Together with this README file, you may get a uuencoded file Composition, ====== =========== which contains what is described below and more. To recover them, please do the following sequentially: uudecode Composition -- to get Composition.tar.Z uncompress Composition.tar.Z -- to get Composition.tar tar xvf Composition.tar -- to get the package Ren-Cang Li, na.rcli@na-net.ornl.gov W. Kahan, wkahan@cs.berkeley.edu Sep. 24, 1996 +-+-+-+-+-+-+-+-+--+-+-+-+-+-+-+-+-+-+-+-+-+-+-+--+-+-+-+-+-+-+-+-+-+-+-+ Purpose: 1) Provide composition constants for palindromic schemes ======= designed 1.1) to raise the orders (and thus efficiency) of unconventional schemes, and 1.2) to approximate exp(t X+t Y). 2) MATHEMATCA and MAPLE codes for computing these constants to arbitrary precisions. Note: A concise paper on doing composition from a eflexive updating ==== formula can be obtained electronically from Ren-Cang Li. The paper will appear in Math. Comp. Subdirectories: ============== BCH-Formula/ --- Baker-Campbell-Hausdorff formulas up to 9-fold brackets (MATHEMATCA) Lindepcy/ --- Codes for finding a basis, linear dependent relations of a space of all possible brackets by a few non-commutative Lie algebra's elements, say, five X's and three Y's (MATHEMATCA) fromRefXV/ --- About composition schemes constructed from a reflexive updating formula Q(h,y0): Q(delta[1] h, Q(delta[2] h, ..., Q(delta[I] h, y0) ... )), where delta[i]=delta[I-i+1]. It is called a I-stage scheme. Denote a I-stage order J scheme by sIodrJ. Set c_i=delta[1]+...+delta[i]. fromXY/ --- About approximating exp(t X+t Y) by composing exp(d t X) and exp(g t Y), where d and g are parameters to be chosen RKN/ --- Dealing with the same problem in fromXY/ with an additional assupmtion: [X,[X,[X,Y]]]=0 The Directory Tree: === ========= ==== / BCH-Formula/ | | / PalinScheme/ | | Palindromic/ | fromRefXV/ -> | | | NPalindromic/ | \ AnExample/ | -> | / PalinScheme/ | fromXY/ -> | Palindromic/ | \ NPalinScheme/ | | / PalinScheme/ | | Palindromic/ | RKN/ -> | | | NPalindromic/ | \ Lindpcy/ | \ Lindepcy/ Remark: Each subdirectory has their own detailed README files. ------ ****** +-+-+-+-+-+-+-+-+--+-+-+-+-+-+-+-+-+-+-+-+-+-+-+--+-+-+-+-+-+-+-+-+-+-+-+ Composition constants up to order 10 for fromRefXV/ (20 correct digits): =========== ========= == == ===== == === ========== == ======= ====== s3odr4: ------ delta's: .13512071919596576340E+1 -.17024143839193152681E+1 c's: .13512071919596576340E+1 s5odr4: ------ delta's: .41449077179437573714 .41449077179437573714 -.65796308717750294857 c's: .41449077179437573714 .82898154358875147428 s5odr4a: ------- delta's: .21132486540518711774 .78867513459481288225 -1 c's: .21132486540518711774 1 s5odr4b: ------- delta's: .78867513459481288225 .21132486540518711774 -1 c's: .78867513459481288225 1 s7odr6: ------ delta's: .78451361047755726382 .23557321335935813368 -.11776799841788710069E+1 .13151863206839112189E+1 c's: .78451361047755726382 .10200868238369153975E+1 -.15759316034195560944 s9odr6a: ------- delta's: .39216144400731413928 .33259913678935943860 -.70624617255763935981 .82213596293550800230E-1 .79854399093482996340 c's: .39216144400731413928 .72476058079667357788 .18514408239034218070E-1 .10072800453258501830 s9odr6b: ------- delta's: .39103020330868478817 .33403728961113601749 -.70622728118756134345 .81877549648059445772E-1 .79856447723936218405 c's: .39103020330868478817 .72506749291982080566 .18840211732259462202E-1 .10071776138031890797 s15odr8: ------- delta's: .74167036435061295345 -.40910082580003159400 .19075471029623837995 -.57386247111608226666 .29906418130365592384 .33462491824529818378 .31529309239676659663 -.79688793935291635402 c's: .74167036435061295345 .33256953855058135945 .52332424884681973940 -.50538222269262527252E-1 .24852595903439339659 .58315087727969158038 .89844396967645817701 s17odr8a: -------- delta's: .13020248308889008088 .56116298177510838456 -.38947496264484728641 .15884190655515560090 -.39590389413323757734 .18453964097831570709 .25837438768632204729 .29501172360931029887 -.60550853383003451170 c's: .13020248308889008088 .69136546486399846544 .30189050221915117903 .46073240877430677993 .64828514641069202593E-1 .24936815561938490968 .50774254330570695698 .80275426691501725585 