|
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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 |
SUBROUTINE SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, INFO ) * * -- LAPACK auxiliary routine (version 3.2) -- * Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N LOGICAL WANTT, WANTZ * .. * .. Array Arguments .. REAL H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * ) * .. * * Purpose * ======= * * SLAHQR is an auxiliary routine called by SHSEQR to update the * eigenvalues and Schur decomposition already computed by SHSEQR, by * dealing with the Hessenberg submatrix in rows and columns ILO to * IHI. * * Arguments * ========= * * WANTT (input) LOGICAL * = .TRUE. : the full Schur form T is required; * = .FALSE.: only eigenvalues are required. * * WANTZ (input) LOGICAL * = .TRUE. : the matrix of Schur vectors Z is required; * = .FALSE.: Schur vectors are not required. * * N (input) INTEGER * The order of the matrix H. N >= 0. * * ILO (input) INTEGER * IHI (input) INTEGER * It is assumed that H is already upper quasi-triangular in * rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless * ILO = 1). SLAHQR works primarily with the Hessenberg * submatrix in rows and columns ILO to IHI, but applies * transformations to all of H if WANTT is .TRUE.. * 1 <= ILO <= max(1,IHI); IHI <= N. * * H (input/output) REAL array, dimension (LDH,N) * On entry, the upper Hessenberg matrix H. * On exit, if INFO is zero and if WANTT is .TRUE., H is upper * quasi-triangular in rows and columns ILO:IHI, with any * 2-by-2 diagonal blocks in standard form. If INFO is zero * and WANTT is .FALSE., the contents of H are unspecified on * exit. The output state of H if INFO is nonzero is given * below under the description of INFO. * * LDH (input) INTEGER * The leading dimension of the array H. LDH >= max(1,N). * * WR (output) REAL array, dimension (N) * WI (output) REAL array, dimension (N) * The real and imaginary parts, respectively, of the computed * eigenvalues ILO to IHI are stored in the corresponding * elements of WR and WI. If two eigenvalues are computed as a * complex conjugate pair, they are stored in consecutive * elements of WR and WI, say the i-th and (i+1)th, with * WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the * eigenvalues are stored in the same order as on the diagonal * of the Schur form returned in H, with WR(i) = H(i,i), and, if * H(i:i+1,i:i+1) is a 2-by-2 diagonal block, * WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i). * * ILOZ (input) INTEGER * IHIZ (input) INTEGER * Specify the rows of Z to which transformations must be * applied if WANTZ is .TRUE.. * 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. * * Z (input/output) REAL array, dimension (LDZ,N) * If WANTZ is .TRUE., on entry Z must contain the current * matrix Z of transformations accumulated by SHSEQR, and on * exit Z has been updated; transformations are applied only to * the submatrix Z(ILOZ:IHIZ,ILO:IHI). * If WANTZ is .FALSE., Z is not referenced. