PKtrXBHmimetypetext/x-wxmathmlPKtrXQdBV55 format.txt This file contains a wxMaxima session in the .wxmx format. .wxmx files are .xml-based files contained in a .zip container like .odt or .docx files. After changing their name to end in .zip the .xml and eventual bitmap files inside them can be extracted using any .zip file viewer. The reason why part of a .wxmx file still might still seem to make sense in a ordinary text viewer is that the text portion of .wxmx by default isn't compressed: The text is typically small and compressing it would mean that changing a single character would (with a high probability) change big parts of the whole contents of the compressed .zip archive. Even if version control tools like git and svn that remember all changes that were ever made to a file can handle binary files compression would make the changed part of the file bigger and therefore seriously reduce the efficiency of version control wxMaxima can be downloaded from https://github.com/wxMaxima-developers/wxmaxima. It also is part of the windows installer for maxima (https://wxmaxima-developers.github.io/wxmaxima/). If a .wxmx file is broken but the content.xml portion of the file can still be viewed using an text editor just save the xml's text as "content.xml" and try to open it using a recent version of wxMaxima. If it is valid XML (the XML header is intact, all opened tags are closed again, the text is saved with the text encoding "UTF8 without BOM" and the few special characters XML requires this for are properly escaped) chances are high that wxMaxima will be able to recover all code and text from the XML file. PKtrX{ll content.xml Kedvenc Matematikai Kísérleteim Alapok Halmazok Számok Határérték Differenciálszámítás Integrálszámítás Lineáris algebra Mátrixok és vektorok Lineáris leképezések Belső szorzat Norma Példák v:matrix([1,2]); (%o1) 12 mat_norm(v,1); (%o5) 2 mat_norm(v,inf); (%o6) 3 mat_norm(v,frobenius); (%o7) 5 v:transpose(v); (%o8) 12 mat_norm(v,1); (%o9) 3 mat_norm(v,inf); (%o10) 2 mat_norm(v,frobenius); (%o11) 5 Belső szorzat Tétel Példa u:[1,2]; v:[%i,4]; (%o7) [1,2](%o8) [%i,4] load(eigen)$ inprod(u,v); (%o10) %i+8 Fizikus belső szorzat! Megjegyzés Szögek, euklideszi tér, unitér tér v:[3,4]; (%o12) [3,4] inprod(u,v)/(inprod(u,u)*inprod(v,v))^(1/2); (%o13) 11532 float(%), numer; (%o14) 0.9838699100999073 Definíció Vegyes szorzat Pitagorasz-tétel Következmény Gram-Schmidt-ortogonalizálás Következmény Következmény Feladat (1) Csak valósban, szimmetrikus (2) Belső szorzat y1:1; y2:t; y3:t^2; (%o53) 1(%o54) t(%o55) t2 u1:y1; x1:u1/(integrate(1*1,t,-1,1))^(1/2); (%o56) 1(%o57) 12 u2:y2-integrate(y2*x1,t,-1,1)*x1; (%o58) t x2:u2/integrate(u2*u2,t,-1,1)^(1/2); (%o59) 3*t2 u3:y3-integrate(y3*x1,t,-1,1)*x1 -integrate(y3*x2,t,-1,1)*x2; (%o60) t213 x3:u3/integrate(u3*u3,t,-1,1)^(1/2); (%o61) 3*5*

t213

232
expand(%); (%o62) 3*5*t22325232 Fizikus belső szorzat! ip(f,g):=integrate(conjugate(f)*g,t,-1,1); (%o79) ip

f,g

:=11f*gdt
o:gramschmidt([1,t,t^2],ip); (%o80) [1,t,3*t213] unitv(x):=x/ip(x,x)^(1/2); (%o81) unitv

x

:=xip

x,x

12
map(uvect,o); (%o82) [12,3*t2,5*

3*t21

232
]
(3) Belső szorzat ip(f,g):=integrate(conjugate(f)*g +conjugate(diff(f,t))*diff(g,t),t,-1,1); (%o83) ip

f,g

:=11f*g+dd*t*f*

dd*t*g

dt
o:gramschmidt([1,t,t^2],ip); (%o84) [1,t,3*t213] map(unitv,o); (%o85) [12,3*t232,5*

3*t21

272
]
(4) Nem pozitív definit (t-1)*(t-2); (%o9)

t2

*

t1

(5) Belső szorzat,stb. (6) Belső szorzat, stb. Feladat (1) acos(3/sqrt(12)); (%o39) %pi6 (2) acos(-2/4); (%o40) 2*%pi3 (3) acos(2/4); (%o41) %pi3 Tétel Az approximáció alapfeladata normált térben A legjobb lineáris approximáció belső szorzat térben Következmény: Bessel-egyenlőtlenség Ortogonális felbontási tétel Következmény Következmény Feladat (1) gramschmidt([[3,0,0],[2,2,0],[1,1,1]]); (%o71) [[3,0,0],[0,2,0],[0,0,1]] unitvector(%); (%o72) [[1,0,0],[0,1,0],[0,0,1]] (2) gramschmidt([[1,1,1],[2,2,0],[3,0,0]]); (%o73) [[1,1,1],[23,23,223],[32,32,0]] unitvector(%); (%o74) [[6133,6133,6133],[4133,4133,8133],[910,910,0]] (3) gramschmidt([[1,2,-1],[2,1,0],[-4,1,5]]); (%o75) [[1,2,1],[223,13,23],[32,3,322]] unitvector(%); (%o76) [[6181,12181,6181],[22118,1118,2118],[9781,18781,27781]] Feladat (1) x:unitvector([1,1,1]); (%o99) [13,13,13] p:[-2,3,-4]; (%o100) [2,3,4] f1:inprod(x,p); (%o101) 3 f1*x; (%o102) [1,1,1] (2) gramschmidt([[1,1,1],[1,1,-1]]); (%o103) [[1,1,1],[23,23,223]] x:map(unitvector,%); (%o104) [[13,13,13],[12*3,12*3,23]] f1:inprod(x[1],p); (%o105) 3 f2:inprod(x[2],p); (%o106) 3322 f1*x[1]+f2*x[2]; (%o107) [12,12,4] (3) pp:[1,1,1]; (%o108) [1,1,1] gramschmidt([[1,0,1],[-1,0,1]]); (%o109) [[1,0,1],[1,0,1]] x:map(unitvector,%); (%o110) [[12,0,12],[12,0,12]] f1:inprod(x[1],p-pp); (%o111) 252 f2:inprod(x[2],p-pp); (%o112) 2 pp+f1*x[1]+f2*x[2]; (%o114) [2,1,4] (4) p:[-2*%i,3*%i,-4-2*%i,5]; (%o115) [2*%i,3*%i,2*%i4,5] pp:[1,1,1,1]; (%o116) [1,1,1,1] gramschmidt([[1,%i,1,0],[%i,-1,0,1]]); (%o117) [[1,%i,1,0],[%i3,13,2*%i3,1]] x:map(unitvector,%); (%o118) [[13,%i3,13,0],[%i3*5,13*5,2*%i3*5,35]] f1:inprod(x[1],p-pp); (%o119) 3*%i+33 f2:inprod(x[2],p-pp); (%o120) 12*5*%i3*5325*3 pp+f1*x[1]+f2*x[2]; (%o121) [%i*

