I’m getting mixed results for Eigen Library eigen solver with a complex float on a 24 x 24 array. Matlab is able to produce eigen values and vectors for the array that are correct. The Eigen library can only correctly identify the last 8 values and vectors. The correctness seems to correspond to the expected eigen value. When the expected eigen value is going to be less than one, then the eigen value returned by eigen library is incorrect. To verify correctness, I’m using the formula for matrix A where:
A*eigen_vector == eigen_value * eigen_vector
C++ Code:
Eigen::ComplexEigenSolver<Eigen::MatrixXcf> ces(A);
eigen_vectors = ces.eigenvectors();
eigen_values = ces.eigenvalues().array().abs();
Eigen::ComputationInfo info = ces.info();
auto vals = ces.eigenvalues();
auto left = (A * eigen_vectors.col(i)).eval();
auto right = (eigen_values(i) * eigen_vectors.col(i)).eval();
The value of info is success(0).
Below are the results for the 1st, 15th, 16th and 24th columns. Again columns 1 through 16, where expected eigen value is less than 1) are wrong and columns 17 through 24 (where expected eigen values are greater than 1) are correct.
The eigen value for the first column is 3.18247221e-05 and the remaining results are:
Eigen Vector: 0.116856-0.101679i -0.053167-0.045308i 0.189751-0.007320i -0.094731-0.100459i 0.031872+0.036732i -0.077513-0.235932i -0.146664-0.057607i 0.240418-0.386918i -0.242117+0.152045i 0.029276-0.015581i -0.163710-0.010984i 0.034757+0.166928i 0.393912+0.028451i -0.244973-0.073360i -0.037887+0.074877i -0.009600-0.200368i -0.115676-0.144717i 0.058291+0.083560i -0.134150+0.110078i
-0.267531-0.144257i 0.111152-0.071048i -0.016644-0.122138i 0.053874+0.033652i 0.084331-0.034665i
Left Result
0.000221-0.000029i 0.000160+0.000070i 0.000084+0.000041i -0.000023+0.000058i -0.000231+0.000050i -0.000078-0.000063i -0.000166-0.000147i -0.000146+0.000069i -0.000067+0.000046i -0.000162-0.000103i -0.000244+0.000027i-0.000177+0.000107i -0.000151+0.000122i -0.000169+0.000244i -0.000070+0.000481i 0.000010+0.000243i 0.000128+0.000313i 0.000086+0.000238i 0.000103+0.000207i 0.000283+0.000114i 0.000381-0.000028i 0.000185-0.000044i 0.000198-0.000041i 0.000137+0.000046i
Right Result
0.000005+0.000001i 0.000000-0.000002i 0.000004+0.000005i 0.000001-0.000004i -0.000000+0.000002i 0.000004-0.000007i -0.000002-0.000005i 0.000014-0.000002i -0.000009-0.000003i 0.000001+0.000000i -0.000003-0.000004i -0.000003+0.000004i 0.000007+0.000010i -0.000003-0.000008i -0.000003+0.000001i 0.000005-0.000004i 0.000001-0.000006i -0.000001+0.000003i -0.000005-0.000001i -0.000002-0.000010i 0.000004+0.000001i 0.000003-0.000003i 0.000000+0.000002i 0.000003+0.000001i
The eigen value for the 16th column is 0.000447395432 and the remaining results are:
Eigen Vector
-0.129462+0.080319i 0.503447-0.030272i -0.250503+0.133473i 0.078905-0.110744i -0.103717-0.274465i 0.002387-0.044809i 0.179838-0.055467i -0.179086+0.207490i 0.254963-0.022882i -0.002364+0.029277i -0.063116+0.014189i -0.029085-0.161171i 0.019494+0.168846i -0.155084-0.110897i 0.196219-0.127527i -0.067199+0.065707i -0.077747+0.136864i 0.177431+0.031691i 0.191456-0.072533i -0.226178-0.096186i -0.081929-0.098818i 0.029809+0.104811i -0.046287-0.080181i 0.033617-0.028921i
Left Result
