Generalized Eigenvalue Problem using MATLAB - matlab
I'm trying to solve a generalized eigenvalue problem. I have two matrices H and S such that:
HX=λSX
I need to find the eigenvalues λ. The matrices H and S are real, asymmetric, contains positive and negative decimal numbers and (can be) singular. I tried this on MATLAB by using the command eig(H,S), but the problem is that I am obtaining real and complex eigenvalues, while in the research paper that I am following the authors have got only real eigenvalues.
After reading about this problem I have got to know that for such matrices, softwares like MATLAB and many other use the QZ Algorithm to solve generalized eigenavalue problem. I am seeking answers to the following questions:
Is there any criteria to determine whether the given matrices will have complex eigenvalues or not? and that the result obtained from MATLAB is correct or not? (though the values don't agree with research paper).
In the research paper the authors have used RGG subroutine which is a part of Fortran library EISPACK. I went through its documentation and got to know that it also uses the QZ algorithm. So my question is can MATLAB and Fortran give different answers to the same problem even when both of them are using the same algorithm?
Here are the matrices:
H=[0,0,0,0,192,1917.04064,10332.51505,40092.51227,125681.1486,338350.2206,811892.8294,1779728.921,3625982.355,6953387.916,12670976.81,22100000,37132930.27,60353006.25,95276316.19,146559937.4,220274060.5,324208411.9,468219308,664618721.8;
0,0,0,0,192,1893.475124,10051.90014,38308.22391,117609.6433,309187.9535,722364.2569,1537115.973,3030677.025,5606681.841,9824567.083,16426180.74,26355891.41,40770017.02,61031174.08,88683247.04,125403139.6,172926323.5,232944447.3,306974887;
0,0,0,0,192,1846.924351,9507.779872,34923.83828,102685.9158,256815.0258,566753.8301,1130503.089,2072244.588,3531888.862,5644710.273,8510616.36,12154554.65,16481872.51,21234784.98,25958063.51,29983201.26,32440241.94,32304892.86,28485362.66;
0,0,0,0,192,1778.534558,8732.953666,30281.20319,83090.50097,191433.6927,383369.9941,681577.7806,1089000.57,1571702.814,2043718.6,2360415.659,2327150.205,1728131.214,376286.8361,-1820993.208,-4791939.112,-8243585.409,-11638086.16,-14220529.21;
0,0,0,0,192,1689.989727,7773.203992,24831.36406,61515.88289,124689.9821,212051.9399,303583.89,356337.2627,308129.1939,94242.56924,-323226.8121,-920239.2345,-1583058.32,-2104105.201,-2214813.175,-1659107.74,-295993.1841,1796496.926,4254923.427;
0,0,0,0,192,1583.470126,6683.555129,19072.50505,40639.69537,66711.57632,81796.22718,60590.29663,-21861.2573,-171429.7577,-355062.5907,-492927.1661,-475398.2758,-208930.4549,320950.1747,997476.1882,1566265.895,1699768.472,1129529.762,-194208.6509;
0,0,0,0,192,1461.598622,5523.845413,13484.72812,22648.14037,23924.65349,4088.400074,-44523.67789,-110266.4937,-154796.0153,-124116.6074,18352.09343,250026.0416,467393.1673,513093.6957,258167.4211,-292568.7362,-936296.041,-1313188.428,-1070939.742;
0,0,0,0,192,1327.376095,4354.022806,8472.235861,8915.430969,-1775.330646,-26379.77787,-53287.95266,-55123.02039,-5131.095738,92963.75123,184430.9087,180646.3185,21137.78348,-252747.0452,-475735.34,-436097.8969,-33470.52706,583046.8515,1024123.527;
0,0,0,0,192,1184.107549,3229.595901,4321.651839,-104.2768198,-12455.61648,-25344.84529,-20852.91355,14642.78081,67162.133,88643.95935,29331.92183,-103721.9759,-215885.8219,-175668.2952,63339.97258,366040.1121,463007.7228,161145.8379,-426631.9431;
0,0,0,0,192,1035.320732,2197.653181,1181.837405,-4776.384006,-12614.95411,-10733.55999,10443.54925,39434.88701,40860.0242,-11919.57845,-90391.52336,-109544.5177,-4764.383231,169447.4417,238928.2291,63214.86651,-273067.0285,-452649.3406,-197417.4769;
0,0,0,0,192,884.6792681,1293.804398,-933.634351,-6033.445815,-7367.269353,3768.214436,22179.4179,22745.61092,-13409.71654,-59759.11695,-51183.66478,39553.74618,132447.0485,93359.81585,-98263.14966,-255824.9417,-144286.083,212696.4921,445204.5408;
0,0,0,0,192,735.8924507,540.3053094,-2124.343367,-5057.868211,-969.9812354,11405.43117,15940.52452,-5622.481002,-37623.47011,-29881.32575,35855.80045,87872.36995,28692.89466,-112944.5884,-154561.6584,24454.28757,254280.9736,202369.5839,-178659.2813;
0,0,0,0,192,592.6239049,-54.49033533,-2567.50857,-2988.296704,3941.876267,11526.38431,2233.953421,-22136.82236,-24139.16168,20459.96219,60268.45865,14532.0698,-88098.13253,-93277.95948,64910.42847,192595.8473,49803.13745,-244897.5404,-254777.9707;
0,0,0,0,192,458.4013779,-495.3380524,-2477.176593,-714.5839156,6392.684212,6873.986266,-9352.41825,-21134.06087,3844.754647,42020.47316,20505.10932,-58980.138,-70793.54333,51327.19794,141815.5672,6229.868714,-208390.0472,-130756.2533,224730.6889;
0,0,0,0,192,336.5298736,-798.086388,-2066.223865,1211.650343,6609.005295,832.2069828,-14542.95731,-9394.727768,23888.17206,29260.48864,-28610.53289,-63533.47519,18266.13986,110340.0964,20413.44929,-159630.1792,-99650.79156,191428.6434,224840.2275;
0,0,0,0,192,230.010273,-986.6727322,-1517.645248,2561.671612,5397.538212,-4248.815769,-13851.25499,3944.496017,28647.88724,2246.348802,-50351.46819,-19936.09886,77156.40402,55808.14847,-103790.0999,-116346.81,120768.0917,206007.3722,-114289.9057;
0,0,0,0,192,141.4654422,-1089.461027,-968.7547191,3346.025684,3627.717551,-7474.097775,-9965.085871,13492.9252,22500.21035,-20544.91549,-44251.15549,26525.3978,78409.54708,-27774.42335,-127872.1365,18879.12551,194650.9903,7368.517829,-279197.6607;
0,0,0,0,192,73.07564879,-1135.312342,-508.5655728,3706.949976,1946.428626,-9020.436512,-5497.089494,18315.25514,12841.39554,-32819.98602,-26305.72752,53592.96081,48907.83539,-81344.97478,-84369.59991,116249.2281,137093.641,-157744.2065,-212101.4947;
0,0,0,0,192,26.52487642,-1149.801347,-185.5323873,3822.418606,714.8963246,-9523.125445,-2036.182875,19914.17359,4805.440196,-36987.23947,-9962.723932,63039.66842,18780.08487,-100646.6018,-32907.74918,152629.087,54418.22672,-222018.2854,-85848.08848;
0,0,0,0,192,2.959359616,-1151.972632,-20.71532046,3839.781056,79.90093794,-9599.042127,-227.8620289,20156.93483,538.572741,-37623.95402,-1118.549352,64493.6094,2112.7619,-103642.0699,-3710.543262,158327.7533,6151.495274,-232190.8611,-9731.39151;
1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0;
0,0,4,0,-16,0,36,0,-64,0,100,0,-144,0,196,0,-256,0,324,0,-400,0,484,0;
0,0,0,24,192,840,2688,7056,16128,33264,63360,113256,192192,312312,489216,742560,1096704,1581408,2232576,3093048,4213440,5653032,7480704,9775920;
