1*> \brief \b DHGEQZ 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download DHGEQZ + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhgeqz.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhgeqz.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhgeqz.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, 22* ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, 23* LWORK, INFO ) 24* 25* .. Scalar Arguments .. 26* CHARACTER COMPQ, COMPZ, JOB 27* INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N 28* .. 29* .. Array Arguments .. 30* DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ), 31* $ H( LDH, * ), Q( LDQ, * ), T( LDT, * ), 32* $ WORK( * ), Z( LDZ, * ) 33* .. 34* 35* 36*> \par Purpose: 37* ============= 38*> 39*> \verbatim 40*> 41*> DHGEQZ computes the eigenvalues of a real matrix pair (H,T), 42*> where H is an upper Hessenberg matrix and T is upper triangular, 43*> using the double-shift QZ method. 44*> Matrix pairs of this type are produced by the reduction to 45*> generalized upper Hessenberg form of a real matrix pair (A,B): 46*> 47*> A = Q1*H*Z1**T, B = Q1*T*Z1**T, 48*> 49*> as computed by DGGHRD. 50*> 51*> If JOB='S', then the Hessenberg-triangular pair (H,T) is 52*> also reduced to generalized Schur form, 53*> 54*> H = Q*S*Z**T, T = Q*P*Z**T, 55*> 56*> where Q and Z are orthogonal matrices, P is an upper triangular 57*> matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2 58*> diagonal blocks. 59*> 60*> The 1-by-1 blocks correspond to real eigenvalues of the matrix pair 61*> (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of 62*> eigenvalues. 63*> 64*> Additionally, the 2-by-2 upper triangular diagonal blocks of P 65*> corresponding to 2-by-2 blocks of S are reduced to positive diagonal 66*> form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0, 67*> P(j,j) > 0, and P(j+1,j+1) > 0. 68*> 69*> Optionally, the orthogonal matrix Q from the generalized Schur 70*> factorization may be postmultiplied into an input matrix Q1, and the 71*> orthogonal matrix Z may be postmultiplied into an input matrix Z1. 72*> If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced 73*> the matrix pair (A,B) to generalized upper Hessenberg form, then the 74*> output matrices Q1*Q and Z1*Z are the orthogonal factors from the 75*> generalized Schur factorization of (A,B): 76*> 77*> A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T. 78*> 79*> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, 80*> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is 81*> complex and beta real. 82*> If beta is nonzero, lambda = alpha / beta is an eigenvalue of the 83*> generalized nonsymmetric eigenvalue problem (GNEP) 84*> A*x = lambda*B*x 85*> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the 86*> alternate form of the GNEP 87*> mu*A*y = B*y. 88*> Real eigenvalues can be read directly from the generalized Schur 89*> form: 90*> alpha = S(i,i), beta = P(i,i). 91*> 92*> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix 93*> Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), 94*> pp. 241--256. 95*> \endverbatim 96* 97* Arguments: 98* ========== 99* 100*> \param[in] JOB 101*> \verbatim 102*> JOB is CHARACTER*1 103*> = 'E': Compute eigenvalues only; 104*> = 'S': Compute eigenvalues and the Schur form. 105*> \endverbatim 106*> 107*> \param[in] COMPQ 108*> \verbatim 109*> COMPQ is CHARACTER*1 110*> = 'N': Left Schur vectors (Q) are not computed; 111*> = 'I': Q is initialized to the unit matrix and the matrix Q 112*> of left Schur vectors of (H,T) is returned; 113*> = 'V': Q must contain an orthogonal matrix Q1 on entry and 114*> the product Q1*Q is returned. 115*> \endverbatim 116*> 117*> \param[in] COMPZ 118*> \verbatim 119*> COMPZ is CHARACTER*1 120*> = 'N': Right Schur vectors (Z) are not computed; 121*> = 'I': Z is initialized to the unit matrix and the matrix Z 122*> of right Schur vectors of (H,T) is returned; 123*> = 'V': Z must contain an orthogonal matrix Z1 on entry and 124*> the product Z1*Z is returned. 125*> \endverbatim 126*> 127*> \param[in] N 128*> \verbatim 129*> N is INTEGER 130*> The order of the matrices H, T, Q, and Z. N >= 0. 131*> \endverbatim 132*> 133*> \param[in] ILO 134*> \verbatim 135*> ILO is INTEGER 136*> \endverbatim 137*> 138*> \param[in] IHI 139*> \verbatim 140*> IHI is INTEGER 141*> ILO and IHI mark the rows and columns of H which are in 142*> Hessenberg form. It is assumed that A is already upper 143*> triangular in rows and columns 1:ILO-1 and IHI+1:N. 144*> If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0. 