12*5*%i3*532

3*532
3*%i+33+1,12*5*%i3*5323*532%i*

3*%i+3

3
+1,2*%i*

12*5*%i3*532

3*532
3*%i+33+1,112*5*%i3*532532]
ratsimp(%); (%o122) [45,%i55,15*%i+85,12*%i205] Előállítási tétel Adjungált leképezés Tétel Adjungált operátor mátrixa Feladat (1) saját maga (2) saját maga (3) az inverze (4) az inverze (5) saját maga (6) saját maga (7) az x tengellyel párhuzamos vetítés az y=x egyenesre (8) (x,y)->(x,x) Önadjungált és normális transzformációk Példa Polarizációs formula Következmény Következmény Tétel Következmény Megjegyzés Tétel Segédtétel Normális transzformáció átlós alakja Következmény Valós önadjungált transzformáció átlós alakja Poláris felbontás Szinguláris érték felbontás Általánosított inverz Feladat invert(matrix([1,1],[0,c])); (%o2) 11c01c Tétel Tétel: QR-felbontás Megjegyzés Feladat (1) a:matrix([0,0,0,0],[1,0,0,0], [0,1,0,0],[0,0,1,0]); (%o226) 0000100001000010 b:ctranspose(a); (%o227) 0100001000010000 a . b; b . a; is(equal(a . b,b . a)); (%o228) 0000010000100001(%o229) 1000010000100000(%o230) false Nem normális (2) a[1,4]:1; (%o231) 1 a; (%o232) 0001100001000010 b:ctranspose(a); (%o233) 0100001000011000 a . b; b . a; is(equal(a . b,b . a)); (%o234) 1000010000100001(%o235) 1000010000100001(%o236) true Ortogonális, unitér eivects(a); (%o237) [[[%i,1,%i,1],[1,1,1,1]],[[[1,%i,1,%i]],[[1,1,1,1]],[[1,%i,1,%i]],[[1,1,1,1]]]] nondiagonalizable; (%o238) false hermitianmatrix:true; (%o239) true eivects(a); (%o240) [[[%i,1,%i,1],[1,1,1,1]],[[[1,%i,1,%i]],[[1,1,1,1]],[[1,%i,1,%i]],[[1,1,1,1]]]] ueivects(a); (%o241) [[[%i,1,%i,1],[1,1,1,1]],[[[12,%i2,12,%i2]],[[12,12,12,12]],[[12,%i2,12,%i2]],[[12,12,12,12]]]] simtran(a); (%o242) [[[%i,1,%i,1],[1,1,1,1]],[[[12,%i2,12,%i2]],[[12,12,12,12]],[[12,%i2,12,%i2]],[[12,12,12,12]]]] leftmatrix; rightmatrix; is(equal(leftmatrix,ctranspose(rightmatrix))); (%o245) 12%i212%i21212121212%i212%i212121212(%o246) 12121212%i212%i21212121212%i212%i212(%o247) true (3) a:matrix([1,1,1,1],[0,0,0,0],[0,0,0,0],[0,0,0,0]); (%o198) 1111000000000000 b:ctranspose(a); (%o199) 1000100010001000 a . b; b . a; is(equal(a . b,b . a)); (%o202) 4000000000000000(%o203) 1111111111111111(%o204) false Nem normális (4) a:matrix([1,1,1,1],[1,1,1,1],[1,1,1,1],[1,1,1,1]); (%o248) 1111111111111111 b:ctranspose(a); (%o249) 1111111111111111 is(equal(a,b)); (%o250) true Szimmetrikus, önadjungált simtran(a); (%o251) [[[0,4],[3,1]],[[[12,0,0,12],[12*3,23,0,12*3],[12*3,12*3,32,12*3]],[[12,12,12,12]]]] leftmatrix; rightmatrix; is(equal(leftmatrix,ctranspose(rightmatrix))); (%o252) 12001212*323012*312*312*33212*312121212(%o253) 1212*312*31202312*3120032121212*312*312(%o254) true (5) a:matrix([1,0,0,-1],[-1,1,0,0], [0,-1,1,0],[0,0,-1,1]); (%o255) 1001110001100011 b:ctranspose(a); (%o256) 1100011000111001 a . b; b . a; is(equal(a . b,b . a)); (%o257) 2101121001211012(%o258) 2101121001211012(%o259) true Normális simtran(a); (%o260) [[[1%i,%i+1,2,0],[1,1,1,1]],[[[12,%i2,12,%i2]],[[12,%i2,12,%i2]],[[12,12,12,12]],[[12,12,12,12]]]] leftmatrix; rightmatrix; is(equal(leftmatrix,ctranspose(rightmatrix))); (%o261) 12%i212%i212%i212%i21212121212121212(%o262) 12121212%i2%i2121212121212%i2%i21212(%o263) true (6) a:1/4*matrix([1,1,1,1],[-1,-1,-1,-1], [1,1,1,1],[-1,-1,-1,-1]); (%o220) 14141414141414141414141414141414 b:ctranspose(a); (%o221) 14141414141414141414141414141414 a . b; b . a; is(equal(a . b,b . a)); (%o222) 14141414141414141414141414141414(%o223) 14141414141414141414141414141414(%o224) false Nem normális Eddig van kész! Feladat kill(all); (%o0) done load("eigen"); (%o1) /usr/share/maxima/5.43.2/share/matrix/eigen.mac h:hilbert_matrix(15); (%o2) 11213141516171819110111112113114115121314151617181911011111211311411511613141516171819110111112113114115116117141516171819110111112113114115116117118151617181911011111211311411511611711811916171819110111112113114115116117118119120171819110111112113114115116117118119120121181911011111211311411511611711811912012112219110111112113114115116117118119120121122123110111112113114115116117118119120121122123124111112113114115116117118119120121122123124125112113114115116117118119120121122123124125126113114115116117118119120121122123124125126127114115116117118119120121122123124125126127128115116117118119120121122123124125126127128129 determinant(h); (%o3) 1944694965363466857137310935123[64 digits]090513203200000000000000000000 x:covect([1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]); (%o4) 111111111111111 b:h.x; (%o5) 11957573603601715839720720237638631225224067868214084080113634299775975201019946717759752030918957258658402839955725865840604691361594914320507040079953542288802374573533126771144400223416543312677114440063306748493803134332005999712899380313433200236266661971332727080400 hf:float(h); (%o7) 1.00.50.33333333333333330.250.20.16666666666666660.14285714285714280.1250.11111111111111110.10.090909090909090910.083333333333333330.076923076923076930.071428571428571430.066666666666666670.50.33333333333333330.250.20.16666666666666660.14285714285714280.1250.11111111111111110.10.090909090909090910.083333333333333330.076923076923076930.071428571428571430.066666666666666670.06250.33333333333333330.250.20.16666666666666660.14285714285714280.1250.11111111111111110.10.090909090909090910.083333333333333330.076923076923076930.071428571428571430.066666666666666670.06250.05882352941176470.250.20.16666666666666660.14285714285714280.1250.11111111111111110.10.090909090909090910.083333333333333330.076923076923076930.071428571428571430.066666666666666670.06250.05882352941176470.055555555555555550.20.16666666666666660.14285714285714280.1250.11111111111111110.10.090909090909090910.083333333333333330.076923076923076930.071428571428571430.066666666666666670.06250.05882352941176470.055555555555555550.052631578947368420.16666666666666660.14285714285714280.1250.11111111111111110.10.090909090909090910.083333333333333330.076923076923076930.071428571428571430.066666666666666670.06250.05882352941176470.055555555555555550.052631578947368420.050.14285714285714280.1250.11111111111111110.10.090909090909090910.083333333333333330.076923076923076930.071428571428571430.066666666666666670.06250.05882352941176470.055555555555555550.052631578947368420.050.047619047619047610.1250.11111111111111110.10.090909090909090910.083333333333333330.076923076923076930