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These are questions asked by gaurav_rs

Differentiation of a matrix...

gaurav_rs 65 Maple 17
differentiation sequence
August 21 2019
1 2

Dear Users,

I have a set of linear equations which can be presented as A(alpha,n) x(alpha)=b(alpha,n), where 'n' is the dimension of the square matric A.

For a particular value of "n" and "alpha", I can solve the unknown vector x. Further, I can differentiate Ax=b with respect to alpha to find out the rate of change of variable x with respect to alpha.

The above exercise reads, Ax'=b'-xA', which gives the unknown vector x', for a given value of alpha and n.

If I chose different values of n while fixing alpha=alpha0, the rate of change of x with alpha ( x' ) does not converge with 'n'. I noticed that x (alpha=alpha0) converges with n, also x(alpha=alpha0+ delta alpha) also converges with 'n'. I am interested in the query why x' does not converge, in spite of the fact that x converges? Any comments regarding the same are highly appreciated.

 

Thanks,

Grv

evaluting integral with real and imagina...

gaurav_rs 65 Maple 17
integration numeric
June 24 2019
1 1

Hi all, I want to evaluate the integral G2. I am using the evalf command, but maple is unable to evaluate it. What am I missing here?

 

G2 := -.5*(int(Re(-(0.8823529412e-2*(-16.19435640-30.82287308*exp(-.2960360476-(1.*I)*theta)))*exp(.1480180238-(1.*I)*(.5*theta-0.6282080040e-1*ln(-1.*sin(-.3926990818+.5*theta)/sin(.3926990818+.5*theta))))*(-.5*cos(.3926990818+.5*theta)*sin(-.3926990818+.5*theta)+.5*sin(.3926990818+.5*theta)*cos(-.3926990818+.5*theta))/sqrt(-1.*sin(.3926990818+.5*theta)*sin(-.3926990818+.5*theta))-0.1764705882e-1*sqrt(-1.*sin(.3926990818+.5*theta)*sin(-.3926990818+.5*theta))*(1.017602550-3.872635115*exp(-.2960360476-(1.*I)*theta))*exp(.1480180238-(1.*I)*(.5*theta-0.6282080040e-1*ln(-1.*sin(-.3926990818+.5*theta)/sin(.3926990818+.5*theta))))-0.1764705882e-1*sqrt(-1.*sin(.3926990818+.5*theta)*sin(-.3926990818+.5*theta))*(-16.19435640-30.82287308*exp(-.2960360476-(1.*I)*theta))*(-0.6282080040e-1-(0.6282080040e-1*I)*(.5*cos(-.3926990818+.5*theta)/sin(.3926990818+.5*theta)+.5*sin(-.3926990818+.5*theta)*cos(.3926990818+.5*theta)/sin(.3926990818+.5*theta)^2)*sin(.3926990818+.5*theta)/sin(-.3926990818+.5*theta))*exp(.1480180238-(1.*I)*(.5*theta-0.6282080040e-1*ln(-1.*sin(-.3926990818+.5*theta)/sin(.3926990818+.5*theta)))))*(-.5-.5*cos(2.*theta))+.5*Im(-(0.8823529412e-2*(-16.19435640-30.82287308*exp(-.2960360476-(1.*I)*theta)))*exp(.1480180238-(1.*I)*(.5*theta-0.6282080040e-1*ln(-1.*sin(-.3926990818+.5*theta)/sin(.3926990818+.5*theta))))*(-.5*cos(.3926990818+.5*theta)*sin(-.3926990818+.5*theta)+.5*sin(.3926990818+.5*theta)*cos(-.3926990818+.5*theta))/sqrt(-1.*sin(.3926990818+.5*theta)*sin(-.3926990818+.5*theta))-0.1764705882e-1*sqrt(-1.*sin(.3926990818+.5*theta)*sin(-.3926990818+.5*theta))*(1.017602550-3.872635115*exp(-.2960360476-(1.*I)*theta))*exp(.1480180238-(1.*I)*(.5*theta-0.6282080040e-1*ln(-1.*sin(-.3926990818+.5*theta)/sin(.3926990818+.5*theta))))-0.1764705882e-1*sqrt(-1.*sin(.3926990818+.5*theta)*sin(-.3926990818+.5*theta))*(-16.19435640-30.82287308*exp(-.2960360476-(1.*I)*theta))*(-0.6282080040e-1-(0.6282080040e-1*I)*(.5*cos(-.3926990818+.5*theta)/sin(.3926990818+.5*theta)+.5*sin(-.3926990818+.5*theta)*cos(.3926990818+.5*theta)/sin(.3926990818+.5*theta)^2)*sin(.3926990818+.5*theta)/sin(-.3926990818+.5*theta))*exp(.1480180238-(1.*I)*(.5*theta-0.6282080040e-1*ln(-1.*sin(-.3926990818+.5*theta)/sin(.3926990818+.5*theta)))))*sin(2.*theta), theta = 0. .. .7853981635))

Numerical intergration ...

gaurav_rs 65 Maple 2017
performance integration numeric
May 14 2019
2 9

 

Hi all,

I am trying to find numerical integration of a complex function (Bessel+ trigonometric function) in (r, theta). MAPLE is unable to solve it due to high memory allocation issues. Function is like this f(r.theta)=Bessel(1,r)+cos(theta)*f(r)+....50 terms.