-0.000309+0.000206i -0.000179+0.000137i 0.000086+0.000257i -0.000004+0.000346i 0.000141+0.000296i -0.000031+0.000223i -0.000126+0.000443i -0.000187+0.000181i -0.000325+0.000122i -0.000282+0.000032i -0.000077-0.000134i 0.000222-0.000206i 0.000171-0.000282i 0.000063-0.000368i 0.000782-0.000860i 0.000485-0.000464i 0.000068-0.000452i 0.000080-0.000582i -0.000048-0.000478i -0.000525-0.000506i -0.000761-0.000059i -0.000264+0.000076i -0.000411+0.000162i -0.000092+0.000224i
Right Result
-0.000062+0.000029i 0.000225+0.000012i -0.000118+0.000047i 0.000041-0.000045i -0.000033-0.000127i 0.000003-0.000020i 0.000083-0.000016i -0.000090+0.000083i 0.000115+0.000002i -0.000003+0.000013i -0.000029+0.000003i -0.000005-0.000073i 0.000000+0.000076i -0.000063-0.000057i 0.000094-0.000047i -0.000033+0.000026i -0.000041+0.000057i 0.000077+0.000023i 0.000089-0.000023i -0.000096-0.000054i -0.000032-0.000048i 0.000008+0.000048i -0.000017-0.000038i 0.000016-0.000011i
The eigen value for the 17th column is 0.000447395432 and the remaining results are:
Eigen Vector
-0.180662+0.000949i 0.175624-0.044660i -0.175235+0.118171i -0.031841+0.032150i 0.103090+0.197154i -0.012064+0.306235i -0.047170-0.072482i 0.026343-0.333177i 0.199390-0.020131i -0.016225+0.141814i -0.035368-0.182202i -0.311619+0.344249i -0.139175-0.048998i -0.147191-0.106527i 0.081654+0.099897i -0.065063+0.054494i -0.072711-0.007868i -0.202887-0.219343i 0.085549-0.128106i -0.032125-0.061846i 0.096094+0.176484i -0.015189-0.226712i 0.056063+0.011951i -0.102028-0.083975i
Left Result
-23.387611+0.122978i 22.735226-5.781250i -22.685196+15.297768i -4.122077+4.161827i 13.345421+25.522297i -1.561729+39.643456i -6.106293-9.383188i 3.410229-43.131386i 25.811787-2.606186i -2.100426+18.358337i -4.578606-23.587095i -40.340446+44.564548i -18.016869-6.343142i -19.054581-13.790518i 10.570524+12.932138i -8.422680+7.054350i -9.412903-1.018623i -26.264793-28.395016i 11.074594-16.583862i -4.158801-8.006237i 12.439541+22.846725i -1.966396-29.348801i 7.257420+1.547224i -13.208191-10.870808i
Right Result
-23.387533+0.122920i 22.735312-5.781419i -22.684887+15.297785i -4.122017+4.161916i 13.345476+25.522495i -1.561712+39.643520i -6.106310-9.383076i 3.410251-43.131256i 25.811895-2.606046i -2.100339+18.358484i -4.578590-23.586895i -40.340446+44.564613i -18.016870-6.343021i -19.054562-13.790393i 10.570497+12.932129i -8.422701+7.054466i -9.412786-1.018598i -26.264687-28.394962i 11.074714-16.583828i -4.158720-8.006290i 12.439752+22.846632i -1.966295-29.348902i 7.257586+1.547073i -13.208040-10.870924i
The eigen value for the 24th column is 19668.7812 and the remaining results are:
Eigen Vector
0.097742+0.000000i 0.080856+0.054807i -0.037769+0.111747i -0.162293+0.061423i -0.282967+0.048335i -0.146718-0.022603i -0.197100-0.133459i -0.163493-0.018498i -0.145976-0.042051i -0.170700-0.097671i -0.129707-0.021901i -0.075060+0.088880i -0.020776+0.123144i -0.050064+0.159086i 0.101916+0.434050i 0.121622+0.270482i 0.208905+0.199689i 0.180265+0.148550i 0.152823+0.123853i 0.227151-0.035251i 0.219237-0.129511i 0.108033-0.057653i 0.100164-0.061891i 0.041450+0.030553i
Left Result
