 2215 2216 2217 2218 2219 2220 2221 2222 2223 2224 2225 2226 2227 2228 2229 2230 2231 2232 2233 2234 2235 2236 2237 2238 2239 2240 2241 2242 2243 2244 2245 2246 2247 2248 2249 2250 2251 2252 2253 2254 2255 2256 2257 2258 2259 2260 2261 2262 2263 2264 2265 2266 2267 2268 2269 2270 2271 2272 2273 2274 2275 2276 2277 2278 2279 2280 2281 2282 2283 2284 2285 2286 2287 2288 2289 2290 2291 2292 2293 2294 2295 2296 2297 2298 2299 2300 2301 2302 2303 2304 2305 2306 2307 2308 2309 2310 2311 2312 2313 2314 2315 2316 2317 2318 2319 2320 2321 2322 2323 2324 2325 2326 2327 2328 2329 2330 2331 2332 2333 2334 2335 2336 2337 2338 2339 2340 2341 2342 2343 2344 2345 2346 2347 2348 2349 2350 2351 2352 2353 2354 2355 2356 2357 2358 2359 2360 2361 2362 2363 2364 2365 2366 2367 2368 2369 2370 2371 2372 2373 2374 2375 2376 2377 2378 2379 2380 2381 2382 2383 2384 2385 2386 2387 2388 2389 2390 2391 2392 2393 2394 2395 2396 2397 2398 2399 2400 2401 2402 2403 2404 2405 2406 2407 2408 2409 2410 2411 2412 2413 2414 2415 2416 2417 2418 2419 2420 2421 2422 2423 2424 2425 2426 2427 2428 2429 2430 2431 2432 2433 2434 2435 2436 2437 2438 2439 2440 2441 2442 2443 2444 2445 2446 2447 2448 2449 2450 2451 2452 2453 2454 2455 2456 2457 2458 2459 2460 2461 2462 2463 2464 2465 2466 2467 2468 2469 2470 2471 2472 2473 2474 2475 2476 2477 2478 2479 2480 2481 2482 2483 2484 2485 2486 2487 2488 2489 2490 2491 2492 2493 2494 2495 2496 2497 2498 2499 2500 2501 2502 2503 2504 2505 2506 2507 2508 2509 2510 2511 2512 2513 2514 2515 2516 2517 2518 2519 2520 2521 2522 2523 2524 2525 2526 2527 2528 2529 2530 2531 2532 2533 2534 2535 2536 2537 2538 2539 2540 2541 2542 2543 2544 2545 2546 2547 2548 2549 2550 2551 2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 2562 2563 2564 2565 2566 2567 2568 2569 2570 2571 2572 2573 2574 2575 2576 2577 2578 2579 2580 2581 2582 2583 2584 2585 2586 2587 2588 2589 2590 2591 2592 2593 2594 2595 2596 2597 2598 2599 2600 2601 2602 2603 2604 2605 2606 2607 2608 2609 2610 2611 2612 2613 2614 2615 2616 2617 2618 2619 2620 2621 2622 2623 2624 2625 2626 2627 2628 2629 2630 2631 2632 2633 2634 2635 2636 2637 2638 2639 2640 2641 2642 2643 2644 2645 2646 2647 2648 2649 2650 2651 2652 2653 2654 2655 2656 2657 2658 2659 2660 2661 2662 2663 2664 2665 2666 2667 2668 2669 2670 2671 2672 2673 2674 2675 2676 2677 2678 2679 2680 2681 2682 2683 2684 2685 2686 2687 2688 2689 2690 2691 2692 2693 2694 2695 2696 2697 2698 2699 2700 2701 2702 2703 2704 2705 2706 2707 2708 2709 2710 2711 2712 2713 2714 2715 2716 2717 2718 2719 2720 2721 2722 2723 2724 2725 2726 2727 2728 2729 2730 2731 2732 2733 2734 2735 2736 2737 2738 2739 2740 2741 2742 2743 2744 2745 2746 2747 2748 2749 2750 2751 2752 2753 2754 2755 2756 2757 2758 2759 2760 2761 2762 2763 2764 2765 2766 2767 2768 2769 2770 2771 2772 2773 2774 2775 2776 2777 2778 2779 2780 2781 2782 2783 2784 2785 2786 2787 2788 2789 2790 2791 2792 2793 2794 2795 2796 2797 2798 2799 2800 2801 2802 2803 2804 2805 2806 2807 2808 2809 2810 2811 2812 2813 2814 2815 2816 2817 2818 2819 2820 2821 2822 2823 2824 2825 2826 2827 2828 2829 2830 2831 2832 2833 2834 2835 2836 2837 2838 2839 2840 2841 2842 2843 2844 2845 2846 2847 2848 2849 2850 2851 2852 2853 2854 2855 2856 2857 2858 2859 2860 2861 2862 2863 2864 2865 2866 2867 2868 2869 2870 2871 2872 2873 2874 2875 2876 2877 2878 2879 2880 2881 2882 2883 2884 2885 2886 2887 2888 2889 2890 2891 2892 2893 2894 2895 2896 2897 2898 2899 2900 2901 2902 2903 2904 2905 2906 2907 2908 2909 2910 2911 2912 2913 2914 2915 2916 2917 2918 2919 2920 2921 2922 2923 2924 2925 2926 2927 2928 2929 2930 2931 2932 2933 2934 2935 2936 2937 2938 2939 2940 2941 2942 2943 2944 2945 2946 2947 2948 2949 2950 2951 2952 2953 2954 2955 2956 2957 2958 2959 2960 2961 2962 2963 2964 2965 2966 2967 2968 2969 2970 2971 2972 2973 2974 2975 2976 2977 2978 2979 2980 2981 2982 2983 2984 2985 2986 2987 2988 2989 2990 2991 2992 2993 2994 2995 2996 2997 2998 2999 3000 3001 3002 3003 3004 3005 3006 3007 3008 3009 3010 3011 3012 3013 3014 3015 3016 3017 3018 3019 3020 3021 3022 3023 3024 3025 3026 3027 3028 3029 3030 3031 3032 3033 3034 3035 3036 3037 3038 3039 3040 3041 3042 3043 3044 3045 3046 3047 3048 3049 3050 3051 3052 3053 3054 3055 3056 3057 3058 3059 3060 3061 3062 3063 3064 3065 3066 3067 3068 3069 3070 3071 3072 3073 3074 3075 3076 3077 3078 3079 3080 3081 3082 3083 3084 3085 3086 3087 3088 3089 3090 3091 3092 3093 3094 3095 3096 3097 3098 3099 3100 3101 3102 3103 3104 3105 3106 3107 3108 3109 3110 3111 3112 3113 3114 3115 3116 3117 3118 3119 3120 3121 3122 3123 3124 3125 3126 3127 3128 3129 3130 3131 3132 3133 3134 3135 3136 3137 3138 3139 3140 3141 3142 3143 3144 3145 3146 3147 3148 3149 3150 3151 3152 3153 3154 3155 3156 3157 3158 3159 3160 3161 3162 3163 3164 3165 3166 3167 3168 3169 3170 3171 3172 3173 3174 3175 3176 3177 3178 3179 3180 3181 3182 3183 3184 3185 3186 3187 3188 3189 3190 3191 3192 3193 3194 3195 3196 3197 3198 3199 3200 3201 3202 3203 3204 3205 3206 3207 3208 3209 3210 3211 3212 3213 3214 3215 3216 3217 3218 3219 3220 3221 3222 3223 3224 3225 3226 3227 3228 3229 3230 3231 3232 3233 3234 3235 3236 3237 3238 3239 3240 3241 3242 3243 3244 3245 3246 3247 3248 3249 3250 3251 3252 3253 3254 3255 3256 3257 3258 3259 3260 3261 3262 3263 3264 3265 3266 3267 3268 3269 3270 3271 3272 3273 3274 3275 3276 3277 3278 3279 3280 3281 3282 3283 3284 3285 3286 3287 3288 3289 3290 3291 3292 3293 3294 3295 3296 3297 3298 3299 3300 3301 3302 3303 3304 3305 3306 3307 3308 3309 3310 3311 3312 3313 3314 3315 3316 3317 3318 3319 3320 3321 3322 3323 3324 3325 3326 3327 3328 3329 3330 3331 3332 3333 3334 3335 3336 3337 3338 3339 3340 3341 3342 3343 3344 3345 3346 3347 3348 3349 3350 3351 3352 3353 3354 3355 3356 3357 3358 3359 3360 3361 3362 3363 3364 3365 3366 3367 3368 3369 3370 3371 3372 3373 3374 3375 3376 3377 3378 3379 3380 3381 3382 3383 3384 3385 3386 3387 3388 3389 3390 3391 3392 3393 3394 3395 3396 3397 3398 3399 3400 3401 3402 3403 3404 3405 3406 3407 3408 3409 3410 3411 3412 3413 3414 3415 3416 3417 3418 3419 3420 3421 3422 3423 3424 3425 3426 3427 3428 3429 3430 3431 3432 3433 3434 3435 3436 3437 3438 3439 3440 3441 3442 3443 3444 3445 3446 3447 3448 3449 3450 3451 3452 3453 3454 3455 3456 3457 3458 3459 3460 3461 3462 3463 3464 3465 3466 3467 3468 3469 3470 3471 3472 3473 3474 3475 3476 3477 3478 3479 3480 3481 3482 3483 3484 3485 3486 3487 3488 3489 3490 3491 3492 3493 3494 3495 3496 3497 3498 3499 3500 3501 3502 3503 3504 3505 3506 3507 3508 3509 3510 3511 3512 3513 3514 3515 3516 3517 3518 3519 3520 3521 3522 3523 3524 3525 3526 3527 3528 3529 3530 3531 3532 3533 3534 3535 3536 3537 3538 3539 3540 3541 3542 3543 3544 3545 3546 3547 3548 3549 3550 3551 3552 3553 3554 3555 3556 3557 3558 3559 3560 3561 3562 3563 3564 3565 3566 3567 3568 3569 3570 3571 3572 3573 3574 3575 3576 3577 3578 3579 3580 3581 3582 3583 3584 3585 3586 3587 3588 3589 3590 3591 3592 3593 3594 3595 3596 3597 3598 3599 3600 3601 3602 3603 3604 3605 3606 3607 3608 3609 3610 3611 3612 3613 3614 3615 3616 3617 3618 3619 3620 3621 3622 3623 3624 3625 3626 3627 3628 3629 3630 3631 3632 3633 3634 3635 3636 3637 3638 3639 3640 3641 3642 3643 3644 3645 3646 3647 3648 3649 3650 3651 3652 3653 3654 3655 3656 3657 3658 3659 3660 3661 3662 3663 3664 3665 3666 3667 3668 3669 3670 3671 3672 3673 3674 3675 3676 3677 3678 3679 3680 3681 3682 3683 3684 3685 3686 3687 3688 3689 3690 3691 3692 3693 3694 3695 3696 3697 3698 3699 3700 3701 3702 3703 3704 3705 3706 3707 3708 3709 3710 3711 3712 3713 3714 3715 3716 3717 3718 3719 3720 3721 3722 3723 3724 3725 3726 3727 3728 3729 3730 3731 3732 3733 3734 3735 3736 3737 3738 3739 3740 3741 3742 3743 3744 3745 3746 3747 3748 3749 3750 3751 3752 | ;+
; NAME:
; MPFIT
;
; AUTHOR:
; Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
; craigm@lheamail.gsfc.nasa.gov
; UPDATED VERSIONs can be found on my WEB PAGE:
; http://cow.physics.wisc.edu/~craigm/idl/idl.html
;
; PURPOSE:
; Perform Levenberg-Marquardt least-squares minimization (MINPACK-1)
;
; MAJOR TOPICS:
; Curve and Surface Fitting
;
; CALLING SEQUENCE:
; parms = MPFIT(MYFUNCT, start_parms, FUNCTARGS=fcnargs, NFEV=nfev,
; MAXITER=maxiter, ERRMSG=errmsg, NPRINT=nprint, QUIET=quiet,
; FTOL=ftol, XTOL=xtol, GTOL=gtol, NITER=niter,
; STATUS=status, ITERPROC=iterproc, ITERARGS=iterargs,
; COVAR=covar, PERROR=perror, BESTNORM=bestnorm,
; PARINFO=parinfo)
;
; DESCRIPTION:
;
; MPFIT uses the Levenberg-Marquardt technique to solve the
; least-squares problem. In its typical use, MPFIT will be used to
; fit a user-supplied function (the "model") to user-supplied data
; points (the "data") by adjusting a set of parameters. MPFIT is
; based upon MINPACK-1 (LMDIF.F) by More' and collaborators.
;
; For example, a researcher may think that a set of observed data
; points is best modelled with a Gaussian curve. A Gaussian curve is
; parameterized by its mean, standard deviation and normalization.
; MPFIT will, within certain constraints, find the set of parameters
; which best fits the data. The fit is "best" in the least-squares
; sense; that is, the sum of the weighted squared differences between
; the model and data is minimized.
;
; The Levenberg-Marquardt technique is a particular strategy for
; iteratively searching for the best fit. This particular
; implementation is drawn from MINPACK-1 (see NETLIB), and seems to
; be more robust than routines provided with IDL. This version
; allows upper and lower bounding constraints to be placed on each
; parameter, or the parameter can be held fixed.
;
; The IDL user-supplied function should return an array of weighted
; deviations between model and data. In a typical scientific problem
; the residuals should be weighted so that each deviate has a
; gaussian sigma of 1.0. If X represents values of the independent
; variable, Y represents a measurement for each value of X, and ERR
; represents the error in the measurements, then the deviates could
; be calculated as follows:
;
; DEVIATES = (Y - F(X)) / ERR
;
; where F is the function representing the model. You are
; recommended to use the convenience functions MPFITFUN and
; MPFITEXPR, which are driver functions that calculate the deviates
; for you. If ERR are the 1-sigma uncertainties in Y, then
;
; TOTAL( DEVIATES^2 )
;
; will be the total chi-squared value. MPFIT will minimize the
; chi-square value. The values of X, Y and ERR are passed through
; MPFIT to the user-supplied function via the FUNCTARGS keyword.
;
; Simple constraints can be placed on parameter values by using the
; PARINFO keyword to MPFIT. See below for a description of this
; keyword.
;
; MPFIT does not perform more general optimization tasks. See TNMIN
; instead. MPFIT is customized, based on MINPACK-1, to the
; least-squares minimization problem.
;
; USER FUNCTION
;
; The user must define a function which returns the appropriate
; values as specified above. The function should return the weighted
; deviations between the model and the data. For applications which
; use finite-difference derivatives -- the default -- the user
; function should be declared in the following way:
;
; FUNCTION MYFUNCT, p, X=x, Y=y, ERR=err
; ; Parameter values are passed in "p"
; model = F(x, p)
; return, (y-model)/err
; END
;
; See below for applications with explicit derivatives.
;
; The keyword parameters X, Y, and ERR in the example above are
; suggestive but not required. Any parameters can be passed to
; MYFUNCT by using the FUNCTARGS keyword to MPFIT. Use MPFITFUN and
; MPFITEXPR if you need ideas on how to do that. The function *must*
; accept a parameter list, P.
;
; In general there are no restrictions on the number of dimensions in
; X, Y or ERR. However the deviates *must* be returned in a
; one-dimensional array, and must have the same type (float or
; double) as the input arrays.
;
; See below for error reporting mechanisms.
;
;
; CHECKING STATUS AND HANNDLING ERRORS
;
; Upon return, MPFIT will report the status of the fitting operation
; in the STATUS and ERRMSG keywords. The STATUS keyword will contain
; a numerical code which indicates the success or failure status.
; Generally speaking, any value 1 or greater indicates success, while
; a value of 0 or less indicates a possible failure. The ERRMSG
; keyword will contain a text string which should describe the error
; condition more fully.
;
; By default, MPFIT will trap fatal errors and report them to the
; caller gracefully. However, during the debugging process, it is
; often useful to halt execution where the error occurred. When you
; set the NOCATCH keyword, MPFIT will not do any special error
; trapping, and execution will stop whereever the error occurred.
;
; MPFIT does not explicitly change the !ERROR_STATE variable
; (although it may be changed implicitly if MPFIT calls MESSAGE). It
; is the caller's responsibility to call MESSAGE, /RESET to ensure
; that the error state is initialized before calling MPFIT.
;
; User functions may also indicate non-fatal error conditions using
; the ERROR_CODE common block variable, as described below under the
; MPFIT_ERROR common block definition (by setting ERROR_CODE to a
; number between -15 and -1). When the user function sets an error
; condition via ERROR_CODE, MPFIT will gracefully exit immediately
; and report this condition to the caller. The ERROR_CODE is
; returned in the STATUS keyword in that case.
;
;
; EXPLICIT DERIVATIVES
;
; In the search for the best-fit solution, MPFIT by default
; calculates derivatives numerically via a finite difference
; approximation. The user-supplied function need not calculate the
; derivatives explicitly. However, the user function *may* calculate
; the derivatives if desired, but only if the model function is
; declared with an additional position parameter, DP, as described
; below. If the user function does not accept this additional
; parameter, MPFIT will report an error. As a practical matter, it
; is often sufficient and even faster to allow MPFIT to calculate the
; derivatives numerically, but this option is available for users who
; wish more control over the fitting process.
;
; There are two ways to enable explicit derivatives. First, the user
; can set the keyword AUTODERIVATIVE=0, which is a global switch for
; all parameters. In this case, MPFIT will request explicit
; derivatives for every free parameter.
;
; Second, the user may request explicit derivatives for specifically
; selected parameters using the PARINFO.MPSIDE=3 (see "CONSTRAINING
; PARAMETER VALUES WITH THE PARINFO KEYWORD" below). In this
; strategy, the user picks and chooses which parameter derivatives
; are computed explicitly versus numerically. When PARINFO[i].MPSIDE
; EQ 3, then the ith parameter derivative is computed explicitly.
;
; The keyword setting AUTODERIVATIVE=0 always globally overrides the
; individual values of PARINFO.MPSIDE. Setting AUTODERIVATIVE=0 is
; equivalent to resetting PARINFO.MPSIDE=3 for all parameters.
;
; Even if the user requests explicit derivatives for some or all
; parameters, MPFIT will not always request explicit derivatives on
; every user function call.
;
; EXPLICIT DERIVATIVES - CALLING INTERFACE
;
; When AUTODERIVATIVE=0, the user function is responsible for
; calculating the derivatives of the *residuals* with respect to each
; parameter. The user function should be declared as follows:
;
; ;
; ; MYFUNCT - example user function
; ; P - input parameter values (N-element array)
; ; DP - upon input, an N-vector indicating which parameters
; ; to compute derivatives for;
; ; upon output, the user function must return
; ; an ARRAY(M,N) of derivatives in this keyword
; ; (keywords) - any other keywords specified by FUNCTARGS
; ; RETURNS - residual values
; ;
; FUNCTION MYFUNCT, p, dp, X=x, Y=y, ERR=err
; model = F(x, p) ;; Model function
; resid = (y - model)/err ;; Residual calculation (for example)
;
; if n_params() GT 1 then begin
; ; Create derivative and compute derivative array
; requested = dp ; Save original value of DP
; dp = make_array(n_elements(x), n_elements(p), value=x[0]*0)
;
; ; Compute derivative if requested by caller
; for i = 0, n_elements(p)-1 do if requested(i) NE 0 then $
; dp(*,i) = FGRAD(x, p, i) / err
; endif
;
; return, resid
; END
;
; where FGRAD(x, p, i) is a model function which computes the
; derivative of the model F(x,p) with respect to parameter P(i) at X.