s17odr8b: -------- delta's: .12713692773487857916 .56170253798880269972 -.38253471994883018888 .16007605629464743119 -.40181637432680696673 .18736671654227849724 .26070870920779240570 .29039738812516162389 -.60607448323584816258 c's: .12713692773487857916 .68883946572368127888 .30630474577485109000 .46638080206949852119 .64564427742691554464E-1 .25193114428497005171 .51263985349276245740 .80303724161792408129 s31odr10a: --------- delta's: -.48159895600253002870 .36303931544595926879E-2 .50180317558723140279 .28298402624506254868 .80702967895372223806 -.26090580538592205447E-1 -.87286590146318071547 -.52373568062510581643 .44521844299952789252 .18612289547097907887 .23137327866438360633 -.52191036590418628905 .74866113714499296793 .66736511890604057532E-1 -.80360324375670830316 .91249037635867994571 c's: -.48159895600253002870 -.47796856284807043601 .23834612739160966776E-1 .30681863898422351546 .11138483179379457535E+1 .10877577373993535481E+1 .21489183593617283260 -.30884384468893298382 .13637459831059490870 .32249749378157398757 .55387077244595759390 .31960406541771304852E-1 .78062154368676427278 .84735805557736833031 .43754811820660027146E-1 s31odr10b: --------- delta's: .27338476926228452782 .44587846502560283997 .83219642847136307126 -.83396868554957942879 .27891843057015194293 .89032738045702532006 .56681514845245709418E-1 -.85737420814978887722 -.46789492554836586111 -.47919009182398264249 .16724074680043708909 -.87443151263376143307 -.49873481853620165786 .58930536608974918851 .83458937790882729775 .28614352562198582747 c's: .27338476926228452782 .71926323428788736779 .15514596627592504390E+1 .71749097720967101026 .99640940777982295318 .18867367882368482732E+1 .19434183030820939827E+1 .10860440949323051054E+1 .61814916938393924433 .13895907755995660185 .30619982436039369094 -.56823168827336774213 -.10669665068095694000E+1 -.47766114071982021148 .35692823718900708627 s33odr10a: --------- delta's: 0.70428877682658066880E-1 0.87415651735353949041 0.55414604963802442707E-1 -0.66800477898797011598E-1 -0.62641308958799555593 0.23682621087528762872 -0.42221063403170054210 0.24222942201040859249 0.47374515478601436594E-1 0.54386826052472423338 -0.93252230928447264311 0.16960179883676464855 0.71608567578450563608 -0.80016730247310573512 0.23778185292256770747 -0.32330301550863943389 0.95529818470370207691 c's: .70428877682658066880E-1 .94458539503619755729 1 .93319952210120298840 .30678643251320743247 .54361264338849506120 .12140200935679451910 .36363143136720311159 .41100594684580454818 .95487420737052878156 .22351898086056138449E-1 .19195369692282078700 .90803937270732642308 .10787207023422068796 .34565392315678839544 .22350907648148961545E-1 s33odr10b: --------- delta's: .12282427644721572094 .77644680890696440342 .14881514553734297479 -.17239125953506067249 -.54745995781852463787 .14512932327306927479 -.31564555153114460562 .12086865089833871979 .17910277517866344258 .44263408813993245949 -.81935337479593697464 .13445474141752884045 .64444239169016646538 -.71930149370201612557 .21036902497348664610 -.26908194941570516294 .83629272067135846284 c's: .12282427644721572094 .89927108535418012436 .10480862308915230991E+1 .87569497135646242666 .32823501353793778878 .47336433681100706358 .15771878527986245795 .27858743617820117774 .45769021135686462033 .90032429949679707981 .80970924700860105168E-1 .21542566611838894562 .85986805780855541100 .14056656410653928542 .35093558908002593152 .81853639664320768579E-1 s33odr10c: --------- delta's: 0.12313526870982994083 0.77644981696937310520 0.14905490079567045613 -0.17250761219393744420 -0.54871240818800177942 0.14289765421841842100 -0.31419193263986861997 0.12670943739561041022 0.17444734584181312998 0.44318544665428572929 -0.81948900568299084419 0.13382545738489583020 0.64509023524410605020 -0.71936337169922060719 0.20951381813463649682 -0.26828113140636051966 0.83647216092348048955 c's: .12313526870982994083 .89958508567920304603 .10486399864748735022E+1 .87613237428093605795 .32741996609293427854 .47031762031135269954 .15612568767148407956 .28283512506709448979 .45728247090890761976 .90046791756319334905 .80978911880202504863E-1 .21480436926509833506 .85989460450920438526 .14053123280998377807 .35004505094462027488 .81763919538259755223E-1