1*> \brief <b> ZGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b> 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download ZGGEVX + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggevx.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggevx.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggevx.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE ZGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, 22* ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, 23* LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, 24* WORK, LWORK, RWORK, IWORK, BWORK, INFO ) 25* 26* .. Scalar Arguments .. 27* CHARACTER BALANC, JOBVL, JOBVR, SENSE 28* INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N 29* DOUBLE PRECISION ABNRM, BBNRM 30* .. 31* .. Array Arguments .. 32* LOGICAL BWORK( * ) 33* INTEGER IWORK( * ) 34* DOUBLE PRECISION LSCALE( * ), RCONDE( * ), RCONDV( * ), 35* $ RSCALE( * ), RWORK( * ) 36* COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), 37* $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ), 38* $ WORK( * ) 39* .. 40* 41* 42*> \par Purpose: 43* ============= 44*> 45*> \verbatim 46*> 47*> ZGGEVX computes for a pair of N-by-N complex nonsymmetric matrices 48*> (A,B) the generalized eigenvalues, and optionally, the left and/or 49*> right generalized eigenvectors. 50*> 51*> Optionally, it also computes a balancing transformation to improve 52*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI, 53*> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for 54*> the eigenvalues (RCONDE), and reciprocal condition numbers for the 55*> right eigenvectors (RCONDV). 56*> 57*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar 58*> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is 59*> singular. It is usually represented as the pair (alpha,beta), as 60*> there is a reasonable interpretation for beta=0, and even for both 61*> being zero. 62*> 63*> The right eigenvector v(j) corresponding to the eigenvalue lambda(j) 64*> of (A,B) satisfies 65*> A * v(j) = lambda(j) * B * v(j) . 66*> The left eigenvector u(j) corresponding to the eigenvalue lambda(j) 67*> of (A,B) satisfies 68*> u(j)**H * A = lambda(j) * u(j)**H * B. 69*> where u(j)**H is the conjugate-transpose of u(j). 70*> 71*> \endverbatim 72* 73* Arguments: 74* ========== 75* 76*> \param[in] BALANC 77*> \verbatim 78*> BALANC is CHARACTER*1 79*> Specifies the balance option to be performed: 80*> = 'N': do not diagonally scale or permute; 81*> = 'P': permute only; 82*> = 'S': scale only; 83*> = 'B': both permute and scale. 84*> Computed reciprocal condition numbers will be for the 85*> matrices after permuting and/or balancing. Permuting does 86*> not change condition numbers (in exact arithmetic), but 87*> balancing does. 88*> \endverbatim 89*> 90*> \param[in] JOBVL 91*> \verbatim 92*> JOBVL is CHARACTER*1 93*> = 'N': do not compute the left generalized eigenvectors; 94*> = 'V': compute the left generalized eigenvectors. 95*> \endverbatim 96*> 97*> \param[in] JOBVR 98*> \verbatim 99*> JOBVR is CHARACTER*1 100*> = 'N': do not compute the right generalized eigenvectors; 101*> = 'V': compute the right generalized eigenvectors. 102*> \endverbatim 103*> 104*> \param[in] SENSE 105*> \verbatim 106*> SENSE is CHARACTER*1 107*> Determines which reciprocal condition numbers are computed. 108*> = 'N': none are computed; 109*> = 'E': computed for eigenvalues only; 110*> = 'V': computed for eigenvectors only; 111*> = 'B': computed for eigenvalues and eigenvectors. 112*> \endverbatim 113*> 114*> \param[in] N 115*> \verbatim 116*> N is INTEGER 117*> The order of the matrices A, B, VL, and VR. N >= 0. 118*> \endverbatim 119*> 120*> \param[in,out] A 121*> \verbatim 122*> A is COMPLEX*16 array, dimension (LDA, N) 123*> On entry, the matrix A in the pair (A,B). 124*> On exit, A has been overwritten. If JOBVL='V' or JOBVR='V' 125*> or both, then A contains the first part of the complex Schur 126*> form of the "balanced" versions of the input A and B. 127*> \endverbatim 128*> 129*> \param[in] LDA 130*> \verbatim 131*> LDA is INTEGER 132*> The leading dimension of A. LDA >= max(1,N). 