1922.458130-0.000084i 1590.333374+1077.984741i -742.875000+2197.920166i -3192.101807+1208.124023i -5565.626465+950.697693i -2885.760986-444.581451i -3876.727051-2624.972900i -3215.713623-363.830933i -2871.160645-827.096252i -3357.463867-1921.066040i -2551.175537-430.774689i -1476.347778+1748.158936i -408.644470+2422.092773i -984.690796+3129.027100i 2004.571167+8537.231445i 2392.158691+5320.046875i 4108.902832+3927.633789i 3545.595459+2921.787842i 3005.838623+2436.031006i 4467.781738-693.343567i 4312.133301-2547.319824i 2124.886963-1133.958252i 1970.101562-1217.314087i 815.273682+600.949463i
Right Result
1922.457886+0.000028i 1590.333618+1077.984741i -742.876587+2197.919678i -3192.101562+1208.123901i -5565.625488+950.697571i -2885.760010-444.581543i -3876.726318-2624.972656i -3215.712646-363.830994i -2871.160400-827.095825i -3357.463623-1921.066406i -2551.175781-430.774475i -1476.347900+1748.158936i -408.644318+2422.092529i -984.690857+3129.026367i 2004.571289+8537.230469i 2392.158691+5320.046875i 4108.902344+3927.633789i 3545.594971+2921.787842i 3005.838623+2436.031006i 4467.781738-693.343872i 4312.133301-2547.319336i 2124.886963-1133.958130i 1970.101440-1217.313721i 815.273743+600.949463i
Complex Float Matrix A 24 x 24
846.034424+0.000000i 692.429993-79.920929i 299.893677-173.686523i 117.934189-249.438538i -397.368805-307.499542i -200.135315-122.347252i -317.231934-34.053032i -228.393478-392.325012i -127.726959-492.413513i -472.311096-325.571991i -510.262024-382.537262i -350.180908-294.300842i -293.310211-433.513275i -292.838348-622.537598i -248.129822-780.637817i 73.105545-369.854614i 311.308044-476.008270i 269.632812-434.859344i 297.421509-367.780273i 422.382996-267.411438i 847.454346+0.086609i 417.832062+135.523224i 634.468994+79.585495i 562.956360+28.823025i
692.429993+79.920929i 729.823608+0.000000i 433.412445-176.004471i 279.240112-333.730011i -247.280304-623.308289i -178.082199-249.672516i -405.823608-229.667175i -221.934280-552.801086i -159.924622-733.100281i -584.464539-557.959167i -513.812439-557.605591i -287.260925-363.278137i -172.812073-393.315796i -126.653442-645.273193i 234.461395-497.079224i 404.682709-167.057907i 440.305817-173.756729i 344.074280-209.670471i 343.704895-161.842102i 330.104797-66.911331i 650.627319+119.730247i 327.168274+228.784500i 448.958160+166.081177i 576.669861+93.079895i
299.893677+173.686523i 433.412445+176.004471i 663.860657+0.000000i 748.176208-369.171478i 416.053192-800.157959i 156.839417-424.814667i -66.856201-704.243286i 48.570087-671.490234i 19.534491-851.697327i -300.561401-772.196716i -252.325562-557.146790i -1.530064-85.880135i 53.092026-8.853506i 118.788445-253.089584i 613.292664+591.216370i 532.384949+546.371887i 237.907715+553.619873i 90.743271+368.473480i 141.670868+360.003815i -323.364014+181.914490i -148.321045+119.432144i -57.208363+161.784821i 30.676718+75.976265i 377.163757+126.714874i
117.934189+249.438538i 279.240112+333.730011i 748.176208+369.171478i 1352.880615+0.000000i 1126.432495-433.352936i 631.408752-354.911865i 683.227661-869.343262i 632.771423-605.632751i 558.547607-982.655090i 227.245529-983.722412i 97.741722-651.704468i 56.844387+134.000076i -10.137331+232.600876i 50.433235+4.382356i -150.310242+1466.185059i -49.551094+1006.306763i -492.348541+747.632629i -513.225464+481.005188i -351.553131+473.534912i -768.257263-210.671112i -438.366119-438.617279i -208.443069-26.301529i -77.328537-41.271057i 327.690460+278.439056i