I am using  evalf( Int(f(r,theta), [r=0..1, theta=0..Pi])).

Will term by term integration be helpful? How to do it in maple?

PS: If I decrease the number of digits, I get the result fast.
 

> 

restart;
> 

F1 := 0.1e10 * (0.55776153956804000740336392666745e0 * r ^ 2 - 0.18915469024923561670746189899598e-134609736 * BesselJ(0.0e0, 0.15157937163140142799278350422223e3 * r) + 0.10159683864017545475828989384714e-98384011 * BesselJ(0.0e0, 0.12958780324510399675374141784136e3 * r) + 0.59829761821461366846048256106725e-56462782 * BesselJ(0.0e0, 0.98170950730790781973537759160851e2 * r) + 0.14811094053601555275542685914404e-80227782 * BesselJ(0.0e0, 0.11702112189889242502757649460146e3 * r) + 0.33892512681723589723181533606428e-7313754 * BesselJ(0.0e0, 0.35332307550083865102634479022519e2 * r) - 0.51262328796358933950059817332311e-2254297 * BesselJ(0.0e0, 0.19615858510468242021125065884138e2 * r) - 0.12881247566594125484600726823569e-19254076 * BesselJ(0.0e0, 0.57327525437901010745090504243751e2 * r) + 0.11118751423887112574088244798447e-31252221 * BesselJ(0.0e0, 0.73036895225573834826506117569092e2 * r) - 0.51777724984261891154172697895593e-33998785 * BesselJ(0.0e0, 0.76178699584641457572852614623535e2 * r) + 0.12182571270348008146031905708415e-42932343 * BesselJ(0.0e0, 0.85604019436350230965949425493380e2 * r) + 0.40737194122764952321439991068058e-36860993 * BesselJ(0.0e0, 0.79320487175476299391184484872488e2 * r) - 0.50622470024129990724764923292822e-6070573 * BesselJ(0.0e0, 0.32189679910974403626622984104460e2 * r) - 0.46336835054606228289459855037304e-46141486 * BesselJ(0.0e0, 0.88745767144926306903735916434854e2 * r) + 0.13326755919882635551499433439984e-71843536 * BesselJ(0.0e0, 0.11073775478089921510860865288827e3 * r) - 0.51549643524094258017297656487619e-15264332 * BesselJ(0.0e0, 0.51043535183571509468733034633224e2 * r) + 0.63020619016879105779529017065422e-17201382 * BesselJ(0.0e0, 0.54185553641061320532099966214534e2 * r) - 0.34143530857990731804462883496266e-75977837 * BesselJ(0.0e0, 0.11387944084759499813488417492843e3 * r) + 0.29817206128159554191843363526765e-49466273 * BesselJ(0.0e0, 0.91887504251694985280553622214490e2 * r) - 0.32466998108445575875801048023258e-52906705 * BesselJ(0.0e0, 0.95029231808044695268050998187174e2 * r) - 0.18661427630098737592148946513116e-60134503 * BesselJ(0.0e0, 0.10131266182303873013714105638865e3 * r) - 0.88067954684538428870806207522441e-67824881 * BesselJ(0.0e0, 0.10759606325950917218267036427761e3 * r) + 0.13287757851408088906808371290053e-1290876``98 * BesselJ(0.0e0, 0.14843772662034223039593927702627e3 * r) - 0.28491383339723867983586755114008e-93671487 * BesselJ(0.0e0, 0.12644613869851659569779448049584e3 * r) + 0.44151440493072282554074854252808e-21422416 * BesselJ(0.0e0, 0.60469457845347491559398749808383e2 * r) - 0.25433459757254658126695515265514e-23706400 * BesselJ(0.0e0, 0.63611356698481232631039762417874e2 * r) + 0.31838472287249562307154488541348e-118390557 * BesselJ(0.0e0, 0.14215442965585902903270090809976e3 * r) + 0.24664036351722993558633516210405e-26106029 * BesselJ(0.0e0, 0.66753226734098493415305259750042e2 * r) - 0.35291670105094410350434844041935e-8672580 * BesselJ(0.0e0, 0.38474766234771615112052197557717e2 * r) + 0.58664491893391140222815167210588e-10147051 * BesselJ(0.0e0, 0.41617094212814450885863516805060e2 * r) - 0.15835272073861680035000959411566e-11737166 * BesselJ(0.0e0, 0.44759318997652821732779352713212e2 * r) + 0.70213789662657167106991346854437e-13442927 * BesselJ(0.0e0, 0.47901460887185447121274008722508e2 * r) + 0.20203042047105171656770921613101e-86016 * BesselJ(0.0e0, 0.38317059702075123156144358863082e1 * r) + 0.45595799288913858149685893872177e-140247419 * BesselJ(0.0e0, 0.15472101451628595352476655565184e3 * r) - 0.18611154629569865685380386607775e-146000746 * BesselJ(0.0e0, 0.15786265540193029780509466960866e3 * r) + 0.98529688671644920915913795962299e-63921870 * BesselJ(0.0e0, 0.10445436579128276007136342813961e3 * r) - 0.15806285101030450527944027463056e-123681305 * BesselJ(0.0e0, 0.14529607934519590723242215085501e3 * r) - 0.40315574736579460691059726643094e-28621303 * BesselJ(0.0e0, 0.69895071837495773969730536435500e2 * r) + 0.62723521218202757338090566184844e-108155995 * BesselJ(0.0e0, 0.13587112236478900059180156821946e3 * r) - 0.10859734567264554119513113490716e-113215453 * BesselJ(0.0e0, 0.13901277738865970417843354613596e3 * r) - 0.54175511325922018873646654014932e-39838846 * BesselJ(0.0e0, 0.82462259914373556453986610648781e2 * r) + 0.11283650227585469604741653680022e-4943036 * BesselJ(0.0e0, 0.29046828534916855066647819883532e2 * r) - 0.61345791140260163801601678872534e-103212181 * BesselJ(0.0e0, 0.13272946438850961588677459735175e3 * r) - 0.10878629914720505255262338938331e-84593372 * BesselJ(0.0e0, 0.12016279832814900375811940782917e3 * r) - 0.35054349658929943485990383440882e-3931145 * BesselJ(0.0e0, 0.25903672087618382625495855445980e2 * r) + 0.13529453916914935758397358737774e-89074607 * BesselJ(0.0e0, 0.12330447048863571801676003206877e3 * r) + 0.13471689526126410315073637771645e-3034898 * BesselJ(0.0e0, 0.22760084380592771898053005152182e2 * r) - 0.21295581245266175979652384428576e-288353 * BesselJ(0.0e0, 0.70155866698156187535370499814765e1 * r) + 0.46293568384524693637583038682636e-606366 * BesselJ(0.0e0, 0.10173468135062722077185711776776e2 * r) - 0.65373336840252622743371660187403e-1040030 * BesselJ(0.0e0, 0.13323691936314223032393684126948e2 * r) + 0.12271878942218097649114096289979e-1589340 * BesselJ(0.0e0, 0.16470630050877632812552460470990e2 * r) + 0.30096533794321654779481815801012e5) * (-0.84195432401461277308031602263610e-5 * r ^ 2 - 0.59149959490724929627371164952978e-2 * r ^ 6 * cos(0.6e1 * theta) + 0.44528672504236299477606103483348e-2 * r ^ 9 * cos(0.9e1 * theta) + 0.2112306765385091377525007041829e-2 * r ^ 25 * cos(0.25e2 * theta) - 0.67200617360940427597733246769568e-3 * r ^ 4 * cos(0.4e1 * theta) + 0.8077651557524848874997646779728e-4 * r ^ 38 * cos(0.38e2 * theta) + 0.6431431133931729186611840353106e-3 * r ^ 39 * cos(0.39e2 * theta) + 0.6638764085868884552072751263020e-3 * r ^ 40 * cos(0.40e2 * theta) + 0.3077586813267194148977094233961e-3 * r ^ 41 * cos(0.41e2 * theta) - 0.1856408707409825202502168626613e-3 * r ^ 42 * cos(0.42e2 * theta) - 0.4195028383398335941571877904622e-3 * r ^ 43 * cos(0.43e2 * theta) - 0.3706398326158304378037548737582e-3 * r ^ 44 * cos(0.44e2 * theta) - 0.7999587757612915190037434403564e-4 * r ^ 45 * cos(0.45e2 * theta) + 0.1737050010593172373976692973078e-3 * r ^ 46 * cos(0.46e2 * theta) + 0.2156346448293426610250334073280e-3 * r ^ 47 * cos(0.47e2 * theta) + 0.8688707406587637755715273073496e-4 * r ^ 48 * cos(0.48e2 * theta) - 0.2566545888070136544474329645476e-4 * r ^ 49 * cos(0.49e2 * theta) + 0.10879633813910334336257501999693e-1 * cos(theta) * r + 0.1887562703232630941270016328998e-2 * r ^ 24 * cos(0.24e2 * theta) + 0.9513343462787182229625573235371e-3 * r ^ 26 * cos(0.26e2 * theta) - 0.6163648649547716429383661026270e-3 * r ^ 27 * cos(0.27e2 * theta) - 0.1638476483444926784339005153548e-2 * r ^ 28 * cos(0.28e2 * theta) - 0.1544747773264052898936010069036e-2 * r ^ 29 * cos(0.29e2 * theta) - 0.5206686266979668543527923877478e-3 * r ^ 30 * cos(0.30e2 * theta) + 0.7031766719478684183248753358164e-3 * r ^ 31 * cos(0.31e2 * theta) + 0.1364403772746535517159915014059e-2 * r ^ 32 * cos(0.32e2 * theta) + 0.10540246948583098852767644351809e-2 * r ^ 33 * cos(0.33e2 * theta) + 0.1949337811874134263703020015791e-3 * r ^ 34 * cos(0.34e2 * theta) - 0.7191715359288498000802128285804e-3 * r ^ 35 * cos(0.35e2 * theta) - 0.10227876151057534138247065986153e-2 * r ^ 36 * cos(0.36e2 * theta) - 0.6867126825080510201446558832207e-3 * r ^ 37 * cos(0.37e2 * theta) - 0.51907452513946892830363140141895e-2 * r ^ 5 * cos(0.5e1 * theta) + 0.15481206149695126077925147166938e-2 * r ^ 11 * cos(0.11e2 * theta) - 0.18891064144929437714573633077525e-2 * r ^ 12 * cos(0.12e2 * theta) - 0.3811736195725823688361734620913e-2 * r ^ 13 * cos(0.13e2 * theta) - 0.32257343081162300403533436479469e-2 * r ^ 14 * cos(0.14e2 * theta) - 0.6456518231629053621129825002098e-3 * r ^ 15 * cos(0.15e2 * theta) + 0.20319096805014454478199422911684e-2 * r ^ 16 * cos(0.16e2 * theta) + 0.3233144446775015541635116158538e-2 * r ^ 17 * cos(0.17e2 * theta) + 0.23137228128708316785559166203584e-2 * r ^ 18 * cos(0.18e2 * theta) + 0.6898483226498941349817978084256e-4 * r ^ 19 * cos(0.19e2 * theta) - 0.20285262491678306920628881668352e-2 * r ^ 20 * cos(0.20e2 * theta) - 0.2671173199674743523515178373090e-2 * r ^ 21 * cos(0.21e2 * theta) - 0.15775142288031750532503075313091e-2 * r ^ 22 * cos(0.22e2 * theta) + 0.3622094777240520457049718035053e-3 * r ^ 23 * cos(0.23e2 * theta) + 0.14579067481459940998484958894370e-2 * r ^ 8 * cos(0.8e1 * theta) + 0.43385218600667457865829805287215e-2 * r ^ 10 * cos(0.10e2 * theta) - 0.29324228962818139404116534560943e-2 * r ^ 7 * cos(0.7e1 * theta) + 0.54771662980043457997274959739776e-2 * r ^ 3 * cos(0.3e1 * theta) - 0.11907324829492592983826593268542e-1 + 0.99737018277250342942042004599405e6 * (0.10375843065514893709650453544669e-7 * r ^ 4 - 0.24066724220589275560649004814238e-8 * r ^ 2) * cos(0.2e1 * theta) / r ^ 2 - 0.18524693450872080736996040590111e-1589345 * BesselJ(0.0e0, 0.16470630050877632812552460470990e2 * r) - 0.20335836094200343189896872255293e-3034903 * BesselJ(0.0e0, 0.22760084380592771898053005152182e2 * r) + 0.32146186927377989454999075542184e-288358 * BesselJ(0.0e0, 0.70155866698156187535370499814765e1 * r) - 0.69881243704258704205303920297122e-606371 * BesselJ(0.0e0, 0.10173468135062722077185711776776e2 * r) + 0.98682608468381340045946744187651e-1040035 * BesselJ(0.0e0, 0.13323691936314223032393684126948e2 * r) - 0.20423032817438260168628393904163e-89074612 * BesselJ(0.0e0, 0.12330447048863571801676003206877e3 * r) + 0.16393027894394588837550747507414e-113215458 * BesselJ(0.0e0, 0.13901277738865970417843354613596e3 * r) + 0.81779224239606095156885663441587e-39838851 * BesselJ(0.0e0, 0.82462259914373556453986610648781e2 * r) - 0.17032938676879018403348115316985e-4943041 * BesselJ(0.0e0, 0.29046828534916855066647819883532e2 * r) + 0.92602932340297485357655867631396e-103212186 * BesselJ(0.0e0, 0.13272946438850961588677459735175e3 * r) + 0.16421550871268572218657911635481e-84593377 * BesselJ(0.0e0, 0.12016279832814900375811940782917e3 * r) + 0.52915375437527581357423578813141e-3931150 * BesselJ(0.0e0, 0.25903672087618382625495855445980e2 * r) + 0.77815414272085141864206462412262e-15264337 * BesselJ(0.0e0, 0.51043535183571509468733034633224e2 * r) - 0.95131124896907983486241420998755e-17201387 * BesselJ(0.0e0, 0.54185553641061320532099966214534e2 * r) + 0.51540472771347914200070162230077e-75977842 * BesselJ(0.0e0, 0.11387944084759499813488417492843e3 * r) - 0.45009782583936088946734982085640e-49466278 * BesselJ(0.0e0, 0.91887504251694985280553622214490e2 * r) + 0.49009706668463083583947296301775e-52906710 * BesselJ(0.0e0, 0.95029231808044695268050998187174e2 * r) + 0.28169869327339522936720076403132e-60134508 * BesselJ(0.0e0, 0.10131266182303873013714105638865e3 * r) + 0.13294067445237467596212175135530e-67824885 * BesselJ(0.0e0, 0.10759606325950917218267036427761e3 * r) - 0.20058186851887448492658350947366e-129087703 * BesselJ(0.0e0, 0.14843772662034223039593927702627e3 * r) + 0.43008421517583172146652387481621e-93671492 * BesselJ(0.0e0, 0.12644613869851659569779448049584e3 * r) - 0.66647650649066255093532041895905e-21422421 * BesselJ(0.0e0, 0.60469457845347491559398749808383e2 * r) + 0.38392413062141555362678468281752e-23706405 * BesselJ(0.0e0, 0.63611356698481232631039762417874e2 * r) - 0.48060931976467196435085585083844e-118390562 * BesselJ(0.0e0, 0.14215442965585902903270090809976e3 * r) - 0.37230950111886614086127736374754e-26106034 * BesselJ(0.0e0, 0.66753226734098493415305259750042e2 * r) + 0.53273616301499657528989740768063e-8672585 * BesselJ(0.0e0, 0.38474766234771615112052197557717e2 * r) - 0.88555447286690435479201942884554e-10147056 * BesselJ(0.0e0, 0.41617094212814450885863516805060e2 * r) + 0.23903720225781678909977638730792e-11737171 * BesselJ(0.0e0, 0.44759318997652821732779352713212e2 * r) - 0.10598938725267772368055360453741e-13442931 * BesselJ(0.0e0, 0.47901460887185447121274008722508e2 * r) - 0.30496972994915901977125629292157e-86021 * BesselJ(0.0e0, 0.38317059702075123156144358863082e1 * r) - 0.68827944640884252540240135035619e-140247424 * BesselJ(0.0e0, 0.15472101451628595352476655565184e3 * r) + 0.28093981036064987725074202641260e-146000751 * BesselJ(0.0e0, 0.15786265540193029780509466960866e3 * r) - 0.14873291099481638062068892057166e-63921874 * BesselJ(0.0e0, 0.10445436579128276007136342813961e3 * r) + 0.23859963700126918177896460503756e-123681310 * BesselJ(0.0e0, 0.14529607934519590723242215085501e3 * r) + 0.60857319959503281138409206408861e-28621308 * BesselJ(0.0e0, 0.69895071837495773969730536435500e2 * r) - 0.94682648696048924172260521336169e-108156000 * BesselJ(0.0e0, 0.13587112236478900059180156821946e3 * r) + 0.28553350861432233569650200943679e-134609741 * BesselJ(0.0e0, 0.15157937163140142799278350422223e3 * r) - 0.15336284689969342456370426833116e-98384016 * BesselJ(0.0e0, 0.12958780324510399675374141784136e3 * r) - 0.90314449987634539477129986599199e-56462787 * BesselJ(0.0e0, 0.98170950730790781973537759160851e2 * r) - 0.22357699119008062011176340166029e-80227787 * BesselJ(0.0e0, 0.11702112189889242502757649460146e3 * r) - 0.51161554857649418772612124539227e-7313759 * BesselJ(0.0e0, 0.35332307550083865102634479022519e2 * r) + 0.77381705849741819343661724258774e-2254302 * BesselJ(0.0e0, 0.19615858510468242021125065884138e2 * r) + 0.19444549898144465612468716205102e-19254081 * BesselJ(0.0e0, 0.57327525437901010745090504243751e2 * r) - 0.16784020006534355647552255243370e-31252226 * BesselJ(0.0e0, 0.73036895225573834826506117569092e2 * r) + 0.78159708666719140456536882061442e-33998790 * BesselJ(0.0e0, 0.76178699584641457572852614623535e2 * r) - 0.18389881393811040868057686236036e-42932348 * BesselJ(0.0e0, 0.85604019436350230965949425493380e2 * r) - 0.61493764461507094694745129374163e-36860998 * BesselJ(0.0e0, 0.79320487175476299391184484872488e2 * r) + 0.76415823798329557427383241351545e-6070578 * BesselJ(0.0e0, 0.32189679910974403626622984104460e2 * r) + 0.69946555772905592227422733556311e-46141491 * BesselJ(0.0e0, 0.88745767144926306903735916434854e2 * r) - 0.20117055364775216522977716192738e-71843541 * BesselJ(0.0e0, 0.11073775478089921510860865288827e3 * r) + 0.24003433134624560908493351044670e-2 * cos(0.2e1 * theta)) * r;