133*> \endverbatim 134*> 135*> \param[in,out] B 136*> \verbatim 137*> B is COMPLEX*16 array, dimension (LDB, N) 138*> On entry, the matrix B in the pair (A,B). 139*> On exit, B has been overwritten. If JOBVL='V' or JOBVR='V' 140*> or both, then B contains the second part of the complex 141*> Schur form of the "balanced" versions of the input A and B. 142*> \endverbatim 143*> 144*> \param[in] LDB 145*> \verbatim 146*> LDB is INTEGER 147*> The leading dimension of B. LDB >= max(1,N). 148*> \endverbatim 149*> 150*> \param[out] ALPHA 151*> \verbatim 152*> ALPHA is COMPLEX*16 array, dimension (N) 153*> \endverbatim 154*> 155*> \param[out] BETA 156*> \verbatim 157*> BETA is COMPLEX*16 array, dimension (N) 158*> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized 159*> eigenvalues. 160*> 161*> Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or 162*> underflow, and BETA(j) may even be zero. Thus, the user 163*> should avoid naively computing the ratio ALPHA/BETA. 164*> However, ALPHA will be always less than and usually 165*> comparable with norm(A) in magnitude, and BETA always less 166*> than and usually comparable with norm(B). 167*> \endverbatim 168*> 169*> \param[out] VL 170*> \verbatim 171*> VL is COMPLEX*16 array, dimension (LDVL,N) 172*> If JOBVL = 'V', the left generalized eigenvectors u(j) are 173*> stored one after another in the columns of VL, in the same 174*> order as their eigenvalues. 175*> Each eigenvector will be scaled so the largest component 176*> will have abs(real part) + abs(imag. part) = 1. 177*> Not referenced if JOBVL = 'N'. 178*> \endverbatim 179*> 180*> \param[in] LDVL 181*> \verbatim 182*> LDVL is INTEGER 183*> The leading dimension of the matrix VL. LDVL >= 1, and 184*> if JOBVL = 'V', LDVL >= N. 185*> \endverbatim 186*> 187*> \param[out] VR 188*> \verbatim 189*> VR is COMPLEX*16 array, dimension (LDVR,N) 190*> If JOBVR = 'V', the right generalized eigenvectors v(j) are 191*> stored one after another in the columns of VR, in the same 192*> order as their eigenvalues. 193*> Each eigenvector will be scaled so the largest component 194*> will have abs(real part) + abs(imag. part) = 1. 195*> Not referenced if JOBVR = 'N'. 196*> \endverbatim 197*> 198*> \param[in] LDVR 199*> \verbatim 200*> LDVR is INTEGER 201*> The leading dimension of the matrix VR. LDVR >= 1, and 202*> if JOBVR = 'V', LDVR >= N. 203*> \endverbatim 204*> 205*> \param[out] ILO 206*> \verbatim 207*> ILO is INTEGER 208*> \endverbatim 209*> 210*> \param[out] IHI 211*> \verbatim 212*> IHI is INTEGER 213*> ILO and IHI are integer values such that on exit 214*> A(i,j) = 0 and B(i,j) = 0 if i > j and 215*> j = 1,...,ILO-1 or i = IHI+1,...,N. 216*> If BALANC = 'N' or 'S', ILO = 1 and IHI = N. 217*> \endverbatim 218*> 219*> \param[out] LSCALE 220*> \verbatim 221*> LSCALE is DOUBLE PRECISION array, dimension (N) 222*> Details of the permutations and scaling factors applied 223*> to the left side of A and B. If PL(j) is the index of the 224*> row interchanged with row j, and DL(j) is the scaling 225*> factor applied to row j, then 226*> LSCALE(j) = PL(j) for j = 1,...,ILO-1 227*> = DL(j) for j = ILO,...,IHI 228*> = PL(j) for j = IHI+1,...,N. 229*> The order in which the interchanges are made is N to IHI+1, 230*> then 1 to ILO-1. 231*> \endverbatim 232*> 233*> \param[out] RSCALE 234*> \verbatim 235*> RSCALE is DOUBLE PRECISION array, dimension (N) 236*> Details of the permutations and scaling factors applied 237*> to the right side of A and B. If PR(j) is the index of the 238*> column interchanged with column j, and DR(j) is the scaling 239*> factor applied to column j, then 240*> RSCALE(j) = PR(j) for j = 1,...,ILO-1 241*> = DR(j) for j = ILO,...,IHI 242*> = PR(j) for j = IHI+1,...,N 243*> The order in which the interchanges are made is N to IHI+1, 244*> then 1 to ILO-1. 245*> \endverbatim 246*> 247*> \param[out] ABNRM 248*> \verbatim 249*> ABNRM is DOUBLE PRECISION 250*> The one-norm of the balanced matrix A. 251*> \endverbatim 252*> 253*> \param[out] BBNRM 254*> \verbatim 255*> BBNRM is DOUBLE PRECISION 256*> The one-norm of the balanced matrix B. 257*> \endverbatim 258*> 259*> \param[out] RCONDE 260*> \verbatim 261*> RCONDE is DOUBLE PRECISION array, dimension (N) 262*> If SENSE = 'E' or 'B', the reciprocal condition numbers of 263*> the eigenvalues, stored in consecutive elements of the array. 