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * .GT. 0: If INFO = i, SLAHQR failed to compute all the * eigenvalues ILO to IHI in a total of 30 iterations * per eigenvalue; elements i+1:ihi of WR and WI * contain those eigenvalues which have been * successfully computed. * * If INFO .GT. 0 and WANTT is .FALSE., then on exit, * the remaining unconverged eigenvalues are the * eigenvalues of the upper Hessenberg matrix rows * and columns ILO thorugh INFO of the final, output * value of H. * * If INFO .GT. 0 and WANTT is .TRUE., then on exit * (*) (initial value of H)*U = U*(final value of H) * where U is an orthognal matrix. The final * value of H is upper Hessenberg and triangular in * rows and columns INFO+1 through IHI. * * If INFO .GT. 0 and WANTZ is .TRUE., then on exit * (final value of Z) = (initial value of Z)*U * where U is the orthogonal matrix in (*) * (regardless of the value of WANTT.) * * Further Details * =============== * * 02-96 Based on modifications by * David Day, Sandia National Laboratory, USA * * 12-04 Further modifications by * Ralph Byers, University of Kansas, USA * This is a modified version of SLAHQR from LAPACK version 3.0. * It is (1) more robust against overflow and underflow and * (2) adopts the more conservative Ahues & Tisseur stopping * criterion (LAWN 122, 1997). * * ========================================================= * * .. Parameters .. INTEGER ITMAX PARAMETER ( ITMAX = 30 ) REAL ZERO, ONE, TWO PARAMETER ( ZERO = 0.0e0, ONE = 1.0e0, TWO = 2.0e0 ) REAL DAT1, DAT2 PARAMETER ( DAT1 = 3.0e0 / 4.0e0, DAT2 = -0.4375e0 ) * .. * .. Local Scalars .. REAL AA, AB, BA, BB, CS, DET, H11, H12, H21, H21S, $ H22, RT1I, RT1R, RT2I, RT2R, RTDISC, S, SAFMAX, $ SAFMIN, SMLNUM, SN, SUM, T1, T2, T3, TR, TST, $ ULP, V2, V3 INTEGER I, I1, I2, ITS, J, K, L, M, NH, NR, NZ * .. * .. Local Arrays .. REAL V( 3 ) * .. * .. External Functions .. REAL SLAMCH EXTERNAL SLAMCH * .. * .. External Subroutines .. EXTERNAL SCOPY, SLABAD, SLANV2, SLARFG, SROT * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, REAL, SQRT * .. * .. Executable Statements .. * INFO = 0 * * Quick return if possible * IF( N.EQ.0 ) $ RETURN IF( ILO.EQ.IHI ) THEN WR( ILO ) = H( ILO, ILO ) WI( ILO ) = ZERO RETURN END IF * * ==== clear out the trash ==== DO 10 J = ILO, IHI - 3 H( J+2, J ) = ZERO H( J+3, J ) = ZERO 10 CONTINUE IF( ILO.LE.IHI-2 ) $ H( IHI, IHI-2 ) = ZERO * NH = IHI - ILO + 1 NZ = IHIZ - ILOZ + 1 * * Set machine-dependent constants for the stopping criterion. * SAFMIN = SLAMCH( 'SAFE MINIMUM' ) SAFMAX = ONE / SAFMIN CALL SLABAD( SAFMIN, SAFMAX ) ULP = SLAMCH( 'PRECISION' ) SMLNUM = SAFMIN*( REAL( NH ) / ULP ) * * I1 and I2 are the indices of the first row and last column of H * to which transformations must be applied. If eigenvalues only are * being computed, I1 and I2 are set inside the main loop. * IF( WANTT ) THEN I1 = 1 I2 = N END IF * * The main loop begins here. I is the loop index and decreases from * IHI to ILO in steps of 1 or 2. Each iteration of the loop works * with the active submatrix in rows and columns L to I. * Eigenvalues I+1 to IHI have already converged. Either L = ILO or * H(L,L-1) is negligible so that the matrix splits. * I = IHI 20 CONTINUE L = ILO IF( I.LT.ILO ) $ GO TO 160 * * Perform QR iterations on rows and columns ILO to I until a * submatrix of order 1 or 2 splits off at the bottom because a * subdiagonal element has become negligible. * DO 140 ITS = 0, ITMAX * * Look for a single small subdiagonal element. * DO 30 K = I, L + 1, -1 IF( ABS( H( K, K-1 ) ).LE.SMLNUM ) $ GO TO 40 TST = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) ) IF( TST.EQ.ZERO ) THEN IF( K-2.GE.ILO ) $ TST = TST + ABS( H( K-1, K-2 ) ) IF( K+1.LE.IHI ) $ TST = TST + ABS( H( K+1, K ) ) END IF * ==== The following is a conservative small subdiagonal * . deflation criterion due to Ahues & Tisseur (LAWN 122, * . 