0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529]
S=[1,0.998458667,0.993839419,0.986156496,0.975433581,0.96170373,0.945009268,0.925401657,0.902941342,0.87769756,0.84974813,0.819179209,0.786085032,0.750567618,0.712736454,0.672708162,0.630606134,0.586560159,0.540706014,0.493185053,0.444143767,0.393733335,0.342109153,0.289430364;
1,0.98618496,0.945121551,0.877944359,0.786509494,0.673343309,0.541572595,0.394838187,0.237194368,0.07299685,-0.093217576,-0.256856394,-0.413398249,-0.558517877,-0.688205612,-0.798878172,-0.887477663,-0.951556077,-0.98934292,-0.999794139,-0.982620967,-0.938297899,-0.868049586,-0.773816993;
1,0.961939766,0.850656228,0.67462034,0.447232036,0.18580022,-0.089774795,-0.35851611,-0.599967012,-0.795748145,-0.930956556,-0.99530012,-0.983880972,-0.897568346,-0.742932397,-0.531744087,-0.280079168,-0.007094493,0.266430219,0.519674138,0.733360219,0.891222577,0.981244656,0.996573933;
1,0.926320082,0.716137789,0.400425549,0.025706666,-0.352800347,-0.679318759,-0.90573287,-0.998678335,-0.944458724,-0.751063831,-0.446992295,-0.077052048,0.304242576,0.640704064,0.882751506,0.994716832,0.960100849,0.784004562,0.492377492,0.128193756,-0.254880591,-0.600395776,-0.857436738;
1,0.880202983,0.549514582,0.087165765,-0.396067449,-0.784405265,-0.984804259,-0.949250027,-0.686261152,-0.258848199,0.230583238,0.664768308,0.939678856,0.989447956,0.802151229,0.422663852,-0.058091262,-0.524928056,-0.86599522,-0.999575095,-0.89366274,-0.573634124,-0.116166194,0.369134463;
1,0.824724024,0.360339432,-0.230362851,-0.740310987,-0.990741662,-0.893865914,-0.483643725,0.096120716,0.642189852,0.963138082,0.946456378,0.597992543,0.039901255,-0.532177495,-0.917700386,-0.981521616,-0.701268527,-0.175184388,0.412310981,0.85526993,0.998412337,0.79155935,0.307223688;
1,0.761249282,0.15900094,-0.51917058,-0.949437402,-0.926346503,-0.460923818,0.224590651,0.802862762,0.997766752,0.716235686,0.092701052,-0.575098468,-0.968287643,-0.899118079,-0.400618342,0.289177228,0.840890257,0.991076981,0.668023024,0.025987114,-0.62845768,-0.98281303,-0.867873748;
1,0.691341716,-0.044093263,-0.75230874,-0.996111568,-0.624998222,0.131936882,0.807425162,0.984476513,0.553794202,-0.218754445,-0.856262349,-0.965185319,-0.4782834,0.303870785,0.8984405,0.93838801,0.399053054,-0.386623964,-0.933631603,-0.904292986,-0.316719326,0.46637042,0.96156198;
1,0.616722682,-0.239306267,-0.911893888,-0.885465021,-0.180278837,0.663100925,0.998177599,0.568096607,-0.297461473,-0.934999082,-0.855808809,-0.120594327,0.707062296,0.992717038,0.517399932,-0.354532491,-0.954696389,-0.823033344,-0.060470273,0.748446566,0.98363822,0.464817436,-0.410311308;
1,0.539229548,-0.418462989,-0.990524765,-0.649777453,0.289766361,0.96227862,0.74801177,-0.155578523,-0.915796843,-0.832070912,0.0184424,0.851960286,0.90036192,0.119043216,-0.771978681,-0.951590646,-0.254272907,0.677367717,0.984786282,0.384684007,-0.569920316,-0.999319756,-0.507805164;
1,0.460770452,-0.575381181,-0.991007746,-0.337872993,0.679643962,0.964192705,0.208899055,-0.771683681,-0.920037132,-0.07616817,0.849845048,0.859335144,-0.057932562,-0.91272237,-0.783178435,0.190991406,0.959184828,0.692936648,-0.320615363,-0.98839682,-0.590232736,0.444473211,0.99983298;
1,0.383277318,-0.706196995,-0.924615899,-0.002571609,0.92264462,0.70982912,-0.378521817,-0.999986774,-0.38802268,0.702546189,0.926562719,0.007714758,-0.920648935,-0.713442468,0.373756304,0.999947095,0.392757778,-0.698876799,-0.928485029,-0.012857704,0.918628896,0.717036943,-0.368980903;
1,0.308658284,-0.809460128,-0.808351431,0.310451397,0.999998222,0.306864073,-0.810565945,-0.807239861,0.312243405,0.999992888,0.305068772,-0.811668881,-0.80612542,0.314034304,0.999983998,0.303272386,-0.81276893,-0.805008112,0.315824085,0.999971552,0.301474921,-0.813866089,-0.803887941;
1,0.238750718,-0.88599619,-0.66181517,0.569978496,0.93398072,-0.124001362,-0.993191548,-0.350249028,0.825947135,0.74463997,-0.47038048,-0.969247324,0.007563492,0.972858903,0.456978031,-0.754651237,-0.81732508,0.364377338,0.991315782,0.10897737,-0.939278931,-0.557484408,0.673079326;
1,0.175275976,-0.938556665,-0.504288846,0.761777225,0.771331339,-0.491385519,-0.943587492,0.160609082,0.999889319,0.18990407,-0.933318077,-0.517080544,0.752054483,0.78071471,-0.478373418,-0.948409446,0.145906635,0.999557301,0.204490127,-0.927872888,-0.529757779,0.742165265,0.789925261;
1,0.119797017,-0.971297349,-0.352514068,0.886837082,0.564994942,-0.751467664,-0.745042111,0.572960019,0.882319914,-0.361561431,-0.968947876,0.1294073,0.999953093,0.110175495,-0.973555702,-0.343433634,0.891271052,0.556976861,-0.757822719,-0.738546663,0.580871344,0.877719972,-0.370574875;
1,0.073679918,-0.989142539,-0.2194398,0.956805927,0.360434564,-0.903692348,-0.49360252,0.830955162,0.616051936,-0.74017385,-0.725123833,0.633319721,0.818449723,-0.512713105,-0.894003042,0.380972963,0.950143155,-0.240960024,-0.985650985,0.095714657,0.999755481,0.051609146,-0.992150366;
1,0.038060234,-0.997102837,-0.113960168,0.988428136,0.18919978,-0.97402616,-0.263343106,0.95398036,0.335960537,-0.928406887,-0.406631304,0.897453922,0.474945916,-0.861300817,-0.540508536,0.820157054,0.602939275,-0.774261035,-0.661876387,0.723878695,0.716978371,-0.669301966,-0.76792595;
1,0.01381504,-0.999618289,-0.041434573,0.998473449,0.069022474,-0.996566352,-0.096557681,0.993898456,0.124019175,-0.990471796,-0.151385989,0.986288989,0.178637233,-0.981353228,-0.2057521,0.975668281,0.232709893,-0.969238489,-0.259490029,0.962068758,0.286072066,-0.954164565,-0.312435708;
1,0.001541333,-0.999995249,-0.004623985,0.999980994,0.007706592,-0.999957238,-0.010789127,0.999923978,0.013871559,-0.999881217,-0.016953859,0.999828954,0.020035998,-0.999767189,-0.023117946,0.999695925,0.026199675,-0.99961516,-0.029281155,0.999524896,0.032362357,-0.999425133,-0.035443251;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
As far as I can tell the matrices you give have 2 complex evals, a complex conjugate pair. I checked using the Fortran program below, which uses both EISPACK and LAPACK, the latter having superseded the former decades ago. Both give the same answers where you can calculate an eval. I say this as note the comment at http://www.netlib.org/lapack/explore-3.1.1-html/dggev.f.html which is the documentation for the LAPACK routine:
* A generalized eigenvalue for a pair of matrices (A,B) is a scalar
* lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
* singular. It is usually represented as the pair (alpha,beta), as
* there is a reasonable interpretation for beta=0, and even for both
* being zero.