145*> \endverbatim 146*> 147*> \param[in,out] H 148*> \verbatim 149*> H is DOUBLE PRECISION array, dimension (LDH, N) 150*> On entry, the N-by-N upper Hessenberg matrix H. 151*> On exit, if JOB = 'S', H contains the upper quasi-triangular 152*> matrix S from the generalized Schur factorization. 153*> If JOB = 'E', the diagonal blocks of H match those of S, but 154*> the rest of H is unspecified. 155*> \endverbatim 156*> 157*> \param[in] LDH 158*> \verbatim 159*> LDH is INTEGER 160*> The leading dimension of the array H. LDH >= max( 1, N ). 161*> \endverbatim 162*> 163*> \param[in,out] T 164*> \verbatim 165*> T is DOUBLE PRECISION array, dimension (LDT, N) 166*> On entry, the N-by-N upper triangular matrix T. 167*> On exit, if JOB = 'S', T contains the upper triangular 168*> matrix P from the generalized Schur factorization; 169*> 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S 170*> are reduced to positive diagonal form, i.e., if H(j+1,j) is 171*> non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and 172*> T(j+1,j+1) > 0. 173*> If JOB = 'E', the diagonal blocks of T match those of P, but 174*> the rest of T is unspecified. 175*> \endverbatim 176*> 177*> \param[in] LDT 178*> \verbatim 179*> LDT is INTEGER 180*> The leading dimension of the array T. LDT >= max( 1, N ). 181*> \endverbatim 182*> 183*> \param[out] ALPHAR 184*> \verbatim 185*> ALPHAR is DOUBLE PRECISION array, dimension (N) 186*> The real parts of each scalar alpha defining an eigenvalue 187*> of GNEP. 188*> \endverbatim 189*> 190*> \param[out] ALPHAI 191*> \verbatim 192*> ALPHAI is DOUBLE PRECISION array, dimension (N) 193*> The imaginary parts of each scalar alpha defining an 194*> eigenvalue of GNEP. 195*> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if 196*> positive, then the j-th and (j+1)-st eigenvalues are a 197*> complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j). 198*> \endverbatim 199*> 200*> \param[out] BETA 201*> \verbatim 202*> BETA is DOUBLE PRECISION array, dimension (N) 203*> The scalars beta that define the eigenvalues of GNEP. 204*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and 205*> beta = BETA(j) represent the j-th eigenvalue of the matrix 206*> pair (A,B), in one of the forms lambda = alpha/beta or 207*> mu = beta/alpha. Since either lambda or mu may overflow, 208*> they should not, in general, be computed. 209*> \endverbatim 210*> 211*> \param[in,out] Q 212*> \verbatim 213*> Q is DOUBLE PRECISION array, dimension (LDQ, N) 214*> On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in 215*> the reduction of (A,B) to generalized Hessenberg form. 216*> On exit, if COMPQ = 'I', the orthogonal matrix of left Schur 217*> vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix 218*> of left Schur vectors of (A,B). 219*> Not referenced if COMPQ = 'N'. 220*> \endverbatim 221*> 222*> \param[in] LDQ 223*> \verbatim 224*> LDQ is INTEGER 225*> The leading dimension of the array Q. LDQ >= 1. 226*> If COMPQ='V' or 'I', then LDQ >= N. 227*> \endverbatim 228*> 229*> \param[in,out] Z 230*> \verbatim 231*> Z is DOUBLE PRECISION array, dimension (LDZ, N) 232*> On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in 233*> the reduction of (A,B) to generalized Hessenberg form. 234*> On exit, if COMPZ = 'I', the orthogonal matrix of 235*> right Schur vectors of (H,T), and if COMPZ = 'V', the 236*> orthogonal matrix of right Schur vectors of (A,B). 237*> Not referenced if COMPZ = 'N'. 238*> \endverbatim 239*> 240*> \param[in] LDZ 241*> \verbatim 242*> LDZ is INTEGER 243*> The leading dimension of the array Z. LDZ >= 1. 244*> If COMPZ='V' or 'I', then LDZ >= N. 245*> \endverbatim 246*> 247*> \param[out] WORK 248*> \verbatim 249*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) 250*> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK. 251*> \endverbatim 252*> 253*> \param[in] LWORK 254*> \verbatim 255*> LWORK is INTEGER 256*> The dimension of the array WORK. LWORK >= max(1,N). 257*> 258*> If LWORK = -1, then a workspace query is assumed; the routine 259*> only calculates the optimal size of the WORK array, returns 260*> this value as the first entry of the WORK array, and no error 261*> message related to LWORK is issued by XERBLA. 262*> \endverbatim 263*> 264*> \param[out] INFO 265*> \verbatim 266*> INFO is INTEGER 267*> = 0: successful exit 268*> < 0: if INFO = -i, the i-th argument had an illegal value 269*> = 1,...,N: the QZ iteration did not converge. (H,T) is not 270*> in Schur form, but ALPHAR(i), ALPHAI(i), and 271*> BETA(i), i=INFO+1,...,N should be correct. 272*> = N+1,...,2*N: the shift calculation failed. (H,T) is not 273*> in Schur form, but ALPHAR(i), ALPHAI(i), and 274*> BETA(i), i=INFO-N+1,...,N should be correct. 