Finding the Largest Eigenvalue Using the Power method

Question

For this problem, I have to take a large 100 by 100 matrix of randomly generated values and an initial x vector and try to obtain the largest absolute value of the eigenvalues and the associated convectors as well as the number of iterations.

The power method as described:

I have to find the absolute val of the largest eigenvalue, the corresponding eigenvector, and the number of iterations.

My code:

%% problem 2b: Power method
% code snippet from class generates A and x
nx = 10;
ex = ones(nx,1);
Dxx = spdiags([ex -2*ex ex], [-1 0 1], nx, nx);
A = kron(Dxx, speye(nx)) + kron(speye(nx), Dxx);
rng(2);
x = rand(nx*nx,1); % initial vector

yk = [] % stores yk values
count = 0; % keeps track of iterations
value = 1; % 1 to ensure while loop is entered
tolerance = 10^(-6); % tol

while (value >= tolerance)
    yk = A*x; % yk = A*xk
    xnext = yk/norm(yk); % Xk+1=yk/2norm(yk)
    rk = dot(xnext,(A*xnext)); % rk = Xk+1 * (A*Xk+1)
    count = count+1; % iterate count
    value = norm(yk - rk*x); % update tolerance check
    x = xnext;
end
value

x % final eigenvector
rk % the largest eignvalue in abs value
count % the number of iterations

According to the answers, my largest eigenvalue—rk—is correct, but the corresponding eigenvector and the number of iterations is incorrect. Can someone clarify this for me? The eigenvalue and the eigenvector agrees when I use:

[C, D] = eigs(A)

Answer 1

Are you sure? It looks correct to me.

When I produce C and D with eigs, I get:

>> [C, D] = eigs(A)

