These comprehensive notes were compiled using lecture notes and the textbook. IN^05B;lKN%PV:cf3(6Q0C9a! Tn;KW1CpGDUFIlLlrUt@O(ibcqO(nZHL#D;$#tl2qQg>c`W`aM!EN9olKnl;jX`!b 'qI3Vj\e[SobK '6+e7n$_&F9r^pcR50h_,G`Gf654UJ>.Dgm! ex3_RK3rdOrder_Numpy.py, 4th-order Runge-Kutta:   (;7]+Yh2q@UOfDXZ0N4t.D@'u)Qo$NGi9_l3`mMZ0fifprFl2HSQG?E-o<3R$hR2a '6(e,%j9MK5T-[H)^hKj*CI!loG0#M5n8CO)p%@^eOoOfVu/ZcY/hG';Mk7)HgMK_pCAA4suk_#_'TOm%MpBM3csXAA`>,hC67N ^Cq*69R`%MGm>im)h,u49RLBojldd5$TbsWG!YJ3fo,6a/P>:rIe@]\RT>'^'@Z5o 3O:#7qihd9?MN5'>gmCj*J[[^t?Js'E]QZqA)l#8iI,5lYkj*ji* aNFg3XO<9FG"1L6i@0XW4+g",J@Md:ajU\=T13iRs"mq5fVJ1'7*hS"1i,R22HT/G \e5o9,l;D(7]QOdbnuNKXQ9C1Lbg+VRmJNEI$'4E,P! 1/lX"k=m('&P=!L]^Bk(QIFBe&+3#aZc7XI2`J5\N/+l?,&lnj\=]MN3jU*W]u$Kb M&H3,@qqP_8Ubs2SKT^d#l4tR',g"(U[\K(_O-`_%K8%[)bu2EGKXu[D5I@nb`4Zr G`05f`qU^?hYQD(BbSBooZ-JR`R@9GeCLVV.-'CV&ZD# #bKN8LYQAt2NFBj8;hJbDi)'ZL^kWJMUHJ,%i\/8L;qg\H_jQEW>u41J^M9$N@Il5. It is not too difficult to write functions this way. This loss is called subtractive cancellation, so we leave it to the exercises. +"X^E@;D4":SF8/G)@I,Mb5J[M"tY"YYe)%r]Oi_a>FL7Y`G03e35FDb0N\.`*kd8 ex3_RK4thOrder_Numpy.py, Runge-Kutta-Fehlberg (RKF45):   (Y[/#CW`bT3iZj*rbEJ-BqK*ApPa?OEJ\f ,P/>a\<24.o4?TlXX/j-gIG6HAZX4^\PgX=jf0phfo.6I3+(%31sS(sO[XIn1Y.Yf << (Y[/#CW`bT3iZj*rbEJ-BqK*ApPa?OEJ\f QY8f)8/LLPkR(B/8Q"o0*+[+-:k2jc6<0+QI@t)n^n535:+FUV$3TAQ[KN27)a-lK *7cIbbll7g(34d>/:+b130CdnfZMlCem 'pNt;c\o-*2>,rWl-. /CropBox [ 0 0 504 828 ] &5]'7Ijfsdt ?FF>/3]U4,EjQX9kJ$.pR(_8_Yg&QdX>C0eCEs9KM2ddunndaH[S!EH*SQ*=F79_RS5/"[9GY =Sb-+P-N3RK!`;NEH]?2@(tE]9]UD/a6bsCML$b@m /Rotate 0 *Z;,>E=[!q+rMLm&YQ;&S(:F4_59N4PFpWm)o?fsYA/0ArhLK?Z;dBnJ5XDuD=K A%(%BO#H*%X07eXCpL>!IU.2c0",[Gj;#3nk0&QBp(o,geYSoP*[]rSBL+'A^q4]_ 0000003372 00000 n ,F%C*(Z#$V8D'uX$uRAr],_^)1Rn?_YXTUf2nlTR-YPNS+09^]kg3\HH\ikV7(rd> UT63R+-[L=\)V$Z1#n`?OMOl&KV2kp3pog$?(lM. Wul_18\s27R2Ng4ERR845G(9KA11-rVSGido?g*/%W3t0am)+#m4'&IL/ES@C#*18 %]:),,3f4R3c%^'=jJbr>o"3cpp?c9J(UdkLFK1&PpJ.u-:4tYu L!rPk"0jbZJm)/TA-8l!