;
; A quirk in the implementation leaves a stray negative sign in the
; definition of DP. The derivative of the *residual* should be
; "-FGRAD(x,p,i) / err" because of how the residual is defined
; ("resid = (data - model) / err"). **HOWEVER** because of the
; implementation quirk, MPFIT expects FGRAD(x,p,i)/err instead,
; i.e. the opposite sign of the gradient of RESID.
;
; Derivatives should be returned in the DP array. DP should be an
; ARRAY(m,n) array, where m is the number of data points and n is the
; number of parameters. -DP[i,j] is the derivative of the ith
; residual with respect to the jth parameter (note the minus sign
; due to the quirk described above).
;
; As noted above, MPFIT may not always request derivatives from the
; user function. In those cases, the parameter DP is not passed.
; Therefore functions can use N_PARAMS() to indicate whether they
; must compute the derivatives or not.
;
; The derivatives with respect to fixed parameters are ignored; zero
; is an appropriate value to insert for those derivatives. Upon
; input to the user function, DP is set to a vector with the same
; length as P, with a value of 1 for a parameter which is free, and a
; value of zero for a parameter which is fixed (and hence no
; derivative needs to be calculated). This input vector may be
; overwritten as needed. In the example above, the original DP
; vector is saved to a variable called REQUESTED, and used as a mask
; to calculate only those derivatives that are required.
;
; If the data is higher than one dimensional, then the *last*
; dimension should be the parameter dimension. Example: fitting a
; 50x50 image, "dp" should be 50x50xNPAR.
;
; EXPLICIT DERIVATIVES - TESTING and DEBUGGING
;
; For reasonably complicated user functions, the calculation of
; explicit derivatives of the correct sign and magnitude can be
; difficult to get right. A simple sign error can cause MPFIT to be
; confused. MPFIT has a derivative debugging mode which will compute
; the derivatives *both* numerically and explicitly, and compare the
; results.
;
; It is expected that during production usage, derivative debugging
; should be disabled for all parameters.
;
; In order to enable derivative debugging mode, set the following
; PARINFO members for the ith parameter.
; PARINFO[i].MPSIDE = 3 ; Enable explicit derivatives
; PARINFO[i].MPDERIV_DEBUG = 1 ; Enable derivative debugging mode
; PARINFO[i].MPDERIV_RELTOL = ?? ; Relative tolerance for comparison
; PARINFO[i].MPDERIV_ABSTOL = ?? ; Absolute tolerance for comparison
; Note that these settings are maintained on a parameter-by-parameter
; basis using PARINFO, so the user can choose which parameters
; derivatives will be tested.
;
; When .MPDERIV_DEBUG is set, then MPFIT first computes the
; derivative explicitly by requesting them from the user function.
; Then, it computes the derivatives numerically via finite
; differencing, and compares the two values. If the difference
; exceeds a tolerance threshold, then the values are printed out to
; alert the user. The tolerance level threshold contains both a
; relative and an absolute component, and is expressed as,
;
; ABS(DERIV_U - DERIV_N) GE (ABSTOL + RELTOL*ABS(DERIV_U))
;
; where DERIV_U and DERIV_N are the derivatives computed explicitly
; and numerically, respectively. Appropriate values
; for most users will be:
;
; PARINFO[i].MPDERIV_RELTOL = 1d-3 ;; Suggested relative tolerance
; PARINFO[i].MPDERIV_ABSTOL = 1d-7 ;; Suggested absolute tolerance
;
; although these thresholds may have to be adjusted for a particular
; problem. When the threshold is exceeded, users can expect to see a
; tabular report like this one:
;
; FJAC DEBUG BEGIN
; # IPNT FUNC DERIV_U DERIV_N DIFF_ABS DIFF_REL
; FJAC PARM 2
; 80 -0.7308 0.04233 0.04233 -5.543E-07 -1.309E-05
; 99 1.370 0.01417 0.01417 -5.518E-07 -3.895E-05
; 118 0.07187 -0.01400 -0.01400 -5.566E-07 3.977E-05
; 137 1.844 -0.04216 -0.04216 -5.589E-07 1.326E-05
; FJAC DEBUG END
;
; The report will be bracketed by FJAC DEBUG BEGIN/END statements.
; Each parameter will be delimited by the statement FJAC PARM n,
; where n is the parameter number. The columns are,
;
; IPNT - data point number (0 ... M-1)
; FUNC - function value at that point
; DERIV_U - explicit derivative value at that point
; DERIV_N - numerical derivative estimate at that point
; DIFF_ABS - absolute difference = (DERIV_U - DERIV_N)
; DIFF_REL - relative difference = (DIFF_ABS)/(DERIV_U)
;
; When prints appear in this report, it is most important to check
; that the derivatives computed in two different ways have the same
; numerical sign and the same order of magnitude, since these are the
; most common programming mistakes.
;
; A line of this form may also appear
;
; # FJAC_MASK = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
;
; This line indicates for which parameters explicit derivatives are
; expected. A list of all-1s indicates all explicit derivatives for
; all parameters are requested from the user function.
;
;
; CONSTRAINING PARAMETER VALUES WITH THE PARINFO KEYWORD
;
; The behavior of MPFIT can be modified with respect to each
; parameter to be fitted. A parameter value can be fixed; simple
; boundary constraints can be imposed; limitations on the parameter
; changes can be imposed; properties of the automatic derivative can
; be modified; and parameters can be tied to one another.
;
; These properties are governed by the PARINFO structure, which is
; passed as a keyword parameter to MPFIT.
;
; PARINFO should be an array of structures, one for each parameter.
; Each parameter is associated with one element of the array, in
; numerical order. The structure can have the following entries
; (none are required):
;
; .VALUE - the starting parameter value (but see the START_PARAMS
; parameter for more information).
;
; .FIXED - a boolean value, whether the parameter is to be held
; fixed or not. Fixed parameters are not varied by
; MPFIT, but are passed on to MYFUNCT for evaluation.
;
; .LIMITED - a two-element boolean array. If the first/second
; element is set, then the parameter is bounded on the
; lower/upper side. A parameter can be bounded on both
; sides. Both LIMITED and LIMITS must be given
; together.
;
; .LIMITS - a two-element float or double array. Gives the
; parameter limits on the lower and upper sides,
; respectively. Zero, one or two of these values can be
; set, depending on the values of LIMITED. Both LIMITED
; and LIMITS must be given together.
;
; .PARNAME - a string, giving the name of the parameter. The
; fitting code of MPFIT does not use this tag in any
; way. However, the default ITERPROC will print the
; parameter name if available.
;
; .STEP - the step size to be used in calculating the numerical
; derivatives. If set to zero, then the step size is
; computed automatically. Ignored when AUTODERIVATIVE=0.
; This value is superceded by the RELSTEP value.
;
; .RELSTEP - the *relative* step size to be used in calculating
; the numerical derivatives. This number is the
; fractional size of the step, compared to the
; parameter value. This value supercedes the STEP
; setting. If the parameter is zero, then a default
; step size is chosen.
;
; .MPSIDE - selector for type of derivative calculation. This
; field can take one of five possible values:
;
; 0 - one-sided derivative computed automatically
; 1 - one-sided derivative (f(x+h) - f(x) )/h
; -1 - one-sided derivative (f(x) - f(x-h))/h
; 2 - two-sided derivative (f(x+h) - f(x-h))/(2*h)
; 3 - explicit derivative used for this parameter
;
; In the first four cases, the derivative is approximated
; numerically by finite difference, with step size
; H=STEP, where the STEP parameter is defined above. The
; last case, MPSIDE=3, indicates to allow the user
; function to compute the derivative explicitly (see
; section on "EXPLICIT DERIVATIVES"). AUTODERIVATIVE=0
; overrides this setting for all parameters, and is
; equivalent to MPSIDE=3 for all parameters. For
; MPSIDE=0, the "automatic" one-sided derivative method
; will chose a direction for the finite difference which
; does not violate any constraints. The other methods
; (MPSIDE=-1 or MPSIDE=1) do not perform this check. The
; two-sided method is in principle more precise, but
; requires twice as many function evaluations. Default:
; 0.
;
; .MPDERIV_DEBUG - set this value to 1 to enable debugging of
; user-supplied explicit derivatives (see "TESTING and
; DEBUGGING" section above). In addition, the
; user must enable calculation of explicit derivatives by
; either setting AUTODERIVATIVE=0, or MPSIDE=3 for the
; desired parameters. When this option is enabled, a
; report may be printed to the console, depending on the
; MPDERIV_ABSTOL and MPDERIV_RELTOL settings.
; Default: 0 (no debugging)
;
;
; .MPDERIV_ABSTOL, .MPDERIV_RELTOL - tolerance settings for
; print-out of debugging information, for each parameter
; where debugging is enabled. See "TESTING and
; DEBUGGING" section above for the meanings of these two
; fields.
;
;
; .MPMAXSTEP - the maximum change to be made in the parameter
; value. During the fitting process, the parameter
; will never be changed by more than this value in
; one iteration.
;
; A value of 0 indicates no maximum. Default: 0.
;
; .TIED - a string expression which "ties" the parameter to other
; free or fixed parameters as an equality constraint. Any
; expression involving constants and the parameter array P
; are permitted.
; Example: if parameter 2 is always to be twice parameter
; 1 then use the following: parinfo[2].tied = '2 * P[1]'.
; Since they are totally constrained, tied parameters are
; considered to be fixed; no errors are computed for them,
; and any LIMITS are not obeyed.
; [ NOTE: the PARNAME can't be used in a TIED expression. ]
;
; .MPPRINT - if set to 1, then the default ITERPROC will print the
; parameter value. If set to 0, the parameter value
; will not be printed. This tag can be used to
; selectively print only a few parameter values out of
; many. Default: 1 (all parameters printed)
;
; .MPFORMAT - IDL format string to print the parameter within
; ITERPROC. Default: '(G20.6)' (An empty string will
; also use the default.)
;
; Future modifications to the PARINFO structure, if any, will involve
; adding structure tags beginning with the two letters "MP".
; Therefore programmers are urged to avoid using tags starting with
; "MP", but otherwise they are free to include their own fields
; within the PARINFO structure, which will be ignored by MPFIT.
;
; PARINFO Example:
; parinfo = replicate({value:0.D, fixed:0, limited:[0,0], $
; limits:[0.D,0]}, 5)
; parinfo[0].fixed = 1
; parinfo[4].limited[0] = 1
; parinfo[4].limits[0] = 50.D
; parinfo[*].value = [5.7D, 2.2, 500., 1.5, 2000.]
;
; A total of 5 parameters, with starting values of 5.7,
; 2.2, 500, 1.5, and 2000 are given. The first parameter
; is fixed at a value of 5.7, and the last parameter is
; constrained to be above 50.
;
;
; RECURSION
;
; Generally, recursion is not allowed. As of version 1.77, MPFIT has
; recursion protection which does not allow a model function to
; itself call MPFIT. Users who wish to perform multi-level
; optimization should investigate the 'EXTERNAL' function evaluation
; methods described below for hard-to-evaluate functions. That
; method places more control in the user's hands. The user can
; design a "recursive" application by taking care.
;
; In most cases the recursion protection should be well-behaved.
; However, if the user is doing debugging, it is possible for the
; protection system to get "stuck." In order to reset it, run the
; procedure:
; MPFIT_RESET_RECURSION
; and the protection system should get "unstuck." It is save to call
; this procedure at any time.
;
;
; COMPATIBILITY
;
; This function is designed to work with IDL 5.0 or greater.
;
; Because TIED parameters and the "(EXTERNAL)" user-model feature use
; the EXECUTE() function, they cannot be used with the free version
; of the IDL Virtual Machine.
;
;
; DETERMINING THE VERSION OF MPFIT
;
; MPFIT is a changing library. Users of MPFIT may also depend on a
; specific version of the library being present. As of version 1.70
; of MPFIT, a VERSION keyword has been added which allows the user to
; query which version is present. The keyword works like this:
;
; RESULT = MPFIT(/query, VERSION=version)
;
; This call uses the /QUERY keyword to query the version number
; without performing any computations. Users of MPFIT can call this
; method to determine which version is in the IDL path before
; actually using MPFIT to do any numerical work. Upon return, the
; VERSION keyword contains the version number of MPFIT, expressed as
; a string of the form 'X.Y' where X and Y are integers.
;
; Users can perform their own version checking, or use the built-in
; error checking of MPFIT. The MIN_VERSION keyword enforces the
; requested minimum version number. For example,
;
; RESULT = MPFIT(/query, VERSION=version, MIN_VERSION='1.70')
;
; will check whether the accessed version is 1.70 or greater, without
; performing any numerical processing.
;
; The VERSION and MIN_VERSION keywords were added in MPFIT
; version 1.70 and later. If the caller attempts to use the VERSION
; or MIN_VERSION keywords, and an *older* version of the code is
; present in the caller's path, then IDL will throw an 'unknown
; keyword' error. Therefore, in order to be robust, the caller, must
; use exception handling. Here is an example demanding at least
; version 1.70.
;
; MPFIT_OK = 0 & VERSION = '<unknown>'
; CATCH, CATCHERR
; IF CATCHERR EQ 0 THEN MPFIT_OK = MPFIT(/query, VERSION=version, $
; MIN_VERSION='1.70')
; CATCH, /CANCEL
;
; IF NOT MPFIT_OK THEN $
; MESSAGE, 'ERROR: you must have MPFIT version 1.70 or higher in '+$
; 'your path (found version '+version+')'
;
; Of course, the caller can also do its own version number
; requirements checking.
;
;
; HARD-TO-COMPUTE FUNCTIONS: "EXTERNAL" EVALUATION
;
; The normal mode of operation for MPFIT is for the user to pass a
; function name, and MPFIT will call the user function multiple times
; as it iterates toward a solution.
;
; Some user functions are particularly hard to compute using the
; standard model of MPFIT. Usually these are functions that depend
; on a large amount of external data, and so it is not feasible, or
; at least highly impractical, to have MPFIT call it. In those cases
; it may be possible to use the "(EXTERNAL)" evaluation option.
;
; In this case the user is responsible for making all function *and
; derivative* evaluations. The function and Jacobian data are passed
; in through the EXTERNAL_FVEC and EXTERNAL_FJAC keywords,
; respectively. The user indicates the selection of this option by
; specifying a function name (MYFUNCT) of "(EXTERNAL)". No
; user-function calls are made when EXTERNAL evaluation is being
; used.
;
; ** SPECIAL NOTE ** For the "(EXTERNAL)" case, the quirk noted above
; does not apply. The gradient matrix, EXTERNAL_FJAC, should be
; comparable to "-FGRAD(x,p)/err", which is the *opposite* sign of
; the DP matrix described above. In other words, EXTERNAL_FJAC
; has the same sign as the derivative of EXTERNAL_FVEC, and the
; opposite sign of FGRAD.
;
; At the end of each iteration, control returns to the user, who must
; reevaluate the function at its new parameter values. Users should
; check the return value of the STATUS keyword, where a value of 9
; indicates the user should supply more data for the next iteration,
; and re-call MPFIT. The user may refrain from calling MPFIT
; further; as usual, STATUS will indicate when the solution has
; converged and no more iterations are required.
;
; Because MPFIT must maintain its own data structures between calls,
; the user must also pass a named variable to the EXTERNAL_STATE
; keyword. This variable must be maintained by the user, but not
; changed, throughout the fitting process. When no more iterations
; are desired, the named variable may be discarded.
;
;
; INPUTS:
; MYFUNCT - a string variable containing the name of the function to
; be minimized. The function should return the weighted
; deviations between the model and the data, as described
; above.
;
; For EXTERNAL evaluation of functions, this parameter
; should be set to a value of "(EXTERNAL)".
;
; START_PARAMS - An one-dimensional array of starting values for each of the
; parameters of the model. The number of parameters
; should be fewer than the number of measurements.
; Also, the parameters should have the same data type
; as the measurements (double is preferred).
;
; This parameter is optional if the PARINFO keyword
; is used (but see PARINFO). The PARINFO keyword
; provides a mechanism to fix or constrain individual
; parameters. If both START_PARAMS and PARINFO are
; passed, then the starting *value* is taken from
; START_PARAMS, but the *constraints* are taken from
; PARINFO.
;
; RETURNS:
;
; Returns the array of best-fit parameters.
; Exceptions:
; * if /QUERY is set (see QUERY).
;
;
; KEYWORD PARAMETERS:
;
; AUTODERIVATIVE - If this is set, derivatives of the function will
; be computed automatically via a finite
; differencing procedure. If not set, then MYFUNCT
; must provide the explicit derivatives.
; Default: set (=1)
; NOTE: to supply your own explicit derivatives,
; explicitly pass AUTODERIVATIVE=0
;
; BESTNORM - upon return, the value of the summed squared weighted
; residuals for the returned parameter values,
; i.e. TOTAL(DEVIATES^2).
;
; BEST_FJAC - upon return, BEST_FJAC contains the Jacobian, or
; partial derivative, matrix for the best-fit model.
; The values are an array,
; ARRAY(N_ELEMENTS(DEVIATES),NFREE) where NFREE is the
; number of free parameters. This array is only
; computed if /CALC_FJAC is set, otherwise BEST_FJAC is
; undefined.
;
; The returned array is such that BEST_FJAC[I,J] is the
; partial derivative of DEVIATES[I] with respect to
; parameter PARMS[PFREE_INDEX[J]]. Note that since
; deviates are (data-model)*weight, the Jacobian of the
; *deviates* will have the opposite sign from the
; Jacobian of the *model*, and may be scaled by a
; factor.
;
; BEST_RESID - upon return, an array of best-fit deviates.
;
; CALC_FJAC - if set, then calculate the Jacobian and return it in
; BEST_FJAC. If not set, then the return value of
; BEST_FJAC is undefined.
;
; COVAR - the covariance matrix for the set of parameters returned
; by MPFIT. The matrix is NxN where N is the number of
; parameters. The square root of the diagonal elements
; gives the formal 1-sigma statistical errors on the
; parameters IF errors were treated "properly" in MYFUNC.
; Parameter errors are also returned in PERROR.
;
; To compute the correlation matrix, PCOR, use this example:
; PCOR = COV * 0
; FOR i = 0, n-1 DO FOR j = 0, n-1 DO $
; PCOR[i,j] = COV[i,j]/sqrt(COV[i,i]*COV[j,j])
; or equivalently, in vector notation,
; PCOR = COV / (PERROR # PERROR)
;
; If NOCOVAR is set or MPFIT terminated abnormally, then
; COVAR is set to a scalar with value !VALUES.D_NAN.
;
; DOF - number of degrees of freedom, computed as
; DOF = N_ELEMENTS(DEVIATES) - NFREE
; Note that this doesn't account for pegged parameters (see
; NPEGGED). It also does not account for data points which
; are assigned zero weight by the user function.
;
; ERRMSG - a string error or warning message is returned.
;
; EXTERNAL_FVEC - upon input, the function values, evaluated at
; START_PARAMS. This should be an M-vector, where M
; is the number of data points.