275*> \endverbatim 276* 277* Authors: 278* ======== 279* 280*> \author Univ. of Tennessee 281*> \author Univ. of California Berkeley 282*> \author Univ. of Colorado Denver 283*> \author NAG Ltd. 284* 285*> \date June 2016 286* 287*> \ingroup doubleGEcomputational 288* 289*> \par Further Details: 290* ===================== 291*> 292*> \verbatim 293*> 294*> Iteration counters: 295*> 296*> JITER -- counts iterations. 297*> IITER -- counts iterations run since ILAST was last 298*> changed. This is therefore reset only when a 1-by-1 or 299*> 2-by-2 block deflates off the bottom. 300*> \endverbatim 301*> 302* ===================================================================== 303 SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, 304 $ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, 305 $ LWORK, INFO ) 306* 307* -- LAPACK computational routine (version 3.7.0) -- 308* -- LAPACK is a software package provided by Univ. of Tennessee, -- 309* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 310* June 2016 311* 312* .. Scalar Arguments .. 313 CHARACTER COMPQ, COMPZ, JOB 314 INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N 315* .. 316* .. Array Arguments .. 317 DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ), 318 $ H( LDH, * ), Q( LDQ, * ), T( LDT, * ), 319 $ WORK( * ), Z( LDZ, * ) 320* .. 321* 322* ===================================================================== 323* 324* .. Parameters .. 325* $ SAFETY = 1.0E+0 ) 326 DOUBLE PRECISION HALF, ZERO, ONE, SAFETY 327 PARAMETER ( HALF = 0.5D+0, ZERO = 0.0D+0, ONE = 1.0D+0, 328 $ SAFETY = 1.0D+2 ) 329* .. 330* .. Local Scalars .. 331 LOGICAL ILAZR2, ILAZRO, ILPIVT, ILQ, ILSCHR, ILZ, 332 $ LQUERY 333 INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST, 334 $ ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER, 335 $ JR, MAXIT 336 DOUBLE PRECISION A11, A12, A1I, A1R, A21, A22, A2I, A2R, AD11, 337 $ AD11L, AD12, AD12L, AD21, AD21L, AD22, AD22L, 338 $ AD32L, AN, ANORM, ASCALE, ATOL, B11, B1A, B1I, 339 $ B1R, B22, B2A, B2I, B2R, BN, BNORM, BSCALE, 340 $ BTOL, C, C11I, C11R, C12, C21, C22I, C22R, CL, 341 $ CQ, CR, CZ, ESHIFT, S, S1, S1INV, S2, SAFMAX, 342 $ SAFMIN, SCALE, SL, SQI, SQR, SR, SZI, SZR, T1, 343 $ TAU, TEMP, TEMP2, TEMPI, TEMPR, U1, U12, U12L, 344 $ U2, ULP, VS, W11, W12, W21, W22, WABS, WI, WR, 345 $ WR2 346* .. 347* .. Local Arrays .. 348 DOUBLE PRECISION V( 3 ) 349* .. 350* .. External Functions .. 351 LOGICAL LSAME 352 DOUBLE PRECISION DLAMCH, DLANHS, DLAPY2, DLAPY3 353 EXTERNAL LSAME, DLAMCH, DLANHS, DLAPY2, DLAPY3 354* .. 355* .. External Subroutines .. 356 EXTERNAL DLAG2, DLARFG, DLARTG, DLASET, DLASV2, DROT, 357 $ XERBLA 358* .. 359* .. Intrinsic Functions .. 360 INTRINSIC ABS, DBLE, MAX, MIN, SQRT 361* .. 362* .. Executable Statements .. 363* 364* Decode JOB, COMPQ, COMPZ 365* 366 IF( LSAME( JOB, 'E' ) ) THEN 367 ILSCHR = .FALSE. 368 ISCHUR = 1 369 ELSE IF( LSAME( JOB, 'S' ) ) THEN 370 ILSCHR = .TRUE. 371 ISCHUR = 2 372 ELSE 373 ISCHUR = 0 374 END IF 375* 376 IF( LSAME( COMPQ, 'N' ) ) THEN 377 ILQ = .FALSE. 378 ICOMPQ = 1 379 ELSE IF( LSAME( COMPQ, 'V' ) ) THEN 380 ILQ = .TRUE. 381 ICOMPQ = 2 382 ELSE IF( LSAME( COMPQ, 'I' ) ) THEN 383 ILQ = .TRUE. 384 ICOMPQ = 3 385 ELSE 386 ICOMPQ = 0 387 END IF 388* 389 IF( LSAME( COMPZ, 'N' ) ) THEN 390 ILZ = .FALSE. 391 ICOMPZ = 1 392 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN 393 ILZ = .TRUE. 394 ICOMPZ = 2 395 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN 396 ILZ = .TRUE. 397 ICOMPZ = 3 398 ELSE 399 ICOMPZ = 0 400 END IF 401* 402* Check Argument Values 403* 404 INFO = 0 405 WORK( 1 ) = MAX( 1, N ) 406 LQUERY = ( LWORK.EQ.-1 ) 407 IF( ISCHUR.EQ.0 ) THEN 408 INFO = -1 409 ELSE IF( ICOMPQ.EQ.0 ) THEN 410 INFO = -2 411 ELSE IF( ICOMPZ.EQ.0 ) THEN 412 INFO = -3 413 ELSE IF( N.LT.0 ) THEN 414 INFO = -4 415 ELSE IF( ILO.LT.1 ) THEN 416 INFO = -5 417 ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN 418 INFO = -6 419 ELSE IF( LDH.LT.N ) THEN 420 INFO = -8 421 ELSE IF( LDT.LT.N ) THEN 422 INFO = -10 423 ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN 424 INFO = -15 425 ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN 426 INFO = -17 427 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN 428 INFO = -19 429 END IF 430 IF( INFO.NE.0 ) THEN 431 CALL XERBLA( 'DHGEQZ', -INFO ) 432 RETURN 433 ELSE IF( LQUERY ) THEN 434 RETURN 435 END IF 436* 437* Quick return if possible 438* 439 IF( N.LE.0 ) THEN 440 WORK( 1 ) = DBLE( 1 ) 441 RETURN 442 END IF 443* 444* Initialize Q and Z 445* 446 IF( ICOMPQ.EQ.3 ) 447 $ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ ) 448 IF( ICOMPZ.EQ.3 ) 449 $ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ ) 450* 451* Machine Constants 452* 453 IN = IHI + 1 - ILO 454 SAFMIN = DLAMCH( 'S' ) 455 SAFMAX = ONE / SAFMIN 456 ULP = DLAMCH( 'E' )*DLAMCH( 'B' ) 