1997). It has better mathematical foundation and * . improves accuracy in some cases. ==== IF( ABS( H( K, K-1 ) ).LE.ULP*TST ) THEN AB = MAX( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) ) BA = MIN( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) ) AA = MAX( ABS( H( K, K ) ), $ ABS( H( K-1, K-1 )-H( K, K ) ) ) BB = MIN( ABS( H( K, K ) ), $ ABS( H( K-1, K-1 )-H( K, K ) ) ) S = AA + AB IF( BA*( AB / S ).LE.MAX( SMLNUM, $ ULP*( BB*( AA / S ) ) ) )GO TO 40 END IF 30 CONTINUE 40 CONTINUE L = K IF( L.GT.ILO ) THEN * * H(L,L-1) is negligible * H( L, L-1 ) = ZERO END IF * * Exit from loop if a submatrix of order 1 or 2 has split off. * IF( L.GE.I-1 ) $ GO TO 150 * * Now the active submatrix is in rows and columns L to I. If * eigenvalues only are being computed, only the active submatrix * need be transformed. * IF( .NOT.WANTT ) THEN I1 = L I2 = I END IF * IF( ITS.EQ.10 ) THEN * * Exceptional shift. * S = ABS( H( L+1, L ) ) + ABS( H( L+2, L+1 ) ) H11 = DAT1*S + H( L, L ) H12 = DAT2*S H21 = S H22 = H11 ELSE IF( ITS.EQ.20 ) THEN * * Exceptional shift. * S = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) ) H11 = DAT1*S + H( I, I ) H12 = DAT2*S H21 = S H22 = H11 ELSE * * Prepare to use Francis' double shift * (i.e. 2nd degree generalized Rayleigh quotient) * H11 = H( I-1, I-1 ) H21 = H( I, I-1 ) H12 = H( I-1, I ) H22 = H( I, I ) END IF S = ABS( H11 ) + ABS( H12 ) + ABS( H21 ) + ABS( H22 ) IF( S.EQ.ZERO ) THEN RT1R = ZERO RT1I = ZERO RT2R = ZERO RT2I = ZERO ELSE H11 = H11 / S H21 = H21 / S H12 = H12 / S H22 = H22 / S TR = ( H11+H22 ) / TWO DET = ( H11-TR )*( H22-TR ) - H12*H21 RTDISC = SQRT( ABS( DET ) ) IF( DET.GE.ZERO ) THEN * * ==== complex conjugate shifts ==== * RT1R = TR*S RT2R = RT1R RT1I = RTDISC*S RT2I = -RT1I ELSE * * ==== real shifts (use only one of them) ==== * RT1R = TR + RTDISC RT2R = TR - RTDISC IF( ABS( RT1R-H22 ).LE.ABS( RT2R-H22 ) ) THEN RT1R = RT1R*S RT2R = RT1R ELSE RT2R = RT2R*S RT1R = RT2R END IF RT1I = ZERO RT2I = ZERO END IF END IF * * Look for two consecutive small subdiagonal elements. * DO 50 M = I - 2, L, -1 * Determine the effect of starting the double-shift QR * iteration at row M, and see if this would make H(M,M-1) * negligible. (The following uses scaling to avoid * overflows and most underflows.) * H21S = H( M+1, M ) S = ABS( H( M, M )-RT2R ) + ABS( RT2I ) + ABS( H21S ) H21S = H( M+1, M ) / S V( 1 ) = H21S*H( M, M+1 ) + ( H( M, M )-RT1R )* $ ( ( H( M, M )-RT2R ) / S ) - RT1I*( RT2I / S ) V( 2 ) = H21S*( H( M, M )+H( M+1, M+1 )-RT1R-RT2R ) V( 3 ) = H21S*H( M+2, M+1 ) S = ABS( V( 1 ) ) + ABS( V( 2 ) ) + ABS( V( 3 ) ) V( 1 ) = V( 1 ) / S V( 2 ) = V( 2 ) / S V( 3 ) = V( 3 ) / S IF( M.EQ.L ) $ GO TO 60 IF( ABS( H( M, M-1 ) )*( ABS( V( 2 ) )+ABS( V( 3 ) ) ).LE. $ ULP*ABS( V( 1 ) )*( ABS( H( M-1, M-1 ) )+ABS( H( M, $ M ) )+ABS( H( M+1, M+1 ) ) ) )GO TO 60 50 CONTINUE 60 CONTINUE * * Double-shift QR step * DO 130 K = M, I - 1 * * The first iteration of this loop determines a reflection G * from the vector V and applies it from left