.071428571428571430.066666666666666670.06250.05882352941176470.055555555555555550.052631578947368420.050.047619047619047610.045454545454545450.11111111111111110.10.090909090909090910.083333333333333330.076923076923076930.071428571428571430.066666666666666670.06250.05882352941176470.055555555555555550.052631578947368420.050.047619047619047610.045454545454545450.043478260869565210.10.090909090909090910.083333333333333330.076923076923076930.071428571428571430.066666666666666670.06250.05882352941176470.055555555555555550.052631578947368420.050.047619047619047610.045454545454545450.043478260869565210.041666666666666660.090909090909090910.083333333333333330.076923076923076930.071428571428571430.066666666666666670.06250.05882352941176470.055555555555555550.052631578947368420.050.047619047619047610.045454545454545450.043478260869565210.041666666666666660.040.083333333333333330.076923076923076930.071428571428571430.066666666666666670.06250.05882352941176470.055555555555555550.052631578947368420.050.047619047619047610.045454545454545450.043478260869565210.041666666666666660.040.038461538461538460.076923076923076930.071428571428571430.066666666666666670.06250.05882352941176470.055555555555555550.052631578947368420.050.047619047619047610.045454545454545450.043478260869565210.041666666666666660.040.038461538461538460.037037037037037030.071428571428571430.066666666666666670.06250.05882352941176470.055555555555555550.052631578947368420.050.047619047619047610.045454545454545450.043478260869565210.041666666666666660.040.038461538461538460.037037037037037030.035714285714285710.066666666666666670.06250.05882352941176470.055555555555555550.052631578947368420.050.047619047619047610.045454545454545450.043478260869565210.041666666666666660.040.038461538461538460.037037037037037030.035714285714285710.03448275862068965 lu_factor(hf,floatfield); Unable to compute the LU factorization −− an error. To debug this try: debugmode(true); A szinguláris érték felbontást QR-felbontásssal fogjuk kiszámolni h2f:hf.hf; (%o9) 1.5804402834449870.93749999999999990.68933823529411740.55215141612200430.46345566735466110.40076453388372890.35381171474437730.31718184083840870.28772432812491190.26347100771597110.2431239069443740.22578969289920840.21083190984896570.19778397764587530.18629553730030440.93749999999999990.58434653344498710.44117647058823510.35947712418300640.30544088980621470.26658066735466110.23707405769325270.21379548097814350.19489923214275660.17921738368046750.16597100771597110.15461866218912920.14477117438068980.13614097578303160.1285105786310970.68933823529411740.44117647058823510.33780674105744380.27777777777777770.23757309941520470.20838206627680310.18604845446950710.16831928305612510.15385302573517670.14179465697935780.13157032485693810.12277890570033970.11513064475993540.10841047625528590.10245508763466890.55215141612200430.35947712418300640.27777777777777770.22978204969941910.19736842105263150.17368421052631580.15547201336675020.14095465937571190.12906807532665650.11913080351295450.11068354586824670.10340404669065990.097058741091286250.091473746103036740.086516661258022660.46345566735466110.30544088980621470.23757309941520470.19736842105263150.17005213280191220.150.13452380952380950.12215007215007210.11199298889516280.10348328000501910.096236066670849280.089980564638949740.084520031096118070.079707671108637330.075431485278561740.40076453388372890.26658066735466110.20838206627680310.17368421052631580.150.13255213280191220.1190476190476190.10822510822510820.09932398519355040.091854100006273940.085483280005019120.079977325412108030.075165749824134920.070921129997216950.06714609475395750.35381171474437730.23707405769325270.18604845446950710.15547201336675020.13452380952380950.1190476190476190.10704192872027950.09740259740259740.089462168266516080.082789593659158890.077092195244369180.07216326668500580.067852104953554230.064045917275730930.060658404217249790.31718184083840870.21379548097814350.16831928305612510.14095465937571190.12215007215007210.10822510822510820.09740259740259740.088699881157452780.081521739130434790.075483091787439620.070322061191626420.065853434005607920.061942006463745580.058486470587919860.055409233762200130.28772432812491190.19489923214275660.15385302573517670.12906807532665650.11199298889516280.09932398519355040.089462168266516080.081521739130434790.074965240325694750.069444444444444450.064722222222222220.060630665630665620.057047073297073290.053879416879416880.051057149534161020.26347100771597110.17921738368046750.14179465697935780.11913080351295450.10348328000501910.091854100006273940.082789593659158890.075483091787439620.069444444444444450.064355672424460170.060.056223776223776220.052914616247949570.049988159988159990.047379690552104340.2431239069443740.16597100771597110.13157032485693810.11068354586824670.096236066670849280.085483280005019120.077092195244369180.070322061191626420.064722222222222220.060.055955672424460160.052447552447552450.049371924371924370.046650879984213310.044224613190130430.22578969289920840.15461866218912920.12277890570033970.10340404669065990.089980564638949740.079977325412108030.07216326668500580.065853434005607920.060630665630665620.056223776223776220.052447552447552450.049170499555371210.046296296296296290.043752543752543750.041483633437656420.21083190984896570.14477117438068980.11513064475993540.097058741091286250.084520031096118070.075165749824134920.067852104953554230.061942006463745580.057047073297073290.052914616247949570.049371924371924370.046296296296296290.043597797223409620.041208791208791210.03907730200833650.19778397764587530.13614097578303160.10841047625528590.091473746103036740.079707671108637330.070921129997216950.064045917275730930.058486470587919860.053879416879416880.049988159988159990.046650879984213310.043752543752543750.041208791208791210.038956147664177640.036945812807881760.18629553730030440.1285105786310970.10245508763466890.086516661258022660.075431485278561740.06714609475395750.060658404217249790.055409233762200130.051057149534161020.047379690552104340.044224613190130430.041483633437656420.03907730200833650.036945812807881760.03504316748994386 Ez lenne az első lépés, de ne indítsuk el, túl sokáig tart! ueivects(h2f); A QR felbontással boldogulhatunk. keepfloat:true; (%o1) true factor(x):=x; (%o2) factor