0.1e10*(30096.533794321654779481815801012+.55776153956804000740336392666745*r^2-0.18915469024923561670746189899598e-134609736*BesselJ(0., 151.57937163140142799278350422223*r)+0.10159683864017545475828989384714e-98384011*BesselJ(0., 129.58780324510399675374141784136*r)+0.59829761821461366846048256106725e-56462782*BesselJ(0., 98.170950730790781973537759160851*r)+0.14811094053601555275542685914404e-80227782*BesselJ(0., 117.02112189889242502757649460146*r)+0.33892512681723589723181533606428e-7313754*BesselJ(0., 35.332307550083865102634479022519*r)-0.51262328796358933950059817332311e-2254297*BesselJ(0., 19.615858510468242021125065884138*r)-0.12881247566594125484600726823569e-19254076*BesselJ(0., 57.327525437901010745090504243751*r)+0.11118751423887112574088244798447e-31252221*BesselJ(0., 73.036895225573834826506117569092*r)-0.51777724984261891154172697895593e-33998785*BesselJ(0., 76.178699584641457572852614623535*r)+0.12182571270348008146031905708415e-42932343*BesselJ(0., 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47.901460887185447121274008722508*r)-0.30496972994915901977125629292157e-86021*BesselJ(0., 3.8317059702075123156144358863082*r)-0.68827944640884252540240135035619e-140247424*BesselJ(0., 154.72101451628595352476655565184*r)+0.28093981036064987725074202641260e-146000751*BesselJ(0., 157.86265540193029780509466960866*r)-0.14873291099481638062068892057166e-63921874*BesselJ(0., 104.45436579128276007136342813961*r)+0.23859963700126918177896460503756e-123681310*BesselJ(0., 145.29607934519590723242215085501*r)+0.60857319959503281138409206408861e-28621308*BesselJ(0., 69.895071837495773969730536435500*r)-0.94682648696048924172260521336169e-108156000*BesselJ(0., 135.87112236478900059180156821946*r)+0.16393027894394588837550747507414e-113215458*BesselJ(0., 139.01277738865970417843354613596*r)+0.81779224239606095156885663441587e-39838851*BesselJ(0., 82.462259914373556453986610648781*r)-0.17032938676879018403348115316985e-4943041*BesselJ(0., 29.046828534916855066647819883532*r)+0.92602932340297485357655867631396e-103212186*BesselJ(0., 132.72946438850961588677459735175*r)+0.16421550871268572218657911635481e-84593377*BesselJ(0., 120.16279832814900375811940782917*r)+0.52915375437527581357423578813141e-3931150*BesselJ(0., 25.903672087618382625495855445980*r)-0.20423032817438260168628393904163e-89074612*BesselJ(0., 123.30447048863571801676003206877*r)-0.20335836094200343189896872255293e-3034903*BesselJ(0., 22.760084380592771898053005152182*r)+0.32146186927377989454999075542184e-288358*BesselJ(0., 7.0155866698156187535370499814765*r)-0.69881243704258704205303920297122e-606371*BesselJ(0., 10.173468135062722077185711776776*r)+0.98682608468381340045946744187651e-1040035*BesselJ(0., 13.323691936314223032393684126948*r)-0.18524693450872080736996040590111e-1589345*BesselJ(0., 16.470630050877632812552460470990*r)+0.24003433134624560908493351044670e-2*cos(2.*theta)+997370.18277250342942042004599405*(0.10375843065514893709650453544669e-7*r^4-0.24066724220589275560649004814238e-8*r^2)*cos(2.*theta)/r^2)*r