;
; EXTERNAL_FJAC - upon input, the Jacobian array of partial
; derivative values. This should be a M x N array,
; where M is the number of data points and N is the
; number of parameters. NOTE: that all FIXED or
; TIED parameters must *not* be included in this
; array.
;
; EXTERNAL_STATE - a named variable to store MPFIT-related state
; information between iterations (used in input and
; output to MPFIT). The user must not manipulate
; or discard this data until the final iteration is
; performed.
;
; FASTNORM - set this keyword to select a faster algorithm to
; compute sum-of-square values internally. For systems
; with large numbers of data points, the standard
; algorithm can become prohibitively slow because it
; cannot be vectorized well. By setting this keyword,
; MPFIT will run faster, but it will be more prone to
; floating point overflows and underflows. Thus, setting
; this keyword may sacrifice some stability in the
; fitting process.
;
; FTOL - a nonnegative input variable. Termination occurs when both
; the actual and predicted relative reductions in the sum of
; squares are at most FTOL (and STATUS is accordingly set to
; 1 or 3). Therefore, FTOL measures the relative error
; desired in the sum of squares. Default: 1D-10
;
; FUNCTARGS - A structure which contains the parameters to be passed
; to the user-supplied function specified by MYFUNCT via
; the _EXTRA mechanism. This is the way you can pass
; additional data to your user-supplied function without
; using common blocks.
;
; Consider the following example:
; if FUNCTARGS = { XVAL:[1.D,2,3], YVAL:[1.D,4,9],
; ERRVAL:[1.D,1,1] }
; then the user supplied function should be declared
; like this:
; FUNCTION MYFUNCT, P, XVAL=x, YVAL=y, ERRVAL=err
;
; By default, no extra parameters are passed to the
; user-supplied function, but your function should
; accept *at least* one keyword parameter. [ This is to
; accomodate a limitation in IDL's _EXTRA
; parameter-passing mechanism. ]
;
; GTOL - a nonnegative input variable. Termination occurs when the
; cosine of the angle between fvec and any column of the
; jacobian is at most GTOL in absolute value (and STATUS is
; accordingly set to 4). Therefore, GTOL measures the
; orthogonality desired between the function vector and the
; columns of the jacobian. Default: 1D-10
;
; ITERARGS - The keyword arguments to be passed to ITERPROC via the
; _EXTRA mechanism. This should be a structure, and is
; similar in operation to FUNCTARGS.
; Default: no arguments are passed.
;
; ITERPRINT - The name of an IDL procedure, equivalent to PRINT,
; that ITERPROC will use to render output. ITERPRINT
; should be able to accept at least four positional
; arguments. In addition, it should be able to accept
; the standard FORMAT keyword for output formatting; and
; the UNIT keyword, to redirect output to a logical file
; unit (default should be UNIT=1, standard output).
; These keywords are passed using the ITERARGS keyword
; above. The ITERPRINT procedure must accept the _EXTRA
; keyword.
; NOTE: that much formatting can be handled with the
; MPPRINT and MPFORMAT tags.
; Default: 'MPFIT_DEFPRINT' (default internal formatter)
;
; ITERPROC - The name of a procedure to be called upon each NPRINT
; iteration of the MPFIT routine. ITERPROC is always
; called in the final iteration. It should be declared
; in the following way:
;
; PRO ITERPROC, MYFUNCT, p, iter, fnorm, FUNCTARGS=fcnargs, $
; PARINFO=parinfo, QUIET=quiet, DOF=dof, PFORMAT=pformat, $
; UNIT=unit, ...
; ; perform custom iteration update
; END
;
; ITERPROC must either accept all three keyword
; parameters (FUNCTARGS, PARINFO and QUIET), or at least
; accept them via the _EXTRA keyword.
;
; MYFUNCT is the user-supplied function to be minimized,
; P is the current set of model parameters, ITER is the
; iteration number, and FUNCTARGS are the arguments to be
; passed to MYFUNCT. FNORM should be the chi-squared
; value. QUIET is set when no textual output should be
; printed. DOF is the number of degrees of freedom,
; normally the number of points less the number of free
; parameters. See below for documentation of PARINFO.
; PFORMAT is the default parameter value format. UNIT is
; passed on to the ITERPRINT procedure, and should
; indicate the file unit where log output will be sent
; (default: standard output).
;
; In implementation, ITERPROC can perform updates to the
; terminal or graphical user interface, to provide
; feedback while the fit proceeds. If the fit is to be
; stopped for any reason, then ITERPROC should set the
; common block variable ERROR_CODE to negative value
; between -15 and -1 (see MPFIT_ERROR common block
; below). In principle, ITERPROC should probably not
; modify the parameter values, because it may interfere
; with the algorithm's stability. In practice it is
; allowed.
;
; Default: an internal routine is used to print the
; parameter values.
;
; ITERSTOP - Set this keyword if you wish to be able to stop the
; fitting by hitting the predefined ITERKEYSTOP key on
; the keyboard. This only works if you use the default
; ITERPROC.
;
; ITERKEYSTOP - A keyboard key which will halt the fit (and if
; ITERSTOP is set and the default ITERPROC is used).
; ITERSTOPKEY may either be a one-character string
; with the desired key, or a scalar integer giving the
; ASCII code of the desired key.
; Default: 7b (control-g)
;
; NOTE: the default value of ASCI 7 (control-G) cannot
; be read in some windowing environments, so you must
; change to a printable character like 'q'.
;
; MAXITER - The maximum number of iterations to perform. If the
; number of calculation iterations exceeds MAXITER, then
; the STATUS value is set to 5 and MPFIT returns.
;
; If MAXITER EQ 0, then MPFIT does not iterate to adjust
; parameter values; however, the user function is evaluated
; and parameter errors/covariance/Jacobian are estimated
; before returning.
; Default: 200 iterations
;
; MIN_VERSION - The minimum requested version number. This must be
; a scalar string of the form returned by the VERSION
; keyword. If the current version of MPFIT does not
; satisfy the minimum requested version number, then,
; MPFIT(/query, min_version='...') returns 0
; MPFIT(...) returns NAN
; Default: no version number check
; NOTE: MIN_VERSION was added in MPFIT version 1.70
;
; NFEV - the number of MYFUNCT function evaluations performed.
;
; NFREE - the number of free parameters in the fit. This includes
; parameters which are not FIXED and not TIED, but it does
; include parameters which are pegged at LIMITS.
;
; NITER - the number of iterations completed.
;
; NOCATCH - if set, then MPFIT will not perform any error trapping.
; By default (not set), MPFIT will trap errors and report
; them to the caller. This keyword will typically be used
; for debugging.
;
; NOCOVAR - set this keyword to prevent the calculation of the
; covariance matrix before returning (see COVAR)
;
; NPEGGED - the number of free parameters which are pegged at a
; LIMIT.
;
; NPRINT - The frequency with which ITERPROC is called. A value of
; 1 indicates that ITERPROC is called with every iteration,
; while 2 indicates every other iteration, etc. Be aware
; that several Levenberg-Marquardt attempts can be made in
; a single iteration. Also, the ITERPROC is *always*
; called for the final iteration, regardless of the
; iteration number.
; Default value: 1
;
; PARINFO - A one-dimensional array of structures.
; Provides a mechanism for more sophisticated constraints
; to be placed on parameter values. When PARINFO is not
; passed, then it is assumed that all parameters are free
; and unconstrained. Values in PARINFO are never
; modified during a call to MPFIT.
;
; See description above for the structure of PARINFO.
;
; Default value: all parameters are free and unconstrained.
;
; PERROR - The formal 1-sigma errors in each parameter, computed
; from the covariance matrix. If a parameter is held
; fixed, or if it touches a boundary, then the error is
; reported as zero.
;
; If the fit is unweighted (i.e. no errors were given, or
; the weights were uniformly set to unity), then PERROR
; will probably not represent the true parameter
; uncertainties.
;
; *If* you can assume that the true reduced chi-squared
; value is unity -- meaning that the fit is implicitly
; assumed to be of good quality -- then the estimated
; parameter uncertainties can be computed by scaling PERROR
; by the measured chi-squared value.
;
; DOF = N_ELEMENTS(X) - N_ELEMENTS(PARMS) ; deg of freedom
; PCERROR = PERROR * SQRT(BESTNORM / DOF) ; scaled uncertainties
;
; PFREE_INDEX - upon return, PFREE_INDEX contains an index array
; which indicates which parameter were allowed to
; vary. I.e. of all the parameters PARMS, only
; PARMS[PFREE_INDEX] were varied.
;
; QUERY - if set, then MPFIT() will return immediately with one of
; the following values:
; 1 - if MIN_VERSION is not set
; 1 - if MIN_VERSION is set and MPFIT satisfies the minimum
; 0 - if MIN_VERSION is set and MPFIT does not satisfy it
; The VERSION output keyword is always set upon return.
; Default: not set.
;
; QUIET - set this keyword when no textual output should be printed
; by MPFIT
;
; RESDAMP - a scalar number, indicating the cut-off value of
; residuals where "damping" will occur. Residuals with
; magnitudes greater than this number will be replaced by
; their logarithm. This partially mitigates the so-called
; large residual problem inherent in least-squares solvers
; (as for the test problem CURVI, http://www.maxthis.com/-
; curviex.htm). A value of 0 indicates no damping.
; Default: 0
;
; Note: RESDAMP doesn't work with AUTODERIV=0
;
; STATUS - an integer status code is returned. All values greater
; than zero can represent success (however STATUS EQ 5 may
; indicate failure to converge). It can have one of the
; following values:
;
; -18 a fatal execution error has occurred. More information
; may be available in the ERRMSG string.
;
; -16 a parameter or function value has become infinite or an
; undefined number. This is usually a consequence of
; numerical overflow in the user's model function, which
; must be avoided.
;
; -15 to -1
; these are error codes that either MYFUNCT or ITERPROC
; may return to terminate the fitting process (see
; description of MPFIT_ERROR common below). If either
; MYFUNCT or ITERPROC set ERROR_CODE to a negative number,
; then that number is returned in STATUS. Values from -15
; to -1 are reserved for the user functions and will not
; clash with MPFIT.
;
; 0 improper input parameters.
;
; 1 both actual and predicted relative reductions
; in the sum of squares are at most FTOL.
;
; 2 relative error between two consecutive iterates
; is at most XTOL
;
; 3 conditions for STATUS = 1 and STATUS = 2 both hold.
;
; 4 the cosine of the angle between fvec and any
; column of the jacobian is at most GTOL in
; absolute value.
;
; 5 the maximum number of iterations has been reached
;
; 6 FTOL is too small. no further reduction in
; the sum of squares is possible.
;
; 7 XTOL is too small. no further improvement in
; the approximate solution x is possible.
;
; 8 GTOL is too small. fvec is orthogonal to the
; columns of the jacobian to machine precision.
;
; 9 A successful single iteration has been completed, and
; the user must supply another "EXTERNAL" evaluation of
; the function and its derivatives. This status indicator
; is neither an error nor a convergence indicator.
;
; VERSION - upon return, VERSION will be set to the MPFIT internal
; version number. The version number will be a string of
; the form "X.Y" where X is a major revision number and Y
; is a minor revision number.
; NOTE: the VERSION keyword was not present before
; MPFIT version number 1.70, therefore, callers must
; use exception handling when using this keyword.
;
; XTOL - a nonnegative input variable. Termination occurs when the
; relative error between two consecutive iterates is at most
; XTOL (and STATUS is accordingly set to 2 or 3). Therefore,
; XTOL measures the relative error desired in the approximate
; solution. Default: 1D-10
;
;
; EXAMPLE:
;
; p0 = [5.7D, 2.2, 500., 1.5, 2000.]
; fa = {X:x, Y:y, ERR:err}
; p = mpfit('MYFUNCT', p0, functargs=fa)
;
; Minimizes sum of squares of MYFUNCT. MYFUNCT is called with the X,
; Y, and ERR keyword parameters that are given by FUNCTARGS. The
; resulting parameter values are returned in p.
;
;
; COMMON BLOCKS:
;
; COMMON MPFIT_ERROR, ERROR_CODE
;
; User routines may stop the fitting process at any time by
; setting an error condition. This condition may be set in either
; the user's model computation routine (MYFUNCT), or in the
; iteration procedure (ITERPROC).
;
; To stop the fitting, the above common block must be declared,
; and ERROR_CODE must be set to a negative number. After the user
; procedure or function returns, MPFIT checks the value of this
; common block variable and exits immediately if the error
; condition has been set. This value is also returned in the
; STATUS keyword: values of -1 through -15 are reserved error
; codes for the user routines. By default the value of ERROR_CODE
; is zero, indicating a successful function/procedure call.
;
; COMMON MPFIT_PROFILE
; COMMON MPFIT_MACHAR
; COMMON MPFIT_CONFIG
;
; These are undocumented common blocks are used internally by
; MPFIT and may change in future implementations.
;
; THEORY OF OPERATION:
;
; There are many specific strategies for function minimization. One
; very popular technique is to use function gradient information to
; realize the local structure of the function. Near a local minimum
; the function value can be taylor expanded about x0 as follows:
;
; f(x) = f(x0) + f'(x0) . (x-x0) + (1/2) (x-x0) . f''(x0) . (x-x0)
; ----- --------------- ------------------------------- (1)
; Order 0th 1st 2nd
;
; Here f'(x) is the gradient vector of f at x, and f''(x) is the
; Hessian matrix of second derivatives of f at x. The vector x is
; the set of function parameters, not the measured data vector. One
; can find the minimum of f, f(xm) using Newton's method, and
; arrives at the following linear equation:
;
; f''(x0) . (xm-x0) = - f'(x0) (2)
;
; If an inverse can be found for f''(x0) then one can solve for
; (xm-x0), the step vector from the current position x0 to the new
; projected minimum. Here the problem has been linearized (ie, the
; gradient information is known to first order). f''(x0) is
; symmetric n x n matrix, and should be positive definite.
;
; The Levenberg - Marquardt technique is a variation on this theme.
; It adds an additional diagonal term to the equation which may aid the
; convergence properties:
;
; (f''(x0) + nu I) . (xm-x0) = -f'(x0) (2a)
;
; where I is the identity matrix. When nu is large, the overall
; matrix is diagonally dominant, and the iterations follow steepest
; descent. When nu is small, the iterations are quadratically
; convergent.
;
; In principle, if f''(x0) and f'(x0) are known then xm-x0 can be
; determined. However the Hessian matrix is often difficult or
; impossible to compute. The gradient f'(x0) may be easier to
; compute, if even by finite difference techniques. So-called
; quasi-Newton techniques attempt to successively estimate f''(x0)
; by building up gradient information as the iterations proceed.
;
; In the least squares problem there are further simplifications
; which assist in solving eqn (2). The function to be minimized is
; a sum of squares:
;
; f = Sum(hi^2) (3)
;
; where hi is the ith residual out of m residuals as described
; above. This can be substituted back into eqn (2) after computing
; the derivatives:
;
; f' = 2 Sum(hi hi')
; f'' = 2 Sum(hi' hj') + 2 Sum(hi hi'') (4)
;
; If one assumes that the parameters are already close enough to a
; minimum, then one typically finds that the second term in f'' is
; negligible [or, in any case, is too difficult to compute]. Thus,
; equation (2) can be solved, at least approximately, using only
; gradient information.
;
; In matrix notation, the combination of eqns (2) and (4) becomes:
;
; hT' . h' . dx = - hT' . h (5)
;
; Where h is the residual vector (length m), hT is its transpose, h'
; is the Jacobian matrix (dimensions n x m), and dx is (xm-x0). The
; user function supplies the residual vector h, and in some cases h'
; when it is not found by finite differences (see MPFIT_FDJAC2,
; which finds h and hT'). Even if dx is not the best absolute step
; to take, it does provide a good estimate of the best *direction*,
; so often a line minimization will occur along the dx vector
; direction.
;
; The method of solution employed by MINPACK is to form the Q . R
; factorization of h', where Q is an orthogonal matrix such that QT .
; Q = I, and R is upper right triangular. Using h' = Q . R and the
; ortogonality of Q, eqn (5) becomes
;
; (RT . QT) . (Q . R) . dx = - (RT . QT) . h
; RT . R . dx = - RT . QT . h (6)
; R . dx = - QT . h
;
; where the last statement follows because R is upper triangular.
; Here, R, QT and h are known so this is a matter of solving for dx.
; The routine MPFIT_QRFAC provides the QR factorization of h, with
; pivoting, and MPFIT_QRSOL;V provides the solution for dx.
;
; REFERENCES:
;
; Markwardt, C. B. 2008, "Non-Linear Least Squares Fitting in IDL
; with MPFIT," in proc. Astronomical Data Analysis Software and
; Systems XVIII, Quebec, Canada, ASP Conference Series, Vol. XXX, eds.
; D. Bohlender, P. Dowler & D. Durand (Astronomical Society of the
; Pacific: San Francisco), p. 251-254 (ISBN: 978-1-58381-702-5)
; http://arxiv.org/abs/0902.2850
; Link to NASA ADS: http://adsabs.harvard.edu/abs/2009ASPC..411..251M
; Link to ASP: http://aspbooks.org/a/volumes/table_of_contents/411
;
; Refer to the MPFIT website as:
; http://purl.com/net/mpfit
;
; MINPACK-1 software, by Jorge More' et al, available from netlib.
; http://www.netlib.org/
;
; "Optimization Software Guide," Jorge More' and Stephen Wright,
; SIAM, *Frontiers in Applied Mathematics*, Number 14.
; (ISBN: 978-0-898713-22-0)
;
; More', J. 1978, "The Levenberg-Marquardt Algorithm: Implementation
; and Theory," in Numerical Analysis, vol. 630, ed. G. A. Watson
; (Springer-Verlag: Berlin), p. 105 (DOI: 10.1007/BFb0067690 )
;
; MODIFICATION HISTORY:
; Translated from MINPACK-1 in FORTRAN, Apr-Jul 1998, CM
; Fixed bug in parameter limits (x vs xnew), 04 Aug 1998, CM
; Added PERROR keyword, 04 Aug 1998, CM
; Added COVAR keyword, 20 Aug 1998, CM
; Added NITER output keyword, 05 Oct 1998
; D.L Windt, Bell Labs, windt@bell-labs.com;
; Made each PARINFO component optional, 05 Oct 1998 CM
; Analytical derivatives allowed via AUTODERIVATIVE keyword, 09 Nov 1998
; Parameter values can be tied to others, 09 Nov 1998
; Fixed small bugs (Wayne Landsman), 24 Nov 1998
; Added better exception error reporting, 24 Nov 1998 CM
; Cosmetic documentation changes, 02 Jan 1999 CM
; Changed definition of ITERPROC to be consistent with TNMIN, 19 Jan 1999 CM
; Fixed bug when AUTDERIVATIVE=0. Incorrect sign, 02 Feb 1999 CM
; Added keyboard stop to MPFIT_DEFITER, 28 Feb 1999 CM
; Cosmetic documentation changes, 14 May 1999 CM
; IDL optimizations for speed & FASTNORM keyword, 15 May 1999 CM
; Tried a faster version of mpfit_enorm, 30 May 1999 CM
; Changed web address to cow.physics.wisc.edu, 14 Jun 1999 CM
; Found malformation of FDJAC in MPFIT for 1 parm, 03 Aug 1999 CM
; Factored out user-function call into MPFIT_CALL. It is possible,
; but currently disabled, to call procedures. The calling format
; is similar to CURVEFIT, 25 Sep 1999, CM
; Slightly changed mpfit_tie to be less intrusive, 25 Sep 1999, CM
; Fixed some bugs associated with tied parameters in mpfit_fdjac, 25
; Sep 1999, CM
; Reordered documentation; now alphabetical, 02 Oct 1999, CM
; Added QUERY keyword for more robust error detection in drivers, 29
; Oct 1999, CM
; Documented PERROR for unweighted fits, 03 Nov 1999, CM
; Split out MPFIT_RESETPROF to aid in profiling, 03 Nov 1999, CM
; Some profiling and speed optimization, 03 Nov 1999, CM
; Worst offenders, in order: fdjac2, qrfac, qrsolv, enorm.
