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.