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Tests Performed on the Symmetric Eigenvalue Routines

Finding the eigenvalues and eigenvectors of a symmetric matrix 16#16 is done in the following stages:

  1. 16#16 is decomposed as 286#286, where 132#132 is unitary, 250#250 is real symmetric tridiagonal, and 136#136 is the conjugate transpose of 132#132. 132#132 is represented as a product of Householder transformations, whose vectors are stored in the first n-1 columns of 245#245, and whose scale factors are in 287#287.

  2. 250#250 is decomposed as 88#88D1288#288, where 88#88 is real orthogonal and 289#289 is a real diagonal matrix of eigenvalues. 290#290 is the matrix of eigenvalues computed when 88#88 is not computed.

  3. The ``PWK'' method is used to compute 291#291, the matrix of eigenvalues, using a square-root-free method which does not compute 88#88.

  4. 250#250 is decomposed as 292#292 293#293 294#294, for a symmetric positive definite tridiagonal matrix. 295#295 is the matrix of eigenvalues computed when 88#88 is not computed.

  5. Selected eigenvalues (296#296, 297#297, and 298#298) are computed and denote eigenvalues computed to high absolute accuracy, with different range options. 299#299 will denote eigenvalues computed to high relative accuracy.

  6. Given the eigenvalues, the eigenvectors of 250#250 are computed in 141#141.

  7. 250#250 is factored as 88#88 289#289 288#288.

To check these calculations, the following test ratios are computed (where banded matrices only compute test ratios 1-4):

300#300 301#301 302#302  
    303#303  
304#304 301#301 305#305  
    306#306  
307#307 301#301 302#302  
    308#308  
309#309 301#301 305#305  
    310#310  

Tests 5-8 are the same as tests 1-4 but for SSPTRD and SOPGTR.


311#311 301#301 312#312  
    313#313  
314#314 301#301 315#315  
    313#313  
316#316 301#301 317#317  
    313#313  
318#318 301#301 319#319  
    320#320  
321#321 301#301 322#322  

For 250#250 positive definite,

323#323 301#301 324#324  
    325#325  
326#326 301#301 327#327  
    325#325  
328#328 301#301 329#329  
    330#330  

When 250#250 is also diagonally dominant by a factor 331#331,

332#332 301#301 333#333  
    334#334  
    335#335  
      (1)
336#336 301#301 337#337  
    335#335  
338#338 301#301 339#339  
    340#340  
341#341 301#301 342#342  
    343#343  
344#344 301#301 345#345  
    343#343  
346#346 301#301 347#347  
    348#348  
349#349 301#301 350#350  
    348#348  
351#351 301#301 347#347  
    352#352  
353#353 301#301 350#350  
    352#352  
354#354 301#301 355#355  
    356#356  
357#357 301#301 358#358  
    334#334  
    359#359  
360#360 301#301 358#358  
    334#334  
    361#361  
362#362 301#301 347#347  
    363#363  
364#364 301#301 350#350  
    363#363  
365#365 301#301 339#339  
    366#366  
367#367 301#301 347#347  
    368#368  
369#369 301#301 350#350  
    368#368  
370#370 301#301 339#339  
    371#371  
372#372 301#301 347#347  
    373#373  
374#374 301#301 350#350  
    373#373  
375#375 301#301 339#339  
    376#376  

where the subscript 133#133 indicates that the eigenvalues and eigenvectors were computed at the same time, and 53#53 that they were computed in separate steps. (All norms are 143#143.) The scalings in the test ratios assure that the ratios will be 124#124 (typically less than 10 or 100), independent of 144#144 and 9#9, and nearly independent of 4#4.

As in the nonsymmetric case, the test ratios for each test matrix are compared to a user-specified threshold 145#145, and a message is printed for each test that exceeds this threshold.

NOTE: Test 27 is disabled at the moment because SSTEGR does not guarantee high relative accuracy. Tests 29 through 34 are disabled at present because SSTEGR does not handle partial spectrum requests.


next up previous contents
Next: Tests Performed on the Up: Testing the Symmetric Eigenvalue Previous: Test Matrices for the   Contents
Susan Blackford 2001-08-13