; fdjac2 depends on user function, qrfac and enorm seem to be
; fully optimized. qrsolv probably could be tweaked a little, but
; is still <10% of total compute time.
; Made sure that !err was set to 0 in MPFIT_DEFITER, 10 Jan 2000, CM
; Fixed small inconsistency in setting of QANYLIM, 28 Jan 2000, CM
; Added PARINFO field RELSTEP, 28 Jan 2000, CM
; Converted to MPFIT_ERROR common block for indicating error
; conditions, 28 Jan 2000, CM
; Corrected scope of MPFIT_ERROR common block, CM, 07 Mar 2000
; Minor speed improvement in MPFIT_ENORM, CM 26 Mar 2000
; Corrected case where ITERPROC changed parameter values and
; parameter values were TIED, CM 26 Mar 2000
; Changed MPFIT_CALL to modify NFEV automatically, and to support
; user procedures more, CM 26 Mar 2000
; Copying permission terms have been liberalized, 26 Mar 2000, CM
; Catch zero value of zero a(j,lj) in MPFIT_QRFAC, 20 Jul 2000, CM
; (thanks to David Schlegel <schlegel@astro.princeton.edu>)
; MPFIT_SETMACHAR is called only once at init; only one common block
; is created (MPFIT_MACHAR); it is now a structure; removed almost
; all CHECK_MATH calls for compatibility with IDL5 and !EXCEPT;
; profiling data is now in a structure too; noted some
; mathematical discrepancies in Linux IDL5.0, 17 Nov 2000, CM
; Some significant changes. New PARINFO fields: MPSIDE, MPMINSTEP,
; MPMAXSTEP. Improved documentation. Now PTIED constraints are
; maintained in the MPCONFIG common block. A new procedure to
; parse PARINFO fields. FDJAC2 now computes a larger variety of
; one-sided and two-sided finite difference derivatives. NFEV is
; stored in the MPCONFIG common now. 17 Dec 2000, CM
; Added check that PARINFO and XALL have same size, 29 Dec 2000 CM
; Don't call function in TERMINATE when there is an error, 05 Jan
; 2000
; Check for float vs. double discrepancies; corrected implementation
; of MIN/MAXSTEP, which I still am not sure of, but now at least
; the correct behavior occurs *without* it, CM 08 Jan 2001
; Added SCALE_FCN keyword, to allow for scaling, as for the CASH
; statistic; added documentation about the theory of operation,
; and under the QR factorization; slowly I'm beginning to
; understand the bowels of this algorithm, CM 10 Jan 2001
; Remove MPMINSTEP field of PARINFO, for now at least, CM 11 Jan
; 2001
; Added RESDAMP keyword, CM, 14 Jan 2001
; Tried to improve the DAMP handling a little, CM, 13 Mar 2001
; Corrected .PARNAME behavior in _DEFITER, CM, 19 Mar 2001
; Added checks for parameter and function overflow; a new STATUS
; value to reflect this; STATUS values of -15 to -1 are reserved
; for user function errors, CM, 03 Apr 2001
; DAMP keyword is now a TANH, CM, 03 Apr 2001
; Added more error checking of float vs. double, CM, 07 Apr 2001
; Fixed bug in handling of parameter lower limits; moved overflow
; checking to end of loop, CM, 20 Apr 2001
; Failure using GOTO, TERMINATE more graceful if FNORM1 not defined,
; CM, 13 Aug 2001
; Add MPPRINT tag to PARINFO, CM, 19 Nov 2001
; Add DOF keyword to DEFITER procedure, and print degrees of
; freedom, CM, 28 Nov 2001
; Add check to be sure MYFUNCT is a scalar string, CM, 14 Jan 2002
; Addition of EXTERNAL_FJAC, EXTERNAL_FVEC keywords; ability to save
; fitter's state from one call to the next; allow '(EXTERNAL)'
; function name, which implies that user will supply function and
; Jacobian at each iteration, CM, 10 Mar 2002
; Documented EXTERNAL evaluation code, CM, 10 Mar 2002
; Corrected signficant bug in the way that the STEP parameter, and
; FIXED parameters interacted (Thanks Andrew Steffl), CM, 02 Apr
; 2002
; Allow COVAR and PERROR keywords to be computed, even in case of
; '(EXTERNAL)' function, 26 May 2002
; Add NFREE and NPEGGED keywords; compute NPEGGED; compute DOF using
; NFREE instead of n_elements(X), thanks to Kristian Kjaer, CM 11
; Sep 2002
; Hopefully PERROR is all positive now, CM 13 Sep 2002
; Documented RELSTEP field of PARINFO (!!), CM, 25 Oct 2002
; Error checking to detect missing start pars, CM 12 Apr 2003
; Add DOF keyword to return degrees of freedom, CM, 30 June 2003
; Always call ITERPROC in the final iteration; add ITERKEYSTOP
; keyword, CM, 30 June 2003
; Correct bug in MPFIT_LMPAR of singularity handling, which might
; likely be fatal for one-parameter fits, CM, 21 Nov 2003
; (with thanks to Peter Tuthill for the proper test case)
; Minor documentation adjustment, 03 Feb 2004, CM
; Correct small error in QR factorization when pivoting; document
; the return values of QRFAC when pivoting, 21 May 2004, CM
; Add MPFORMAT field to PARINFO, and correct behavior of interaction
; between MPPRINT and PARNAME in MPFIT_DEFITERPROC (thanks to Tim
; Robishaw), 23 May 2004, CM
; Add the ITERPRINT keyword to allow redirecting output, 26 Sep
; 2004, CM
; Correct MAXSTEP behavior in case of a negative parameter, 26 Sep
; 2004, CM
; Fix bug in the parsing of MINSTEP/MAXSTEP, 10 Apr 2005, CM
; Fix bug in the handling of upper/lower limits when the limit was
; negative (the fitting code would never "stick" to the lower
; limit), 29 Jun 2005, CM
; Small documentation update for the TIED field, 05 Sep 2005, CM
; Convert to IDL 5 array syntax (!), 16 Jul 2006, CM
; If MAXITER equals zero, then do the basic parameter checking and
; uncertainty analysis, but do not adjust the parameters, 15 Aug
; 2006, CM
; Added documentation, 18 Sep 2006, CM
; A few more IDL 5 array syntax changes, 25 Sep 2006, CM
; Move STRICTARR compile option inside each function/procedure, 9 Oct 2006
; Bug fix for case of MPMAXSTEP and fixed parameters, thanks
; to Huib Intema (who found it from the Python translation!), 05 Feb 2007
; Similar fix for MPFIT_FDJAC2 and the MPSIDE sidedness of
; derivatives, also thanks to Huib Intema, 07 Feb 2007
; Clarify documentation on user-function, derivatives, and PARINFO,
; 27 May 2007
; Change the wording of "Analytic Derivatives" to "Explicit
; Derivatives" in the documentation, CM, 03 Sep 2007
; Further documentation tweaks, CM, 13 Dec 2007
; Add COMPATIBILITY section and add credits to copyright, CM, 13 Dec
; 2007
; Document and enforce that START_PARMS and PARINFO are 1-d arrays,
; CM, 29 Mar 2008
; Previous change for 1-D arrays wasn't correct for
; PARINFO.LIMITED/.LIMITS; now fixed, CM, 03 May 2008
; Documentation adjustments, CM, 20 Aug 2008
; Change some minor FOR-loop variables to type-long, CM, 03 Sep 2008
; Change error handling slightly, document NOCATCH keyword,
; document error handling in general, CM, 01 Oct 2008
; Special case: when either LIMITS is zero, and a parameter pushes
; against that limit, the coded that 'pegged' it there would not
; work since it was a relative condition; now zero is handled
; properly, CM, 08 Nov 2008
; Documentation of how TIED interacts with LIMITS, CM, 21 Dec 2008
; Better documentation of references, CM, 27 Feb 2009
; If MAXITER=0, then be sure to set STATUS=5, which permits the
; the covariance matrix to be computed, CM, 14 Apr 2009
; Avoid numerical underflow while solving for the LM parameter,
; (thanks to Sergey Koposov) CM, 14 Apr 2009
; Use individual functions for all possible MPFIT_CALL permutations,
; (and make sure the syntax is right) CM, 01 Sep 2009
; Correct behavior of MPMAXSTEP when some parameters are frozen,
; thanks to Josh Destree, CM, 22 Nov 2009
; Update the references section, CM, 22 Nov 2009
; 1.70 - Add the VERSION and MIN_VERSION keywords, CM, 22 Nov 2009
; 1.71 - Store pre-calculated revision in common, CM, 23 Nov 2009
; 1.72-1.74 - Documented alternate method to compute correlation matrix,
; CM, 05 Feb 2010
; 1.75 - Enforce TIED constraints when preparing to terminate the
; routine, CM, 2010-06-22
; 1.76 - Documented input keywords now are not modified upon output,
; CM, 2010-07-13
; 1.77 - Upon user request (/CALC_FJAC), compute Jacobian matrix and
; return in BEST_FJAC; also return best residuals in
; BEST_RESID; also return an index list of free parameters as
; PFREE_INDEX; add a fencepost to prevent recursion
; CM, 2010-10-27
; 1.79 - Documentation corrections. CM, 2011-08-26
; 1.81 - Fix bug in interaction of AUTODERIVATIVE=0 and .MPSIDE=3;
; Document FJAC_MASK. CM, 2012-05-08
; 1.83 - Trap more overflow conditions (ref. Nirmal Iyer), CM 2013-12-23
; 1.84 - More robust handling of FNORM, CM 2016-05-19
; 1.85 - Add MPFORMAT_PARNAME for explicit formatting of printed
; parms, CM, 2017-01-03
;
; $Id: mpfit.pro,v 1.85 2017/01/03 19:08:14 cmarkwar Exp $
;-
; Original MINPACK by More' Garbow and Hillstrom, translated with permission
; Modifications and enhancements are:
; Copyright (C) 1997-2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2017 Craig Markwardt
; This software is provided as is without any warranty whatsoever.
; Permission to use, copy, modify, and distribute modified or
; unmodified copies is granted, provided this copyright and disclaimer
; are included unchanged.
;-
pro mpfit_dummy
;; Enclose in a procedure so these are not defined in the main level
COMPILE_OPT strictarr
FORWARD_FUNCTION mpfit_fdjac2, mpfit_enorm, mpfit_lmpar, mpfit_covar, $
mpfit, mpfit_call
COMMON mpfit_error, error_code ;; For error passing to user function
COMMON mpfit_config, mpconfig ;; For internal error configrations
end
;; Reset profiling registers for another run. By default, and when
;; uncommented, the profiling registers simply accumulate.
pro mpfit_resetprof
COMPILE_OPT strictarr
common mpfit_profile, mpfit_profile_vals
mpfit_profile_vals = { status: 1L, fdjac2: 0D, lmpar: 0D, mpfit: 0D, $
qrfac: 0D, qrsolv: 0D, enorm: 0D}
return
end
;; Following are machine constants that can be loaded once. I have
;; found that bizarre underflow messages can be produced in each call
;; to MACHAR(), so this structure minimizes the number of calls to
;; one.
pro mpfit_setmachar, double=isdouble
COMPILE_OPT strictarr
common mpfit_profile, profvals
if n_elements(profvals) EQ 0 then mpfit_resetprof
common mpfit_machar, mpfit_machar_vals
;; In earlier versions of IDL, MACHAR itself could produce a load of
;; error messages. We try to mask some of that out here.
if (!version.release) LT 5 then dummy = check_math(1, 1)
mch = 0.
mch = machar(double=keyword_set(isdouble))
dmachep = mch.eps
dmaxnum = mch.xmax
dminnum = mch.xmin
dmaxlog = alog(mch.xmax)
dminlog = alog(mch.xmin)
if keyword_set(isdouble) then $
dmaxgam = 171.624376956302725D $
else $
dmaxgam = 171.624376956302725
drdwarf = sqrt(dminnum*1.5) * 10
drgiant = sqrt(dmaxnum) * 0.1
mpfit_machar_vals = {machep: dmachep, maxnum: dmaxnum, minnum: dminnum, $
maxlog: dmaxlog, minlog: dminlog, maxgam: dmaxgam, $
rdwarf: drdwarf, rgiant: drgiant}
if (!version.release) LT 5 then dummy = check_math(0, 0)
return
end
; Call user function with no _EXTRA parameters
function mpfit_call_func_noextra, fcn, x, fjac, _EXTRA=extra
if n_params() EQ 2 then begin
return, call_function(fcn, x)
endif else begin
return, call_function(fcn, x, fjac)
endelse
end
; Call user function with _EXTRA parameters
function mpfit_call_func_extra, fcn, x, fjac, _EXTRA=extra
if n_params() EQ 2 then begin
return, call_function(fcn, x, _EXTRA=extra)
endif else begin
return, call_function(fcn, x, fjac, _EXTRA=extra)
endelse
end
; Call user procedure with no _EXTRA parameters
function mpfit_call_pro_noextra, fcn, x, fjac, _EXTRA=extra
if n_params() EQ 2 then begin
call_procedure, fcn, x, f
endif else begin
call_procedure, fcn, x, f, fjac
endelse
return, f
end
; Call user procedure with _EXTRA parameters
function mpfit_call_pro_extra, fcn, x, fjac, _EXTRA=extra
if n_params() EQ 2 then begin
call_procedure, fcn, x, f, _EXTRA=extra
endif else begin
call_procedure, fcn, x, f, fjac, _EXTRA=extra
endelse
return, f
end
;; Call user function or procedure, with _EXTRA or not, with
;; derivatives or not.
function mpfit_call, fcn, x, fjac, _EXTRA=extra
COMPILE_OPT strictarr
common mpfit_config, mpconfig
if keyword_set(mpconfig.qanytied) then mpfit_tie, x, mpconfig.ptied
;; Decide whether we are calling a procedure or function, and
;; with/without FUNCTARGS
proname = 'MPFIT_CALL'
proname = proname + ((mpconfig.proc) ? '_PRO' : '_FUNC')
proname = proname + ((n_elements(extra) GT 0) ? '_EXTRA' : '_NOEXTRA')
if n_params() EQ 2 then begin
f = call_function(proname, fcn, x, _EXTRA=extra)
endif else begin
f = call_function(proname, fcn, x, fjac, _EXTRA=extra)
endelse
mpconfig.nfev = mpconfig.nfev + 1
if n_params() EQ 2 AND mpconfig.damp GT 0 then begin
damp = mpconfig.damp[0]
;; Apply the damping if requested. This replaces the residuals
;; with their hyperbolic tangent. Thus residuals larger than
;; DAMP are essentially clipped.
f = tanh(f/damp)
endif
return, f
end
function mpfit_fdjac2, fcn, x, fvec, step, ulimited, ulimit, dside, $
iflag=iflag, epsfcn=epsfcn, autoderiv=autoderiv, $
FUNCTARGS=fcnargs, xall=xall, ifree=ifree, dstep=dstep, $
deriv_debug=ddebug, deriv_reltol=ddrtol, deriv_abstol=ddatol
COMPILE_OPT strictarr
common mpfit_machar, machvals
common mpfit_profile, profvals
common mpfit_error, mperr
; prof_start = systime(1)
MACHEP0 = machvals.machep
DWARF = machvals.minnum
if n_elements(epsfcn) EQ 0 then epsfcn = MACHEP0
if n_elements(xall) EQ 0 then xall = x
if n_elements(ifree) EQ 0 then ifree = lindgen(n_elements(xall))
if n_elements(step) EQ 0 then step = x * 0.
if n_elements(ddebug) EQ 0 then ddebug = intarr(n_elements(xall))
if n_elements(ddrtol) EQ 0 then ddrtol = x * 0.
if n_elements(ddatol) EQ 0 then ddatol = x * 0.
has_debug_deriv = max(ddebug)
if keyword_set(has_debug_deriv) then begin
;; Header for debugging
print, 'FJAC DEBUG BEGIN'
print, "IPNT", "FUNC", "DERIV_U", "DERIV_N", "DIFF_ABS", "DIFF_REL", $
format='("# ",A10," ",A10," ",A10," ",A10," ",A10," ",A10)'
endif
nall = n_elements(xall)
eps = sqrt(max([epsfcn, MACHEP0]));
m = n_elements(fvec)
n = n_elements(x)
;; Compute analytical derivative if requested
;; Two ways to enable computation of explicit derivatives:
;; 1. AUTODERIVATIVE=0
;; 2. AUTODERIVATIVE=1, but P[i].MPSIDE EQ 3
if keyword_set(autoderiv) EQ 0 OR max(dside[ifree] EQ 3) EQ 1 then begin
fjac_mask = intarr(nall)
;; Specify which parameters need derivatives
;; ---- Case 2 ------ ----- Case 1 -----
fjac_mask[ifree] = (dside[ifree] EQ 3) OR (keyword_set(autoderiv) EQ 0)
if has_debug_deriv then $
print, fjac_mask, format='("# FJAC_MASK = ",100000(I0," ",:))'
fjac = fjac_mask ;; Pass the mask to the calling function as FJAC
mperr = 0
fp = mpfit_call(fcn, xall, fjac, _EXTRA=fcnargs)
iflag = mperr
if n_elements(fjac) NE m*nall then begin
message, /cont, 'ERROR: Derivative matrix was not computed properly.'
iflag = 1
; profvals.fdjac2 = profvals.fdjac2 + (systime(1) - prof_start)
return, 0
endif
;; This definition is consistent with CURVEFIT (WRONG, see below)
;; Sign error found (thanks Jesus Fernandez <fernande@irm.chu-caen.fr>)
;; ... and now I regret doing this sign flip since it's not
;; strictly correct. The definition should be RESID =
;; (Y-F)/SIGMA, so d(RESID)/dP should be -dF/dP. My response to
;; Fernandez was unfounded because he was trying to supply
;; dF/dP. Sigh. (CM 31 Aug 2007)
fjac = reform(-temporary(fjac), m, nall, /overwrite)
;; Select only the free parameters
if n_elements(ifree) LT nall then $
fjac = reform(fjac[*,ifree], m, n, /overwrite)
;; Are we done computing derivatives? The answer is, YES, if we
;; computed explicit derivatives for all free parameters, EXCEPT
;; when we are going on to compute debugging derivatives.
if min(fjac_mask[ifree]) EQ 1 AND NOT has_debug_deriv then begin
return, fjac
endif
endif
;; Final output array, if it was not already created above
if n_elements(fjac) EQ 0 then begin
fjac = make_array(m, n, value=fvec[0]*0.)
fjac = reform(fjac, m, n, /overwrite)
endif
h = eps * abs(x)
;; if STEP is given, use that
;; STEP includes the fixed parameters
if n_elements(step) GT 0 then begin
stepi = step[ifree]
wh = where(stepi GT 0, ct)
if ct GT 0 then h[wh] = stepi[wh]
endif
;; if relative step is given, use that
;; DSTEP includes the fixed parameters
if n_elements(dstep) GT 0 then begin
dstepi = dstep[ifree]
wh = where(dstepi GT 0, ct)
if ct GT 0 then h[wh] = abs(dstepi[wh]*x[wh])
endif
;; In case any of the step values are zero
wh = where(h EQ 0, ct)
if ct GT 0 then h[wh] = eps
;; Reverse the sign of the step if we are up against the parameter
;; limit, or if the user requested it.