C =

    0.0520    0.0170    0.0531    0.0158   -0.0358   -0.0144
   -0.0788   -0.0433   -0.0894   -0.0327    0.0626    0.0277
    0.0717    0.0801    0.0973    0.0510   -0.0741   -0.0387
   -0.0437   -0.1180   -0.0743   -0.0694    0.0684    0.0466
    0.0200    0.1424    0.0277    0.0855   -0.0486   -0.0507
   -0.0200   -0.1424    0.0277   -0.0959    0.0217    0.0507
    0.0437    0.1180   -0.0743    0.0974    0.0038   -0.0466
   -0.0717   -0.0801    0.0973   -0.0876   -0.0204    0.0387
    0.0788    0.0433   -0.0894    0.0664    0.0243   -0.0277
   -0.0520   -0.0170    0.0531   -0.0358   -0.0158    0.0144
   -0.0892   -0.0116   -0.0894   -0.0243    0.0664    0.0277
    0.1308    0.0427    0.1504    0.0511   -0.1157   -0.0531
   -0.1090   -0.0972   -0.1637   -0.0814    0.1359    0.0743
    0.0494    0.1585    0.1250    0.1135   -0.1236   -0.0894
   -0.0009   -0.1993   -0.0466   -0.1427    0.0849    0.0973
    0.0009    0.1993   -0.0466    0.1627   -0.0333   -0.0973
   -0.0494   -0.1585    0.1250   -0.1671   -0.0149    0.0894
    0.1090    0.0972   -0.1637    0.1517    0.0456   -0.0743
   -0.1308   -0.0427    0.1504   -0.1157   -0.0511    0.0531
    0.0892    0.0116   -0.0894    0.0626    0.0327   -0.0277
    0.1051   -0.0223    0.0973    0.0204   -0.0876   -0.0387
   -0.1453    0.0142   -0.1637   -0.0456    0.1517    0.0743
    0.0999    0.0326    0.1781    0.0777   -0.1760   -0.1038
   -0.0060   -0.0973   -0.1360   -0.1151    0.1560    0.1250
   -0.0674    0.1433    0.0507    0.1521   -0.1004   -0.1360
    0.0674   -0.1433    0.0507   -0.1801    0.0282    0.1360
    0.0060    0.0973   -0.1360    0.1902    0.0377   -0.1250
   -0.0999   -0.0326    0.1781   -0.1760   -0.0777    0.1038
    0.1453   -0.0142   -0.1637    0.1359    0.0814   -0.0743
   -0.1051    0.0223    0.0973   -0.0741   -0.0510    0.0387
   -0.1049    0.0695   -0.0743   -0.0038    0.0974    0.0466
    0.1334   -0.0988    0.1250    0.0149   -0.1671   -0.0894
   -0.0622    0.0750   -0.1360   -0.0377    0.1902    0.1250
   -0.0626   -0.0205    0.1038    0.0714   -0.1617   -0.1504
    0.1571   -0.0229   -0.0387   -0.1100    0.0926    0.1637
   -0.1571    0.0229   -0.0387    0.1437   -0.0057   -0.1637
    0.0626    0.0205    0.1038   -0.1617   -0.0714    0.1504
    0.0622   -0.0750   -0.1360    0.1560    0.1151   -0.1250
   -0.1334    0.0988    0.1250   -0.1236   -0.1135    0.0894
    0.1049   -0.0695   -0.0743    0.0684    0.0694   -0.0466
    0.1002   -0.1031    0.0277   -0.0217   -0.0959   -0.0507
   -0.1184    0.1604   -0.0466    0.0333    0.1627    0.0973
    0.0302   -0.1555    0.0507   -0.0282   -0.1801   -0.1360
    0.1133    0.1112   -0.0387    0.0057    0.1437    0.1637
   -0.2201   -0.0719    0.0144    0.0290   -0.0656   -0.1781
    0.2201    0.0719    0.0144   -0.0656   -0.0290    0.1781
   -0.1133   -0.1112   -0.0387    0.0926    0.1100   -0.1637
   -0.0302    0.1555    0.0507   -0.1004   -0.1521    0.1360
    0.1184   -0.1604   -0.0466    0.0849    0.1427   -0.0973
   -0.1002    0.1031    0.0277   -0.0486   -0.0855    0.0507