In your case beta is zero in four cases, I am assuming due to the 4 columns of your matrices which contain zeros. In these cases you can't calculate lambda, so I quote the real and imaginary parts of alpha, and beta in the results below, before calculating lambda where possible.
LAPACK was the one acquired by apt get on my machine. The relevant parts of EISPACK were downloaded from http://www.netlib.org/cgi-bin/netlibfiles.pl?filename=/eispack/rgg.f and the compiled into a little library with
ijb#ijb-Latitude-5410:~/Downloads/eispack$ unzip netlibfiles.zip
Archive: netlibfiles.zip
inflating: eispack/epslon.f
inflating: eispack/qzhes.f
inflating: eispack/qzit.f
inflating: eispack/qzval.f
inflating: eispack/qzvec.f
inflating: eispack/rgg.f
ijb#ijb-Latitude-5410:~/Downloads/eispack$ cd eispack/
ijb#ijb-Latitude-5410:~/Downloads/eispack/eispack$ gfortran -c -O *f
qzvec.f:92:72:
92 | 610 r = r + (betm * a(i,j) - alfm * b(i,j)) * b(j,en)
| 1
Warning: Fortran 2018 deleted feature: DO termination statement which is not END DO or CONTINUE with label 610 at (1)
qzvec.f:207:72:
207 | do 880 i = 1, n
| 1
Warning: Fortran 2018 deleted feature: Shared DO termination label 880 at (1)
qzvec.f:211:72:
211 | 860 zz = zz + z(i,k) * b(k,j)
| 1
Warning: Fortran 2018 deleted feature: DO termination statement which is not END DO or CONTINUE with label 860 at (1)
qzvec.f:228:72:
228 | 900 z(i,j) = z(i,j) / d
| 1
Warning: Fortran 2018 deleted feature: DO termination statement which is not END DO or CONTINUE with label 900 at (1)
ijb#ijb-Latitude-5410:~/Downloads/eispack/eispack$ ar r eispack.a *.o
ar: creating eispack.a
ijb#ijb-Latitude-5410:~/Downloads/eispack/eispack$
The data file was created from what you have above:
ijb#ijb-Latitude-5410:~/work/stack$ cat eig.dat
0 0 0 0 192 1917.04064 10332.51505 40092.51227 125681.1486 338350.2206 811892.8294 1779728.921 3625982.355 6953387.916 12670976.81 22100000 37132930.27 60353006.25 95276316.19 146559937.4 220274060.5 324208411.9 468219308 664618721.8
0 0 0 0 192 1893.475124 10051.90014 38308.22391 117609.6433 309187.9535 722364.2569 1537115.973 3030677.025 5606681.841 9824567.083 16426180.74 26355891.41 40770017.02 61031174.08 88683247.04 125403139.6 172926323.5 232944447.3 306974887
0 0 0 0 192 1846.924351 9507.779872 34923.83828 102685.9158 256815.0258 566753.8301 1130503.089 2072244.588 3531888.862 5644710.273 8510616.36 12154554.65 16481872.51 21234784.98 25958063.51 29983201.26 32440241.94 32304892.86 28485362.66
0 0 0 0 192 1778.534558 8732.953666 30281.20319 83090.50097 191433.6927 383369.9941 681577.7806 1089000.57 1571702.814 2043718.6 2360415.659 2327150.205 1728131.214 376286.8361 -1820993.208 -4791939.112 -8243585.409 -11638086.16 -14220529.21
0 0 0 0 192 1689.989727 7773.203992 24831.36406 61515.88289 124689.9821 212051.9399 303583.89 356337.2627 308129.1939 94242.56924 -323226.8121 -920239.2345 -1583058.32 -2104105.201 -2214813.175 -1659107.74 -295993.1841 1796496.926 4254923.427
0 0 0 0 192 1583.470126 6683.555129 19072.50505 40639.69537 66711.57632 81796.22718 60590.29663 -21861.2573 -171429.7577 -355062.5907 -492927.1661 -475398.2758 -208930.4549 320950.1747 997476.1882 1566265.895 1699768.472 1129529.762 -194208.6509
0 0 0 0 192 1461.598622 5523.845413 13484.72812 22648.14037 23924.65349 4088.400074 -44523.67789 -110266.4937 -154796.0153 -124116.6074 18352.09343 250026.0416 467393.1673 513093.6957 258167.4211 -292568.7362 -936296.041 -1313188.428 -1070939.742
0 0 0 0 192 1327.376095 4354.022806 8472.235861 8915.430969 -1775.330646 -26379.77787 -53287.95266 -55123.02039 -5131.095738 92963.75123 184430.9087 180646.3185 21137.78348 -252747.0452 -475735.34 -436097.8969 -33470.52706 583046.8515 1024123.527
0 0 0 0 192 1184.107549 3229.595901 4321.651839 -104.2768198 -12455.61648 -25344.84529 -20852.91355 14642.78081 67162.133 88643.95935 29331.92183 -103721.9759 -215885.8219 -175668.2952 63339.97258 366040.1121 463007.7228 161145.8379 -426631.9431
0 0 0 0 192 1035.320732 2197.653181 1181.837405 -4776.384006 -12614.95411 -10733.55999 10443.54925 39434.88701 40860.0242 -11919.57845 -90391.52336 -109544.5177 -4764.383231 169447.4417 238928.2291 63214.86651 -273067.0285 -452649.3406 -197417.4769
0 0 0 0 192 884.6792681 1293.804398 -933.634351 -6033.445815 -7367.269353 3768.214436 22179.4179 22745.61092 -13409.71654 -59759.11695 -51183.66478 39553.74618 132447.0485 93359.81585 -98263.14966 -255824.9417 -144286.083 212696.4921 445204.5408
0 0 0 0 192 735.8924507 540.3053094 -2124.343367 -5057.868211 -969.9812354 11405.43117 15940.52452 -5622.481002 -37623.47011 -29881.32575 35855.80045 87872.36995 28692.89466 -112944.5884 -154561.6584 24454.28757 254280.9736 202369.5839 -178659.2813
0 0 0 0 192 592.6239049 -54.49033533 -2567.50857 -2988.296704 3941.876267 11526.38431 2233.953421 -22136.82236 -24139.16168 20459.96219 60268.45865 14532.0698 -88098.13253 -93277.95948 64910.42847 192595.8473 49803.13745 -244897.5404 -254777.9707
0 0 0 0 192 458.4013779 -495.3380524 -2477.176593 -714.5839156 6392.684212 6873.986266 -9352.41825 -21134.06087 3844.754647 42020.47316 20505.10932 -58980.138 -70793.54333 51327.19794 141815.5672 6229.868714 -208390.0472 -130756.2533 224730.6889