264*> If SENSE = 'N' or 'V', RCONDE is not referenced. 265*> \endverbatim 266*> 267*> \param[out] RCONDV 268*> \verbatim 269*> RCONDV is DOUBLE PRECISION array, dimension (N) 270*> If JOB = 'V' or 'B', the estimated reciprocal condition 271*> numbers of the eigenvectors, stored in consecutive elements 272*> of the array. If the eigenvalues cannot be reordered to 273*> compute RCONDV(j), RCONDV(j) is set to 0; this can only occur 274*> when the true value would be very small anyway. 275*> If SENSE = 'N' or 'E', RCONDV is not referenced. 276*> \endverbatim 277*> 278*> \param[out] WORK 279*> \verbatim 280*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) 281*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 282*> \endverbatim 283*> 284*> \param[in] LWORK 285*> \verbatim 286*> LWORK is INTEGER 287*> The dimension of the array WORK. LWORK >= max(1,2*N). 288*> If SENSE = 'E', LWORK >= max(1,4*N). 289*> If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N). 290*> 291*> If LWORK = -1, then a workspace query is assumed; the routine 292*> only calculates the optimal size of the WORK array, returns 293*> this value as the first entry of the WORK array, and no error 294*> message related to LWORK is issued by XERBLA. 295*> \endverbatim 296*> 297*> \param[out] RWORK 298*> \verbatim 299*> RWORK is DOUBLE PRECISION array, dimension (lrwork) 300*> lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B', 301*> and at least max(1,2*N) otherwise. 302*> Real workspace. 303*> \endverbatim 304*> 305*> \param[out] IWORK 306*> \verbatim 307*> IWORK is INTEGER array, dimension (N+2) 308*> If SENSE = 'E', IWORK is not referenced. 309*> \endverbatim 310*> 311*> \param[out] BWORK 312*> \verbatim 313*> BWORK is LOGICAL array, dimension (N) 314*> If SENSE = 'N', BWORK is not referenced. 315*> \endverbatim 316*> 317*> \param[out] INFO 318*> \verbatim 319*> INFO is INTEGER 320*> = 0: successful exit 321*> < 0: if INFO = -i, the i-th argument had an illegal value. 322*> = 1,...,N: 323*> The QZ iteration failed. No eigenvectors have been 324*> calculated, but ALPHA(j) and BETA(j) should be correct 325*> for j=INFO+1,...,N. 326*> > N: =N+1: other than QZ iteration failed in ZHGEQZ. 327*> =N+2: error return from ZTGEVC. 328*> \endverbatim 329* 330* Authors: 331* ======== 332* 333*> \author Univ. of Tennessee 334*> \author Univ. of California Berkeley 335*> \author Univ. of Colorado Denver 336*> \author NAG Ltd. 337* 338*> \ingroup complex16GEeigen 339* 340*> \par Further Details: 341* ===================== 342*> 343*> \verbatim 344*> 345*> Balancing a matrix pair (A,B) includes, first, permuting rows and 346*> columns to isolate eigenvalues, second, applying diagonal similarity 347*> transformation to the rows and columns to make the rows and columns 348*> as close in norm as possible. The computed reciprocal condition 349*> numbers correspond to the balanced matrix. Permuting rows and columns 350*> will not change the condition numbers (in exact arithmetic) but 351*> diagonal scaling will. For further explanation of balancing, see 352*> section 4.11.1.2 of LAPACK Users' Guide. 353*> 354*> An approximate error bound on the chordal distance between the i-th 355*> computed generalized eigenvalue w and the corresponding exact 356*> eigenvalue lambda is 357*> 358*> chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I) 359*> 360*> An approximate error bound for the angle between the i-th computed 361*> eigenvector VL(i) or VR(i) is given by 362*> 363*> EPS * norm(ABNRM, BBNRM) / DIF(i). 364*> 365*> For further explanation of the reciprocal condition numbers RCONDE 366*> and RCONDV, see section 4.11 of LAPACK User's Guide. 