 (1)
> 

evalf(subs(r=1,theta=Pi/4,F1))

0.7135632392e12

 (2)
> 

Digits:=16;

16

 (3)
> 

int_F1:=evalf(Int(F1,[theta=Pi/4..2*Pi-Pi/4,r=0..1]));

Warning,  computation interrupted

  
> 

``


 

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Float undefined while solving system of ...

gaurav_rs 65 Maple 2017
linearalgebra linearsolve
March 31 2019
2 2

Dear Users,

I am solving a large system of linear equation with the Hybrid solver. Up to 2400 equations, I get a solution, but if I change the number of equations to 3000, I don't get any solution in fact solution entry reads Float(undefined). All the entries in matrix and vector are floating point decimal with 32 digits of accuracy. Any idea how to overcome this?

 

thanks.

 

derivative of an implicit function...

gaurav_rs 65 Maple 2017
differentiation constraint energy
January 09 2019
1 3

Dear all,

Following the comments I am editing this post:

I have a function F of variables (r1,r2,theta1,theta2,r,theta,a). r1, r2,theta1,theta2 are function of r,  theta and a. I want to take derivative of F with respect to a. r and theta are independent of a . I expressed everything in terms of 'a' as a function of 'a' at first. Then I use diff(F, a). I see there is an error in the final expression G .There is a restriction that theta1 should lie between -Pi to Pi and theta2 between 0 to 2*Pi. I speculate this is the source of error. Work sheet is attached. Reason: value of G: integration in 0 to pi/4 gives some  value but for 0 to pi it evaluates to zero and so is the case with 0 to 2*Pi. As "G "physically represents energy it must be a positive value.
 


> 

restart;
> 

theta1 := unapply(arctan(r*sin(theta)/(r*cos(theta)-a)), a);

proc (a) options operator, arrow; arctan(r*sin(theta)/(r*cos(theta)-a)) end proc

 (1)
> 

 

> 

## theta1 -->[-Pi,Pi] and theta2-->[0,2*Pi]
> 

NULL
> 

theta2 := unapply(arctan(r*sin(theta)/(r*cos(theta)+a)), a);

proc (a) options operator, arrow; arctan(r*sin(theta)/(r*cos(theta)+a)) end proc

 (2)
> 

``
> 

r1:=unapply(sqrt((r*cos(theta)-a)^2+r^2*(sin(theta))^2),a);r2:=unapply(sqrt((r*cos(theta)+a)^2+r^2*(sin(theta))^2),a);

proc (a) options operator, arrow; ((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2) end proc

  

proc (a) options operator, arrow; ((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2) end proc

 (3)
> 

sigma12:=0;sigma22:=sigma;

0

  

sigma

 (4)
> 

plot(arctan(tan(x)), x = (1/2)*Pi .. Pi)