   -0.1002    0.1031    0.0277    0.0486    0.0855    0.0507
    0.1184   -0.1604   -0.0466   -0.0849   -0.1427   -0.0973
   -0.0302    0.1555    0.0507    0.1004    0.1521    0.1360
   -0.1133   -0.1112   -0.0387   -0.0926   -0.1100   -0.1637
    0.2201    0.0719    0.0144    0.0656    0.0290    0.1781
   -0.2201   -0.0719    0.0144   -0.0290    0.0656   -0.1781
    0.1133    0.1112   -0.0387   -0.0057   -0.1437    0.1637
    0.0302   -0.1555    0.0507    0.0282    0.1801   -0.1360
   -0.1184    0.1604   -0.0466   -0.0333   -0.1627    0.0973
    0.1002   -0.1031    0.0277    0.0217    0.0959   -0.0507
    0.1049   -0.0695   -0.0743   -0.0684   -0.0694   -0.0466
   -0.1334    0.0988    0.1250    0.1236    0.1135    0.0894
    0.0622   -0.0750   -0.1360   -0.1560   -0.1151   -0.1250
    0.0626    0.0205    0.1038    0.1617    0.0714    0.1504
   -0.1571    0.0229   -0.0387   -0.1437    0.0057   -0.1637
    0.1571   -0.0229   -0.0387    0.1100   -0.0926    0.1637
   -0.0626   -0.0205    0.1038   -0.0714    0.1617   -0.1504
   -0.0622    0.0750   -0.1360    0.0377   -0.1902    0.1250
    0.1334   -0.0988    0.1250   -0.0149    0.1671   -0.0894
   -0.1049    0.0695   -0.0743    0.0038   -0.0974    0.0466
   -0.1051    0.0223    0.0973    0.0741    0.0510    0.0387
    0.1453   -0.0142   -0.1637   -0.1359   -0.0814   -0.0743
   -0.0999   -0.0326    0.1781    0.1760    0.0777    0.1038
    0.0060    0.0973   -0.1360   -0.1902   -0.0377   -0.1250
    0.0674   -0.1433    0.0507    0.1801   -0.0282    0.1360
   -0.0674    0.1433    0.0507   -0.1521    0.1004   -0.1360
   -0.0060   -0.0973   -0.1360    0.1151   -0.1560    0.1250
    0.0999    0.0326    0.1781   -0.0777    0.1760   -0.1038
   -0.1453    0.0142   -0.1637    0.0456   -0.1517    0.0743
    0.1051   -0.0223    0.0973   -0.0204    0.0876   -0.0387
    0.0892    0.0116   -0.0894   -0.0626   -0.0327   -0.0277
   -0.1308   -0.0427    0.1504    0.1157    0.0511    0.0531
    0.1090    0.0972   -0.1637   -0.1517   -0.0456   -0.0743
   -0.0494   -0.1585    0.1250    0.1671    0.0149    0.0894
    0.0009    0.1993   -0.0466   -0.1627    0.0333   -0.0973
   -0.0009   -0.1993   -0.0466    0.1427   -0.0849    0.0973
    0.0494    0.1585    0.1250   -0.1135    0.1236   -0.0894
   -0.1090   -0.0972   -0.1637    0.0814   -0.1359    0.0743
    0.1308    0.0427    0.1504   -0.0511    0.1157   -0.0531
   -0.0892   -0.0116   -0.0894    0.0243   -0.0664    0.0277
   -0.0520   -0.0170    0.0531    0.0358    0.0158    0.0144
    0.0788    0.0433   -0.0894   -0.0664   -0.0243   -0.0277
   -0.0717   -0.0801    0.0973    0.0876    0.0204    0.0387
    0.0437    0.1180   -0.0743   -0.0974   -0.0038   -0.0466
   -0.0200   -0.1424    0.0277    0.0959   -0.0217    0.0507
    0.0200    0.1424    0.0277   -0.0855    0.0486   -0.0507
   -0.0437   -0.1180   -0.0743    0.0694   -0.0684    0.0466
    0.0717    0.0801    0.0973   -0.0510    0.0741   -0.0387
   -0.0788   -0.0433   -0.0894    0.0327   -0.0626    0.0277
    0.0520    0.0170    0.0531   -0.0158    0.0358   -0.0144