and right to H, * thus creating a nonzero bulge below the subdiagonal. * * Each subsequent iteration determines a reflection G to * restore the Hessenberg form in the (K-1)th column, and thus * chases the bulge one step toward the bottom of the active * submatrix. NR is the order of G. * NR = MIN( 3, I-K+1 ) IF( K.GT.M ) $ CALL SCOPY( NR, H( K, K-1 ), 1, V, 1 ) CALL SLARFG( NR, V( 1 ), V( 2 ), 1, T1 ) IF( K.GT.M ) THEN H( K, K-1 ) = V( 1 ) H( K+1, K-1 ) = ZERO IF( K.LT.I-1 ) $ H( K+2, K-1 ) = ZERO ELSE IF( M.GT.L ) THEN * ==== Use the following instead of * . H( K, K-1 ) = -H( K, K-1 ) to * . avoid a bug when v(2) and v(3) * . underflow. ==== H( K, K-1 ) = H( K, K-1 )*( ONE-T1 ) END IF V2 = V( 2 ) T2 = T1*V2 IF( NR.EQ.3 ) THEN V3 = V( 3 ) T3 = T1*V3 * * Apply G from the left to transform the rows of the matrix * in columns K to I2. * DO 70 J = K, I2 SUM = H( K, J ) + V2*H( K+1, J ) + V3*H( K+2, J ) H( K, J ) = H( K, J ) - SUM*T1 H( K+1, J ) = H( K+1, J ) - SUM*T2 H( K+2, J ) = H( K+2, J ) - SUM*T3 70 CONTINUE * * Apply G from the right to transform the columns of the * matrix in rows I1 to min(K+3,I). * DO 80 J = I1, MIN( K+3, I ) SUM = H( J, K ) + V2*H( J, K+1 ) + V3*H( J, K+2 ) H( J, K ) = H( J, K ) - SUM*T1 H( J, K+1 ) = H( J, K+1 ) - SUM*T2 H( J, K+2 ) = H( J, K+2 ) - SUM*T3 80 CONTINUE * IF( WANTZ ) THEN * * Accumulate transformations in the matrix Z * DO 90 J = ILOZ, IHIZ SUM = Z( J, K ) + V2*Z( J, K+1 ) + V3*Z( J, K+2 ) Z( J, K ) = Z( J, K ) - SUM*T1 Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2 Z( J, K+2 ) = Z( J, K+2 ) - SUM*T3 90 CONTINUE END IF ELSE IF( NR.EQ.2 ) THEN * * Apply G from the left to transform the rows of the matrix * in columns K to I2. * DO 100 J = K, I2 SUM = H( K, J ) + V2*H( K+1, J ) H( K, J ) = H( K, J ) - SUM*T1 H( K+1, J ) = H( K+1, J ) - SUM*T2 100 CONTINUE * * Apply G from the right to transform the columns of the * matrix in rows I1 to min(K+3,I). * DO 110 J = I1, I SUM = H( J, K ) + V2*H( J, K+1 ) H( J, K ) = H( J, K ) - SUM*T1 H( J, K+1 ) = H( J, K+1 ) - SUM*T2 110 CONTINUE * IF( WANTZ ) THEN * * Accumulate transformations in the matrix Z * DO 120 J = ILOZ, IHIZ SUM = Z( J, K ) + V2*Z( J, K+1 ) Z( J, K ) = Z( J, K ) - SUM*T1 Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2 120 CONTINUE END IF END IF 130 CONTINUE * 140 CONTINUE * * Failure to converge in remaining number of iterations * INFO = I RETURN * 150 CONTINUE * IF( L.EQ.I ) THEN * * H(I,I-1) is negligible: one eigenvalue has converged. * WR( I ) = H( I, I ) WI( I ) = ZERO ELSE IF( L.EQ.I-1 ) THEN * * H(I-1,I-2) is negligible: a pair of eigenvalues have converged. * * Transform the 2-by-2 submatrix to standard Schur form, * and compute and store the eigenvalues. * CALL SLANV2( H( I-1, I-1 ), H( I-1, I ), H( I, I-1 ), $ H( I, I ), WR( I-1 ), WI( I-1 ), WR( I ), WI( I ), $ CS, SN ) * IF( WANTT ) THEN * * Apply the transformation to the rest of H. * IF( I2.GT.I ) $ CALL SROT( I2-I, H( I-1, I+1 ), LDH, H( I, I+1 ), LDH, $ CS, SN ) CALL SROT( I-I1-1, H( I1, I-1 ), 1, H( I1, I ), 1, CS, SN ) END IF IF( WANTZ ) THEN * * Apply the transformation to Z. * CALL SROT( NZ, Z( ILOZ, I-1 ), 1, Z( ILOZ, I ), 1, CS, SN ) END IF END IF * * return to start of the main loop with new value of I. * I = L - 1 GO TO 20 * 160 CONTINUE RETURN * * End of SLAHQR * END |