-397.368805+307.499542i -247.280304+623.308289i 416.053192+800.157959i 1126.432495+433.352936i 1862.093872+0.000000i 893.866394-257.745972i 1074.619385-936.400269i 1080.759033-393.798340i 1080.445312-524.300049i 1011.214233-959.518372i 751.320740-563.206177i 421.583740+289.984283i 192.584259+499.200745i 437.832153+596.648315i -242.677963+2380.033691i -500.537079+1650.186523i -927.506714+1268.838135i -824.274353+888.214966i -694.109680+779.420105i -1105.235107-118.333450i -1049.157837-437.560486i -587.285156-124.098953i -431.681488-151.020599i -80.391388+461.592102i
-200.135315+122.347252i -178.082199+249.672516i 156.839417+424.814667i 631.408752+354.911865i 893.866394+257.745972i 543.277100+0.000000i 730.056763-328.768524i 586.875366-16.146624i 604.010132-139.252884i 594.211060-330.463440i 402.430389-166.598511i 155.124969+239.680237i 14.554134+282.846649i 67.461212+333.344971i -544.192322+1092.563965i -489.398132+682.415710i -660.956177+435.380310i -592.071167+283.615387i -481.770996+273.152588i -578.364807-284.839905i -432.705444-467.157928i -252.863144-177.831146i -184.364960-105.140976i -70.373978+192.619568i
-317.231934+34.053032i -405.823608+229.667175i -66.856201+704.243286i 683.227661+869.343262i 1074.619385+936.400269i 730.056763+328.768524i 1366.968384+0.000000i 925.953979+357.572357i 987.470398+227.252899i 1120.328857-116.214615i 720.144531+113.828911i 93.614426+436.813782i -174.339905+460.500732i -125.334450+536.489380i -1566.807861+1231.646362i -1241.660278+685.666565i -1289.289917+110.175888i -1030.719727+10.466843i -849.454102+15.663177i -637.056580-776.948059i -307.042267-905.982361i -203.836639-352.417480i -150.287170-344.785004i -200.414276+181.432739i
-228.393478+392.325012i -221.934280+552.801086i 48.570087+671.490234i 632.771423+605.632751i 1080.759033+393.798340i 586.875366+16.146624i 925.953979-357.572357i 1023.432495+0.000000i 1085.747070-106.506378i 951.967957-544.681274i 813.398071-377.059845i 315.068604+17.699055i 180.264069+150.657867i 375.880463+189.927933i -601.740784+1099.747559i -647.649109+755.437317i -722.275391+403.219574i -489.099426+294.049866i -442.473572+232.180328i -365.307831-185.589325i -361.585815-195.537323i -255.364120+12.100807i -154.270142+108.992714i -145.517654+461.467590i
-127.726959+492.413513i -159.924622+733.100281i 19.534491+851.697327i 558.547607+982.655090i 1080.445312+524.300049i 604.010132+139.252884i 987.470398-227.252899i 1085.747070+106.506378i 1459.037720+0.000000i 1145.059692-616.668396i 975.515930-451.436035i 297.028229-125.700050i 179.042145-5.695498i 471.569214+52.771812i -827.958557+694.131958i -721.609009+679.779724i -680.237000+173.968353i -418.373901+94.836716i -401.116058+90.706520i -130.420578-310.272217i 16.708679-108.416885i -171.285599+47.552147i -109.981544+160.078156i -182.209366+615.671204i
-472.311096+325.571991i -584.464539+557.959167i -300.561401+772.196716i 227.245529+983.722412i 1011.214233+959.518372i 594.211060+330.463440i 1120.328857+116.214615i 951.967957+544.681274i 1145.059692+616.668396i 1466.600464+0.000000i 1097.442139+148.505630i 390.618164+268.185699i 96.993210+299.075256i 384.726807+593.591919i -1139.338013+891.152893i -1071.154663+551.454407i -913.621521+200.369247i -644.001709+140.031479i -533.023804+126.435074i -359.098083-336.750305i -339.591888-220.743805i -412.548798-126.666504i -311.889130-30.953690i -458.171875+459.022369i