;; DSIDE includes the fixed parameters (ULIMITED/ULIMIT have only
;; varying ones)
mask = dside[ifree] EQ -1
if n_elements(ulimited) GT 0 AND n_elements(ulimit) GT 0 then $
mask = mask OR (ulimited AND (x GT ulimit-h))
wh = where(mask, ct)
if ct GT 0 then h[wh] = -h[wh]
;; Loop through parameters, computing the derivative for each
for j=0L, n-1 do begin
dsidej = dside[ifree[j]]
ddebugj = ddebug[ifree[j]]
;; Skip this parameter if we already computed its derivative
;; explicitly, and we are not debugging.
if (dsidej EQ 3) AND (ddebugj EQ 0) then continue
if (dsidej EQ 3) AND (ddebugj EQ 1) then $
print, ifree[j], format='("FJAC PARM ",I0)'
xp = xall
xp[ifree[j]] = xp[ifree[j]] + h[j]
mperr = 0
fp = mpfit_call(fcn, xp, _EXTRA=fcnargs)
iflag = mperr
if iflag LT 0 then return, !values.d_nan
if ((dsidej GE -1) AND (dsidej LE 1)) OR (dsidej EQ 3) then begin
;; COMPUTE THE ONE-SIDED DERIVATIVE
;; Note optimization fjac(0:*,j)
fjacj = (fp-fvec)/h[j]
endif else begin
;; COMPUTE THE TWO-SIDED DERIVATIVE
xp[ifree[j]] = xall[ifree[j]] - h[j]
mperr = 0
fm = mpfit_call(fcn, xp, _EXTRA=fcnargs)
iflag = mperr
if iflag LT 0 then return, !values.d_nan
;; Note optimization fjac(0:*,j)
fjacj = (fp-fm)/(2*h[j])
endelse
;; Debugging of explicit derivatives
if (dsidej EQ 3) AND (ddebugj EQ 1) then begin
;; Relative and absolute tolerances
dr = ddrtol[ifree[j]] & da = ddatol[ifree[j]]
;; Explicitly calculated
fjaco = fjac[*,j]
;; If tolerances are zero, then any value for deriv triggers print...
if (da EQ 0 AND dr EQ 0) then $
diffj = (fjaco NE 0 OR fjacj NE 0)
;; ... otherwise the difference must be a greater than tolerance
if (da NE 0 OR dr NE 0) then $
diffj = (abs(fjaco-fjacj) GT (da+abs(fjaco)*dr))
for k = 0L, m-1 do if diffj[k] then begin
print, k, fvec[k], fjaco[k], fjacj[k], fjaco[k]-fjacj[k], $
(fjaco[k] EQ 0)?(0):((fjaco[k]-fjacj[k])/fjaco[k]), $
format='(" ",I10," ",G10.4," ",G10.4," ",G10.4," ",G10.4," ",G10.4)'
endif
endif
;; Store final results in output array
fjac[0,j] = fjacj
endfor
if has_debug_deriv then print, 'FJAC DEBUG END'
; profvals.fdjac2 = profvals.fdjac2 + (systime(1) - prof_start)
return, fjac
end
function mpfit_enorm, vec
COMPILE_OPT strictarr
;; NOTE: it turns out that, for systems that have a lot of data
;; points, this routine is a big computing bottleneck. The extended
;; computations that need to be done cannot be effectively
;; vectorized. The introduction of the FASTNORM configuration
;; parameter allows the user to select a faster routine, which is
;; based on TOTAL() alone.
common mpfit_profile, profvals
; prof_start = systime(1)
common mpfit_config, mpconfig
; Very simple-minded sum-of-squares
if n_elements(mpconfig) GT 0 then if mpconfig.fastnorm then begin
ans = sqrt(total(vec^2))
goto, TERMINATE
endif
common mpfit_machar, machvals
agiant = machvals.rgiant / n_elements(vec)
adwarf = machvals.rdwarf * n_elements(vec)
;; This is hopefully a compromise between speed and robustness.
;; Need to do this because of the possibility of over- or underflow.
mx = max(vec, min=mn)
mx = max(abs([mx,mn]))
if mx EQ 0 then return, vec[0]*0.
if mx GT agiant OR mx LT adwarf then ans = mx * sqrt(total((vec/mx)^2))$
else ans = sqrt( total(vec^2) )
TERMINATE:
; profvals.enorm = profvals.enorm + (systime(1) - prof_start)
return, ans
end
; **********
;
; subroutine qrfac
;
; this subroutine uses householder transformations with column
; pivoting (optional) to compute a qr factorization of the
; m by n matrix a. that is, qrfac determines an orthogonal
; matrix q, a permutation matrix p, and an upper trapezoidal
; matrix r with diagonal elements of nonincreasing magnitude,
; such that a*p = q*r. the householder transformation for
; column k, k = 1,2,...,min(m,n), is of the form
;
; t
; i - (1/u(k))*u*u
;
; where u has zeros in the first k-1 positions. the form of
; this transformation and the method of pivoting first
; appeared in the corresponding linpack subroutine.
;
; the subroutine statement is
;
; subroutine qrfac(m,n,a,lda,pivot,ipvt,lipvt,rdiag,acnorm,wa)
;
; where
;
; m is a positive integer input variable set to the number
; of rows of a.
;
; n is a positive integer input variable set to the number
; of columns of a.
;
; a is an m by n array. on input a contains the matrix for
; which the qr factorization is to be computed. on output
; the strict upper trapezoidal part of a contains the strict
; upper trapezoidal part of r, and the lower trapezoidal
; part of a contains a factored form of q (the non-trivial
; elements of the u vectors described above).
;
; lda is a positive integer input variable not less than m
; which specifies the leading dimension of the array a.
;
; pivot is a logical input variable. if pivot is set true,
; then column pivoting is enforced. if pivot is set false,
; then no column pivoting is done.
;
; ipvt is an integer output array of length lipvt. ipvt
; defines the permutation matrix p such that a*p = q*r.
; column j of p is column ipvt(j) of the identity matrix.
; if pivot is false, ipvt is not referenced.
;
; lipvt is a positive integer input variable. if pivot is false,
; then lipvt may be as small as 1. if pivot is true, then
; lipvt must be at least n.
;
; rdiag is an output array of length n which contains the
; diagonal elements of r.
;
; acnorm is an output array of length n which contains the
; norms of the corresponding columns of the input matrix a.
; if this information is not needed, then acnorm can coincide
; with rdiag.
;
; wa is a work array of length n. if pivot is false, then wa
; can coincide with rdiag.
;
; subprograms called
;
; minpack-supplied ... dpmpar,enorm
;
; fortran-supplied ... dmax1,dsqrt,min0
;
; argonne national laboratory. minpack project. march 1980.
; burton s. garbow, kenneth e. hillstrom, jorge j. more
;
; **********
;
; PIVOTING / PERMUTING:
;
; Upon return, A(*,*) is in standard parameter order, A(*,IPVT) is in
; permuted order.
;
; RDIAG is in permuted order.
;
; ACNORM is in standard parameter order.
;
; NOTE: in IDL the factors appear slightly differently than described
; above. The matrix A is still m x n where m >= n.
;
; The "upper" triangular matrix R is actually stored in the strict
; lower left triangle of A under the standard notation of IDL.
;
; The reflectors that generate Q are in the upper trapezoid of A upon
; output.
;
; EXAMPLE: decompose the matrix [[9.,2.,6.],[4.,8.,7.]]
; aa = [[9.,2.,6.],[4.,8.,7.]]
; mpfit_qrfac, aa, aapvt, rdiag, aanorm
; IDL> print, aa
; 1.81818* 0.181818* 0.545455*
; -8.54545+ 1.90160* 0.432573*
; IDL> print, rdiag
; -11.0000+ -7.48166+
;
; The components marked with a * are the components of the
; reflectors, and those marked with a + are components of R.
;
; To reconstruct Q and R we proceed as follows. First R.
; r = fltarr(m, n)
; for i = 0, n-1 do r(0:i,i) = aa(0:i,i) ; fill in lower diag
; r(lindgen(n)*(m+1)) = rdiag
;
; Next, Q, which are composed from the reflectors. Each reflector v
; is taken from the upper trapezoid of aa, and converted to a matrix
; via (I - 2 vT . v / (v . vT)).
;
; hh = ident ;; identity matrix
; for i = 0, n-1 do begin
; v = aa(*,i) & if i GT 0 then v(0:i-1) = 0 ;; extract reflector
; hh = hh ## (ident - 2*(v # v)/total(v * v)) ;; generate matrix
; endfor
;
; Test the result:
; IDL> print, hh ## transpose(r)
; 9.00000 4.00000
; 2.00000 8.00000
; 6.00000 7.00000
;
; Note that it is usually never necessary to form the Q matrix
; explicitly, and MPFIT does not.
pro mpfit_qrfac, a, ipvt, rdiag, acnorm, pivot=pivot
COMPILE_OPT strictarr
sz = size(a)
m = sz[1]
n = sz[2]
common mpfit_machar, machvals
common mpfit_profile, profvals
; prof_start = systime(1)
MACHEP0 = machvals.machep
DWARF = machvals.minnum
;; Compute the initial column norms and initialize arrays
acnorm = make_array(n, value=a[0]*0.)
for j = 0L, n-1 do $
acnorm[j] = mpfit_enorm(a[*,j])
rdiag = acnorm
wa = rdiag
ipvt = lindgen(n)
;; Reduce a to r with householder transformations
minmn = min([m,n])
for j = 0L, minmn-1 do begin
if NOT keyword_set(pivot) then goto, HOUSE1
;; Bring the column of largest norm into the pivot position
rmax = max(rdiag[j:*])
kmax = where(rdiag[j:*] EQ rmax, ct) + j
if ct LE 0 then goto, HOUSE1
kmax = kmax[0]
;; Exchange rows via the pivot only. Avoid actually exchanging
;; the rows, in case there is lots of memory transfer. The
;; exchange occurs later, within the body of MPFIT, after the
;; extraneous columns of the matrix have been shed.
if kmax NE j then begin
temp = ipvt[j] & ipvt[j] = ipvt[kmax] & ipvt[kmax] = temp
rdiag[kmax] = rdiag[j]
wa[kmax] = wa[j]
endif
HOUSE1:
;; Compute the householder transformation to reduce the jth
;; column of A to a multiple of the jth unit vector
lj = ipvt[j]
ajj = a[j:*,lj]
ajnorm = mpfit_enorm(ajj)
if ajnorm EQ 0 then goto, NEXT_ROW
if a[j,lj] LT 0 then ajnorm = -ajnorm
ajj = ajj / ajnorm
ajj[0] = ajj[0] + 1
;; *** Note optimization a(j:*,j)
a[j,lj] = ajj
;; Apply the transformation to the remaining columns
;; and update the norms
;; NOTE to SELF: tried to optimize this by removing the loop,
;; but it actually got slower. Reverted to "for" loop to keep
;; it simple.
if j+1 LT n then begin
for k=j+1, n-1 do begin
lk = ipvt[k]
ajk = a[j:*,lk]
;; *** Note optimization a(j:*,lk)
;; (corrected 20 Jul 2000)
if a[j,lj] NE 0 then $
a[j,lk] = ajk - ajj * total(ajk*ajj)/a[j,lj]
if keyword_set(pivot) AND rdiag[k] NE 0 then begin
temp = a[j,lk]/rdiag[k]
rdiag[k] = rdiag[k] * sqrt((1.-temp^2) > 0)
temp = rdiag[k]/wa[k]
if 0.05D*temp*temp LE MACHEP0 then begin
rdiag[k] = mpfit_enorm(a[j+1:*,lk])
wa[k] = rdiag[k]
endif
endif
endfor
endif
NEXT_ROW:
rdiag[j] = -ajnorm
endfor
; profvals.qrfac = profvals.qrfac + (systime(1) - prof_start)
return
end
; **********
;
; subroutine qrsolv
;
; given an m by n matrix a, an n by n diagonal matrix d,
; and an m-vector b, the problem is to determine an x which
; solves the system
;
; a*x = b , d*x = 0 ,
;
; in the least squares sense.
;
; this subroutine completes the solution of the problem
; if it is provided with the necessary information from the
; qr factorization, with column pivoting, of a. that is, if
; a*p = q*r, where p is a permutation matrix, q has orthogonal
; columns, and r is an upper triangular matrix with diagonal
; elements of nonincreasing magnitude, then qrsolv expects
; the full upper triangle of r, the permutation matrix p,
; and the first n components of (q transpose)*b. the system
; a*x = b, d*x = 0, is then equivalent to
;
; t t
; r*z = q *b , p *d*p*z = 0 ,
;
; where x = p*z. if this system does not have full rank,
; then a least squares solution is obtained. on output qrsolv
; also provides an upper triangular matrix s such that
;
; t t t
; p *(a *a + d*d)*p = s *s .
;
; s is computed within qrsolv and may be of separate interest.
;
; the subroutine statement is
;
; subroutine qrsolv(n,r,ldr,ipvt,diag,qtb,x,sdiag,wa)
;
; where
;
; n is a positive integer input variable set to the order of r.
;
; r is an n by n array. on input the full upper triangle
; must contain the full upper triangle of the matrix r.
; on output the full upper triangle is unaltered, and the
; strict lower triangle contains the strict upper triangle
; (transposed) of the upper triangular matrix s.
;
; ldr is a positive integer input variable not less than n
; which specifies the leading dimension of the array r.
;
; ipvt is an integer input array of length n which defines the
; permutation matrix p such that a*p = q*r. column j of p
; is column ipvt(j) of the identity matrix.
;
; diag is an input array of length n which must contain the
; diagonal elements of the matrix d.
;
; qtb is an input array of length n which must contain the first
; n elements of the vector (q transpose)*b.
;
; x is an output array of length n which contains the least
; squares solution of the system a*x = b, d*x = 0.
;
; sdiag is an output array of length n which contains the
; diagonal elements of the upper triangular matrix s.
;
; wa is a work array of length n.
;
; subprograms called
;
; fortran-supplied ... dabs,dsqrt
;
; argonne national laboratory. minpack project. march 1980.
; burton s. garbow, kenneth e. hillstrom, jorge j. more
;
pro mpfit_qrsolv, r, ipvt, diag, qtb, x, sdiag
COMPILE_OPT strictarr
sz = size(r)
m = sz[1]
n = sz[2]
delm = lindgen(n) * (m+1) ;; Diagonal elements of r
common mpfit_profile, profvals
; prof_start = systime(1)
;; copy r and (q transpose)*b to preserve input and initialize s.
;; in particular, save the diagonal elements of r in x.
for j = 0L, n-1 do $
r[j:n-1,j] = r[j,j:n-1]
x = r[delm]
wa = qtb
;; Below may look strange, but it's so we can keep the right precision
zero = qtb[0]*0.
half = zero + 0.5
quart = zero + 0.25
;; Eliminate the diagonal matrix d using a givens rotation
for j = 0L, n-1 do begin
l = ipvt[j]
if diag[l] EQ 0 then goto, STORE_RESTORE
sdiag[j:*] = 0
sdiag[j] = diag[l]
;; The transformations to eliminate the row of d modify only a
;; single element of (q transpose)*b beyond the first n, which
;; is initially zero.
qtbpj = zero
for k = j, n-1 do begin
if sdiag[k] EQ 0 then goto, ELIM_NEXT_LOOP
if abs(r[k,k]) LT abs(sdiag[k]) then begin
cotan = r[k,k]/sdiag[k]
sine = half/sqrt(quart + quart*cotan*cotan)
cosine = sine*cotan
endif else begin
tang = sdiag[k]/r[k,k]
cosine = half/sqrt(quart + quart*tang*tang)
sine = cosine*tang
endelse
;; Compute the modified diagonal element of r and the
;; modified element of ((q transpose)*b,0).
r[k,k] = cosine*r[k,k] + sine*sdiag[k]
temp = cosine*wa[k] + sine*qtbpj
qtbpj = -sine*wa[k] + cosine*qtbpj
wa[k] = temp
;; Accumulate the transformation in the row of s
if n GT k+1 then begin
temp = cosine*r[k+1:n-1,k] + sine*sdiag[k+1:n-1]
sdiag[k+1:n-1] = -sine*r[k+1:n-1,k] + cosine*sdiag[k+1:n-1]
r[k+1:n-1,k] = temp
endif
ELIM_NEXT_LOOP:
endfor
STORE_RESTORE:
sdiag[j] = r[j,j]
r[j,j] = x[j]
endfor
;; Solve the triangular system for z. If the system is singular
;; then obtain a least squares solution
nsing = n
wh = where(sdiag EQ 0, ct)
if ct GT 0 then begin
nsing = wh[0]
wa[nsing:*] = 0
endif
if nsing GE 1 then begin
wa[nsing-1] = wa[nsing-1]/sdiag[nsing-1] ;; Degenerate case
;; *** Reverse loop ***
for j=nsing-2,0,-1 do begin
sum = total(r[j+1:nsing-1,j]*wa[j+1:nsing-1])
wa[j] = (wa[j]-sum)/sdiag[j]
endfor
endif
;; Permute the components of z back to components of x
x[ipvt] = wa
; profvals.qrsolv = profvals.qrsolv + (systime(1) - prof_start)
return
end
;
; subroutine lmpar
;
; given an m by n matrix a, an n by n nonsingular diagonal
; matrix d, an m-vector b, and a positive number delta,
; the problem is to determine a value for the parameter
; par such that if x solves the system
;
; a*x = b , sqrt(par)*d*x = 0 ,
;
; in the least squares sense, and dxnorm is the euclidean
; norm of d*x, then either par is zero and
;
; (dxnorm-delta) .le. 0.1*delta ,
;
; or par is positive and
;
; abs(dxnorm-delta) .le. 0.1*delta .
;
; this subroutine completes the solution of the problem
; if it is provided with the necessary information from the
; qr factorization, with column pivoting, of a. that is, if
; a*p = q*r, where p is a permutation matrix, q has orthogonal
; columns, and r is an upper triangular matrix with diagonal
; elements of nonincreasing magnitude, then lmpar expects
; the full upper triangle of r, the permutation matrix p,
; and the first n components of (q transpose)*b. on output
; lmpar also provides an upper triangular matrix s such that
;
; t t t
; p *(a *a + par*d*d)*p = s *s .
;
; s is employed within lmpar and may be of separate interest.