D =

   -7.2287         0         0         0         0         0
         0   -7.2287         0         0         0         0
         0         0   -7.3650         0         0         0
         0         0         0   -7.6015         0         0
         0         0         0         0   -7.6015         0
         0         0         0         0         0   -7.8380

Also, rk from your Power Method gives:

>> rk

rk =

   -7.8380

rk is the last eigenvalue produced by eigs, and that corresponds to the last eigenvector / last column in C. If we compare x and the last column of C, we get:

>> format long g;
>> [x C(:,6)]

ans =

       -0.0144315780109194       -0.0144314970153476
        0.0276939697943544         0.027693839969929
       -0.0387127177669401       -0.0387125927117545
        0.0465951447011548        0.0465950814247274
       -0.0507026743311123        -0.050702713752288
        0.0507025614842252        0.0507027137522879
       -0.0465948419885827       -0.0465950814247267
        0.0387123212977858        0.0387125927117535
        -0.027693605444773        -0.027693839969928
        0.0144313614593288         0.014431497015347
        0.0276940018504475        0.0276938399699289
       -0.0531443512380279        -0.053144089727102
        0.0742891786790794        0.0742889213946561
       -0.0894154487195339       -0.0894153064640425
        0.0972977421934561        0.0972977951770151
       -0.0972975256418808        -0.097297795177015
        0.0894148678184032        0.0894153064640416
       -0.0742884178603978        -0.074288921394655
        0.0531436520563511        0.0531440897271008
       -0.0276935862910177       -0.0276938399699281
       -0.0387128332167354       -0.0387125927117545
         0.074289314234918         0.074288921394656
        -0.103847201265651         -0.10384680347939
         0.124991879901604         0.124991635146944
         -0.13601036376947        -0.136010387888769
         0.136010061056953         0.136010387888769
        -0.124991067873139        -0.124991635146943
         0.103846137734539         0.103846803479389
       -0.0742883368647708       -0.0742889213946555
        0.0387122523156699         0.038712592711754
        0.0465953928808543        0.0465950814247275
       -0.0894158214731441       -0.0894153064640426
         0.124992172891791         0.124991635146944
        -0.150442250463584        -0.150441884904116
         0.163704276053648         0.163704227858698
        -0.163703911704186        -0.163704227858698
         0.150441273093578         0.150441884904117
        -0.124990892809312        -0.124991635146944
        0.0894146450953342        0.0894153064640427
       -0.0465946936993567       -0.0465950814247274
       -0.0507030799451604       -0.0507027137522887
        0.0972984079372535        0.0972977951770161
        -0.136011046217971         -0.13601038788877
         0.163704713723789         0.163704227858698
         -0.17813587316184        -0.178135724874045
         0.178135476692881         0.178135724874045
        -0.163703650193004        -0.163704227858699
         0.136009653288871          0.13601038788877
       -0.0972971278549844       -0.0972977951770159
        0.0507023191267988        0.0507027137522887
        0.0507031083768884        0.0507027137522886
       -0.0972984624973554        -0.097297795177016
         0.136011122486338          0.13601038788877
         -0.16370480552164        -0.163704227858699
          0.17813597305228         0.178135724874046
        -0.178135576583388        -0.178135724874046
         0.163703741991037         0.163704227858699
        -0.136009729557476         -0.13601038788877
        0.0972971824153056        0.0972977951770162
       -0.0507023475586572       -0.0507027137522885
       -0.0465954691491656       -0.0465950814247277
         0.089415967831003        0.0894153064640427
        -0.124992377482224        -0.124991635146945
         0.150442496711966         0.150441884904117
        -0.163704544010458        -0.163704227858698
         0.163704179661178         0.163704227858698
        -0.150441519342448        -0.150441884904117
         0.124991097400385         0.124991635146944
       -0.0894147914537812       -0.0894153064640427
        0.0465947699680175        0.0465950814247277
          0.03871293310698        0.0387125927117539
       -0.0742895059229478       -0.0742889213946556
         0.103847469222133          0.10384680347939
        -0.124992202418397        -0.124991635146944
         0.136010714718256          0.13601038788877
        -0.136010412005978        -0.136010387888769
         0.124991390390572         0.124991635146944
        -0.103846405691859        -0.103846803479389
        0.0742885285535708        0.0742889213946558
       -0.0387123522063723       -0.0387125927117542
       -0.0276940936481785       -0.0276938399699281
        0.0531445273966335        0.0531440897271008
       -0.0742894249273207       -0.0742889213946552
         0.089415745107934        0.0894153064640417
        -0.097298064710459       -0.0972977951770146
         0.097297848159103        0.0972977951770148
       -0.0894151642073914       -0.0894153064640414
        0.0742886641094094        0.0742889213946553
       -0.0531438282156646        -0.053144089727101
        0.0276936780891695        0.0276938399699285
        0.0144316325710059        0.0144314970153468
       -0.0276940744944232       -0.0276938399699279
        0.0387128641248642        0.0387125927117535
       -0.0465953208599396       -0.0465950814247267
        0.0507028660194621        0.0507027137522877
       -0.0507027531727052       -0.0507027137522875
         0.046595018147717        0.0465950814247267
       -0.0387124676561676       -0.0387125927117538
        0.0276937101452626        0.0276938399699282
       -0.0144314160196653       -0.0144314970153473

Also looks fine. Shall we do another test? Let's find the maximum deviation between x and the last eigenvector of C:

>> max(abs(x - C(:,6)))

ans =

      7.42337632364531e-07

That also agrees with your tolerance you specified at the beginning, where it's < 1e-6.

---

As such, I don't see anything wrong with your code. You should probably double-check your theory on the Power Method because it produces what you expect.