x

:=x
Lhh:gramschmidt(hf); (%o3) gramschmidt

hf

Lhhn:map(uvect,Lhh); (%o11) [[0.7954464621776844,0.3977232310888422,0.2651488207258948,0.1988616155444211,0.1590892924355369,0.1325744103629474,0.1136352088825263,0.09943080777221056,0.08838294024196493,0.07954464621776845,0.07231331474342587,0.0662872051814737,0.06118818939828342,0.05681760444126317,0.05302976414517896],[0.5546213470272648,0.2186532287877113,0.3110901199589224,0.3077121753143932,0.2858468524356219,0.2618230390440348,0.2396023161898261,0.2199846079640424,0.2028894871132429,0.1880110536997513,0.1750180149526776,0.1636124361572709,0.1535423192709617,0.1445993532099802,0.136612609336986],[0.229664736568945,0.6290301313637544,0.320489566192404,0.09183303364190503,0.04538812690865278,0.1262649943214054,0.1739298219970169,0.2017198378254623,0.2173313100358541,0.2253180086287803,0.2284519851968622,0.2284658894500764,0.2264679337997995,0.2231807966491856,0.2190832242554918],[0.07923994351140343,0.5332801122075641,0.227112945002206,0.3925577229485682,0.3396153424991413,0.229748820172458,0.1176492788335072,0.01966079323140554,0.06114245024817727,0.1260326981973648,0.1774371039929316,0.2178227123654638,0.2493476741238371,0.2737932094643157,0.2925922291132093],[0.02423507982566053,0.3040130155605244,0.5475639805231896,0.2201210844443791,0.1128674880593017,0.2787351718636597,0.3139475548779166,0.2716594869666314,0.1898315460337975,0.09164613008674155,0.009719842482282627,0.1072103387767532,0.1973566725702416,0.2787370779312713,0.3510658866867713],[0.006700165678285149,0.1364046038902443,0.5031146376659508,0.267339787935784,0.3962755188540781,0.1932929729486373,0.05009207604503958,0.2186479121126369,0.2898225375828978,0.2762030930415487,0.1993572161168715,0.07987431170673783,0.06560303147400443,0.2246449983432625,0.388411914052083],[0.001685371239056343,0.05114913629920098,0.3141211534222173,0.519752384813437,0.08298213756997347,0.2967461097595311,0.3355169854436573,0.1630555624427659,0.05882535195643359,0.2275652399615219,0.2969033676137711,0.2565985466473441,0.1154219903141673,0.1097100950204573,0.3998298474905255],[3.859073029347743*10−4,0.01647522914802311,0.1517311641360511,0.461689732264799,0.3565989267633488,0.3137444498288468,0.08283384884984349,0.3184101466017855,0.2819810892691094,0.07339256448117562,0.1614565140933972,0.303347557405721,0.2778447600301825,0.05065837521940038,0.3833433044935025],[8.027654935826121*10−5,0.004613852141191162,0.05971455504719362,0.2826088937658508,0.5089510530891869,0.1231171820392693,0.3839783645952384,0.106120920699577,0.234269843494423,0.3245106225708402,0.1434773634039511,0.1496515760296827,0.3380714458561697,0.2234156430405787,0.3409946303896302],[1.954887215673738*10−5,0.001126538905949473,0.01958685625010521,0.1322571383770798,0.3966335669142013,0.4602257315146551,0.07760609106214359,0.355617846036452,0.2195021757845533,0.1676418724762389,0.3404295623123791,0.1385747915788956,0.240335616352116,0.3652853597666297,0.2791694859966491],[2.45100722954479*10−4,2.25311616240184*10−4,0.005423968452241508,0.04914951229723174,0.216417594976803,0.4650758117576639,0.3699574705583536,0.202774232576877,0.3113769656881233,0.2641953631341954,0.1723947107555635,0.3545776193863263,0.001402217328984934,0.4371454839399082,0.2076983875223859],[0.01033895603430265,0.002199946340255639,0.004511813790644226,0.0109023068768082,0.0922173454131146,0.2849622324896053,0.4958625149531238,0.2956975116508665,0.2497778398282359,0.3152428473650322,0.2438196346333835,0.2801909742338002,0.278750077015639,0.4253313094527148,0.1356264649570611],[0.2205040878067062,0.1794387986455909,0.2214095241620738,0.2188790794865408,0.2276708581600753,0.1006732267502905,0.4305919616725367,0.2354241967226601,0.4111125080236628,0.310656243558579,0.2560758424534185,0.2383975632323073,0.3193791751347981,0.07736124201494046,0.153393211847176],[0.2494434959034687,0.3079322218838642,0.3259953685117375,0.2856477158491706,0.2244514872398295,0.2825940298054032,0.01856256016334323,0.4581047052833332,0.03994384762626675,0.04393962179112478,0.2418198195696231,0.2826532023450533,0.2570800645226497,0.3243545412629754,0.03812949640335246],[0.7987123446833323,0.1708663907724019,0.1105737237711453,0.04385169390511283,0.01495832838072749,0.04504317862856612,0.03905766089746032,0.07428231387271417,0.1289454278340789,0.1712282017183705,0.09448298494749254,0.2980359249017081,0.3298254409991203,0.2385043434879045,0.007080547820562795]] Lhf:args(hf); (%o15) [[1.0,0.5,0.3333333333333333,0.25,0.2,0.1666666666666666,0.1428571428571428,0.125,0.1111111111111111,0.1,0.09090909090909091,0.08333333333333333,0.07692307692307693,0.07142857142857143,0.06666666666666667],[0.5,0.3333333333333333,0.25,0.2,0.1666666666666666,0.1428571428571428,0.125,0.1111111111111111,0.1,0.09090909090909091,0.08333333333333333,0.07692307692307693,0.07142857142857143,0.06666666666666667,0.0625],[0.3333333333333333,0.25,0.2,0.1666666666666666,0.1428571428571428,0.125,0.1111111111111111,0.1,0.09090909090909091,0.08333333333333333,0.07692307692307693,0.07142857142857143,