0 0 0 0 192 336.5298736 -798.086388 -2066.223865 1211.650343 6609.005295 832.2069828 -14542.95731 -9394.727768 23888.17206 29260.48864 -28610.53289 -63533.47519 18266.13986 110340.0964 20413.44929 -159630.1792 -99650.79156 191428.6434 224840.2275
0 0 0 0 192 230.010273 -986.6727322 -1517.645248 2561.671612 5397.538212 -4248.815769 -13851.25499 3944.496017 28647.88724 2246.348802 -50351.46819 -19936.09886 77156.40402 55808.14847 -103790.0999 -116346.81 120768.0917 206007.3722 -114289.9057
0 0 0 0 192 141.4654422 -1089.461027 -968.7547191 3346.025684 3627.717551 -7474.097775 -9965.085871 13492.9252 22500.21035 -20544.91549 -44251.15549 26525.3978 78409.54708 -27774.42335 -127872.1365 18879.12551 194650.9903 7368.517829 -279197.6607
0 0 0 0 192 73.07564879 -1135.312342 -508.5655728 3706.949976 1946.428626 -9020.436512 -5497.089494 18315.25514 12841.39554 -32819.98602 -26305.72752 53592.96081 48907.83539 -81344.97478 -84369.59991 116249.2281 137093.641 -157744.2065 -212101.4947
0 0 0 0 192 26.52487642 -1149.801347 -185.5323873 3822.418606 714.8963246 -9523.125445 -2036.182875 19914.17359 4805.440196 -36987.23947 -9962.723932 63039.66842 18780.08487 -100646.6018 -32907.74918 152629.087 54418.22672 -222018.2854 -85848.08848
0 0 0 0 192 2.959359616 -1151.972632 -20.71532046 3839.781056 79.90093794 -9599.042127 -227.8620289 20156.93483 538.572741 -37623.95402 -1118.549352 64493.6094 2112.7619 -103642.0699 -3710.543262 158327.7533 6151.495274 -232190.8611 -9731.39151
1 0 -1 0 1 0 -1 0 1 0 -1 0 1 0 -1 0 1 0 -1 0 1 0 -1 0
0 0 4 0 -16 0 36 0 -64 0 100 0 -144 0 196 0 -256 0 324 0 -400 0 484 0
0 0 0 24 192 840 2688 7056 16128 33264 63360 113256 192192 312312 489216 742560 1096704 1581408 2232576 3093048 4213440 5653032 7480704 9775920
0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 441 484 529
1 0.998458667 0.993839419 0.986156496 0.975433581 0.96170373 0.945009268 0.925401657 0.902941342 0.87769756 0.84974813 0.819179209 0.786085032 0.750567618 0.712736454 0.672708162 0.630606134 0.586560159 0.540706014 0.493185053 0.444143767 0.393733335 0.342109153 0.289430364
1 0.98618496 0.945121551 0.877944359 0.786509494 0.673343309 0.541572595 0.394838187 0.237194368 0.07299685 -0.093217576 -0.256856394 -0.413398249 -0.558517877 -0.688205612 -0.798878172 -0.887477663 -0.951556077 -0.98934292 -0.999794139 -0.982620967 -0.938297899 -0.868049586 -0.773816993
1 0.961939766 0.850656228 0.67462034 0.447232036 0.18580022 -0.089774795 -0.35851611 -0.599967012 -0.795748145 -0.930956556 -0.99530012 -0.983880972 -0.897568346 -0.742932397 -0.531744087 -0.280079168 -0.007094493 0.266430219 0.519674138 0.733360219 0.891222577 0.981244656 0.996573933
1 0.926320082 0.716137789 0.400425549 0.025706666 -0.352800347 -0.679318759 -0.90573287 -0.998678335 -0.944458724 -0.751063831 -0.446992295 -0.077052048 0.304242576 0.640704064 0.882751506 0.994716832 0.960100849 0.784004562 0.492377492 0.128193756 -0.254880591 -0.600395776 -0.857436738
1 0.880202983 0.549514582 0.087165765 -0.396067449 -0.784405265 -0.984804259 -0.949250027 -0.686261152 -0.258848199 0.230583238 0.664768308 0.939678856 0.989447956 0.802151229 0.422663852 -0.058091262 -0.524928056 -0.86599522 -0.999575095 -0.89366274 -0.573634124 -0.116166194 0.369134463
1 0.824724024 0.360339432 -0.230362851 -0.740310987 -0.990741662 -0.893865914 -0.483643725 0.096120716 0.642189852 0.963138082 0.946456378 0.597992543 0.039901255 -0.532177495 -0.917700386 -0.981521616 -0.701268527 -0.175184388 0.412310981 0.85526993 0.998412337 0.79155935 0.307223688
1 0.761249282 0.15900094 -0.51917058 -0.949437402 -0.926346503 -0.460923818 0.224590651 0.802862762 0.997766752 0.716235686 0.092701052 -0.575098468 -0.968287643 -0.899118079 -0.400618342 0.289177228 0.840890257 0.991076981 0.668023024 0.025987114 -0.62845768 -0.98281303 -0.867873748
1 0.691341716 -0.044093263 -0.75230874 -0.996111568 -0.624998222 0.131936882 0.807425162 0.984476513 0.553794202 -0.218754445 -0.856262349 -0.965185319 -0.4782834 0.303870785 0.8984405 0.93838801 0.399053054 -0.386623964 -0.933631603 -0.904292986 -0.316719326 0.46637042 0.96156198
1 0.616722682 -0.239306267 -0.911893888 -0.885465021 -0.180278837 0.663100925 0.998177599 0.568096607 -0.297461473 -0.934999082 -0.855808809 -0.120594327 0.707062296 0.992717038 0.517399932 -0.354532491 -0.954696389 -0.823033344 -0.060470273 0.748446566 0.98363822 0.464817436 -0.410311308
1 0.539229548 -0.418462989 -0.990524765 -0.649777453 0.289766361 0.96227862 0.74801177 -0.155578523 -0.915796843 -0.832070912 0.0184424 0.851960286 0.90036192 0.119043216 -0.771978681 -0.951590646 -0.254272907 0.677367717 0.984786282 0.384684007 -0.569920316 -0.999319756 -0.507805164
1 0.460770452 -0.575381181 -0.991007746 -0.337872993 0.679643962 0.964192705 0.208899055 -0.771683681 -0.920037132 -0.07616817 0.849845048 0.859335144 -0.057932562 -0.91272237 -0.783178435 0.190991406 0.959184828 0.692936648 -0.320615363 -0.98839682 -0.590232736 0.444473211 0.99983298