457 ANORM = DLANHS( 'F', IN, H( ILO, ILO ), LDH, WORK ) 458 BNORM = DLANHS( 'F', IN, T( ILO, ILO ), LDT, WORK ) 459 ATOL = MAX( SAFMIN, ULP*ANORM ) 460 BTOL = MAX( SAFMIN, ULP*BNORM ) 461 ASCALE = ONE / MAX( SAFMIN, ANORM ) 462 BSCALE = ONE / MAX( SAFMIN, BNORM ) 463* 464* Set Eigenvalues IHI+1:N 465* 466 DO 30 J = IHI + 1, N 467 IF( T( J, J ).LT.ZERO ) THEN 468 IF( ILSCHR ) THEN 469 DO 10 JR = 1, J 470 H( JR, J ) = -H( JR, J ) 471 T( JR, J ) = -T( JR, J ) 472 10 CONTINUE 473 ELSE 474 H( J, J ) = -H( J, J ) 475 T( J, J ) = -T( J, J ) 476 END IF 477 IF( ILZ ) THEN 478 DO 20 JR = 1, N 479 Z( JR, J ) = -Z( JR, J ) 480 20 CONTINUE 481 END IF 482 END IF 483 ALPHAR( J ) = H( J, J ) 484 ALPHAI( J ) = ZERO 485 BETA( J ) = T( J, J ) 486 30 CONTINUE 487* 488* If IHI < ILO, skip QZ steps 489* 490 IF( IHI.LT.ILO ) 491 $ GO TO 380 492* 493* MAIN QZ ITERATION LOOP 494* 495* Initialize dynamic indices 496* 497* Eigenvalues ILAST+1:N have been found. 498* Column operations modify rows IFRSTM:whatever. 499* Row operations modify columns whatever:ILASTM. 500* 501* If only eigenvalues are being computed, then 502* IFRSTM is the row of the last splitting row above row ILAST; 503* this is always at least ILO. 504* IITER counts iterations since the last eigenvalue was found, 505* to tell when to use an extraordinary shift. 506* MAXIT is the maximum number of QZ sweeps allowed. 507* 508 ILAST = IHI 509 IF( ILSCHR ) THEN 510 IFRSTM = 1 511 ILASTM = N 512 ELSE 513 IFRSTM = ILO 514 ILASTM = IHI 515 END IF 516 IITER = 0 517 ESHIFT = ZERO 518 MAXIT = 30*( IHI-ILO+1 ) 519* 520 DO 360 JITER = 1, MAXIT 521* 522* Split the matrix if possible. 523* 524* Two tests: 525* 1: H(j,j-1)=0 or j=ILO 526* 2: T(j,j)=0 527* 528 IF( ILAST.EQ.ILO ) THEN 529* 530* Special case: j=ILAST 531* 532 GO TO 80 533 ELSE 534 IF( ABS( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN 535 H( ILAST, ILAST-1 ) = ZERO 536 GO TO 80 537 END IF 538 END IF 539* 540 IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN 541 T( ILAST, ILAST ) = ZERO 542 GO TO 70 543 END IF 544* 545* General case: j<ILAST 546* 547 DO 60 J = ILAST - 1, ILO, -1 548* 549* Test 1: for H(j,j-1)=0 or j=ILO 550* 551 IF( J.EQ.ILO ) THEN 552 ILAZRO = .TRUE. 553 ELSE 554 IF( ABS( H( J, J-1 ) ).LE.ATOL ) THEN 555 H( J, J-1 ) = ZERO 556 ILAZRO = .TRUE. 557 ELSE 558 ILAZRO = .FALSE. 559 END IF 560 END IF 561* 562* Test 2: for T(j,j)=0 563* 564 IF( ABS( T( J, J ) ).LT.BTOL ) THEN 565 T( J, J ) = ZERO 566* 567* Test 1a: Check for 2 consecutive small subdiagonals in A 568* 569 ILAZR2 = .FALSE. 570 IF( .NOT.ILAZRO ) THEN 571 TEMP = ABS( H( J, J-1 ) ) 572 TEMP2 = ABS( H( J, J ) ) 573 TEMPR = MAX( TEMP, TEMP2 ) 574 IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN 575 TEMP = TEMP / TEMPR 576 TEMP2 = TEMP2 / TEMPR 577 END IF 578 IF( TEMP*( ASCALE*ABS( H( J+1, J ) ) ).LE.TEMP2* 579 $ ( ASCALE*ATOL ) )ILAZR2 = .TRUE. 580 END IF 581* 582* If both tests pass (1 & 2), i.e., the leading diagonal 583* element of B in the block is zero, split a 1x1 block off 584* at the top. (I.e., at the J-th row/column) The leading 585* diagonal element of the remainder can also be zero, so 586* this may have to be done repeatedly. 587* 588 IF( ILAZRO .OR. ILAZR2 ) THEN 589 DO 40 JCH = J, ILAST - 1 590 TEMP = H( JCH, JCH ) 591 CALL DLARTG( TEMP, H( JCH+1, JCH ), C, S, 592 $ H( JCH, JCH ) ) 593 H( JCH+1, JCH ) = ZERO 594 CALL DROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH, 595 $ H( JCH+1, JCH+1 ), LDH, C, S ) 596 CALL DROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT, 597 $ T( JCH+1, JCH+1 ), LDT, C, S ) 598 IF( ILQ ) 599 $ CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1, 600 $ C, S ) 601 IF( ILAZR2 ) 602 $ H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C 603 ILAZR2 = .FALSE. 604 IF( ABS( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN 605 IF( JCH+1.GE.ILAST ) THEN 606 GO TO 80 607 ELSE 608 IFIRST = JCH + 1 609 GO TO 110 610 END IF 611 END IF 612 T( JCH+1, JCH+1 ) = ZERO 613 40 CONTINUE 614 GO TO 70 615 ELSE 616* 617* Only test 2 passed -- chase the zero to T(ILAST,ILAST) 618* Then process as in the case T(ILAST,ILAST)=0 619* 620 DO 50 JCH = J, ILAST - 1 621 TEMP = T( JCH, JCH+1 ) 622 CALL DLARTG( TEMP, T( JCH+1, JCH+1 ), C, S, 623 $ T( JCH, JCH+1 ) ) 624 T( JCH+1, JCH+1 ) = ZERO 625 IF( JCH.LT.ILASTM-1 ) 626 $ CALL DROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT, 627 $ T( JCH+1, JCH+2 ), LDT, C, S ) 628 CALL DROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH, 629 $ H( JCH+1, JCH-1 ), LDH, C, S ) 630 IF( ILQ ) 631 $ CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1, 632 $ C, S ) 633 TEMP = H( JCH+1, JCH ) 634 CALL DLARTG( TEMP, H( JCH+1, JCH-1 ), C, S, 635 $ H( JCH+1, JCH ) ) 636 H( JCH+1, JCH-1 ) = ZERO 637 CALL DROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1, 638 $ H( IFRSTM, JCH-1 ), 1, C, S ) 639 CALL DROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1, 640 $ T( IFRSTM, JCH-1 ), 1, C, S ) 641 IF( ILZ ) 642 $ CALL DROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1, 643 $ C, S ) 644 50 CONTINUE 645 GO TO 70 646 END IF 647 ELSE IF( ILAZRO ) THEN 648* 649* Only test 1 passed -- work on J:ILAST 650* 651 IFIRST = J 652 GO TO 110 653 END IF 654* 655* Neither test passed -- try next J 656* 657 60 CONTINUE 658* 659* (Drop-through is "impossible") 660* 661 INFO = N + 1 662 GO TO 420 663* 664* T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a 665* 1x1 block. 666* 667 70 CONTINUE 668 TEMP = H( ILAST, ILAST ) 669 CALL DLARTG( TEMP, H( ILAST, ILAST-1 ), C, S, 670 $ H( ILAST, ILAST ) ) 671 H( ILAST, ILAST-1 ) = ZERO 672 CALL DROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1, 673 $ H( IFRSTM, ILAST-1 ), 1, C, S ) 674 CALL DROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1, 675 $ T( IFRSTM, ILAST-1 ), 1, C, S ) 676 IF( ILZ ) 677 $ CALL DROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S ) 678* 679* H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHAR, ALPHAI, 680* and BETA 681* 682 80 CONTINUE 683 IF( T( ILAST, ILAST ).LT.ZERO ) THEN 684 IF( ILSCHR ) THEN 685 DO 90 J = IFRSTM, ILAST 686 H( J, ILAST ) = -H( J, ILAST ) 687 T( J, ILAST ) = -T( J, ILAST ) 688 90 CONTINUE 689 ELSE 690 H( ILAST, ILAST ) = -H( ILAST, ILAST ) 691 T( ILAST, ILAST ) = -T( ILAST, ILAST ) 692 END IF 693 IF( ILZ ) THEN 694 DO 100 J = 1, N 695 Z( J, ILAST ) = -Z( J, ILAST ) 696 100 CONTINUE 697 END IF 698 END IF 699 ALPHAR( ILAST ) = H( ILAST, ILAST ) 700 ALPHAI( ILAST ) = ZERO 701 BETA( ILAST ) = T( ILAST, ILAST ) 702* 703* Go to next block -- exit if finished. 704* 705 ILAST = ILAST - 1 706 IF( ILAST.LT.ILO ) 707 $ GO TO 380 708* 709* Reset counters 710* 711 IITER = 0 712 ESHIFT = ZERO 713 IF( .NOT.ILSCHR ) THEN 714 ILASTM = ILAST 715 IF( IFRSTM.GT.ILAST ) 716 $ IFRSTM = ILO 717 END IF 718 GO TO 350 719* 720* QZ step 721* 722* This iteration only involves rows/columns IFIRST:ILAST. We 723* assume IFIRST < ILAST, and that the diagonal of B is non-zero. 724* 725 110 CONTINUE 726 IITER = IITER + 1 727 IF( .NOT.ILSCHR ) THEN 728 IFRSTM = IFIRST 729 END IF 730* 731* Compute single shifts. 732* 733* At this point, IFIRST < ILAST, and the diagonal elements of 734* T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in 735* magnitude) 736* 737 IF( ( IITER / 10 )*10.EQ.IITER ) THEN 738* 739* Exceptional shift. Chosen for no particularly good reason. 740* (Single shift only.) 741* 742 IF( ( DBLE( MAXIT )*SAFMIN )*ABS( H( ILAST, ILAST-1 ) ).LT. 743 $ ABS( T( ILAST-1, ILAST-1 ) ) ) THEN 744 ESHIFT = H( ILAST, ILAST-1 ) / 745 $ T( ILAST-1, ILAST-1 ) 746 ELSE 747 ESHIFT = ESHIFT + ONE / ( SAFMIN*DBLE( MAXIT ) ) 748 END IF 749 S1 = ONE 750 WR = ESHIFT 751* 752 ELSE 753* 754* Shifts based on the generalized eigenvalues of the 755* bottom-right 2x2 block of A and B. The first eigenvalue 756* returned by DLAG2 is the Wilkinson shift (AEP p.512), 757* 758 CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH, 759 $ T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1, 760 $ S2, WR, WR2, WI ) 761* 762 IF ( ABS( (WR/S1)*T( ILAST, ILAST ) - H( ILAST, ILAST ) ) 763 $ .GT. ABS( (WR2/S2)*T( ILAST, ILAST ) 764 $ - H( ILAST, ILAST ) ) ) THEN 765 TEMP = WR 766 WR = WR2 767 WR2 = TEMP 768 TEMP = S1 769 S1 = S2 770 S2 = TEMP 771 END IF 772 TEMP = MAX( S1, SAFMIN*MAX( ONE, ABS( WR ), ABS( WI ) ) ) 773 IF( WI.NE.ZERO ) 774 $ GO TO 200 775 END IF 776* 777* Fiddle with shift to avoid overflow 778* 779 TEMP = MIN( ASCALE, ONE )*( HALF*SAFMAX ) 780 IF( S1.GT.TEMP ) THEN 781 SCALE = TEMP / S1 782 ELSE 783 SCALE = ONE 784 END IF 785* 786 TEMP = MIN( BSCALE, ONE )*( HALF*SAFMAX ) 787 IF( ABS( WR ).GT.TEMP ) 788 $ SCALE = MIN( SCALE, TEMP / ABS( WR ) ) 789 S1 = SCALE*S1 790 WR = SCALE*WR 791* 792* Now check for two consecutive small subdiagonals. 793* 794 DO 120 J = ILAST - 1, IFIRST + 1, -1 795 ISTART = J 796 TEMP = ABS( S1*H( J, J-1 ) ) 797 TEMP2 = ABS( S1*H( J, J )-WR*T( J, J ) ) 798 TEMPR = MAX( TEMP, TEMP2 ) 799 IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN 800 TEMP = TEMP / TEMPR 801 TEMP2 = TEMP2 / TEMPR 802 END IF 803 IF( ABS( ( ASCALE*H( J+1, J ) )*TEMP ).LE.( ASCALE*ATOL )* 804 $ TEMP2 )GO TO 130 805 120 CONTINUE 806* 807 ISTART = IFIRST 808 130 CONTINUE 809* 810* Do an implicit single-shift QZ sweep. 