-510.262024+382.537262i -513.812439+557.605591i -252.325562+557.146790i 97.741722+651.704468i 751.320740+563.206177i 402.430389+166.598511i 720.144531-113.828911i 813.398071+377.059845i 975.515930+451.436035i 1097.442139-148.505630i 1074.651611+0.000000i 498.815552+56.508560i 367.963562+203.591965i 615.645020+403.514679i -350.488464+733.823608i -504.463165+504.889038i -504.662231+343.320801i -278.440002+305.153259i -279.109680+218.992584i -215.620941+113.587891i -397.410400+153.275726i -364.260803+47.977142i -448.225433+178.677902i -488.958679+429.138702i
-350.180908+294.300842i -287.260925+363.278137i -1.530064+85.880135i 56.844387-134.000076i 421.583740-289.984283i 155.124969-239.680237i 93.614426-436.813782i 315.068604-17.699055i 297.028229+125.700050i 390.618164-268.185699i 498.815552-56.508560i 665.190491+0.000000i 518.804199+168.709900i 671.654785+350.977234i 719.247009+684.347839i 248.052338+503.694885i 134.574005+564.267334i 110.810654+599.050964i 56.786274+445.408478i -318.087860+609.023926i -716.544556+473.061127i -408.837891+194.474060i -481.766632+230.151093i -246.862701+160.150284i
-293.310211+433.513275i -172.812073+393.315796i 53.092026+8.853506i -10.137331-232.600876i 192.584259-499.200745i 14.554134-282.846649i -174.339905-460.500732i 180.264069-150.657867i 179.042145+5.695498i 96.993210-299.075256i 367.963562-203.591965i 518.804199-168.709900i 596.381104+0.000000i 658.443665+61.984833i 1171.753784+223.644882i 596.661865+292.638000i 469.755066+528.914429i 391.641907+546.784607i 274.620239+424.423920i -46.843994+753.871216i -508.842896+737.788269i -259.924133+301.629944i -407.842957+388.554260i -225.513992+205.022705i
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I tried using complex doubles and increasing iterations with no change to the returned values. I've also tried normalizing the Eigen Vectors to further identify they do not match what Matlab produces.
The complex float matrix is N by N hermitian, and therefore must have N valid eigen values. After going back and forth with a person responding to my Eigen Library bug submittal, there was no progress on resolving the issue. What they do not tell you is that the Eigen solver has extremely limited precision. So limited that it's useless for complex matrixes. It appears that the only way this minimally works is if all the returned values happen to be within 8 decimal places. They say 8, but it looks more like 6. There is definitely a huge lack of proper resolution in the internals of the method they came up with which unfortunately means that this is kind of useless as an Eigen solver for a complex matrix. This issue is compounded because when they exceed their internal extremely limited precision level, they still report success while returning incorrect values. The position from the person responding to the reported bug, is that it is not a bug because the results are exactly what they expect from their undocumented extremely limited precision. It appears there just another group trying to make people believe they have a valid solution when they do not. I did have some luck producing my own QR decomposition method to determine Eigen values but unfortunately that is an estimation algorithm and even though I got better precision than Eigen Library but it still wasn't good enough for me to accept it. If anyone has a functional complex Eigen decomposition method out there that actually works, feel free to respond.