1 0.383277318 -0.706196995 -0.924615899 -0.002571609 0.92264462 0.70982912 -0.378521817 -0.999986774 -0.38802268 0.702546189 0.926562719 0.007714758 -0.920648935 -0.713442468 0.373756304 0.999947095 0.392757778 -0.698876799 -0.928485029 -0.012857704 0.918628896 0.717036943 -0.368980903
1 0.308658284 -0.809460128 -0.808351431 0.310451397 0.999998222 0.306864073 -0.810565945 -0.807239861 0.312243405 0.999992888 0.305068772 -0.811668881 -0.80612542 0.314034304 0.999983998 0.303272386 -0.81276893 -0.805008112 0.315824085 0.999971552 0.301474921 -0.813866089 -0.803887941
1 0.238750718 -0.88599619 -0.66181517 0.569978496 0.93398072 -0.124001362 -0.993191548 -0.350249028 0.825947135 0.74463997 -0.47038048 -0.969247324 0.007563492 0.972858903 0.456978031 -0.754651237 -0.81732508 0.364377338 0.991315782 0.10897737 -0.939278931 -0.557484408 0.673079326
1 0.175275976 -0.938556665 -0.504288846 0.761777225 0.771331339 -0.491385519 -0.943587492 0.160609082 0.999889319 0.18990407 -0.933318077 -0.517080544 0.752054483 0.78071471 -0.478373418 -0.948409446 0.145906635 0.999557301 0.204490127 -0.927872888 -0.529757779 0.742165265 0.789925261
1 0.119797017 -0.971297349 -0.352514068 0.886837082 0.564994942 -0.751467664 -0.745042111 0.572960019 0.882319914 -0.361561431 -0.968947876 0.1294073 0.999953093 0.110175495 -0.973555702 -0.343433634 0.891271052 0.556976861 -0.757822719 -0.738546663 0.580871344 0.877719972 -0.370574875
1 0.073679918 -0.989142539 -0.2194398 0.956805927 0.360434564 -0.903692348 -0.49360252 0.830955162 0.616051936 -0.74017385 -0.725123833 0.633319721 0.818449723 -0.512713105 -0.894003042 0.380972963 0.950143155 -0.240960024 -0.985650985 0.095714657 0.999755481 0.051609146 -0.992150366
1 0.038060234 -0.997102837 -0.113960168 0.988428136 0.18919978 -0.97402616 -0.263343106 0.95398036 0.335960537 -0.928406887 -0.406631304 0.897453922 0.474945916 -0.861300817 -0.540508536 0.820157054 0.602939275 -0.774261035 -0.661876387 0.723878695 0.716978371 -0.669301966 -0.76792595
1 0.01381504 -0.999618289 -0.041434573 0.998473449 0.069022474 -0.996566352 -0.096557681 0.993898456 0.124019175 -0.990471796 -0.151385989 0.986288989 0.178637233 -0.981353228 -0.2057521 0.975668281 0.232709893 -0.969238489 -0.259490029 0.962068758 0.286072066 -0.954164565 -0.312435708
1 0.001541333 -0.999995249 -0.004623985 0.999980994 0.007706592 -0.999957238 -0.010789127 0.999923978 0.013871559 -0.999881217 -0.016953859 0.999828954 0.020035998 -0.999767189 -0.023117946 0.999695925 0.026199675 -0.99961516 -0.029281155 0.999524896 0.032362357 -0.999425133 -0.035443251
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
The program, its compilation are
ijb#ijb-Latitude-5410:~/work/stack$ cat eig.f90
Program eig
Use, Intrinsic :: iso_fortran_env, Only : wp => real64
Implicit None
Integer, Parameter :: n = 24
Real( wp ), Dimension( 1:n, 1:n ) :: H
Real( wp ), Dimension( 1:n, 1:n ) :: S
Integer :: unit
Open( newunit = unit, file = 'eig.dat' )
Read( unit, * ) H
Read( unit, * ) S
Close( unit )
Write( *, * ) 'Lapack: '
Call lapack( H, S )
Write( *, * ) 'Eispack: '
Call eispack( H, S )
Contains
Subroutine lapack( H, S )
Use, Intrinsic :: iso_fortran_env, Only : wp => real64
Implicit None
Real( wp ), Dimension( :, : ), Intent( In ) :: H
Real( wp ), Dimension( :, : ), Intent( In ) :: S
Complex( wp ) :: lambda
Real( wp ), Dimension( :, : ), Allocatable :: A
Real( wp ), Dimension( :, : ), Allocatable :: B
Real( wp ), Dimension( :, : ), Allocatable :: QR
Real( wp ), Dimension( 1:1, 1:1 ) :: Ql_dummy
Real( wp ), Dimension( : ), Allocatable :: alphar
Real( wp ), Dimension( : ), Allocatable :: alphai
Real( wp ), Dimension( : ), Allocatable :: beta
Real( wp ), Dimension( : ), Allocatable :: work
Real( wp ), Dimension( 1:1 ) :: twork
Integer :: n
Integer :: lwork
Integer :: info
Integer :: i
n = Size( H, Dim = 1 )
! H and S would be overwritten
A = H
B = S
! Space for evals and evecs
Allocate( alphar( 1:n ) )
Allocate( alphai( 1:n ) )
Allocate( beta( 1:n ) )
Allocate( QR( 1:n, 1:n ) )
! Calculate and allocate the worksize
Call dggev( 'N', 'V', n, A, Size( A, Dim = 1 ), B, Size( B, Dim = 1 ), &
alphar, alphai, beta, &
QL_dummy, Size( QL_dummy, Dim = 1 ), &
QR, Size( QR, Dim = 1 ), &
twork, -1, info )
! Allocate workspace
lwork = Nint( twork( 1 ) )
Allocate( work( 1:lwork ) )
! Solve eval problem
Call dggev( 'N', 'V', n, A, Size( A, Dim = 1 ), B, Size( B, Dim = 1 ), &
alphar, alphai, beta, &
QL_dummy, Size( QL_dummy, Dim = 1 ), &
QR, Size( QR, Dim = 1 ), &
work, Size( work ), info )
Write( *, * ) 'info = ', info
! Report results
Write( *, * ) 'Evals - last column reports lambda where appropriate'
Do i = 1, n
If( Abs( beta( i ) ) > 1.0e-16_wp ) Then
lambda = Cmplx( alphar( i ), alphai( i ), wp ) / beta( i )
Write( *, '( i2, 1x, 3( e18.6, 1x ), 5x, e18.6, " + ", e18.6, "i" )' ) &
i, alphar( i ), alphai( i ), beta( i ), lambda
Else
Write( *, '( i2, 1x, 3( e18.6, 1x ) )' ) i, alphar( i ), alphai( i ), beta( i )
End If
End Do
Write( *, * )
End Subroutine lapack
Subroutine eispack( H, S )
Use, Intrinsic :: iso_fortran_env, Only : wp => real64
Implicit None
Real( wp ), Dimension( :, : ), Intent( In ) :: H