 
> 

## I have to use a constraint that

> 

assume(theta1(a) < Pi, theta1(a) > -Pi, theta2(a) > 0, theta2(a) < 2*Pi, a>0,r>0)
> 

u1:=(1+nu)*sigma22*sqrt(r1(a)*r2(a))*(4*(1-2*nu)*cos((theta1(a)+(theta2(a)))/2)-4*r*(1-nu)*cos(theta)/sqrt(r1(a)*r2(a))-2*r^2/(r1(a)*r2(a))*(cos((theta1(a)+(theta2(a)))/2)-cos(2*theta-theta1(a)/2-(theta2(a))/2)))/(4*E)+(1+nu)*sigma12*sqrt(r1(a)*r2(a))*(2*(1-2*nu)*sin((theta1(a)+(theta2(a)))/2)-2*r*(1-nu)*sin(theta)/sqrt(r1(a)*r2(a))+1*r^2/(r1(a)*r2(a))*sin(theta)*cos(theta-theta1(a)/2-(theta2(a))/2))/(E);
> 

 

(1/4)*(1+nu)*sigma*(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2))^(1/2)*(4*(1-2*nu)*cos((1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))+(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))-4*r*(1-nu)*cos(theta)/(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2))^(1/2)-2*r^2*(cos((1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))+(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))-cos(2*theta-(1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a))))/(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2)))/E

 (5)
> 

NULL
> 

u2:=(1+nu)*sigma*sqrt(r1(a)*r2(a))*(8*(1-nu)*sin((theta1(a)+(theta2(a)))/2)-4*r*(nu)*sin(theta)/sqrt(r1(a)*r2(a))-2*r^2/(r1(a)*r2(a))*(sin((theta1(a)+(theta2(a)))/2)+sin(2*theta-theta1(a)/2-(theta2(a))/2)))/(4*E)+(1+nu)*sigma12*sqrt(r1(a)*r2(a))*((1-2*nu)*cos((theta1(a)+theta2(a))/2)+2*r*(1-nu)*cos(theta)/sqrt(r1(a)*r2(a))-1*r^2/(r1(a)*r2(a))*sin(theta)*sin(theta-theta1(a)/2-theta2(a)/2))/(E);
> 

 

(1/4)*(1+nu)*sigma*(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2))^(1/2)*(8*(1-nu)*sin((1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))+(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))-4*r*nu*sin(theta)/(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2))^(1/2)-2*r^2*(sin((1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))+(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))+sin(2*theta-(1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a))))/(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2)))/E

 (6)
> 

 

> 

## get u_r and u_theta as u[1] and u[2]
> 

u[1] := u1*cos(theta)+u2*sin(theta);

(1/4)*(1+nu)*sigma*(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2))^(1/2)*(4*(1-2*nu)*cos((1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))+(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))-4*r*(1-nu)*cos(theta)/(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2))^(1/2)-2*r^2*(cos((1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))+(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))-cos(2*theta-(1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a))))/(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2)))*cos(theta)/E+(1/4)*(1+nu)*sigma*(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2))^(1/2)*(8*(1-nu)*sin((1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))+(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))-4*r*nu*sin(theta)/(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2))^(1/2)-2*r^2*(sin((1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))+(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))+sin(2*theta-(1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a))))/(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2)))*sin(theta)/E

 (7)
> 

u[2] := -sin(theta)*u1+cos(theta)*u2;

-(1/4)*sin(theta)*(1+nu)*sigma*(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2))^(1/2)*(4*(1-2*nu)*cos((1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))+(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))-4*r*(1-nu)*cos(theta)/(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2))^(1/2)-2*r^2*(cos((1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))+(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))-cos(2*theta-(1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a))))/(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2)))/E+(1/4)*cos(theta)*(1+nu)*sigma*(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2))^(1/2)*(8*(1-nu)*sin((1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))+(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))-4*r*nu*sin(theta)/(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2))^(1/2)-2*r^2*(sin((1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))+(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))+sin(2*theta-(1/2)*arctan(r*sin(theta)/(r*cos(theta)-a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a))))/(((r*cos(theta)-a)^2+r^2*sin(theta)^2)^(1/2)*((r*cos(theta)+a)^2+r^2*sin(theta)^2)^(1/2)))/E

 (8)
> 

Diff_ur := simplify(diff(u[1], a));

(1/2)*sigma*(1+nu)*a*(-(-2*cos(theta)*a*r+a^2+r^2)^(1/2)*(2*cos(theta)*a*r+a^2+r^2)^(1/2)*r^2*cos(theta)*(a-r)*(a+r)*cos(2*theta+(1/2)*arctan(r*sin(theta)/(-r*cos(theta)+a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))+(-2*cos(theta)*a*r+a^2+r^2)^(1/2)*(2*cos(theta)*a*r+a^2+r^2)^(1/2)*r^2*sin(theta)*(a^2+r^2)*sin(2*theta+(1/2)*arctan(r*sin(theta)/(-r*cos(theta)+a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))+((a^2*r^2-r^4)*(2*cos(theta)*a*r+a^2+r^2)^(1/2)*(-2*cos(theta)*a*r+a^2+r^2)^(1/2)-4*(-2*cos(theta)*a*r+a^2+r^2)*(2*cos(theta)*a*r+a^2+r^2)*(-4*r^2*(nu-3/4)*cos(theta)^2+(3*nu-5/2)*r^2+a^2*(nu-1/2)))*cos(theta)*cos((1/2)*arctan(r*sin(theta)/(-r*cos(theta)+a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))-sin(theta)*((2*cos(theta)*a*r+a^2+r^2)^(1/2)*r^2*(a^2+r^2)*(-2*cos(theta)*a*r+a^2+r^2)^(1/2)-4*(-2*cos(theta)*a*r+a^2+r^2)*(2*cos(theta)*a*r+a^2+r^2)*(-4*r^2*(nu-3/4)*cos(theta)^2+(a^2+r^2)*(nu-1)))*sin((1/2)*arctan(r*sin(theta)/(-r*cos(theta)+a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a))))/((-2*cos(theta)*a*r+a^2+r^2)^(3/2)*(2*cos(theta)*a*r+a^2+r^2)^(3/2)*((-2*cos(theta)*a*r+a^2+r^2)^(1/2)*(2*cos(theta)*a*r+a^2+r^2)^(1/2))^(1/2)*E)