811* 812* Initial Q 813* 814 TEMP = S1*H( ISTART, ISTART ) - WR*T( ISTART, ISTART ) 815 TEMP2 = S1*H( ISTART+1, ISTART ) 816 CALL DLARTG( TEMP, TEMP2, C, S, TEMPR ) 817* 818* Sweep 819* 820 DO 190 J = ISTART, ILAST - 1 821 IF( J.GT.ISTART ) THEN 822 TEMP = H( J, J-1 ) 823 CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) ) 824 H( J+1, J-1 ) = ZERO 825 END IF 826* 827 DO 140 JC = J, ILASTM 828 TEMP = C*H( J, JC ) + S*H( J+1, JC ) 829 H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC ) 830 H( J, JC ) = TEMP 831 TEMP2 = C*T( J, JC ) + S*T( J+1, JC ) 832 T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC ) 833 T( J, JC ) = TEMP2 834 140 CONTINUE 835 IF( ILQ ) THEN 836 DO 150 JR = 1, N 837 TEMP = C*Q( JR, J ) + S*Q( JR, J+1 ) 838 Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 ) 839 Q( JR, J ) = TEMP 840 150 CONTINUE 841 END IF 842* 843 TEMP = T( J+1, J+1 ) 844 CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) ) 845 T( J+1, J ) = ZERO 846* 847 DO 160 JR = IFRSTM, MIN( J+2, ILAST ) 848 TEMP = C*H( JR, J+1 ) + S*H( JR, J ) 849 H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J ) 850 H( JR, J+1 ) = TEMP 851 160 CONTINUE 852 DO 170 JR = IFRSTM, J 853 TEMP = C*T( JR, J+1 ) + S*T( JR, J ) 854 T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J ) 855 T( JR, J+1 ) = TEMP 856 170 CONTINUE 857 IF( ILZ ) THEN 858 DO 180 JR = 1, N 859 TEMP = C*Z( JR, J+1 ) + S*Z( JR, J ) 860 Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J ) 861 Z( JR, J+1 ) = TEMP 862 180 CONTINUE 863 END IF 864 190 CONTINUE 865* 866 GO TO 350 867* 868* Use Francis double-shift 869* 870* Note: the Francis double-shift should work with real shifts, 871* but only if the block is at least 3x3. 872* This code may break if this point is reached with 873* a 2x2 block with real eigenvalues. 874* 875 200 CONTINUE 876 IF( IFIRST+1.EQ.ILAST ) THEN 877* 878* Special case -- 2x2 block with complex eigenvectors 879* 880* Step 1: Standardize, that is, rotate so that 881* 882* ( B11 0 ) 883* B = ( ) with B11 non-negative. 884* ( 0 B22 ) 885* 886 CALL DLASV2( T( ILAST-1, ILAST-1 ), T( ILAST-1, ILAST ), 887 $ T( ILAST, ILAST ), B22, B11, SR, CR, SL, CL ) 888* 889 IF( B11.LT.ZERO ) THEN 890 CR = -CR 891 SR = -SR 892 B11 = -B11 893 B22 = -B22 894 END IF 895* 896 CALL DROT( ILASTM+1-IFIRST, H( ILAST-1, ILAST-1 ), LDH, 897 $ H( ILAST, ILAST-1 ), LDH, CL, SL ) 898 CALL DROT( ILAST+1-IFRSTM, H( IFRSTM, ILAST-1 ), 1, 899 $ H( IFRSTM, ILAST ), 1, CR, SR ) 900* 901 IF( ILAST.LT.ILASTM ) 902 $ CALL DROT( ILASTM-ILAST, T( ILAST-1, ILAST+1 ), LDT, 903 $ T( ILAST, ILAST+1 ), LDT, CL, SL ) 904 IF( IFRSTM.LT.ILAST-1 ) 905 $ CALL DROT( IFIRST-IFRSTM, T( IFRSTM, ILAST-1 ), 1, 906 $ T( IFRSTM, ILAST ), 1, CR, SR ) 907* 908 IF( ILQ ) 909 $ CALL DROT( N, Q( 1, ILAST-1 ), 1, Q( 1, ILAST ), 1, CL, 910 $ SL ) 911 IF( ILZ ) 912 $ CALL DROT( N, Z( 1, ILAST-1 ), 1, Z( 1, ILAST ), 1, CR, 913 $ SR ) 914* 915 T( ILAST-1, ILAST-1 ) = B11 916 T( ILAST-1, ILAST ) = ZERO 917 T( ILAST, ILAST-1 ) = ZERO 918 T( ILAST, ILAST ) = B22 919* 920* If B22 is negative, negate column ILAST 921* 922 IF( B22.LT.ZERO ) THEN 923 DO 210 J = IFRSTM, ILAST 924 H( J, ILAST ) = -H( J, ILAST ) 925 T( J, ILAST ) = -T( J, ILAST ) 926 210 CONTINUE 927* 928 IF( ILZ ) THEN 929 DO 220 J = 1, N 930 Z( J, ILAST ) = -Z( J, ILAST ) 931 220 CONTINUE 932 END IF 933 B22 = -B22 934 END IF 935* 936* Step 2: Compute ALPHAR, ALPHAI, and BETA (see refs.) 937* 938* Recompute shift 939* 940 CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH, 941 $ T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1, 942 $ TEMP, WR, TEMP2, WI ) 943* 944* If standardization has perturbed the shift onto real line, 945* do another (real single-shift) QR step. 946* 947 IF( WI.EQ.ZERO ) 948 $ GO TO 350 949 S1INV = ONE / S1 950* 951* Do EISPACK (QZVAL) computation of alpha and beta 952* 953 A11 = H( ILAST-1, ILAST-1 ) 954 A21 = H( ILAST, ILAST-1 ) 955 A12 = H( ILAST-1, ILAST ) 956 A22 = H( ILAST, ILAST ) 957* 958* Compute complex Givens rotation on right 959* (Assume some element of C = (sA - wB) > unfl ) 960* __ 961* (sA - wB) ( CZ -SZ ) 962* ( SZ CZ ) 963* 964 C11R = S1*A11 - WR*B11 965 C11I = -WI*B11 966 C12 = S1*A12 967 C21 = S1*A21 968 C22R = S1*A22 - WR*B22 969 C22I = -WI*B22 970* 971 IF( ABS( C11R )+ABS( C11I )+ABS( C12 ).GT.ABS( C21 )+ 972 $ ABS( C22R )+ABS( C22I ) ) THEN 973 T1 = DLAPY3( C12, C11R, C11I ) 974 CZ = C12 / T1 975 SZR = -C11R / T1 976 SZI = -C11I / T1 977 ELSE 978 CZ = DLAPY2( C22R, C22I ) 979 IF( CZ.LE.SAFMIN ) THEN 980 CZ = ZERO 981 SZR = ONE 982 SZI = ZERO 983 ELSE 984 TEMPR = C22R / CZ 985 TEMPI = C22I / CZ 986 T1 = DLAPY2( CZ, C21 ) 987 CZ = CZ / T1 988 SZR = -C21*TEMPR / T1 989 SZI = C21*TEMPI / T1 990 END IF 991 END IF 992* 993* Compute Givens rotation on left 994* 995* ( CQ SQ ) 996* ( __ ) A or B 997* ( -SQ CQ ) 998* 999 AN = ABS( A11 ) + ABS( A12 ) + ABS( A21 ) + ABS( A22 ) 1000 BN = ABS( B11 ) + ABS( B22 ) 1001 WABS = ABS( WR ) + ABS( WI ) 1002 IF( S1*AN.GT.WABS*BN ) THEN 1003 CQ = CZ*B11 1004 SQR = SZR*B22 1005 SQI = -SZI*B22 1006 ELSE 1007 A1R = CZ*A11 + SZR*A12 1008 A1I = SZI*A12 1009 A2R = CZ*A21 + SZR*A22 1010 A2I = SZI*A22 1011 CQ = DLAPY2( A1R, A1I ) 1012 IF( CQ.LE.SAFMIN ) THEN 1013 CQ = ZERO 1014 SQR = ONE 1015 SQI = ZERO 1016 ELSE 1017 TEMPR = A1R / CQ 1018 TEMPI = A1I / CQ 1019 SQR = TEMPR*A2R + TEMPI*A2I 1020 SQI = TEMPI*A2R - TEMPR*A2I 1021 END IF 1022 END IF 1023 T1 = DLAPY3( CQ, SQR, SQI ) 1024 CQ = CQ / T1 1025 SQR = SQR / T1 1026 SQI = SQI / T1 1027* 1028* Compute diagonal elements of QBZ 1029* 1030 TEMPR = SQR*SZR - SQI*SZI 1031 TEMPI = SQR*SZI + SQI*SZR 1032 B1R = CQ*CZ*B11 + TEMPR*B22 1033 B1I = TEMPI*B22 1034 B1A = DLAPY2( B1R, B1I ) 1035 B2R = CQ*CZ*B22 + TEMPR*B11 1036 B2I = -TEMPI*B11 1037 B2A = DLAPY2( B2R, B2I ) 1038* 1039* Normalize so beta > 0, and Im( alpha1 ) > 0 1040* 1041 BETA( ILAST-1 ) = B1A 1042 BETA( ILAST ) = B2A 1043 ALPHAR( ILAST-1 ) = ( WR*B1A )*S1INV 1044 ALPHAI( ILAST-1 ) = ( WI*B1A )*S1INV 1045 ALPHAR( ILAST ) = ( WR*B2A )*S1INV 1046 ALPHAI( ILAST ) = -( WI*B2A )*S1INV 1047* 1048* Step 3: Go to next block -- exit if finished. 1049* 1050 ILAST = IFIRST - 1 1051 IF( ILAST.LT.ILO ) 1052 $ GO TO 380 1053* 1054* Reset counters 1055* 1056 IITER = 0 1057 ESHIFT = ZERO 1058 IF( .NOT.ILSCHR ) THEN 1059 ILASTM = ILAST 1060 IF( IFRSTM.GT.ILAST ) 1061 $ IFRSTM = ILO 1062 END IF 1063 GO TO 350 1064 ELSE 1065* 1066* Usual case: 3x3 or larger block, using Francis implicit 1067* double-shift 1068* 1069* 2 1070* Eigenvalue equation is w - c w + d = 0, 1071* 1072* -1 2 -1 1073* so compute 1st column of (A B ) - c A B + d 1074* using the formula in QZIT (from EISPACK) 1075* 1076* We assume that the block is at least 3x3 1077* 1078 AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) / 1079 $ ( BSCALE*T( ILAST-1, ILAST-1 ) ) 1080 AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) / 1081 $ ( BSCALE*T( ILAST-1, ILAST-1 ) ) 1082 AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) / 1083 $ ( BSCALE*T( ILAST, ILAST ) ) 1084 AD22 = ( ASCALE*H( ILAST, ILAST ) ) / 1085 $ ( BSCALE*T( ILAST, ILAST ) ) 1086 U12 = T( ILAST-1, ILAST ) / T( ILAST, ILAST ) 1087 AD11L = ( ASCALE*H( IFIRST, IFIRST ) ) / 1088 $ ( BSCALE*T( IFIRST, IFIRST ) ) 1089 AD21L = ( ASCALE*H( IFIRST+1, IFIRST ) ) / 1090 $ ( BSCALE*T( IFIRST, IFIRST ) ) 1091 AD12L = ( ASCALE*H( IFIRST, IFIRST+1 ) ) / 1092 $ ( BSCALE*T( IFIRST+1, IFIRST+1 ) ) 1093 AD22L = ( ASCALE*H( IFIRST+1, IFIRST+1 ) ) / 1094 $ ( BSCALE*T( IFIRST+1, IFIRST+1 ) ) 1095 AD32L = ( ASCALE*H( IFIRST+2, IFIRST+1 ) ) / 1096 $ ( BSCALE*T( IFIRST+1, IFIRST+1 ) ) 1097 U12L = T( IFIRST, IFIRST+1 ) / T( IFIRST+1, IFIRST+1 ) 1098* 1099 V( 1 ) = ( AD11-AD11L )*( AD22-AD11L ) - AD12*AD21 + 1100 $ AD21*U12*AD11L + ( AD12L-AD11L*U12L )*AD21L 1101 V( 2 ) = ( ( AD22L-AD11L )-AD21L*U12L-( AD11-AD11L )- 1102 $ ( AD22-AD11L )+AD21*U12 )*AD21L 1103 V( 3 ) = AD32L*AD21L 1104* 1105 ISTART = IFIRST 1106* 1107 CALL DLARFG( 3, V( 1 ), V( 2 ), 1, TAU ) 1108 V( 1 ) = ONE 1109* 1110* Sweep 1111* 1112 DO 290 J = ISTART, ILAST - 2 1113* 1114* All but last elements: use 3x3 Householder transforms. 1115* 1116* Zero (j-1)st column of A 1117* 1118 IF( J.GT.ISTART ) THEN 1119 V( 1 ) = H( J, J-1 ) 1120 V( 2 ) = H( J+1, J-1 ) 1121 V( 3 ) = H( J+2, J-1 ) 1122* 1123 CALL DLARFG( 3, H( J, J-1 ), V( 2 ), 1, TAU ) 1124 V( 1 ) = ONE 1125 H( J+1, J-1 ) = ZERO 1126 H( J+2, J-1 ) = ZERO 1127 END IF 1128* 1129 DO 230 JC = J, ILASTM 1130 TEMP = TAU*( H( J, JC )+V( 2 )*H( J+1, JC )+V( 3 )* 1131 $ H( J+2, JC ) ) 1132 H( J, JC ) = H( J, JC ) - TEMP 1133 H( J+1, JC ) = H( J+1, JC ) - TEMP*V( 2 ) 1134 H( J+2, JC ) = H( J+2, JC ) - TEMP*V( 3 ) 1135 TEMP2 = TAU*( T( J, JC )+V( 2 )*T( J+1, JC )+V( 3 )* 1136 $ T( J+2, JC ) ) 1137 T( J, JC ) = T( J, JC ) - TEMP2 1138 T( J+1, JC ) = T( J+1, JC ) - TEMP2*V( 2 ) 1139 T( J+2, JC ) = T( J+2, JC ) - TEMP2*V( 3 ) 1140 230 CONTINUE 1141 IF( ILQ ) THEN 1142 DO 240 JR = 1, N 1143 TEMP = TAU*( Q( JR, J )+V( 2 )*Q( JR, J+1 )+V( 3 )* 1144 $ Q( JR, J+2 ) ) 1145 Q( JR, J ) = Q( JR, J ) - TEMP 1146 Q( JR, J+1 ) = Q( JR, J+1 ) - TEMP*V( 2 ) 1147 Q( JR, J+2 ) = Q( JR, J+2 ) - TEMP*V( 3 ) 1148 240 CONTINUE 1149 END IF 1150* 1151* Zero j-th column of B (see DLAGBC for details) 1152* 1153* Swap rows to pivot 1154* 1155 ILPIVT = .FALSE. 