Real( wp ), Dimension( :, : ), Intent( In ) :: S
Complex( wp ) :: lambda
Real( wp ), Dimension( :, : ), Allocatable :: A
Real( wp ), Dimension( :, : ), Allocatable :: B
Real( wp ), Dimension( :, : ), Allocatable :: QR
Real( wp ), Dimension( : ), Allocatable :: alphar
Real( wp ), Dimension( : ), Allocatable :: alphai
Real( wp ), Dimension( : ), Allocatable :: beta
Integer :: n
Integer :: info
Integer :: i
n = Size( H, Dim = 1 )
! H and S would be overwritten
A = H
B = S
! Space for evals and evecs
Allocate( alphar( 1:n ) )
Allocate( alphai( 1:n ) )
Allocate( beta( 1:n ) )
Allocate( QR( 1:n, 1:n ) )
! Calc evals and evecs by eispack
Call rgg( Size( A, Dim = 1 ), n, A, B, alphar, alphai, beta, 1, QR, info )
Write( *, * ) 'info = ', info
! Report results
Write( *, * ) 'Evals - last column reports lambda where appropriate'
Do i = 1, n
If( Abs( beta( i ) ) > 1.0e-16_wp ) Then
lambda = Cmplx( alphar( i ), alphai( i ), wp ) / beta( i )
Write( *, '( i2, 1x, 3( e18.6, 1x ), 5x, e18.6, " + ", e18.6, "i" )' ) &
i, alphar( i ), alphai( i ), beta( i ), lambda
Else
Write( *, '( i2, 1x, 3( e18.6, 1x ) )' ) i, alphar( i ), alphai( i ), beta( i )
End If
End Do
Write( *, * )
End Subroutine eispack
End Program eig
ijb#ijb-Latitude-5410:~/work/stack$ gfortran -Wall -Wextra -fcheck=all -O -g eig.f90 eispack.a -llapack
And the results are (note the complex conjugate pair)
ijb#ijb-Latitude-5410:~/work/stack$ ./a.out
Lapack:
info = 0
Evals - last column reports lambda where appropriate
1 -0.608116E+01 0.000000E+00 0.187802E-11 -0.323806E+13 + 0.000000E+00i
2 0.370614E+03 0.000000E+00 0.262593E-05 0.141137E+09 + 0.000000E+00i
3 -0.166285E+05 0.000000E+00 0.150764E-02 -0.110295E+08 + 0.000000E+00i
4 0.356929E+06 0.000000E+00 0.387651E-01 0.920748E+07 + 0.000000E+00i
5 0.376034E+06 0.000000E+00 0.654174E-01 0.574823E+07 + 0.000000E+00i
6 0.881731E+05 0.129854E+06 0.121455E+00 0.725976E+06 + 0.106916E+07i
7 0.154526E+06 -0.227573E+06 0.212853E+00 0.725976E+06 + -0.106916E+07i
8 0.400747E+05 0.000000E+00 0.327119E-01 0.122508E+07 + 0.000000E+00i
9 0.353978E+05 0.000000E+00 0.623272E-01 0.567936E+06 + 0.000000E+00i
10 0.177191E+05 0.000000E+00 0.342541E-01 0.517282E+06 + 0.000000E+00i
11 0.703041E+04 0.000000E+00 0.224366E-01 0.313346E+06 + 0.000000E+00i
12 0.250241E+04 0.000000E+00 0.142464E-01 0.175653E+06 + 0.000000E+00i
13 0.816411E+02 0.000000E+00 0.815119E-03 0.100159E+06 + 0.000000E+00i
14 0.127044E+04 0.000000E+00 0.141856E-01 0.895583E+05 + 0.000000E+00i
15 0.396171E+04 0.000000E+00 0.991827E-01 0.399436E+05 + 0.000000E+00i
16 0.461457E+05 0.000000E+00 0.315689E+01 0.146175E+05 + 0.000000E+00i
17 0.109954E+05 0.000000E+00 0.288969E+01 0.380504E+04 + 0.000000E+00i
18 0.188721E+02 0.000000E+00 0.309986E+01 0.608807E+01 + 0.000000E+00i
19 0.123631E-03 0.000000E+00 0.139487E-06 0.886324E+03 + 0.000000E+00i
20 0.158220E+04 0.000000E+00 0.320845E+01 0.493134E+03 + 0.000000E+00i
21 0.899768E+03 0.000000E+00 0.000000E+00
22 0.376730E-01 0.000000E+00 0.000000E+00
23 0.366102E-01 0.000000E+00 0.000000E+00
24 0.960100E-01 0.000000E+00 0.000000E+00
Eispack:
info = 0
Evals - last column reports lambda where appropriate
1 -0.346410E+01 0.000000E+00 0.000000E+00
2 -0.546079E+03 0.000000E+00 0.000000E+00
3 0.144715E+08 0.000000E+00 0.000000E+00
4 0.528475E+03 0.000000E+00 0.000000E+00
5 -0.233202E+08 0.000000E+00 0.719984E-05 -0.323898E+13 + 0.000000E+00i
6 0.278909E+06 0.000000E+00 0.197617E-02 0.141136E+09 + 0.000000E+00i
7 -0.250582E+05 0.000000E+00 0.227192E-02 -0.110295E+08 + 0.000000E+00i
8 0.491054E+04 0.000000E+00 0.533321E-03 0.920746E+07 + 0.000000E+00i
9 0.777637E+05 0.000000E+00 0.135281E-01 0.574829E+07 + 0.000000E+00i
10 0.199724E+04 0.294137E+04 0.275110E-02 0.725977E+06 + 0.106916E+07i
11 0.335291E+04 -0.493790E+04 0.461849E-02 0.725977E+06 + -0.106916E+07i
12 0.358837E+04 0.000000E+00 0.292908E-02 0.122508E+07 + 0.000000E+00i
13 0.981532E+03 0.000000E+00 0.172823E-02 0.567941E+06 + 0.000000E+00i
14 0.138597E+03 0.000000E+00 0.267934E-03 0.517279E+06 + 0.000000E+00i
15 0.286093E+03 0.000000E+00 0.913029E-03 0.313345E+06 + 0.000000E+00i
16 0.518162E+02 0.000000E+00 0.294991E-03 0.175653E+06 + 0.000000E+00i
17 0.957124E+00 0.000000E+00 0.955594E-05 0.100160E+06 + 0.000000E+00i
18 0.104269E+01 0.000000E+00 0.116426E-04 0.895582E+05 + 0.000000E+00i
19 0.874474E+02 0.000000E+00 0.218927E-02 0.399436E+05 + 0.000000E+00i
20 0.185986E+04 0.000000E+00 0.127235E+00 0.146175E+05 + 0.000000E+00i
21 0.968367E+04 0.000000E+00 0.254496E+01 0.380504E+04 + 0.000000E+00i
22 0.171890E+02 0.000000E+00 0.282339E+01 0.608807E+01 + 0.000000E+00i
23 0.100843E-04 0.000000E+00 0.113655E-07 0.887279E+03 + 0.000000E+00i
24 0.234040E+03 0.000000E+00 0.474598E+00 0.493134E+03 + 0.000000E+00i
Given all this I suggest one of the following is occurring:
The original paper is wrong
The data you quote is wrong
I am misunderstanding something you are saying
I don't know if it is possible a priori to say whether a given problem has a complex eval without solving the problem itself.
Related
How to decompose a 9x9 matrix to 3x3 matrices?