0.06666666666666667,0.0625,0.0588235294117647],[0.25,0.2,0.1666666666666666,0.1428571428571428,0.125,0.1111111111111111,0.1,0.09090909090909091,0.08333333333333333,0.07692307692307693,0.07142857142857143,0.06666666666666667,0.0625,0.0588235294117647,0.05555555555555555],[0.2,0.1666666666666666,0.1428571428571428,0.125,0.1111111111111111,0.1,0.09090909090909091,0.08333333333333333,0.07692307692307693,0.07142857142857143,0.06666666666666667,0.0625,0.0588235294117647,0.05555555555555555,0.05263157894736842],[0.1666666666666666,0.1428571428571428,0.125,0.1111111111111111,0.1,0.09090909090909091,0.08333333333333333,0.07692307692307693,0.07142857142857143,0.06666666666666667,0.0625,0.0588235294117647,0.05555555555555555,0.05263157894736842,0.05],[0.1428571428571428,0.125,0.1111111111111111,0.1,0.09090909090909091,0.08333333333333333,0.07692307692307693,0.07142857142857143,0.06666666666666667,0.0625,0.0588235294117647,0.05555555555555555,0.05263157894736842,0.05,0.04761904761904761],[0.125,0.1111111111111111,0.1,0.09090909090909091,0.08333333333333333,0.07692307692307693,0.07142857142857143,0.06666666666666667,0.0625,0.0588235294117647,0.05555555555555555,0.05263157894736842,0.05,0.04761904761904761,0.04545454545454545],[0.1111111111111111,0.1,0.09090909090909091,0.08333333333333333,0.07692307692307693,0.07142857142857143,0.06666666666666667,0.0625,0.0588235294117647,0.05555555555555555,0.05263157894736842,0.05,0.04761904761904761,0.04545454545454545,0.04347826086956521],[0.1,0.09090909090909091,0.08333333333333333,0.07692307692307693,0.07142857142857143,0.06666666666666667,0.0625,0.0588235294117647,0.05555555555555555,0.05263157894736842,0.05,0.04761904761904761,0.04545454545454545,0.04347826086956521,0.04166666666666666],[0.09090909090909091,0.08333333333333333,0.07692307692307693,0.07142857142857143,0.06666666666666667,0.0625,0.0588235294117647,0.05555555555555555,0.05263157894736842,0.05,0.04761904761904761,0.04545454545454545,0.04347826086956521,0.04166666666666666,0.04],[0.08333333333333333,0.07692307692307693,0.07142857142857143,0.06666666666666667,0.0625,0.0588235294117647,0.05555555555555555,0.05263157894736842,0.05,0.04761904761904761,0.04545454545454545,0.04347826086956521,0.04166666666666666,0.04,0.03846153846153846],[0.07692307692307693,0.07142857142857143,0.06666666666666667,0.0625,0.0588235294117647,0.05555555555555555,0.05263157894736842,0.05,0.04761904761904761,0.04545454545454545,0.04347826086956521,0.04166666666666666,0.04,0.03846153846153846,0.03703703703703703],[0.07142857142857143,0.06666666666666667,0.0625,0.0588235294117647,0.05555555555555555,0.05263157894736842,0.05,0.04761904761904761,0.04545454545454545,0.04347826086956521,0.04166666666666666,0.04,0.03846153846153846,0.03703703703703703,0.03571428571428571],[0.06666666666666667,0.0625,0.0588235294117647,0.05555555555555555,0.05263157894736842,0.05,0.04761904761904761,0.04545454545454545,0.04347826086956521,0.04166666666666666,0.04,0.03846153846153846,0.03703703703703703,0.03571428571428571,0.03448275862068965]] n:15; (%o27) 15 R:zeromatrix(n,n); (%o36) 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 for i:1 thru n do for j:1 thru n do R[j,i]:innerproduct(Lhf[i],Lhhn[j]); (%o42) done R; (%o43) 1.2571556321494110.74573105829157930.54833166050851410.43920689054064680.3686541709734630.31878673064410090.2814382767704350.25230117316191760.22886929888941240.20957708097405860.19339205164971880.17960361241286120.16770549680352430.15732656529384170.1481881260650187.389922007661198*10−160.16802298100054230.19204825976952630.1901341393953320.18166577310321120.17171163831470280.16186353528265410.1526378115373220.14417246692834920.13646493566227630.12946174531909060.12309429405629350.11729334077235030.11199461905163510.1071406449712218.746475765875062*10−155.656933255160368*10−150.016018647287342830.026956307823516660.033695111584454250.037724008980935420.040036249917914170.041247911598345810.041743328476018820.041767244063945540.041478980061427980.040984341550329820.040354795993337080.039639288000108680.038871695800253851.801857274497109*10−131.095963597652627*10−138.170200627155566*10−140.0013128236575359260.0029003955939362920.0043701535333771040.0056134652259763020.0066233522689255790.0074254339203774260.008052613904353360.008536197705261260.0089032226286182870.0091761037824966020.009373080147270290.0095088707542252492.215418820616932*10−121.290037521251008*10−129.402062461916217*10−137.491889053579115*10−139.523289052033868*10−52.594242411312342*10−44.564415412986258*10−46.611061941109303*10−48.587405168138506*10−40.0010417590394984070.0012068791436859930.0013532853916018120.0014815061351267240.0015927487424371180.0016885150958613294.204749048941636*10−112.442937865887451*10−111.777268263114528*10−111.41418647625624*10−111.18162216122819*10−116.164613492412596*10−61.990546505780538*10−53.997497860067089*10−56.438896914277848*10−59.128011864554739*10−51.191641367377753*10−41.469606673462475*10−41.739313229249667*10−41.996015838252283*10−42.236904523913001*10−49.119575858362872*10−105.385041544625