1156 TEMP = MAX( ABS( T( J+1, J+1 ) ), ABS( T( J+1, J+2 ) ) ) 1157 TEMP2 = MAX( ABS( T( J+2, J+1 ) ), ABS( T( J+2, J+2 ) ) ) 1158 IF( MAX( TEMP, TEMP2 ).LT.SAFMIN ) THEN 1159 SCALE = ZERO 1160 U1 = ONE 1161 U2 = ZERO 1162 GO TO 250 1163 ELSE IF( TEMP.GE.TEMP2 ) THEN 1164 W11 = T( J+1, J+1 ) 1165 W21 = T( J+2, J+1 ) 1166 W12 = T( J+1, J+2 ) 1167 W22 = T( J+2, J+2 ) 1168 U1 = T( J+1, J ) 1169 U2 = T( J+2, J ) 1170 ELSE 1171 W21 = T( J+1, J+1 ) 1172 W11 = T( J+2, J+1 ) 1173 W22 = T( J+1, J+2 ) 1174 W12 = T( J+2, J+2 ) 1175 U2 = T( J+1, J ) 1176 U1 = T( J+2, J ) 1177 END IF 1178* 1179* Swap columns if nec. 1180* 1181 IF( ABS( W12 ).GT.ABS( W11 ) ) THEN 1182 ILPIVT = .TRUE. 1183 TEMP = W12 1184 TEMP2 = W22 1185 W12 = W11 1186 W22 = W21 1187 W11 = TEMP 1188 W21 = TEMP2 1189 END IF 1190* 1191* LU-factor 1192* 1193 TEMP = W21 / W11 1194 U2 = U2 - TEMP*U1 1195 W22 = W22 - TEMP*W12 1196 W21 = ZERO 1197* 1198* Compute SCALE 1199* 1200 SCALE = ONE 1201 IF( ABS( W22 ).LT.SAFMIN ) THEN 1202 SCALE = ZERO 1203 U2 = ONE 1204 U1 = -W12 / W11 1205 GO TO 250 1206 END IF 1207 IF( ABS( W22 ).LT.ABS( U2 ) ) 1208 $ SCALE = ABS( W22 / U2 ) 1209 IF( ABS( W11 ).LT.ABS( U1 ) ) 1210 $ SCALE = MIN( SCALE, ABS( W11 / U1 ) ) 1211* 1212* Solve 1213* 1214 U2 = ( SCALE*U2 ) / W22 1215 U1 = ( SCALE*U1-W12*U2 ) / W11 1216* 1217 250 CONTINUE 1218 IF( ILPIVT ) THEN 1219 TEMP = U2 1220 U2 = U1 1221 U1 = TEMP 1222 END IF 1223* 1224* Compute Householder Vector 1225* 1226 T1 = SQRT( SCALE**2+U1**2+U2**2 ) 1227 TAU = ONE + SCALE / T1 1228 VS = -ONE / ( SCALE+T1 ) 1229 V( 1 ) = ONE 1230 V( 2 ) = VS*U1 1231 V( 3 ) = VS*U2 1232* 1233* Apply transformations from the right. 1234* 1235 DO 260 JR = IFRSTM, MIN( J+3, ILAST ) 1236 TEMP = TAU*( H( JR, J )+V( 2 )*H( JR, J+1 )+V( 3 )* 1237 $ H( JR, J+2 ) ) 1238 H( JR, J ) = H( JR, J ) - TEMP 1239 H( JR, J+1 ) = H( JR, J+1 ) - TEMP*V( 2 ) 1240 H( JR, J+2 ) = H( JR, J+2 ) - TEMP*V( 3 ) 1241 260 CONTINUE 1242 DO 270 JR = IFRSTM, J + 2 1243 TEMP = TAU*( T( JR, J )+V( 2 )*T( JR, J+1 )+V( 3 )* 1244 $ T( JR, J+2 ) ) 1245 T( JR, J ) = T( JR, J ) - TEMP 1246 T( JR, J+1 ) = T( JR, J+1 ) - TEMP*V( 2 ) 1247 T( JR, J+2 ) = T( JR, J+2 ) - TEMP*V( 3 ) 1248 270 CONTINUE 1249 IF( ILZ ) THEN 1250 DO 280 JR = 1, N 1251 TEMP = TAU*( Z( JR, J )+V( 2 )*Z( JR, J+1 )+V( 3 )* 1252 $ Z( JR, J+2 ) ) 1253 Z( JR, J ) = Z( JR, J ) - TEMP 1254 Z( JR, J+1 ) = Z( JR, J+1 ) - TEMP*V( 2 ) 1255 Z( JR, J+2 ) = Z( JR, J+2 ) - TEMP*V( 3 ) 1256 280 CONTINUE 1257 END IF 1258 T( J+1, J ) = ZERO 1259 T( J+2, J ) = ZERO 1260 290 CONTINUE 1261* 1262* Last elements: Use Givens rotations 1263* 1264* Rotations from the left 1265* 1266 J = ILAST - 1 1267 TEMP = H( J, J-1 ) 1268 CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) ) 1269 H( J+1, J-1 ) = ZERO 1270* 1271 DO 300 JC = J, ILASTM 1272 TEMP = C*H( J, JC ) + S*H( J+1, JC ) 1273 H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC ) 1274 H( J, JC ) = TEMP 1275 TEMP2 = C*T( J, JC ) + S*T( J+1, JC ) 1276 T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC ) 1277 T( J, JC ) = TEMP2 1278 300 CONTINUE 1279 IF( ILQ ) THEN 1280 DO 310 JR = 1, N 1281 TEMP = C*Q( JR, J ) + S*Q( JR, J+1 ) 1282 Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 ) 1283 Q( JR, J ) = TEMP 1284 310 CONTINUE 1285 END IF 1286* 1287* Rotations from the right. 1288* 1289 TEMP = T( J+1, J+1 ) 1290 CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) ) 1291 T( J+1, J ) = ZERO 1292* 1293 DO 320 JR = IFRSTM, ILAST 1294 TEMP = C*H( JR, J+1 ) + S*H( JR, J ) 1295 H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J ) 1296 H( JR, J+1 ) = TEMP 1297 320 CONTINUE 1298 DO 330 JR = IFRSTM, ILAST - 1 1299 TEMP = C*T( JR, J+1 ) + S*T( JR, J ) 1300 T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J ) 1301 T( JR, J+1 ) = TEMP 1302 330 CONTINUE 1303 IF( ILZ ) THEN 1304 DO 340 JR = 1, N 1305 TEMP = C*Z( JR, J+1 ) + S*Z( JR, J ) 1306 Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J ) 1307 Z( JR, J+1 ) = TEMP 1308 340 CONTINUE 1309 END IF 1310* 1311* End of Double-Shift code 1312* 1313 END IF 1314* 1315 GO TO 350 1316* 1317* End of iteration loop 1318* 1319 350 CONTINUE 1320 360 CONTINUE 1321* 1322* Drop-through = non-convergence 1323* 1324 INFO = ILAST 1325 GO TO 420 1326* 1327* Successful completion of all QZ steps 1328* 1329 380 CONTINUE 1330* 1331* Set Eigenvalues 1:ILO-1 1332* 1333 DO 410 J = 1, ILO - 1 1334 IF( T( J, J ).LT.ZERO ) THEN 1335 IF( ILSCHR ) THEN 1336 DO 390 JR = 1, J 1337 H( JR, J ) = -H( JR, J ) 1338 T( JR, J ) = -T( JR, J ) 1339 390 CONTINUE 1340 ELSE 1341 H( J, J ) = -H( J, J ) 1342 T( J, J ) = -T( J, J ) 1343 END IF 1344 IF( ILZ ) THEN 1345 DO 400 JR = 1, N 1346 Z( JR, J ) = -Z( JR, J ) 1347 400 CONTINUE 1348 END IF 1349 END IF 1350 ALPHAR( J ) = H( J, J ) 1351 ALPHAI( J ) = ZERO 1352 BETA( J ) = T( J, J ) 1353 410 CONTINUE 1354* 1355* Normal Termination 1356* 1357 INFO = 0 1358* 1359* Exit (other than argument error) -- return optimal workspace size 1360* 1361 420 CONTINUE 1362 WORK( 1 ) = DBLE( N ) 1363 RETURN 1364* 1365* End of DHGEQZ 1366* 1367 END 1368