I'm trying to decompose a 9x9 matrix into 9 3x3 matrices. I've already used the command reshape, but the matrices it returns are the ones as transforming the columns of the 9x9 matrix to 3x3 matrices, but that isn't what I need. This is 9x9 the matrix M 0 0 0 0 0 0 0 7 6 0 2 0 9 0 0 0 0 0 0 3 8 0 5 4 1 0 0 9 0 0 5 0 0 0 3 0 0 0 0 0 1 8 0 6 7 4 0 0 0 0 0 2 0 0 0 1 0 8 0 0 0 5 0 0 0 0 0 0 7 0 0 0 6 0 2 0 3 0 8 9 4 And I need it in form 0 0 0 0 0 0 0 7 6 0 2 0 9 0 0 0 0 0 0 3 8 0 5 4 1 0 0 etc... This code generates the matrices exactly as I need them, but only saving the last one: for i=[1 4 7] for j=[1 4 7] v(:,:)=M([i:i+2],[j:j+2]) end end
You make a new v of size 3x3 every loop iteration, i.e. you overwrite it. What you can easily do is to add a third dimension, and store all your separate matrices along there: n = 3; % Amount of rows of the sub-matrices m = 3; % Amount of columns of the sub-matrices v = zeros(size(M,1)./n,size(M,2)./m,(size(M,1)*size(M,2))./(n*m)); % Initialise v kk = 1; % Initialise page counter for ii = 1:n:size(M,1) for jj = 1:m:size(M,2) v(:,:,kk) = M(ii:ii+2, jj:jj+2); % Save on page kk kk = kk+1; % Increase page counter end end A few other notes: In order to generalise the code I used size(M) everywhere, such that you can easily extend M to larger sizes. Don't use i or j as indices/variables names, as they denote the imaginary unit. Using those can easily lead to hard to debug errors. ii:ii+2 already is an array, adding square brackets is superfluous Initialise v, i.e. tell MATLAB how large it will be before going into the loop. Otherwise, MATLAB will have to increase its size every iteration,which is very slow. This is called preallocation. n and m obviously need to be integers, but not only that. You need an integer amount of sub-matrices to fill each dimension, i.e. size(M,1)/n and size(M,2)/m both need to be integers. If this is not the case, this code will error out on the initialisation of v.
You can also use mat2cell function. You need to declare the input_size and the chunk_size. Then you need to declare the number of chunks in each dimension (sc variable).
For example:
M = [0 0 0 0 0 0 0 7 6
0 2 0 9 0 0 0 0 0
0 3 8 0 5 4 1 0 0
9 0 0 5 0 0 0 3 0
0 0 0 0 1 8 0 6 7
4 0 0 0 0 0 2 0 0
0 1 0 8 0 0 0 5 0
0 0 0 0 0 7 0 0 0
6 0 2 0 3 0 8 9 4];
% Z = zeros(size(M, 1), size(M, 2));
input_size = size(M);
chunk_size = [3 3];
sc = input_size ./ chunk_size;
C = mat2cell(M, chunk_size(1)*ones(sc(1), 1),...
chunk_size(2)*ones(sc(2), 1));
celldisp(C')
Output:
ans{1,1} =
0 0 0
0 2 0
0 3 8
ans{2,1} =
0 0 0
9 0 0
0 5 4
ans{3,1} =
0 7 6
0 0 0
1 0 0
ans{1,2} =
9 0 0
0 0 0
4 0 0
ans{2,2} =
5 0 0
0 1 8
0 0 0
ans{3,2} =
0 3 0
0 6 7
2 0 0
ans{1,3} =
0 1 0
0 0 0
6 0 2
ans{2,3} =
8 0 0
0 0 7
0 3 0
ans{3,3} =
0 5 0
0 0 0
8 9 4
You can create a tiled index matrix, then use this to get the 3*3 arrays idx = repelem( reshape(1:9,3,3), 3, 3 ); out = arrayfun( #(x) reshape(A(idx==x),3,3), 1:9, 'uni', 0 ); Explanation: idx is a tiled array as shown: idx = 1 1 1 4 4 4 7 7 7 1 1 1 4 4 4 7 7 7 1 1 1 4 4 4 7 7 7 2 2 2 5 5 5 8 8 8 2 2 2 5 5 5 8 8 8 2 2 2 5 5 5 8 8 8 3 3 3 6 6 6 9 9 9 3 3 3 6 6 6 9 9 9 3 3 3 6 6 6 9 9 9 Then the arrayfun line loops through the values 1 .. 9, and extracts the matrix where idx matches the indexing value of the loop. We have to use a final reshape here because the logical index turns the array 1D. The output is a 9 by 1 cell array, where each cell is a submatrix.
Using only linear indexing and implicit expansion: % Random example matrix M = randi(9,[9,12]) % Block size n = 3; % Indexing s = size(M,1) ind = [1:n].'+[0:n-1]*s+reshape(floor((0:n:(numel(M)/n)-1)/s)*n*s+mod(1:n:numel(M)/n,s)-1,1,1,[]) % Split each block into a third dimension. B = M(ind) Where: [1:n].'+[0:n-1]*s = 1 10 19 2 11 20 3 12 21 % The linear index of the first block And: reshape(floor((0:n:(numel(M)/n)-1)/s)*n*s+mod(1:n:numel(M)/n,s)-1,1,1,[]) = ans(:,:,1) = 0 ans(:,:,2) = 3 ans(:,:,3) = 6 ans(:,:,4) = 27 ans(:,:,5) = 30 ans(:,:,6) = 33 ans(:,:,7) = 54 ... % The shifting number for each block Noticed that the length of each dimension of the matrix M has to be divisible by n
#java
// method decompose a 9*9 matrix to 3*3 matrices
public static int[][][] decompose(int[][] matrix) {
int[][][] result = new int[9][3][3];
for (int i = 0; i < 9; ) {
for (int j = 0; j < 9; j += 3) {
for (int k = 0; k < 9; k += 3) {
for (int l = j; l < j + 3; l++) {
for (int m = k; m < k + 3; m++) {
result[i][l % 3][m % 3] = (matrix[l][m]);
}
}
i++;
}
}
}
return result;
}
For example:
int[][] matrix = {
{1, 2, 3, 10, 11, 12, 19, 20, 21},
{4, 5, 6, 13, 14, 15, 22, 23, 24},
{7, 8, 9, 16, 17, 18, 25, 26, 27},
{28, 29, 30, 37, 38, 39, 46, 47, 48},
{31, 32, 33, 40, 41, 42, 49, 50, 51},
{34, 35, 36, 43, 44, 45, 52, 53, 54},
{55, 56, 57, 64, 65, 66, 73, 74, 75},
{58, 59, 60, 67, 68, 69, 76, 77, 78},
{61, 62, 63, 70, 71, 72, 79, 80, 81}
};
int[][][] test = decompose(matrix);
for (int[][] grid : test) {
for (int[] gridLine : grid) {
for (int i = 0; i < 3; i++) {
System.out.print(gridLine[i] + " ");
}
System.out.println();
}
System.out.println("========");
}
Output:
1 2 3
4 5 6
7 8 9
========
10 11 12
13 14 15
16 17 18
========
19 20 21
22 23 24
25 26 27
========
28 29 30
31 32 33
34 35 36
========
37 38 39
40 41 42
43 44 45
========
46 47 48
49 50 51
52 53 54
========
55 56 57
58 59 60
61 62 63
========
64 65 66
67 68 69
70 71 72
========
73 74 75
76 77 78
79 80 81
========
Taking a matrix and retrieving both the diagonals keeping the dimensions of the original matrix in MATLAB
Given the matrix A = magic(5) you get: A = 17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9 I want to use commands such as rot90, diag, triu, tril and matrices sum to get the matrix: A = 17 0 0 0 15 0 5 0 14 0 0 0 13 0 0 0 12 0 21 0 11 0 0 0 9 Please, if you can't think of a way to solve this without the commands I wrote, it's OK to do it your own way.