491*10−103.950030545707417*10−103.159113179385109*10−102.648858701825851*10−102.288788533699737*10−103.570258618121324*10−71.329483114466401*10−62.995771057431873*10−65.306667014395167*10−68.14921828562118*10−61.13905603896236*10−51.490232596302286*10−51.85719723743219*10−52.230652361622821*10−51.442909750362653*10−88.578168522732632*10−96.318112082126515*10−95.067606799469271*10−94.258363165560164*10−93.685859627927668*10−93.256707552939941*10−91.552511803837308*10−87.50913284226773*10−81.915918582395814*10−73.723269820093966*10−76.170634967057886*10−79.204416777826296*10−71.274211240954309*10−61.66891204641216*10−61.880037320475813*10−71.103080073706908*10−78.12143382093733*10−86.531954287181119*10−85.508325446462736*10−84.784952294459299*10−84.242275418453456*10−83.817809847081466*10−83.560263000211561*10−83.591146520605381*10−84.044603766316379*10−85.027366322199467*10−86.602751841632004*10−88.790531080653252*10−81.157446256876831*10−75.787540384896705*10−63.353878842527608*10−62.477050896444038*10−62.004849920090798*10−61.701917950111542*10−61.487621483695772*10−61.326319090073219*10−61.199601160274677*10−61.096897718450701*10−61.0116222126412*10−69.393955479296279*10−78.77160464977933*10−78.227071729774438*10−77.744002899767732*10−77.310102580117966*10−73.392096299259118*10−42.028168429197389*10−41.520179061347531*10−41.240617229947317*10−41.058608789599249*10−49.285099713262843*10−58.298395254135657*10−57.51883947248972*10−56.884265066348094*10−56.355791353543361*10−55.907684797252198*10−55.522142448871788*10−55.186402287909661*10−54.891045270521716*10−54.628948380927707*10−50.01752646203367250.011052342538454030.0084880455798791570.0070223073444322980.0060438965205912880.0053322779907614310.0047857247263212940.0043498137406993110.0039923765129293840.0036929846205452020.0034379439623418220.0032176785592635680.0030252611230858850.0028555390088626140.0027045894194081970.6357442607244660.4427797632908010.35460464475478520.30017600810677440.26209161247830450.23350873593596040.21106297763937310.19286473566195310.17775395538829640.16497141281585470.15399560419228250.14445447655588790.13607436146728510.12864882111075150.12201880161586630.28262883184884570.29044827081371630.26197678031687010.2349085354386130.21215365576887560.19324403061549090.17740512216330990.16398219673861320.1524727796849410.1424975165671930.13376855153049640.12606489891363560.11921462485818480.11308216530786840.1075592759952280.69550241122559920.33332656725804920.2191906276936510.16400761638690120.13155929546428930.1101804174539720.095006308043713840.083656885083514340.074832457737400060.067763738777960270.061966366003107780.05712008337992520.053004564575002550.049463168302842390.04638140950426561 R:zeromatrix(n,n)$ for i:1 thru n do for j:1 thru i do R[j,i]:innerproduct(Lhf[i],Lhhn[j]); (%o45) done R; (%o46) 1.2571556321494110.74573105829157930.54833166050851410.43920689054064680.3686541709734630.31878673064410090.2814382767704350.25230117316191760.22886929888941240.20957708097405860.19339205164971880.17960361241286120.16770549680352430.15732656529384170.14818812606501800.16802298100054230.19204825976952630.1901341393953320.18166577310321120.17171163831470280.16186353528265410.1526378115373220.14417246692834920.13646493566227630.12946174531909060.12309429405629350.11729334077235030.11199461905163510.107140644971221000.016018647287342830.026956307823516660.033695111584454250.037724008980935420.040036249917914170.041247911598345810.041743328476018820.041767244063945540.041478980061427980.040984341550329820.040354795993337080.039639288000108680.038871695800253850000.0013128236575359260.0029003955939362920.0043701535333771040.0056134652259763020.0066233522689255790.0074254339203774260.008052613904353360.008536197705261260.0089032226286182870.0091761037824966020.009373080147270290.00950887075422524900009.523289052033868*10−52.594242411312342*10−44.564415412986258*10−46.611061941109303*10−48.587405168138506*10−40.0010417590394984070.0012068791436859930.0013532853916018120.0014815061351267240.0015927487424371180.001688515095861329000006.164613492412596*10−61.990546505780538*10−53.997497860067089*10−56.438896914277848*10−59.128011864554739*10−51.191641367377753*10−41.469606673462475*10−41.739313229249667*10−41.996015838252283*10−42.236904523913001*10−40000003.570258618121324*10−71.329483114466401*10−62.995771057431873*10−65.306667014395167*10−68.14921828562118*10−61.13905603896236*10−51.490232596302286*10−51.85719723743219*10−52.230652361622821*10−500000001.552511803837308*10−87.50913284226773*10−81.915918582395814*10−73.723269820093966*10−76.170634967057886*10−79.204416777826296*10−71.274211240954309*10−61.66891204641216*10−6000000003.560263000211561*10−83.591146520605381*10−84.044603766316379*10−85.027366322199467*10−86.602751841632004*10−88.790531080653252*10−81.157446256876831*10−70000000001.0116222126412*10−69.393955479296279*10−78.77160464977933*10−78.227071729774438*10−77.744002899767732*10−77.310102580117966*10−700000000005.907684797252198*10−55.522142448871788*10−55.186402287909661*10−54.891045270521716*10−54.628948380927707*10−5000000000000.0032176785592635680.0030252611230858850.0028555390088626140.0027045894194081970000000000000.13607436146728510.12864882111075150.122018801615866300000000000000.11308216530786840.107559275995228000000000000000.04638140950426561 