 (9)
> 

``
> 

 

> 

Diff_ut := simplify(diff(u[2], a));

-(1/2)*sigma*(1+nu)*a*(-(-2*cos(theta)*a*r+a^2+r^2)^(1/2)*(2*cos(theta)*a*r+a^2+r^2)^(1/2)*r^2*sin(theta)*(a^2+r^2)*cos(2*theta+(1/2)*arctan(r*sin(theta)/(-r*cos(theta)+a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))-(-2*cos(theta)*a*r+a^2+r^2)^(1/2)*(2*cos(theta)*a*r+a^2+r^2)^(1/2)*r^2*cos(theta)*(a-r)*(a+r)*sin(2*theta+(1/2)*arctan(r*sin(theta)/(-r*cos(theta)+a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))+sin(theta)*((2*cos(theta)*a*r+a^2+r^2)^(1/2)*r^2*(a^2+r^2)*(-2*cos(theta)*a*r+a^2+r^2)^(1/2)-4*(-2*cos(theta)*a*r+a^2+r^2)*(-4*r^2*(nu-3/4)*cos(theta)^2+(a^2+r^2)*(nu-1/2))*(2*cos(theta)*a*r+a^2+r^2))*cos((1/2)*arctan(r*sin(theta)/(-r*cos(theta)+a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))+((a^2*r^2-r^4)*(2*cos(theta)*a*r+a^2+r^2)^(1/2)*(-2*cos(theta)*a*r+a^2+r^2)^(1/2)-4*(-2*cos(theta)*a*r+a^2+r^2)*(-4*r^2*(nu-3/4)*cos(theta)^2+(3*nu-2)*r^2+a^2*(nu-1))*(2*cos(theta)*a*r+a^2+r^2))*sin((1/2)*arctan(r*sin(theta)/(-r*cos(theta)+a))-(1/2)*arctan(r*sin(theta)/(r*cos(theta)+a)))*cos(theta))/((-2*cos(theta)*a*r+a^2+r^2)^(3/2)*(2*cos(theta)*a*r+a^2+r^2)^(3/2)*((-2*cos(theta)*a*r+a^2+r^2)^(1/2)*(2*cos(theta)*a*r+a^2+r^2)^(1/2))^(1/2)*E)

 (10)
> 

``
> 

# find the limiting case
> 

Att := limit(Diff_ut*r*sin(2*theta), r = infinity);

(2*a*sigma*cos(theta)^3*sin(theta)^2*nu+2*a*sigma*cos(theta)*sin(theta)^4*nu+2*a*sigma*cos(theta)^3*sin(theta)^2+2*a*sigma*cos(theta)*sin(theta)^4-8*a*sigma*cos(theta)*sin(theta)^2*nu^2-6*a*sigma*cos(theta)*sin(theta)^2*nu+2*a*sigma*cos(theta)*sin(theta)^2)/(((cos(theta)^2+sin(theta)^2)/cos(theta)^2)^(1/2)*E)

 (11)
> 

Arr := limit(Diff_ur*r*(1-cos(2*theta)), r = infinity);

(-16*a*sigma*cos(theta)^6*nu^2-16*a*sigma*cos(theta)^4*sin(theta)^2*nu^2-6*a*sigma*cos(theta)^6*nu-6*a*sigma*cos(theta)^4*nu*sin(theta)^2+10*a*sigma*cos(theta)^6+10*a*sigma*sin(theta)^2*cos(theta)^4+28*a*sigma*cos(theta)^4*nu^2+20*a*sigma*cos(theta)^2*sin(theta)^2*nu^2+10*a*sigma*cos(theta)^4*nu+4*a*sigma*cos(theta)^2*sin(theta)^2*nu-18*a*sigma*cos(theta)^4-16*a*sigma*sin(theta)^2*cos(theta)^2-12*a*sigma*cos(theta)^2*nu^2-4*a*sigma*sin(theta)^2*nu^2-4*a*sigma*cos(theta)^2*nu+2*a*sigma*sin(theta)^2*nu+8*a*sigma*cos(theta)^2+6*a*sigma*sin(theta)^2)/(cos(theta)*((cos(theta)^2+sin(theta)^2)/cos(theta)^2)^(1/2)*E)

 (12)
> 

G := (1/8)*(int(Arr+Att, theta = 0 .. Pi/2))*sigma*4;

-(1/8)*Pi*a*sigma^2*(4*nu^2-nu-5)/E

 (13)
> 

simplify(G)

-(1/8)*Pi*a*sigma^2*(4*nu^2-nu-5)/E

 (14)
> 

 


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