;
; only a few iterations are generally needed for convergence
; of the algorithm. if, however, the limit of 10 iterations
; is reached, then the output par will contain the best
; value obtained so far.
;
; the subroutine statement is
;
; subroutine lmpar(n,r,ldr,ipvt,diag,qtb,delta,par,x,sdiag,
; wa1,wa2)
;
; where
;
; n is a positive integer input variable set to the order of r.
;
; r is an n by n array. on input the full upper triangle
; must contain the full upper triangle of the matrix r.
; on output the full upper triangle is unaltered, and the
; strict lower triangle contains the strict upper triangle
; (transposed) of the upper triangular matrix s.
;
; ldr is a positive integer input variable not less than n
; which specifies the leading dimension of the array r.
;
; ipvt is an integer input array of length n which defines the
; permutation matrix p such that a*p = q*r. column j of p
; is column ipvt(j) of the identity matrix.
;
; diag is an input array of length n which must contain the
; diagonal elements of the matrix d.
;
; qtb is an input array of length n which must contain the first
; n elements of the vector (q transpose)*b.
;
; delta is a positive input variable which specifies an upper
; bound on the euclidean norm of d*x.
;
; par is a nonnegative variable. on input par contains an
; initial estimate of the levenberg-marquardt parameter.
; on output par contains the final estimate.
;
; x is an output array of length n which contains the least
; squares solution of the system a*x = b, sqrt(par)*d*x = 0,
; for the output par.
;
; sdiag is an output array of length n which contains the
; diagonal elements of the upper triangular matrix s.
;
; wa1 and wa2 are work arrays of length n.
;
; subprograms called
;
; minpack-supplied ... dpmpar,enorm,qrsolv
;
; fortran-supplied ... dabs,dmax1,dmin1,dsqrt
;
; argonne national laboratory. minpack project. march 1980.
; burton s. garbow, kenneth e. hillstrom, jorge j. more
;
function mpfit_lmpar, r, ipvt, diag, qtb, delta, x, sdiag, par=par
COMPILE_OPT strictarr
common mpfit_machar, machvals
common mpfit_profile, profvals
; prof_start = systime(1)
MACHEP0 = machvals.machep
DWARF = machvals.minnum
sz = size(r)
m = sz[1]
n = sz[2]
delm = lindgen(n) * (m+1) ;; Diagonal elements of r
;; Compute and store in x the gauss-newton direction. If the
;; jacobian is rank-deficient, obtain a least-squares solution
nsing = n
wa1 = qtb
rthresh = max(abs(r[delm]))*MACHEP0
wh = where(abs(r[delm]) LT rthresh, ct)
if ct GT 0 then begin
nsing = wh[0]
wa1[wh[0]:*] = 0
endif
if nsing GE 1 then begin
;; *** Reverse loop ***
for j=nsing-1,0,-1 do begin
wa1[j] = wa1[j]/r[j,j]
if (j-1 GE 0) then $
wa1[0:(j-1)] = wa1[0:(j-1)] - r[0:(j-1),j]*wa1[j]
endfor
endif
;; Note: ipvt here is a permutation array
x[ipvt] = wa1
;; Initialize the iteration counter. Evaluate the function at the
;; origin, and test for acceptance of the gauss-newton direction
iter = 0L
wa2 = diag * x
dxnorm = mpfit_enorm(wa2)
fp = dxnorm - delta
if fp LE 0.1*delta then goto, TERMINATE
;; If the jacobian is not rank deficient, the newton step provides a
;; lower bound, parl, for the zero of the function. Otherwise set
;; this bound to zero.
zero = wa2[0]*0.
parl = zero
if nsing GE n then begin
wa1 = diag[ipvt]*wa2[ipvt]/dxnorm
wa1[0] = wa1[0] / r[0,0] ;; Degenerate case
for j=1L, n-1 do begin ;; Note "1" here, not zero
sum = total(r[0:(j-1),j]*wa1[0:(j-1)])
wa1[j] = (wa1[j] - sum)/r[j,j]
endfor
temp = mpfit_enorm(wa1)
parl = ((fp/delta)/temp)/temp
endif
;; Calculate an upper bound, paru, for the zero of the function
for j=0L, n-1 do begin
sum = total(r[0:j,j]*qtb[0:j])
wa1[j] = sum/diag[ipvt[j]]
endfor
gnorm = mpfit_enorm(wa1)
paru = gnorm/delta
if paru EQ 0 then paru = DWARF/min([delta,0.1])
;; If the input par lies outside of the interval (parl,paru), set
;; par to the closer endpoint
par = max([par,parl])
par = min([par,paru])
if par EQ 0 then par = gnorm/dxnorm
;; Beginning of an interation
ITERATION:
iter = iter + 1
;; Evaluate the function at the current value of par
if par EQ 0 then par = max([DWARF, paru*0.001])
temp = sqrt(par)
wa1 = temp * diag
mpfit_qrsolv, r, ipvt, wa1, qtb, x, sdiag
wa2 = diag*x
dxnorm = mpfit_enorm(wa2)
temp = fp
fp = dxnorm - delta
if (abs(fp) LE 0.1D*delta) $
OR ((parl EQ 0) AND (fp LE temp) AND (temp LT 0)) $
OR (iter EQ 10) then goto, TERMINATE
;; Compute the newton correction
wa1 = diag[ipvt]*wa2[ipvt]/dxnorm
for j=0L,n-2 do begin
wa1[j] = wa1[j]/sdiag[j]
wa1[j+1:n-1] = wa1[j+1:n-1] - r[j+1:n-1,j]*wa1[j]
endfor
wa1[n-1] = wa1[n-1]/sdiag[n-1] ;; Degenerate case
temp = mpfit_enorm(wa1)
parc = ((fp/delta)/temp)/temp
;; Depending on the sign of the function, update parl or paru
if fp GT 0 then parl = max([parl,par])
if fp LT 0 then paru = min([paru,par])
;; Compute an improved estimate for par
par = max([parl, par+parc])
;; End of an iteration
goto, ITERATION
TERMINATE:
;; Termination
; profvals.lmpar = profvals.lmpar + (systime(1) - prof_start)
if iter EQ 0 then return, par[0]*0.
return, par
end
;; Procedure to tie one parameter to another.
pro mpfit_tie, p, _ptied
COMPILE_OPT strictarr
if n_elements(_ptied) EQ 0 then return
if n_elements(_ptied) EQ 1 then if _ptied[0] EQ '' then return
for _i = 0L, n_elements(_ptied)-1 do begin
if _ptied[_i] EQ '' then goto, NEXT_TIE
_cmd = 'p['+strtrim(_i,2)+'] = '+_ptied[_i]
_err = execute(_cmd)
if _err EQ 0 then begin
message, 'ERROR: Tied expression "'+_cmd+'" failed.'
return
endif
NEXT_TIE:
endfor
end
;; Default print procedure
pro mpfit_defprint, p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, $
p11, p12, p13, p14, p15, p16, p17, p18, $
format=format, unit=unit0, _EXTRA=extra
COMPILE_OPT strictarr
if n_elements(unit0) EQ 0 then unit = -1 else unit = round(unit0[0])
if n_params() EQ 0 then printf, unit, '' $
else if n_params() EQ 1 then printf, unit, p1, format=format $
else if n_params() EQ 2 then printf, unit, p1, p2, format=format $
else if n_params() EQ 3 then printf, unit, p1, p2, p3, format=format $
else if n_params() EQ 4 then printf, unit, p1, p2, p4, format=format
return
end
;; Default procedure to be called every iteration. It simply prints
;; the parameter values.
pro mpfit_defiter, fcn, x, iter, fnorm, FUNCTARGS=fcnargs, $
quiet=quiet, iterstop=iterstop, iterkeybyte=iterkeybyte, $
parinfo=parinfo, iterprint=iterprint0, $
format=fmt, pformat=pformat, dof=dof0, $
iterparnameformat=iterparnameformat0, $
_EXTRA=iterargs
COMPILE_OPT strictarr
common mpfit_error, mperr
mperr = 0
if keyword_set(quiet) then goto, DO_ITERSTOP
if n_params() EQ 3 then begin
fvec = mpfit_call(fcn, x, _EXTRA=fcnargs)
fnorm = mpfit_enorm(fvec)^2
endif
;; Determine which parameters to print
nprint = n_elements(x)
iprint = lindgen(nprint)
if n_elements(iterprint0) EQ 0 then iterprint = 'MPFIT_DEFPRINT' $
else iterprint = strtrim(iterprint0[0],2)
if n_elements(iterparnameformat0) EQ 0 then iterparnameformat = 'A25' $
else iterparnameformat = iterparnameformat0
if n_elements(dof0) EQ 0 then dof = 1L else dof = floor(dof0[0])
call_procedure, iterprint, iter, fnorm, dof, $
format='("Iter ",I6," CHI-SQUARE = ",G15.8," DOF = ",I0)', $
_EXTRA=iterargs
if n_elements(fmt) GT 0 then begin
call_procedure, iterprint, x, format=fmt, _EXTRA=iterargs
endif else begin
if n_elements(pformat) EQ 0 then pformat = '(G20.6)'
parname = 'P('+strtrim(iprint,2)+')'
parnamefmt = strarr(nprint) + iterparnameformat
pformats = strarr(nprint) + pformat
pardesc = strarr(nprint)
if n_elements(parinfo) GT 0 then begin
parinfo_tags = tag_names(parinfo)
wh = where(parinfo_tags EQ 'PARNAME', ct)
if ct EQ 1 then begin
wh = where(parinfo.parname NE '', ct)
if ct GT 0 then $
parname[wh] = parinfo[wh].parname
endif
wh = where(parinfo_tags EQ 'MPPRINT', ct)
if ct EQ 1 then begin
iprint = where(parinfo.mpprint EQ 1, nprint)
if nprint EQ 0 then goto, DO_ITERSTOP
endif
wh = where(parinfo_tags EQ 'MPFORMAT', ct)
if ct EQ 1 then begin
wh = where(parinfo.mpformat NE '', ct)
if ct GT 0 then pformats[wh] = parinfo[wh].mpformat
endif
wh = where(parinfo_tags EQ 'MPFORMAT_PARNAME', ct)
if ct EQ 1 then begin
wh = where(parinfo.mpformat_parname NE '', ct)
if ct GT 0 then parnamefmt[wh] = parinfo[wh].mpformat_parname
endif
endif
for i = 0L, nprint-1 do begin
call_procedure, iterprint, parname[iprint[i]], x[iprint[i]], $
format='(" ",'+parnamefmt[i]+'," = ",'+pformats[iprint[i]]+')', $
_EXTRA=iterargs
endfor
endelse
DO_ITERSTOP:
if n_elements(iterkeybyte) EQ 0 then iterkeybyte = 7b
if keyword_set(iterstop) then begin
k = get_kbrd(0)
if k EQ string(iterkeybyte[0]) then begin
message, 'WARNING: minimization not complete', /info
print, 'Do you want to terminate this procedure? (y/n)', $
format='(A,$)'
k = ''
read, k
if strupcase(strmid(k,0,1)) EQ 'Y' then begin
message, 'WARNING: Procedure is terminating.', /info
mperr = -1
endif
endif
endif
return
end
;; Procedure to parse the parameter values in PARINFO
pro mpfit_parinfo, parinfo, tnames, tag, values, default=def, status=status, $
n_param=n
COMPILE_OPT strictarr
status = 0
if n_elements(n) EQ 0 then n = n_elements(parinfo)
if n EQ 0 then begin
if n_elements(def) EQ 0 then return
values = def
status = 1
return
endif
if n_elements(parinfo) EQ 0 then goto, DO_DEFAULT
if n_elements(tnames) EQ 0 then tnames = tag_names(parinfo)
wh = where(tnames EQ tag, ct)
if ct EQ 0 then begin
DO_DEFAULT:
if n_elements(def) EQ 0 then return
values = make_array(n, value=def[0])
values[0] = def
endif else begin
values = parinfo.(wh[0])
np = n_elements(parinfo)
nv = n_elements(values)
values = reform(values[*], nv/np, np)
endelse
status = 1
return
end
; **********
;
; subroutine covar
;
; given an m by n matrix a, the problem is to determine
; the covariance matrix corresponding to a, defined as
;
; t
; inverse(a *a) .
;
; this subroutine completes the solution of the problem
; if it is provided with the necessary information from the
; qr factorization, with column pivoting, of a. that is, if
; a*p = q*r, where p is a permutation matrix, q has orthogonal
; columns, and r is an upper triangular matrix with diagonal
; elements of nonincreasing magnitude, then covar expects
; the full upper triangle of r and the permutation matrix p.
; the covariance matrix is then computed as
;
; t t
; p*inverse(r *r)*p .
;
; if a is nearly rank deficient, it may be desirable to compute
; the covariance matrix corresponding to the linearly independent
; columns of a. to define the numerical rank of a, covar uses
; the tolerance tol. if l is the largest integer such that
;
; abs(r(l,l)) .gt. tol*abs(r(1,1)) ,
;
; then covar computes the covariance matrix corresponding to
; the first l columns of r. for k greater than l, column
; and row ipvt(k) of the covariance matrix are set to zero.
;
; the subroutine statement is
;
; subroutine covar(n,r,ldr,ipvt,tol,wa)
;
; where
;
; n is a positive integer input variable set to the order of r.
;
; r is an n by n array. on input the full upper triangle must
; contain the full upper triangle of the matrix r. on output
; r contains the square symmetric covariance matrix.
;
; ldr is a positive integer input variable not less than n
; which specifies the leading dimension of the array r.
;
; ipvt is an integer input array of length n which defines the
; permutation matrix p such that a*p = q*r. column j of p
; is column ipvt(j) of the identity matrix.
;
; tol is a nonnegative input variable used to define the
; numerical rank of a in the manner described above.
;
; wa is a work array of length n.
;
; subprograms called
;
; fortran-supplied ... dabs
;
; argonne national laboratory. minpack project. august 1980.
; burton s. garbow, kenneth e. hillstrom, jorge j. more
;
; **********
function mpfit_covar, rr, ipvt, tol=tol
COMPILE_OPT strictarr
sz = size(rr)
if sz[0] NE 2 then begin
message, 'ERROR: r must be a two-dimensional matrix'
return, -1L
endif
n = sz[1]
if n NE sz[2] then begin
message, 'ERROR: r must be a square matrix'
return, -1L
endif
zero = rr[0] * 0.
one = zero + 1.
if n_elements(ipvt) EQ 0 then ipvt = lindgen(n)
r = rr
r = reform(rr, n, n, /overwrite)
;; Form the inverse of r in the full upper triangle of r
l = -1L
if n_elements(tol) EQ 0 then tol = one*1.E-14
tolr = tol * abs(r[0,0])
for k = 0L, n-1 do begin
if abs(r[k,k]) LE tolr then goto, INV_END_LOOP
r[k,k] = one/r[k,k]
for j = 0L, k-1 do begin
temp = r[k,k] * r[j,k]
r[j,k] = zero
r[0,k] = r[0:j,k] - temp*r[0:j,j]
endfor
l = k
endfor
INV_END_LOOP:
;; Form the full upper triangle of the inverse of (r transpose)*r
;; in the full upper triangle of r
if l GE 0 then $
for k = 0L, l do begin
for j = 0L, k-1 do begin
temp = r[j,k]
r[0,j] = r[0:j,j] + temp*r[0:j,k]
endfor
temp = r[k,k]
r[0,k] = temp * r[0:k,k]
endfor
;; Form the full lower triangle of the covariance matrix
;; in the strict lower triangle of r and in wa
wa = replicate(r[0,0], n)
for j = 0L, n-1 do begin
jj = ipvt[j]
sing = j GT l
for i = 0L, j do begin
if sing then r[i,j] = zero
ii = ipvt[i]
if ii GT jj then r[ii,jj] = r[i,j]
if ii LT jj then r[jj,ii] = r[i,j]
endfor
wa[jj] = r[j,j]
endfor
;; Symmetrize the covariance matrix in r
for j = 0L, n-1 do begin
r[0:j,j] = r[j,0:j]
r[j,j] = wa[j]
endfor
return, r
end
;; Parse the RCSID revision number
function mpfit_revision
;; NOTE: this string is changed every time an RCS check-in occurs
revision = '$Revision: 1.85 $'
;; Parse just the version number portion
revision = stregex(revision,'\$'+'Revision: *([0-9.]+) *'+'\$',/extract,/sub)
revision = revision[1]
return, revision
end
;; Parse version numbers of the form 'X.Y'
function mpfit_parse_version, version
sz = size(version)
if sz[sz[0]+1] NE 7 then return, 0
s = stregex(version[0], '^([0-9]+)\.([0-9]+)$', /extract,/sub)
if s[0] NE version[0] then return, 0
return, long(s[1:2])
end
;; Enforce a minimum version number
function mpfit_min_version, version, min_version
mv = mpfit_parse_version(min_version)
if mv[0] EQ 0 then return, 0
v = mpfit_parse_version(version)
;; Compare version components
if v[0] LT mv[0] then return, 0
if v[1] LT mv[1] then return, 0
return, 1
end
; Manually reset recursion fencepost if the user gets in trouble
pro mpfit_reset_recursion
common mpfit_fencepost, mpfit_fencepost_active
mpfit_fencepost_active = 0
end
; **********
;
; subroutine lmdif
;
; the purpose of lmdif is to minimize the sum of the squares of
; m nonlinear functions in n variables by a modification of
; the levenberg-marquardt algorithm. the user must provide a
; subroutine which calculates the functions. the jacobian is
; then calculated by a forward-difference approximation.
;
; the subroutine statement is
;
; subroutine lmdif(fcn,m,n,x,fvec,ftol,xtol,gtol,maxfev,epsfcn,
; diag,mode,factor,nprint,info,nfev,fjac,
; ldfjac,ipvt,qtf,wa1,wa2,wa3,wa4)
;
; where
;
; fcn is the name of the user-supplied subroutine which
; calculates the functions. fcn must be declared
; in an external statement in the user calling
; program, and should be written as follows.
;
; subroutine fcn(m,n,x,fvec,iflag)
; integer m,n,iflag
; double precision x(n),fvec(m)
; ----------
; calculate the functions at x and
; return this vector in fvec.
; ----------
; return
; end
;
; the value of iflag should not be changed by fcn unless
; the user wants to terminate execution of lmdif.
; in this case set iflag to a negative integer.
;
; m is a positive integer input variable set to the number
; of functions.
;
; n is a positive integer input variable set to the number
; of variables. n must not exceed m.
;
; x is an array of length n. on input x must contain
; an initial estimate of the solution vector. on output x
; contains the final estimate of the solution vector.
;
; fvec is an output array of length m which contains
; the functions evaluated at the output x.
;
; ftol is a nonnegative input variable. termination
; occurs when both the actual and predicted relative
; reductions in the sum of squares are at most ftol.
; therefore, ftol measures the relative error desired
; in the sum of squares.
;
; xtol is a nonnegative input variable. termination
; occurs when the relative error between two consecutive
; iterates is at most xtol. therefore, xtol measures the
; relative error desired in the approximate solution.
;
; gtol is a nonnegative input variable. termination
; occurs when the cosine of the angle between fvec and
; any column of the jacobian is at most gtol in absolute
; value. therefore, gtol measures the orthogonality
; desired between the function vector and the columns
; of the jacobian.