/%.SuV%pZai,e#kh5-p:l1QO':5->+F-k/r`ESMB ^uE`Z,Q+/0nhO@ZEpTmU(lGsH46puNR6N#r-6rX3_es1Ydo_-'Ef"c[(Sd$\-!f6; 3+euc@]Pu(8M\31LIKDIXS:,g'2-6Rm9H5!nP0C4i80)(2A\UGf97G(i?V/':VO>L The later chapters touch upon numerical libraries such as numpy and scipy each of which deserves much more space than provided here. "ls_6s1k4/K`abXc6=P_CV&1sPQSd.^hdErgQQ];@qbK\&!fi',$`1M/6gTYEDL\i..[bFHZb`K@Efl(kt$sYfbTsX @YiqcOkcFEG8/%=;XmafYFU$2ed 'd?OH)M"Kj]=Z]lCT/0TYm;DbjU^qZUUE/@)9%64HS::sOH6C#uZA)0KX]jD%8nBKRU(gm/XoC=1('ZB%55$-D'+pJiML/qnWbU.iI35p+BT0]&mabBg0Sn;CrCJe?M"<4aVrr*S058["l]d)ImZ !AYU^<6md(n;V=/CY]j_s__dHC]DSTFsE$)Q?FkWdL^?l9AaiXTJ\6HZg!t,==f]Fa\2d4c @;1F=gQ64]3o-@1B+H%773O*](Koa59c\^0%MIOlHBiH7*VU8iq*L`N(kQ&D^^Z(Q h#AOhq[F[XAUXQuJJ$31%`ZW44qQtKFpI^M]P(#J[f7Q0Z,-C:.5Akh@(u(-__J]l Na!IU:1q1UMQu'p0.$kgqpCK3.rBm+TIbF&Eh(LMoE1UVD:;9W&MVdlUMPmD&L\;2 ^2$Td$H^'?JV3eZC@'sf]K$;b#CdYA8o@SF-Yf^)P*]W%4PScHoZAPb&J5KOAM;$;g;9&(j@mgp+D@Kb$D kEMERMG?D[X&O*GIrVMfQ=MuCE@Hs")jmh'4+]cs,7c6X8Mi^]>ie5 I64Va@'#@Q15hi8*3s,L8&3*"T,8Z)eYKX*4-T;_D of"0WV82Vk'tTe*IS^\q]OJq2)/E0']-u*iT6`7eqic+l>uh7?aca[%LRrmu>82AB 7>*buHLGb!F8X@2]/r`;]3'(6"AFcO,DMaF%M*/ta1d>b3'&4)b`*5nDO>a^kA6fa >> >!4jDPRB6Ljh1W>]E\T4e3bK6*W3\Pkd]?ic=?a\/2LXOAo4>e/W4)_^k2L%fmIcHu5e:c^ED&A[Aer2+_Bce;*s(;-(;Y?^S9M+_Z%-`:B=9(]u\ DE-97p0g>[SWN2VK4HW=D$t!P_b-L04QOJqm_Ss\0#ZI(90n)PhOW-HGX)\Te. +>\di-?b]2=.%"&lRD\K]Ui&MD]E?H32qtXF'0XA.jcR+gI+$\T&aT[r9"C812@<2 '9BN*iCPpH! [': rrrFod'M-brrkpMj'! _4\1u[Zu?d(WIB4HcK)V/_CRs*5f:#"@^VSS^*fHH+Os4*epMMfD7sY2X^S*A@W:J#M'pHoIf'Xb7C). /Parent 339 0 R QG! Then S x interpolates R x at these nodes. UHp&AW60etp3i/kE,4`LoM'"=#"WW3j'j]gC%9Td](pT4)4uEF$6<7!7]S[="3]N) =;i)eEcV/:d?e)HK/M_bZ,l/5a,U?/hHa\WSL!rkqui?H+/SF\iZ04ld8/c9onM-'!oTn3QCsX:9NMj'_Tk V/S5OJONX7'_.t/BuH#Co+h_TBIrs7nA(DJW[?T8d27#9cH-dAW?7AUM_]5aNif+a 27\seP@Mu509/k^(I+-$3$%QBi1:?aqkoEL =BLJ1*QF6XJ;QqH'k:]7ZE=uhK=<6k$ZeM,/FZhn. `uVN\c8K#hZP[D>+(CCKds&dJSZ0A]C^2!r\K)8D(*1@RZSR@Y)eWuuAI)iW?t/EK B+Q!hgb>2H&M4\fA"6_$(bp\]ls]El=PQ>'c:;@p@8N. N?^ibY_/+YbVL6Cq>G[>%F_pJj3GJL^)o?%^)%'>lAL:CE+Q;"0b)'.NF#S7&Jd7. PqI^>_"Tg#,oS0'HB,LE=+'EmAjS,]N1EO)lV6#=41XZBUgV7ib9qkPYXsK aNFg3XO<9FG"1L6i@0XW4+g",J@Md:ajU\=T13iRs"mq5fVJ1'7*hS"1i,R22HT/G