Q:transpose(apply('matrix,Lhhn)); (%o49) 0.79544646217768440.55462134702726480.2296647365689450.079239943511403430.024235079825660530.0067001656782851490.0016853712390563433.859073029347743*10−48.027654935826121*10−51.954887215673738*10−52.45100722954479*10−40.010338956034302650.22050408780670620.24944349590346870.79871234468333230.39772323108884220.21865322878771130.62903013136375440.53328011220756410.30401301556052440.13640460389024430.051149136299200980.016475229148023110.0046138521411911620.0011265389059494732.25311616240184*10−40.0021999463402556390.17943879864559090.30793222188386420.17086639077240190.26514882072589480.31109011995892240.3204895661924040.2271129450022060.54756398052318960.50311463766595080.31412115342221730.15173116413605110.059714555047193620.019586856250105210.0054239684522415080.0045118137906442260.22140952416207380.32599536851173750.11057372377114530.19886161554442110.30771217531439320.091833033641905030.39255772294856820.22012108444437910.2673397879357840.5197523848134370.4616897322647990.28260889376585080.13225713837707980.049149512297231740.01090230687680820.21887907948654080.28564771584917060.043851693905112830.15908929243553690.28584685243562190.045388126908652780.33961534249914130.11286748805930170.39627551885407810.082982137569973470.35659892676334880.50895105308918690.39663356691420130.2164175949768030.09221734541311460.22767085816007530.22445148723982950.014958328380727490.13257441036294740.26182303904403480.12626499432140540.2297488201724580.27873517186365970.19329297294863730.29674610975953110.31374444982884680.12311718203926930.46022573151465510.46507581175766390.28496223248960530.10067322675029050.28259402980540320.045043178628566120.11363520888252630.23960231618982610.17392982199701690.11764927883350720.31394755487791660.050092076045039580.33551698544365730.082833848849843490.38397836459523840.077606091062143590.36995747055835360.49586251495312380.43059196167253670.018562560163343230.039057660897460320.099430807772210560.21998460796404240.20171983782546230.019660793231405540.27165948696663140.21864791211263690.16305556244276590.31841014660178550.1061209206995770.3556178460364520.2027742325768770.29569751165086650.23542419672266010.45810470528333320.074282313872714170.088382940241964930.20288948711324290.21733131003585410.061142450248177270.18983154603379750.28982253758289780.058825351956433590.28198108926910940.2342698434944230.21950217578455330.31137696568812330.24977783982823590.41111250802366280.039943847626266750.12894542783407890.079544646217768450.18801105369975130.22531800862878030.12603269819736480.091646130086741550.27620309304154870.22756523996152190.073392564481175620.32451062257084020.16764187247623890.26419536313419540.31524284736503220.3106562435585790.043939621791124780.17122820171837050.072313314743425870.17501801495267760.22845198519686220.17743710399293160.0097198424822826270.19935721611687150.29690336761377110.16145651409339720.14347736340395110.34042956231237910.17239471075556350.24381963463338350.25607584245341850.24181981956962310.094482984947492540.06628720518147370.16361243615727090.22846588945007640.21782271236546380.10721033877675320.079874311706737830.25659854664734410.3033475574057210.14965157602968270.13857479157889560.35457761938632630.28019097423380020.23839756323230730.28265320234505330.29803592490170810.061188189398283420.15354231927096170.22646793379979950.24934767412383710.19735667257024160.065603031474004430.11542199031416730.27784476003018250.33807144585616970.2403356163521160.0014022173289849340.2787500770156390.31937917513479810.25708006452264970.32982544099912030.056817604441263170.14459935320998020.22318079664918560.27379320946431570.27873707793127130.22464499834326250.10971009502045730.050658375219400380.22341564304057870.36528535976662970.43714548393990820.42533130945271480.077361242014940460.32435454126297540.23850434348790450.053029764145178960.1366126093369860.21908322425549180.29259222911320930.35106588668677130.3884119140520830.39982984749052550.38334330449350250.34099463038963020.27916948599664910.20769838752238590.13562646495706110.1533932118471760.038129496403352460.007080547820562795 Q.R-hf; (%o50) 