;
; maxfev is a positive integer input variable. termination
; occurs when the number of calls to fcn is at least
; maxfev by the end of an iteration.
;
; epsfcn is an input variable used in determining a suitable
; step length for the forward-difference approximation. this
; approximation assumes that the relative errors in the
; functions are of the order of epsfcn. if epsfcn is less
; than the machine precision, it is assumed that the relative
; errors in the functions are of the order of the machine
; precision.
;
; diag is an array of length n. if mode = 1 (see
; below), diag is internally set. if mode = 2, diag
; must contain positive entries that serve as
; multiplicative scale factors for the variables.
;
; mode is an integer input variable. if mode = 1, the
; variables will be scaled internally. if mode = 2,
; the scaling is specified by the input diag. other
; values of mode are equivalent to mode = 1.
;
; factor is a positive input variable used in determining the
; initial step bound. this bound is set to the product of
; factor and the euclidean norm of diag*x if nonzero, or else
; to factor itself. in most cases factor should lie in the
; interval (.1,100.). 100. is a generally recommended value.
;
; nprint is an integer input variable that enables controlled
; printing of iterates if it is positive. in this case,
; fcn is called with iflag = 0 at the beginning of the first
; iteration and every nprint iterations thereafter and
; immediately prior to return, with x and fvec available
; for printing. if nprint is not positive, no special calls
; of fcn with iflag = 0 are made.
;
; info is an integer output variable. if the user has
; terminated execution, info is set to the (negative)
; value of iflag. see description of fcn. otherwise,
; info is set as follows.
;
; info = 0 improper input parameters.
;
; info = 1 both actual and predicted relative reductions
; in the sum of squares are at most ftol.
;
; info = 2 relative error between two consecutive iterates
; is at most xtol.
;
; info = 3 conditions for info = 1 and info = 2 both hold.
;
; info = 4 the cosine of the angle between fvec and any
; column of the jacobian is at most gtol in
; absolute value.
;
; info = 5 number of calls to fcn has reached or
; exceeded maxfev.
;
; info = 6 ftol is too small. no further reduction in
; the sum of squares is possible.
;
; info = 7 xtol is too small. no further improvement in
; the approximate solution x is possible.
;
; info = 8 gtol is too small. fvec is orthogonal to the
; columns of the jacobian to machine precision.
;
; nfev is an integer output variable set to the number of
; calls to fcn.
;
; fjac is an output m by n array. the upper n by n submatrix
; of fjac contains an upper triangular matrix r with
; diagonal elements of nonincreasing magnitude such that
;
; t t t
; p *(jac *jac)*p = r *r,
;
; where p is a permutation matrix and jac is the final
; calculated jacobian. column j of p is column ipvt(j)
; (see below) of the identity matrix. the lower trapezoidal
; part of fjac contains information generated during
; the computation of r.
;
; ldfjac is a positive integer input variable not less than m
; which specifies the leading dimension of the array fjac.
;
; ipvt is an integer output array of length n. ipvt
; defines a permutation matrix p such that jac*p = q*r,
; where jac is the final calculated jacobian, q is
; orthogonal (not stored), and r is upper triangular
; with diagonal elements of nonincreasing magnitude.
; column j of p is column ipvt(j) of the identity matrix.
;
; qtf is an output array of length n which contains
; the first n elements of the vector (q transpose)*fvec.
;
; wa1, wa2, and wa3 are work arrays of length n.
;
; wa4 is a work array of length m.
;
; subprograms called
;
; user-supplied ...... fcn
;
; minpack-supplied ... dpmpar,enorm,fdjac2,lmpar,qrfac
;
; fortran-supplied ... dabs,dmax1,dmin1,dsqrt,mod
;
; argonne national laboratory. minpack project. march 1980.
; burton s. garbow, kenneth e. hillstrom, jorge j. more
;
; **********
function mpfit, fcn, xall, FUNCTARGS=fcnargs, SCALE_FCN=scalfcn, $
ftol=ftol0, xtol=xtol0, gtol=gtol0, epsfcn=epsfcn, $
resdamp=damp0, $
nfev=nfev, maxiter=maxiter, errmsg=errmsg, $
factor=factor0, nprint=nprint0, STATUS=info, $
iterproc=iterproc0, iterargs=iterargs, iterstop=ss,$
iterkeystop=iterkeystop, $
niter=iter, nfree=nfree, npegged=npegged, dof=dof, $
diag=diag, rescale=rescale, autoderivative=autoderiv0, $
pfree_index=ifree, $
perror=perror, covar=covar, nocovar=nocovar, $
bestnorm=fnorm, best_resid=fvec, $
best_fjac=output_fjac, calc_fjac=calc_fjac, $
parinfo=parinfo, quiet=quiet, nocatch=nocatch, $
fastnorm=fastnorm0, proc=proc, query=query, $
external_state=state, external_init=extinit, $
external_fvec=efvec, external_fjac=efjac, $
version=version, min_version=min_version0
COMPILE_OPT strictarr
info = 0L
errmsg = ''
;; Compute the revision number, to be returned in the VERSION and
;; QUERY keywords.
common mpfit_revision_common, mpfit_revision_str
if n_elements(mpfit_revision_str) EQ 0 then $
mpfit_revision_str = mpfit_revision()
version = mpfit_revision_str
if keyword_set(query) then begin
if n_elements(min_version0) GT 0 then $
if mpfit_min_version(version, min_version0[0]) EQ 0 then $
return, 0
return, 1
endif
if n_elements(min_version0) GT 0 then $
if mpfit_min_version(version, min_version0[0]) EQ 0 then begin
message, 'ERROR: minimum required version '+min_version0[0]+' not satisfied', /info
return, !values.d_nan
endif
if n_params() EQ 0 then begin
message, "USAGE: PARMS = MPFIT('MYFUNCT', START_PARAMS, ... )", /info
return, !values.d_nan
endif
;; Use of double here not a problem since f/x/gtol are all only used
;; in comparisons
if n_elements(ftol0) EQ 0 then ftol = 1.D-10 else ftol = ftol0[0]
if n_elements(xtol0) EQ 0 then xtol = 1.D-10 else xtol = xtol0[0]
if n_elements(gtol0) EQ 0 then gtol = 1.D-10 else gtol = gtol0[0]
if n_elements(factor0) EQ 0 then factor = 100. else factor = factor0[0]
if n_elements(nprint0) EQ 0 then nprint = 1 else nprint = nprint0[0]
if n_elements(iterproc0) EQ 0 then iterproc = 'MPFIT_DEFITER' else iterproc = iterproc0[0]
if n_elements(autoderiv0) EQ 0 then autoderiv = 1 else autoderiv = autoderiv0[0]
if n_elements(fastnorm0) EQ 0 then fastnorm = 0 else fastnorm = fastnorm0[0]
if n_elements(damp0) EQ 0 then damp = 0 else damp = damp0[0]
;; These are special configuration parameters that can't be easily
;; passed by MPFIT directly.
;; FASTNORM - decide on which sum-of-squares technique to use (1)
;; is fast, (0) is slower
;; PROC - user routine is a procedure (1) or function (0)
;; QANYTIED - set to 1 if any parameters are TIED, zero if none
;; PTIED - array of strings, one for each parameter
common mpfit_config, mpconfig
mpconfig = {fastnorm: keyword_set(fastnorm), proc: 0, nfev: 0L, damp: damp}
common mpfit_machar, machvals
iflag = 0L
catch_msg = 'in MPFIT'
nfree = 0L
npegged = 0L
dof = 0L
output_fjac = 0L
;; Set up a persistent fencepost that prevents recursive calls
common mpfit_fencepost, mpfit_fencepost_active
if n_elements(mpfit_fencepost_active) EQ 0 then mpfit_fencepost_active = 0
if mpfit_fencepost_active then begin
errmsg = 'ERROR: recursion detected; you cannot run MPFIT recursively'
goto, TERMINATE
endif
;; Only activate the fencepost if we are not in debugging mode
if NOT keyword_set(nocatch) then mpfit_fencepost_active = 1
;; Parameter damping doesn't work when user is providing their own
;; gradients.
if damp NE 0 AND NOT keyword_set(autoderiv) then begin
errmsg = 'ERROR: keywords DAMP and AUTODERIV are mutually exclusive'
goto, TERMINATE
endif
;; Process the ITERSTOP and ITERKEYSTOP keywords, and turn this into
;; a set of keywords to pass to MPFIT_DEFITER.
if strupcase(iterproc) EQ 'MPFIT_DEFITER' AND n_elements(iterargs) EQ 0 $
AND keyword_set(ss) then begin
if n_elements(iterkeystop) GT 0 then begin
sz = size(iterkeystop)
tp = sz[sz[0]+1]
if tp EQ 7 then begin
;; String - convert first char to byte
iterkeybyte = (byte(iterkeystop[0]))[0]
endif
if (tp GE 1 AND tp LE 3) OR (tp GE 12 AND tp LE 15) then begin
;; Integer - convert to byte
iterkeybyte = byte(iterkeystop[0])
endif
if n_elements(iterkeybyte) EQ 0 then begin
errmsg = 'ERROR: ITERKEYSTOP must be either a BYTE or STRING'
goto, TERMINATE
endif
iterargs = {iterstop: 1, iterkeybyte: iterkeybyte}
endif else begin
iterargs = {iterstop: 1, iterkeybyte: 7b}
endelse
endif
;; Handle error conditions gracefully
if NOT keyword_set(nocatch) then begin
catch, catcherror
if catcherror NE 0 then begin ;; An error occurred!!!
catch, /cancel
mpfit_fencepost_active = 0
err_string = ''+!error_state.msg
message, /cont, 'Error detected while '+catch_msg+':'
message, /cont, err_string
message, /cont, 'Error condition detected. Returning to MAIN level.'
if errmsg EQ '' then $
errmsg = 'Error detected while '+catch_msg+': '+err_string
if info EQ 0 then info = -18
return, !values.d_nan
endif
endif
mpconfig = create_struct(mpconfig, 'NOCATCH', keyword_set(nocatch))
;; Parse FCN function name - be sure it is a scalar string
sz = size(fcn)
if sz[0] NE 0 then begin
FCN_NAME:
errmsg = 'ERROR: MYFUNCT must be a scalar string'
goto, TERMINATE
endif
if sz[sz[0]+1] NE 7 then goto, FCN_NAME
isext = 0
if fcn EQ '(EXTERNAL)' then begin
if n_elements(efvec) EQ 0 OR n_elements(efjac) EQ 0 then begin
errmsg = 'ERROR: when using EXTERNAL function, EXTERNAL_FVEC '+$
'and EXTERNAL_FJAC must be defined'
goto, TERMINATE
endif
nv = n_elements(efvec)
nj = n_elements(efjac)
if (nj MOD nv) NE 0 then begin
errmsg = 'ERROR: the number of values in EXTERNAL_FJAC must be '+ $
'a multiple of the number of values in EXTERNAL_FVEC'
goto, TERMINATE
endif
isext = 1
endif
;; Parinfo:
;; --------------- STARTING/CONFIG INFO (passed in to routine, not changed)
;; .value - starting value for parameter
;; .fixed - parameter is fixed
;; .limited - a two-element array, if parameter is bounded on
;; lower/upper side
;; .limits - a two-element array, lower/upper parameter bounds, if
;; limited vale is set
;; .step - step size in Jacobian calc, if greater than zero
catch_msg = 'parsing input parameters'
;; Parameters can either be stored in parinfo, or x. Parinfo takes
;; precedence if it exists.
if n_elements(xall) EQ 0 AND n_elements(parinfo) EQ 0 then begin
errmsg = 'ERROR: must pass parameters in P or PARINFO'
goto, TERMINATE
endif
;; Be sure that PARINFO is of the right type
if n_elements(parinfo) GT 0 then begin
;; Make sure the array is 1-D
parinfo = parinfo[*]
parinfo_size = size(parinfo)
if parinfo_size[parinfo_size[0]+1] NE 8 then begin
errmsg = 'ERROR: PARINFO must be a structure.'
goto, TERMINATE
endif
if n_elements(xall) GT 0 AND n_elements(xall) NE n_elements(parinfo) $
then begin
errmsg = 'ERROR: number of elements in PARINFO and P must agree'
goto, TERMINATE
endif
endif
;; If the parameters were not specified at the command line, then
;; extract them from PARINFO
if n_elements(xall) EQ 0 then begin
mpfit_parinfo, parinfo, tagnames, 'VALUE', xall, status=status
if status EQ 0 then begin
errmsg = 'ERROR: either P or PARINFO[*].VALUE must be supplied.'
goto, TERMINATE
endif
sz = size(xall)
;; Convert to double if parameters are not float or double
if sz[sz[0]+1] NE 4 AND sz[sz[0]+1] NE 5 then $
xall = double(xall)
endif
xall = xall[*] ;; Make sure the array is 1-D
npar = n_elements(xall)
zero = xall[0] * 0.
one = zero + 1.
fnorm = -one
fnorm1 = -one
;; TIED parameters?
mpfit_parinfo, parinfo, tagnames, 'TIED', ptied, default='', n=npar
ptied = strtrim(ptied, 2)
wh = where(ptied NE '', qanytied)
qanytied = qanytied GT 0
mpconfig = create_struct(mpconfig, 'QANYTIED', qanytied, 'PTIED', ptied)
;; FIXED parameters ?
mpfit_parinfo, parinfo, tagnames, 'FIXED', pfixed, default=0, n=npar
pfixed = pfixed EQ 1
pfixed = pfixed OR (ptied NE '');; Tied parameters are also effectively fixed
;; Finite differencing step, absolute and relative, and sidedness of deriv.
mpfit_parinfo, parinfo, tagnames, 'STEP', step, default=zero, n=npar
mpfit_parinfo, parinfo, tagnames, 'RELSTEP', dstep, default=zero, n=npar
mpfit_parinfo, parinfo, tagnames, 'MPSIDE', dside, default=0, n=npar
;; Debugging parameters
mpfit_parinfo, parinfo, tagnames, 'MPDERIV_DEBUG', ddebug, default=0, n=npar
mpfit_parinfo, parinfo, tagnames, 'MPDERIV_RELTOL', ddrtol, default=zero, n=npar
mpfit_parinfo, parinfo, tagnames, 'MPDERIV_ABSTOL', ddatol, default=zero, n=npar
;; Maximum and minimum steps allowed to be taken in one iteration
mpfit_parinfo, parinfo, tagnames, 'MPMAXSTEP', maxstep, default=zero, n=npar
mpfit_parinfo, parinfo, tagnames, 'MPMINSTEP', minstep, default=zero, n=npar
qmin = minstep * 0 ;; Remove minstep for now!!
qmax = maxstep NE 0
wh = where(qmin AND qmax AND maxstep LT minstep, ct)
if ct GT 0 then begin
errmsg = 'ERROR: MPMINSTEP is greater than MPMAXSTEP'
goto, TERMINATE
endif
;; Finish up the free parameters
ifree = where(pfixed NE 1, nfree)
if nfree EQ 0 then begin
errmsg = 'ERROR: no free parameters'
goto, TERMINATE
endif
;; An external Jacobian must be checked against the number of
;; parameters
if isext then begin
if (nj/nv) NE nfree then begin
errmsg = string(nv, nfree, nfree, $
format=('("ERROR: EXTERNAL_FJAC must be a ",I0," x ",I0,' + $
'" array, where ",I0," is the number of free parameters")'))
goto, TERMINATE
endif
endif
;; Compose only VARYING parameters
xnew = xall ;; xnew is the set of parameters to be returned
x = xnew[ifree] ;; x is the set of free parameters
; Same for min/max step diagnostics
qmin = qmin[ifree] & minstep = minstep[ifree]
qmax = qmax[ifree] & maxstep = maxstep[ifree]
wh = where(qmin OR qmax, ct)
qminmax = ct GT 0
;; LIMITED parameters ?
mpfit_parinfo, parinfo, tagnames, 'LIMITED', limited, status=st1
mpfit_parinfo, parinfo, tagnames, 'LIMITS', limits, status=st2
if st1 EQ 1 AND st2 EQ 1 then begin
;; Error checking on limits in parinfo
wh = where((limited[0,*] AND xall LT limits[0,*]) OR $
(limited[1,*] AND xall GT limits[1,*]), ct)
if ct GT 0 then begin
errmsg = 'ERROR: parameters are not within PARINFO limits'
goto, TERMINATE
endif
wh = where(limited[0,*] AND limited[1,*] AND $
limits[0,*] GE limits[1,*] AND $
pfixed EQ 0, ct)
if ct GT 0 then begin
errmsg = 'ERROR: PARINFO parameter limits are not consistent'
goto, TERMINATE
endif
;; Transfer structure values to local variables
qulim = limited[1, ifree]
ulim = limits [1, ifree]
qllim = limited[0, ifree]
llim = limits [0, ifree]
wh = where(qulim OR qllim, ct)
if ct GT 0 then qanylim = 1 else qanylim = 0
endif else begin
;; Fill in local variables with dummy values
qulim = lonarr(nfree)
ulim = x * 0.
qllim = qulim
llim = x * 0.
qanylim = 0
endelse
;; Initialize the number of parameters pegged at a hard limit value
wh = where((qulim AND (x EQ ulim)) OR (qllim AND (x EQ llim)), npegged)
n = n_elements(x)
if n_elements(maxiter) EQ 0 then maxiter = 200L
;; Check input parameters for errors
if (n LE 0) OR (ftol LE 0) OR (xtol LE 0) OR (gtol LE 0) $
OR (maxiter LT 0) OR (factor LE 0) then begin
errmsg = 'ERROR: input keywords are inconsistent'
goto, TERMINATE
endif
if keyword_set(rescale) then begin
errmsg = 'ERROR: DIAG parameter scales are inconsistent'
if n_elements(diag) LT n then goto, TERMINATE
wh = where(diag LE 0, ct)
if ct GT 0 then goto, TERMINATE
errmsg = ''
endif
if n_elements(state) NE 0 AND NOT keyword_set(extinit) then begin
szst = size(state)
if szst[szst[0]+1] NE 8 then begin
errmsg = 'EXTERNAL_STATE keyword was not preserved'
status = 0
goto, TERMINATE
endif
if nfree NE n_elements(state.ifree) then begin
BAD_IFREE:
errmsg = 'Number of free parameters must not change from one '+$
'external iteration to the next'
status = 0
goto, TERMINATE
endif
wh = where(ifree NE state.ifree, ct)
if ct GT 0 then goto, BAD_IFREE
tnames = tag_names(state)
for i = 0L, n_elements(tnames)-1 do begin
dummy = execute(tnames[i]+' = state.'+tnames[i])
endfor
wa4 = reform(efvec, n_elements(efvec))
goto, RESUME_FIT
endif
common mpfit_error, mperr
if NOT isext then begin
mperr = 0
catch_msg = 'calling '+fcn
fvec = mpfit_call(fcn, xnew, _EXTRA=fcnargs)
iflag = mperr
if iflag LT 0 then begin
errmsg = 'ERROR: first call to "'+fcn+'" failed'
goto, TERMINATE
endif
endif else begin
fvec = reform(efvec, n_elements(efvec))
endelse
catch_msg = 'calling MPFIT_SETMACHAR'
sz = size(fvec[0])
isdouble = (sz[sz[0]+1] EQ 5)
mpfit_setmachar, double=isdouble
common mpfit_profile, profvals
; prof_start = systime(1)
MACHEP0 = machvals.machep
DWARF = machvals.minnum
szx = size(x)
;; The parameters and the squared deviations should have the same
;; type. Otherwise the MACHAR-based evaluation will fail.
catch_msg = 'checking parameter data'
tp = szx[szx[0]+1]
if tp NE 4 AND tp NE 5 then begin
if NOT keyword_set(quiet) then begin
message, 'WARNING: input parameters must be at least FLOAT', /info
message, ' (converting parameters to FLOAT)', /info
endif
x = float(x)
xnew = float(x)
szx = size(x)
endif
if isdouble AND tp NE 5 then begin
if NOT keyword_set(quiet) then begin
message, 'WARNING: data is DOUBLE but parameters are FLOAT', /info
message, ' (converting parameters to DOUBLE)', /info
endif
x = double(x)
xnew = double(xnew)
endif
m = n_elements(fvec)
if (m LT n) then begin
errmsg = 'ERROR: number of parameters must not exceed data'
goto, TERMINATE
endif
fnorm = mpfit_enorm(fvec)
if finite(fnorm) EQ 0 then goto, FAIL_OVERFLOW
;; Initialize Levelberg-Marquardt parameter and iteration counter
par = zero
iter = 1L
qtf = x * 0.
