Robust Optimization Multidimensional Marquardt Parameters Multidimensional Optimization The
Robust Optimization Multi-dimensional Marquardt Parameters
Multi-dimensional Optimization
The approximating function • A linear fitting function would be In this work the fitting function is essentially
Expand the first derivatives about the value c 0.
The result is a new estimate for the values of c
The usual problem is the inversion of the matrix • If a constant cn is such that its effect is a linear multiple of cm - that is if f. A has parts cn Pn and cm. Pm with Pn and Pm effectively the same for all xi - the matrix has no inverse owing to the fact that there are no unique values for cn and cm
The Marquardt Parameter • The solution is to request small constants. In particular add c∙c to 2
This results in constant estimates that depend on
The general result for any single constant is
Relative Smoothers
Define p 2 and Fr
Fr versus log
Fr values • If on a minimization step the actual 2 equals p 2, not brave enough • Fr (Fr)3 • If the actual 2 > the last 2 then • Fr (9+Fr)/10 • If the values are slowly changing from one side Aitkin’s extrapolation.
At the beginning • • • • • FR, CHI, CHB, CHL 0. 999990000 FR, CHI, CHB, CHL 0. 999970000 FR, CHI, CHB, CHL 0. 999946001 FR, CHI, CHB, CHL 0. 999902805 FR, CHI, CHB, CHL 0. 999825058 FR, CHI, CHB, CHL 0. 999685131 FR, CHI, CHB, CHL 0. 999433325 FR, CHI, CHB, CHL 0. 998980275 FR, CHI, CHB, CHL 0. 998165430 FR, CHI, CHB, CHL 0. 996700804 FR, CHI, CHB, CHL 0. 994071243 FR, CHI, CHB, CHL 0. 989359873 FR, CHI, CHB, CHL 0. 980949663 FR, CHI, CHB, CHL 0. 966036016 ierr, iloop 1 0 FR, CHI, CHB, CHL 0. 993990303 FR, CHI, CHB, CHL 0. 989215050 FR, CHI, CHB, CHL 0. 980691773 118785. 585 118785. 230 118784. 590 118783. 438 118781. 362 118777. 623 118770. 875 118758. 679 118736. 558 118696. 154 118621. 239 118476. 802 118128. 813 116688. 910 0. 10000 E+67 0. 50000 E+ 118782. 022 118785. 585 118778. 816 118785. 230 118773. 045 118784. 590 118762. 657 118783. 438 118743. 962 118781. 362 118710. 315 118777. 623 118649. 761 118770. 875 118540. 808 118758. 679 118344. 823 118736. 558 117992. 433 118696. 154 117359. 094 118621. 239 116219. 779 118476. 802 114116. 688 118128. 813 116611. 223 116441. 523 115680. 541 115987. 645 115353. 576 114193. 243 116688. 910 116611. 223 116441. 523
Aitkin’s extrapolation • • • • FR, CHI, CHB, CHL 0. 996558072 FR, CHI, CHB, CHL 0. 993815191 FR, CHI, CHB, CHL 0. 988901771 ierr, iloop 1 0 AITKIN'S EXTRAPOLATION FR, CHI, CHB, CHL 0. 998890177 FR, CHI, CHB, CHL 0. 998003427 FR, CHI, CHB, CHL 0. 996409757 115637. 868 115550. 779 115232. 958 114790. 155 114243. 910 113659. 796 113455. 616 112988. 531 115282. 377 114922. 670 114268. 370 113954. 076 113516. 188 112976. 005 112398. 373 112196. 459 115680. 541 115637. 868 115550. 779 115232. 958 114790. 155 114243. 910 113659. 796 113455. 616 112951. 863 112875. 210 112716. 318 112863. 134 112726. 346 112469. 960 112988. 531 112951. 863 112875. 210
Non-Linear Irevs • • • • • FR, CHI, CHB, CHL 0. 999968166 FR, CHI, CHB, CHL 0. 999942699 FR, CHI, CHB, CHL 0. 999896862 FR, CHI, CHB, CHL 0. 999814361 IREV= 1 IEND= 151 FR, CHI, CHB, CHL 0. 999953590 FR, CHI, CHB, CHL 0. 999860777 IREV= 1 IEND= 150 FR, CHI, CHB, CHL 0. 999965194 FR, CHI, CHB, CHL 0. 999906028 ierr, iloop 1 0 FR, CHI, CHB, CHL 0. 999983086 FR, CHI, CHB, CHL 0. 999969555 FR, CHI, CHB, CHL 0. 999945199 FR, CHI, CHB, CHL 0. 999901361 IREV= 1 IEND= 145 FR, CHI, CHB, CHL 0. 999975340 FR, CHI, CHB, CHL 0. 999926023 IREV= 1 IEND= 144 112081. 893 112080. 061 112075. 828 115490. 184 112078. 975 112075. 471 112068. 501 112055. 022 112082. 543 112081. 893 112080. 061 112075. 828 112070. 874 113183. 400 112070. 627 112055. 271 112075. 828 112070. 874 112067. 371 112064. 902 112066. 973 112056. 840 112070. 874 112067. 371 112064. 676 112063. 861 112061. 774 114265. 708 112063. 007 112061. 264 112057. 720 112050. 720 112064. 902 112064. 676 112063. 861 112061. 774 112059. 109 113792. 644 112059. 010 112050. 819 112061. 774 112059. 109
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