0.00.01.054711873393898*10−156.050715484207103*10−152.034483692625599*10−148.570921750106209*10−144.158895450245836*10−131.50007783972228*10−124.151207155800307*10−122.358654938028337*10−111.450166818572906*10−83.32809924020877*10−50.030036243722045731.895156060160163*10−40.037149061053101020.01.665334536937734*10−162.55351295663786*10−153.735900477863652*10−142.22211138378725*10−131.581013098217454*10−121.172183183850705*10−115.863624863433614*10−112.135639026734637*10−101.335719326811535*10−91.455048395659908*10−87.065120654486945*10−60.024423662588723220.0579125023152270.047096788529003730.02.220446049250313*10−161.165734175856414*10−151.398881011027697*10−143.549938121238938*10−135.429878768836716*10−126.821694598091455*10−115.04422600910992*10−102.533134257909353*10−92.214102730324629*10−83.409857341624134*10−71.483627945689236*10−50.030142108357915350.065361502339525870.056963855186069780.01.665334536937734*10−162.498001805406602*10−162.5729418595688*10−141.597055820923287*10−132.584640834690788*10−121.072372607380245*10−101.444975639142143*10−91.113335539038029*10−81.440229800281356*10−73.037322742038828*10−63.791911128094016*10−50.029748182357831690.060426551418227520.055365655261023990.01.387778780781445*10−163.330669073875469*10−162.270406085358445*10−145.152822613041508*10−144.079916959831564*10−122.035044355253035*10−111.023621479245839*10−91.861734084951383*10−84.183571542665243*10−71.317375902289141*10−53.090395154157904*10−40.031270713054501170.05494528399055290.051487907942172420.01.387778780781445*10−166.938893903907228*10−161.570965579844596*10−141.613986722048821*10−132.107578001009358*10−125.807052061435058*10−119.686834667732569*10−103.399610246090922*10−94.687155640142881*10−72.791052250512155*10−59.430056433593972*10−40.012812457487390710.04407100911081690.043976227533824530.01.387778780781445*10−168.743006318923108*10−168.479328350574633*10−151.895983370303611*10−133.466255060757817*10−136.806119556834744*10−111.810240152222419*10−101.35092273478854*10−89.092029739504737*10−82.17715200625107*10−50.001615876951435280.060111753353086760.05473003359268540.050090353963081010.01.110223024625156*10−169.853229343548265*10−162.081668171172168*10−151.684763439868675*10−132.081321226476973*10−123.476098575649899*10−118.991327266061333*10−104.504318446008071*10−93.556273305074797*10−71.230956567355168*10−59.399535012817872*10−40.032918953085658930.020682241687865350.023202454764681920.01.110223024625156*10−161.040834085586084*10−153.219646771412954*10−151.216665657111093*10−132.846500812836439*10−129.37301625203446*10−128.322478470268457*10−107.386432722411573*10−92.28848037048257*10−71.818264805740488*10−58.207009708099677*10−40.055170265902347290.047643898409966080.051158001114906670.08.326672684688674*10−171.068589661201713*10−157.466249840604178*10−156.37406794012918*10−142.757752359805465*10−124.352734839230265*10−112.534326290493105*10−101.104809555646513*10−81.594463739928997*10−71.575594750272135*10−59.996225178275226*10−40.043212209747990040.045821490926875050.035530468238292611.387778780781445*10−178.326672684688674*10−171.068589661201713*10−151.086630785351872*10−143.080868893334809*10−152.027489287570461*10−125.817288317722102*10−114.195881003754564*10−105.367344416140618*10−93.394681433385726*10−79.869238574600203*10−67.937585458694801*10−40.034099074112510260.0048939387195161351.866832496244391*10−40.08.326672684688674*10−171.068589661201713*10−151.353084311261909*10−145.576095141179849*10−148.614428614883707*10−135.106028100332338*10−118.405165244362856*10−104.438042246202034*10−91.442688210007148*10−72.081335028374792*10−58.821093547507486*10−40.031610417679839620.061849737173229240.0725727919441030.08.326672684688674*10−171.054711873393898*10−151.563332796550298*10−141.104810687380108*10−135.72861202918773*10−132.381335406642648*10−117.904232754429863*10−101.103191227258548*10−82.330012208381915*10−71.335928508372497*10−78.968033752403889*10−40.042616142659782790.011220712677655960.0047326503323453440.06.938893903907228*10−171.026956297778269*10−151.726396803292118*10−141.601149768326593*10−132.149704025899979*10−122.026246531672271*10−111.661642143324648*10−107.613816681295571*10−93.625487534908433*10−72.548843759597047*10−50.0013447542212786720.0092625201329605660.027919728846104160.037640355964679560.06.938893903907228*10−171.013078509970455*10−151.851990782952839*10−142.043712421517796*10−133.779296320338687*10−127.741579599596094*10−111.050615164310819*10−91.089728961234381*10−82.724108954729742*10−71.201713359023914*10−54.251693618090094*10−40.020473128026674520.023668284618619430.02213223350042021 Feladat kill(all); (%o0) done load("eigen"); (%o1) /usr/share/maxima/5.43.2/share/matrix/eigen.mac Az első esetben: x1:1; x:covect([1,0]); (%o2) 1(%o3) 10 x1:1; x2:2*10^13; x:covect([x1,x2]); (%o12) 1(%o13) 20000000000000(%o14) 120000000000000 Tyihonov-regularizáció Feladat kill(all); (%o0) done load("eigen"); (%o1) /usr/share/maxima/5.43.2/share/matrix/eigen.mac A:matrix([1,1],[0,c]); (%o2) 110c b:covect([a,a]); (%o3) aa x:covect([x1,x2]); (%o4) x1x2 A.x-b; (%o8) x2+x1ac*x2a f:(%[1]^2+%[2]^2)[1]; (%o9)

c*x2a

2
+

x2+x1a

2
eqf1:diff(f,x1); (%o19) 2*

x2+x1a

eqf2:diff(f,x2); (%o20) 2*c*

c*x2a

+2*

x2+x1a

linsolve([eqf1,eqf2],[x1,x2]); (%o21) [x1=a*

c1

c
,x2=ac]
g:f+δ*(x1^2+x2^2); (%o22)

x22+x12

*δ+

c*x2a

2
+

x2+x1a

2
eqg1:diff(g,x1); (%o23) 2*x1*δ+2*

x2+x1a

eqg2:diff(g,x2); (%o24) 2*x2*δ+2*c*

c*x2a

+2*

x2+x1a

linsolve([eqg1,eqg2],[x1,x2]); (%o25) [x1=a*δ+a*

c2c

δ2+

c2+2

*δ+c2
,x2=a*

c+1

*δ+a*c
δ2+

c2+2

*δ+c2
]
limit(%,δ,0); (%o26) [x1=a*cac,x2=ac]
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