;; Beginning of the outer loop
OUTER_LOOP:
;; If requested, call fcn to enable printing of iterates
xnew[ifree] = x
if qanytied then mpfit_tie, xnew, ptied
dof = (n_elements(fvec) - nfree) > 1L
if nprint GT 0 AND iterproc NE '' then begin
catch_msg = 'calling '+iterproc
iflag = 0L
if (iter-1) MOD nprint EQ 0 then begin
mperr = 0
xnew0 = xnew
call_procedure, iterproc, fcn, xnew, iter, fnorm^2, $
FUNCTARGS=fcnargs, parinfo=parinfo, quiet=quiet, $
dof=dof, _EXTRA=iterargs
iflag = mperr
;; Check for user termination
if iflag LT 0 then begin
errmsg = 'WARNING: premature termination by "'+iterproc+'"'
goto, TERMINATE
endif
;; If parameters were changed (grrr..) then re-tie
if max(abs(xnew0-xnew)) GT 0 then begin
if qanytied then mpfit_tie, xnew, ptied
x = xnew[ifree]
endif
endif
endif
;; Calculate the jacobian matrix
iflag = 2
if NOT isext then begin
catch_msg = 'calling MPFIT_FDJAC2'
;; NOTE! If you change this call then change the one during
;; clean-up as well!
fjac = mpfit_fdjac2(fcn, x, fvec, step, qulim, ulim, dside, $
iflag=iflag, epsfcn=epsfcn, $
autoderiv=autoderiv, dstep=dstep, $
FUNCTARGS=fcnargs, ifree=ifree, xall=xnew, $
deriv_debug=ddebug, deriv_reltol=ddrtol, deriv_abstol=ddatol)
if iflag LT 0 then begin
errmsg = 'WARNING: premature termination by FDJAC2'
goto, TERMINATE
endif
endif else begin
fjac = reform(efjac,n_elements(fvec),npar, /overwrite)
endelse
;; Rescale the residuals and gradient, for use with "alternative"
;; statistics such as the Cash statistic.
catch_msg = 'prescaling residuals and gradient'
if n_elements(scalfcn) GT 0 then begin
call_procedure, strtrim(scalfcn[0],2), fvec, fjac
endif
;; Determine if any of the parameters are pegged at the limits
npegged = 0L
if qanylim then begin
catch_msg = 'zeroing derivatives of pegged parameters'
whlpeg = where(qllim AND (x EQ llim), nlpeg)
whupeg = where(qulim AND (x EQ ulim), nupeg)
npegged = nlpeg + nupeg
;; See if any "pegged" values should keep their derivatives
if (nlpeg GT 0) then begin
;; Total derivative of sum wrt lower pegged parameters
;; Note: total(fvec*fjac[*,i]) is d(CHI^2)/dX[i]
for i = 0L, nlpeg-1 do begin
sum = total(fvec * fjac[*,whlpeg[i]])
if sum GT 0 then fjac[*,whlpeg[i]] = 0
endfor
endif
if (nupeg GT 0) then begin
;; Total derivative of sum wrt upper pegged parameters
for i = 0L, nupeg-1 do begin
sum = total(fvec * fjac[*,whupeg[i]])
if sum LT 0 then fjac[*,whupeg[i]] = 0
endfor
endif
endif
;; Save a copy of the Jacobian if the user requests it...
if keyword_set(calc_fjac) then output_fjac = fjac
;; =====================
;; Compute the QR factorization of the jacobian
catch_msg = 'calling MPFIT_QRFAC'
;; IN: Jacobian
;; OUT: Hh Vects Permutation RDIAG ACNORM
mpfit_qrfac, fjac, ipvt, wa1, wa2, /pivot
;; Jacobian - jacobian matrix computed by mpfit_fdjac2
;; Hh vects - house holder vectors from QR factorization & R matrix
;; Permutation - permutation vector for pivoting
;; RDIAG - diagonal elements of R matrix
;; ACNORM - norms of input Jacobian matrix before factoring
;; =====================
;; On the first iteration if "diag" is unspecified, scale
;; according to the norms of the columns of the initial jacobian
catch_msg = 'rescaling diagonal elements'
if (iter EQ 1) then begin
;; Input: WA2 = root sum of squares of original Jacobian matrix
;; DIAG = user-requested diagonal (not documented!)
;; FACTOR = user-requested norm factor (not documented!)
;; Output: DIAG = Diagonal scaling values
;; XNORM = sum of squared scaled residuals
;; DELTA = rescaled XNORM
if NOT keyword_set(rescale) OR (n_elements(diag) LT n) then begin
diag = wa2 ;; Calculated from original Jacobian
wh = where (diag EQ 0, ct) ;; Handle zero values
if ct GT 0 then diag[wh] = one
endif
;; On the first iteration, calculate the norm of the scaled x
;; and initialize the step bound delta
wa3 = diag * x ;; WA3 is temp variable
xnorm = mpfit_enorm(wa3)
delta = factor*xnorm
if delta EQ zero then delta = zero + factor
endif
;; Form (q transpose)*fvec and store the first n components in qtf
catch_msg = 'forming (q transpose)*fvec'
wa4 = fvec
for j=0L, n-1 do begin
lj = ipvt[j]
temp3 = fjac[j,lj]
if temp3 NE 0 then begin
fj = fjac[j:*,lj]
wj = wa4[j:*]
;; *** optimization wa4(j:*)
wa4[j] = wj - fj * total(fj*wj) / temp3
endif
fjac[j,lj] = wa1[j]
qtf[j] = wa4[j]
endfor
;; From this point on, only the square matrix, consisting of the
;; triangle of R, is needed.
fjac = fjac[0:n-1, 0:n-1]
fjac = reform(fjac, n, n, /overwrite)
fjac = fjac[*, ipvt] ;; Convert to permuted order
fjac = reform(fjac, n, n, /overwrite)
;; Check for overflow. This should be a cheap test here since FJAC
;; has been reduced to a (small) square matrix, and the test is
;; O(N^2).
wh = where(finite(fjac) EQ 0, ct)
if ct GT 0 then goto, FAIL_OVERFLOW
;; Compute the norm of the scaled gradient
catch_msg = 'computing the scaled gradient'
gnorm = zero
if fnorm NE 0 then begin
for j=0L, n-1 do begin
l = ipvt[j]
if wa2[l] NE 0 then begin
sum = total(fjac[0:j,j]*qtf[0:j])/fnorm
gnorm = max([gnorm,abs(sum/wa2[l])])
endif
endfor
endif
;; Test for convergence of the gradient norm
if gnorm LE gtol then info = 4
if info NE 0 then goto, TERMINATE
if maxiter EQ 0 then begin
info = 5
goto, TERMINATE
endif
;; Rescale if necessary
if NOT keyword_set(rescale) then $
diag = diag > wa2
;; Beginning of the inner loop
INNER_LOOP:
;; Determine the levenberg-marquardt parameter
catch_msg = 'calculating LM parameter (MPFIT_LMPAR)'
par = mpfit_lmpar(fjac, ipvt, diag, qtf, delta, wa1, wa2, par=par)
;; Store the direction p and x+p. Calculate the norm of p
wa1 = -wa1
if qanylim EQ 0 AND qminmax EQ 0 then begin
;; No parameter limits, so just move to new position WA2
alpha = one
wa2 = x + wa1
endif else begin
;; Respect the limits. If a step were to go out of bounds, then
;; we should take a step in the same direction but shorter distance.
;; The step should take us right to the limit in that case.
alpha = one
if qanylim EQ 1 then begin
;; Do not allow any steps out of bounds
catch_msg = 'checking for a step out of bounds'
if nlpeg GT 0 then wa1[whlpeg] = wa1[whlpeg] > 0
if nupeg GT 0 then wa1[whupeg] = wa1[whupeg] < 0
dwa1 = abs(wa1) GT MACHEP0
whl = where(dwa1 AND qllim AND (x + wa1 LT llim), lct)
if lct GT 0 then $
alpha = min([alpha, (llim[whl]-x[whl])/wa1[whl]])
whu = where(dwa1 AND qulim AND (x + wa1 GT ulim), uct)
if uct GT 0 then $
alpha = min([alpha, (ulim[whu]-x[whu])/wa1[whu]])
endif
;; Obey any max step values.
if qminmax EQ 1 then begin
nwa1 = wa1 * alpha
whmax = where(qmax AND maxstep GT 0, ct)
if ct GT 0 then begin
mrat = max(abs(nwa1[whmax])/abs(maxstep[whmax]))
if mrat GT 1 then alpha = alpha / mrat
endif
endif
;; Scale the resulting vector
wa1 = wa1 * alpha
wa2 = x + wa1
;; Adjust the final output values. If the step put us exactly
;; on a boundary, make sure we peg it there.
sgnu = (ulim GE 0)*2d - 1d
sgnl = (llim GE 0)*2d - 1d
;; Handles case of
;; ... nonzero *LIM ... ... zero *LIM ...
ulim1 = ulim*(1-sgnu*MACHEP0) - (ulim EQ 0)*MACHEP0
llim1 = llim*(1+sgnl*MACHEP0) + (llim EQ 0)*MACHEP0
wh = where(qulim AND (wa2 GE ulim1), ct)
if ct GT 0 then wa2[wh] = ulim[wh]
wh = where(qllim AND (wa2 LE llim1), ct)
if ct GT 0 then wa2[wh] = llim[wh]
endelse
wa3 = diag * wa1
pnorm = mpfit_enorm(wa3)
;; On the first iteration, adjust the initial step bound
if iter EQ 1 then delta = min([delta,pnorm])
xnew[ifree] = wa2
if isext then goto, SAVE_STATE
;; Evaluate the function at x+p and calculate its norm
mperr = 0
catch_msg = 'calling '+fcn
wa4 = mpfit_call(fcn, xnew, _EXTRA=fcnargs)
iflag = mperr
if iflag LT 0 then begin
errmsg = 'WARNING: premature termination by "'+fcn+'"'
goto, TERMINATE
endif
RESUME_FIT:
fnorm1 = mpfit_enorm(wa4)
if finite(fnorm1) EQ 0 then goto, FAIL_OVERFLOW
;; Compute the scaled actual reduction
catch_msg = 'computing convergence criteria'
actred = -one
if 0.1D * fnorm1 LT fnorm then actred = - (fnorm1/fnorm)^2 + 1.
;; Compute the scaled predicted reduction and the scaled directional
;; derivative
for j = 0L, n-1 do begin
wa3[j] = 0
wa3[0:j] = wa3[0:j] + fjac[0:j,j]*wa1[ipvt[j]]
endfor
;; Remember, alpha is the fraction of the full LM step actually
;; taken
temp1 = mpfit_enorm(alpha*wa3)/fnorm
temp2 = (sqrt(alpha*par)*pnorm)/fnorm
half = zero + 0.5
prered = temp1*temp1 + (temp2*temp2)/half
dirder = -(temp1*temp1 + temp2*temp2)
;; Compute the ratio of the actual to the predicted reduction.
ratio = zero
tenth = zero + 0.1
if prered NE 0 then ratio = actred/prered
;; Update the step bound
if ratio LE 0.25D then begin
if actred GE 0 then temp = half $
else temp = half*dirder/(dirder + half*actred)
if ((0.1D*fnorm1) GE fnorm) OR (temp LT 0.1D) then temp = tenth
delta = temp*min([delta,pnorm/tenth])
par = par/temp
endif else begin
if (par EQ 0) OR (ratio GE 0.75) then begin
delta = pnorm/half
par = half*par
endif
endelse
;; Test for successful iteration
if ratio GE 0.0001 then begin
;; Successful iteration. Update x, fvec, and their norms
x = wa2
wa2 = diag * x
fvec = wa4
xnorm = mpfit_enorm(wa2)
fnorm = fnorm1
iter = iter + 1
endif
;; Tests for convergence
if (abs(actred) LE ftol) AND (prered LE ftol) $
AND (0.5D * ratio LE 1) then info = 1
if delta LE xtol*xnorm then info = 2
if (abs(actred) LE ftol) AND (prered LE ftol) $
AND (0.5D * ratio LE 1) AND (info EQ 2) then info = 3
if info NE 0 then goto, TERMINATE
;; Tests for termination and stringent tolerances
if iter GE maxiter then info = 5
if (abs(actred) LE MACHEP0) AND (prered LE MACHEP0) $
AND (0.5*ratio LE 1) then info = 6
if delta LE MACHEP0*xnorm then info = 7
if gnorm LE MACHEP0 then info = 8
if info NE 0 then goto, TERMINATE
;; End of inner loop. Repeat if iteration unsuccessful
if ratio LT 0.0001 then begin
goto, INNER_LOOP
endif
;; Check for over/underflow
wh = where(finite(wa1) EQ 0 OR finite(wa2) EQ 0 OR finite(x) EQ 0, ct)
if ct GT 0 OR finite(ratio) EQ 0 then begin
FAIL_OVERFLOW:
errmsg = ('ERROR: parameter or function value(s) have become '+$
'infinite; check model function for over- '+$
'and underflow')
info = -16
goto, TERMINATE
endif
;; End of outer loop.
goto, OUTER_LOOP
TERMINATE:
catch_msg = 'in the termination phase'
;; Termination, either normal or user imposed.
if iflag LT 0 then info = iflag
iflag = 0
if n_elements(xnew) EQ 0 then goto, FINAL_RETURN
if nfree EQ 0 then xnew = xall else xnew[ifree] = x
if n_elements(qanytied) GT 0 then if qanytied then mpfit_tie, xnew, ptied
dof = n_elements(fvec) - nfree
;; Call the ITERPROC at the end of the fit, if the fit status is
;; okay. Don't call it if the fit failed for some reason.
if info GT 0 then begin
mperr = 0
xnew0 = xnew
call_procedure, iterproc, fcn, xnew, iter, fnorm^2, $
FUNCTARGS=fcnargs, parinfo=parinfo, quiet=quiet, $
dof=dof, _EXTRA=iterargs
iflag = mperr
if iflag LT 0 then begin
errmsg = 'WARNING: premature termination by "'+iterproc+'"'
endif else begin
;; If parameters were changed (grrr..) then re-tie
if max(abs(xnew0-xnew)) GT 0 then begin
if qanytied then mpfit_tie, xnew, ptied
x = xnew[ifree]
endif
endelse
endif
;; Initialize the number of parameters pegged at a hard limit value
npegged = 0L
if n_elements(qanylim) GT 0 then if qanylim then begin
wh = where((qulim AND (x EQ ulim)) OR $
(qllim AND (x EQ llim)), npegged)
endif
;; Calculate final function value (FNORM) and residuals (FVEC)
if isext EQ 0 AND nprint GT 0 AND info GT 0 then begin
catch_msg = 'calling '+fcn
fvec = mpfit_call(fcn, xnew, _EXTRA=fcnargs)
catch_msg = 'in the termination phase'
fnorm = mpfit_enorm(fvec)
endif
if n_elements(fnorm) EQ 0 AND n_elements(fnorm1) GT 0 then fnorm = fnorm1
if n_elements(fnorm) GT 0 then fnorm = fnorm^2.
covar = !values.d_nan
;; (very carefully) set the covariance matrix COVAR
if info GT 0 AND NOT keyword_set(nocovar) $
AND n_elements(n) GT 0 $
AND n_elements(fjac) GT 0 AND n_elements(ipvt) GT 0 then begin
sz = size(fjac)
if n GT 0 AND sz[0] GT 1 AND sz[1] GE n AND sz[2] GE n $
AND n_elements(ipvt) GE n then begin
catch_msg = 'computing the covariance matrix'
if n EQ 1 then $
cv = mpfit_covar(reform([fjac[0,0]],1,1), ipvt[0]) $
else $
cv = mpfit_covar(fjac[0:n-1,0:n-1], ipvt[0:n-1])
cv = reform(cv, n, n, /overwrite)
nn = n_elements(xall)
;; Fill in actual covariance matrix, accounting for fixed
;; parameters.
covar = replicate(zero, nn, nn)
for i = 0L, n-1 do begin
covar[ifree, ifree[i]] = cv[*,i]
end
;; Compute errors in parameters
catch_msg = 'computing parameter errors'
i = lindgen(nn)
perror = replicate(abs(covar[0])*0., nn)
wh = where(covar[i,i] GE 0, ct)
if ct GT 0 then $
perror[wh] = sqrt(covar[wh, wh])
endif
endif
; catch_msg = 'returning the result'
; profvals.mpfit = profvals.mpfit + (systime(1) - prof_start)
FINAL_RETURN:
mpfit_fencepost_active = 0
nfev = mpconfig.nfev
if n_elements(xnew) EQ 0 then return, !values.d_nan
return, xnew
;; ------------------------------------------------------------------
;; Alternate ending if the user supplies the function and gradients
;; externally
;; ------------------------------------------------------------------
SAVE_STATE:
catch_msg = 'saving MPFIT state'
;; Names of variables to save
varlist = ['alpha', 'delta', 'diag', 'dwarf', 'factor', 'fnorm', $
'fjac', 'gnorm', 'nfree', 'ifree', 'ipvt', 'iter', $
'm', 'n', 'machvals', 'machep0', 'npegged', $
'whlpeg', 'whupeg', 'nlpeg', 'nupeg', $
'mpconfig', 'par', 'pnorm', 'qtf', $
'wa1', 'wa2', 'wa3', 'xnorm', 'x', 'xnew']
cmd = ''
;; Construct an expression that will save them
for i = 0L, n_elements(varlist)-1 do begin
ival = 0
dummy = execute('ival = n_elements('+varlist[i]+')')
if ival GT 0 then begin
cmd = cmd + ',' + varlist[i]+':'+varlist[i]
endif
endfor
cmd = 'state = create_struct({'+strmid(cmd,1)+'})'
state = 0
if execute(cmd) NE 1 then $
message, 'ERROR: could not save MPFIT state'
;; Set STATUS keyword to prepare for next iteration, and reset init
;; so we do not init the next time
info = 9
extinit = 0
return, xnew
end
|