You can use eye function for indexing >> A(~eye(size(A)) & ~flipud(eye(size(A))))=0 A = 17 0 0 0 15 0 5 0 14 0 0 0 13 0 0 0 12 0 21 0 11 0 0 0 9
You can simply use linear indexing to access the diagonals: n = size(A,1); B = zeros(n); B( 1:(n+1):end ) = A( 1:(n+1):end ); %// main diagonal B( n:(n-1):(end-n+1) ) = A( n:(n-1):(end-n+1) ) And you get B = 17 0 0 0 15 0 5 0 14 0 0 0 13 0 0 0 12 0 21 0 11 0 0 0 9
Another approach is: mDiag = diag(diag(A)); aDiag = rot90(diag(diag(rot90(A))))'; overlap = A.*((diag(diag(A)) ~= 0) & (rot90(diag(diag(rot90(A)))) ~= 0)); solution = mDiag + aDiag - overlap And than: solution = 17 0 0 0 15 0 5 0 14 0 0 0 13 0 0 0 12 0 21 0 11 0 0 0 9
Using bsxfun out = A.*bsxfun(#(x,y) x == y | x+y == size(A,1)+1,(1:size(A,1)).',1:size(A,1)) %//'
All combinations for matrix in matlab
I'm trying find all combinations of an n-by-n matrix without repetitions. For example, I have a matrix like this: A = [321 319 322; ... 320 180 130; ... 299 100 310]; I want the following result: (321 180 310) (321 130 100) (319 320 310) (319 139 299) (322 320 100) (322 180 299) I have tried using ndgrid, but it takes the row or the column twice.
Here's a simpler (native) solution with perms and meshgrid: N = size(A, 1); X = perms(1:N); % # Permuations of column indices Y = meshgrid(1:N, 1:factorial(N)); % # Row indices idx = (X - 1) * N + Y; % # Convert to linear indexing C = A(idx) % # Extract combinations The result is a matrix, each row containing a different combination of elements: C = 321 180 310 319 320 310 321 130 100 319 130 299 322 320 100 322 180 299 This solution can also be shortened to: C = A((perms(1:N) - 1) * N + meshgrid(1:N, 1:factorial(N)))
ALLCOMB is the key to your question E.g. I am not front of a MATLAB machine so, I took a sample from web. x = allcomb([1 3 5],[-3 8],[],[0 1]) ; ans 1 -3 0 1 -3 1 1 8 0 ... 5 -3 1 5 8 0 5 8 1
You can use perms to permute the columns as follows: % A is given m x n matrix row = 1:size( A, 1 ); col = perms( 1:size( A, 2 ) ); B = zeros( size( col, 1 ), length( row )); % Allocate memory for storage % Simple for-loop (this should be vectorized) % for c = 1:size( B, 2 ) % for r = 1:size( B, 1 ) % B( r, c ) = A( row( c ), col( r, c )); % end % end % Simple for-loop (further vectorization possible) r = 1:size( B, 1 ); for c = 1:size( B, 2 ) B( r, c ) = A( row( c ), col( r, c )); end
Matlab, linear programming
I want to solve this linear programming (simplex) problem using MATLAB 7, but it returns Exiting: the problem is unbounded. This function f = 2(15 s0 + 8s1 + 2576s2 + 744s3 + 427s4 + 8s5) Should be minimized in such a way that two constraints for each observation are satisfied 0.1s0 + 0.1s1 + 14.5s2 + 4s3 + 2.4s4 – a0 − a1 − 145a2 − 40a3 − 24a4 ≥ −2.2 0.1s0 + 0.1s1 + 14.5s2 + 4s3 + 2.4s4 + a0 + a1 + 145a2 + 40a3 + 24a4 ≥ 2.2 S5 and a5 are 0. I used f = [15 8 2576 744 427 8 15 8 2576 744 427 8]; b = [-2.2; 2.2]; a = [0.1 0.1 14.5 0.4 2.4 0 -1 -1 -145 -40 -24 0 ; 0.1 0.1 14.5 4 2.4 0 1 1 145 40 24 0]; [x, fval, exitflag, output, lambda] = linprog(f, a, b) What is the right way to solve this problem?
You didn't constrain your s5 and a5 to actually be zero, since you set the corresponding coefficients in the a matrix to zero. Thus, they can take on any value, and the LP is unbounded. To fix, add an equality constraint: beq = [0; 0]; aeq = [0 0 0 0 0 1 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 1]; [x,fval,exitflag,output,lambda] = linprog(f,a,b,aeq,beq) Or, just drop s5 and a5 from the LP since they don't contribute at all.
MATLAB vectorize
I was wondering if anyone could help me vectorize this piece of code. fr_bw is a matrix. for i=1:height for j=1:width [min_w, min_w_index] = min(w(i,j,:)); mean(i,j,min_w_index) = double(fr_bw(i,j)); sd(i,j,min_w_index) = sd_init; end end
I can't help you with this sif (match == 0) stuff -- if it's supposed to be if (match == 0) you're not changing match so it could be brought outside the loop. Otherwise, how about this: [min_w, min_w_index] = min(w, [], 3); r = repmat((1:height)',1,width); c = repmat(1:width,height,1); ind = sub2ind(size(w),r(:),c(:),min_w_index(:)); w_mean(ind) = double(fr_bw); w_sd(ind) = repmat(sd_init,height,width); (Please note that mean is a built-in function so I renamed your variables to w_mean and w_sd.) The sub2ind call gives you linear indices that correspond to subscripts. (Direct subscripts won't work; z([a1 a2 a3],[b1 b2 b3],[c1 c2 c3]) refers to 27 elements in the z array with subscripts that are the cartesian product of the specified subscripts, rather than z(a1,b1,c1) and z(a2,b2,c2) and z(a3,b3,c3) that you might expect.) Here's an illustration of this technique: >> height = 6; width = 4; >> w = randi(1000,height,width,2) w(:,:,1) = 426 599 69 719 313 471 320 969 162 696 531 532 179 700 655 326 423 639 408 106 95 34 820 611 w(:,:,2) = 779 441 638 696 424 528 958 68 91 458 241 255 267 876 677 225 154 519 290 668 282 944 672 845 >> [min_w, min_w_index] = min(w, [], 3); >> min_w_index min_w_index = 1 2 1 2 1 1 1 2 2 2 2 2 1 1 1 2 2 2 2 1 1 1 2 1 >> z = zeros(height,width,2); >> r = repmat((1:height)',1,width); >> c = repmat(1:width,height,1); >> ind = sub2ind(size(w),r(:),c(:),min_w_index(:)); >> z(ind) = 1 z(:,:,1) = 1 0 1 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 1 z(:,:,2) = 0 1 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 0
A few comments on your code: Did you mean if rather than sif? The code isn't replicable at the moment, since you haven't provided examples of the variables w, fr_bw and sd_init. This makes it tricky to give an exact answer. It looks like you are assigning things to a variable named mean. This will shadow the mean function, and probably cause you grief. I'm just guessing, but I don't think double does what you think it does. You don't need to convert individual elements of a numeric matrix to type double; they are already the correct type. (On the other hand, if fr_bw is a different type, say integers, then you should create a new variable dbl_fr_bw = double(fr_bw); before the loops. You might need to adjust the dimension over which you calculate the minimums, but the first line of the loop can be replaced with [min_w, min_w_index] = min(w, [], 3) The second line with mean_values(:, :, min_w_index) = double(fr_bw) Not sure about the third line, since I don't know what sd_init is/does.