367*> \endverbatim 368*> 369* ===================================================================== 370 SUBROUTINE ZGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, 371 $ ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, 372 $ LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, 373 $ WORK, LWORK, RWORK, IWORK, BWORK, INFO ) 374* 375* -- LAPACK driver routine -- 376* -- LAPACK is a software package provided by Univ. of Tennessee, -- 377* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 378* 379* .. Scalar Arguments .. 380 CHARACTER BALANC, JOBVL, JOBVR, SENSE 381 INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N 382 DOUBLE PRECISION ABNRM, BBNRM 383* .. 384* .. Array Arguments .. 385 LOGICAL BWORK( * ) 386 INTEGER IWORK( * ) 387 DOUBLE PRECISION LSCALE( * ), RCONDE( * ), RCONDV( * ), 388 $ RSCALE( * ), RWORK( * ) 389 COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), 390 $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ), 391 $ WORK( * ) 392* .. 393* 394* ===================================================================== 395* 396* .. Parameters .. 397 DOUBLE PRECISION ZERO, ONE 398 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 399 COMPLEX*16 CZERO, CONE 400 PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), 401 $ CONE = ( 1.0D+0, 0.0D+0 ) ) 402* .. 403* .. Local Scalars .. 404 LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL, 405 $ WANTSB, WANTSE, WANTSN, WANTSV 406 CHARACTER CHTEMP 407 INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS, 408 $ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK, MINWRK 409 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, 410 $ SMLNUM, TEMP 411 COMPLEX*16 X 412* .. 413* .. Local Arrays .. 414 LOGICAL LDUMMA( 1 ) 415* .. 416* .. External Subroutines .. 417 EXTERNAL DLABAD, DLASCL, XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, 418 $ ZGGHRD, ZHGEQZ, ZLACPY, ZLASCL, ZLASET, ZTGEVC, 419 $ ZTGSNA, ZUNGQR, ZUNMQR 420* .. 421* .. External Functions .. 422 LOGICAL LSAME 423 INTEGER ILAENV 424 DOUBLE PRECISION DLAMCH, ZLANGE 425 EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE 426* .. 427* .. Intrinsic Functions .. 428 INTRINSIC ABS, DBLE, DIMAG, MAX, SQRT 429* .. 430* .. Statement Functions .. 431 DOUBLE PRECISION ABS1 432* .. 433* .. Statement Function definitions .. 434 ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) ) 435* .. 436* .. Executable Statements .. 437* 438* Decode the input arguments 439* 440 IF( LSAME( JOBVL, 'N' ) ) THEN 441 IJOBVL = 1 442 ILVL = .FALSE. 443 ELSE IF( LSAME( JOBVL, 'V' ) ) THEN 444 IJOBVL = 2 445 ILVL = .TRUE. 446 ELSE 447 IJOBVL = -1 448 ILVL = .FALSE. 449 END IF 450* 451 IF( LSAME( JOBVR, 'N' ) ) THEN 452 IJOBVR = 1 453 ILVR = .FALSE. 454 ELSE IF( LSAME( JOBVR, 'V' ) ) THEN 455 IJOBVR = 2 456 ILVR = .TRUE. 457 ELSE 458 IJOBVR = -1 459 ILVR = .FALSE. 460 END IF 461 ILV = ILVL .OR. ILVR 462* 463 NOSCL = LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'P' ) 464 WANTSN = LSAME( SENSE, 'N' ) 465 WANTSE = LSAME( SENSE, 'E' ) 466 WANTSV = LSAME( SENSE, 'V' ) 467 WANTSB = LSAME( SENSE, 'B' ) 468* 469* Test the input arguments 470* 471 INFO = 0 472 LQUERY = ( LWORK.EQ.-1 ) 473 IF( .NOT.( NOSCL .OR. LSAME( BALANC,'S' ) .OR. 474 $ LSAME( BALANC, 'B' ) ) ) THEN 475 INFO = -1 476 ELSE IF( IJOBVL.LE.0 ) THEN 477 INFO = -2 478 ELSE IF( IJOBVR.LE.0 ) THEN 479 INFO = -3 480 ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) ) 481 $ THEN 482 INFO = -4 483 ELSE IF( N.LT.0 ) THEN 484 INFO = -5 485 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 486 INFO = -7 487 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 488 INFO = -9 489 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN 490 INFO = -13 491 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN 492 INFO = -15 493 END IF 494* 495* Compute workspace 496* (Note: Comments in the code beginning "Workspace:" describe the 497* minimal amount of workspace needed at that point in the code, 498* as well as the preferred amount for good performance. 499* NB refers to the optimal block size for the immediately 500* following subroutine, as returned by ILAENV. The workspace is 501* computed assuming ILO = 1 and IHI = N, the worst case.) 502* 503 IF( INFO.EQ.0 ) THEN 504 IF( N.EQ.0 ) THEN 505 MINWRK = 1 506 MAXWRK = 1 507 ELSE 508 MINWRK = 2*N 509 IF( WANTSE ) THEN 510 MINWRK = 4*N 511 ELSE IF( WANTSV .OR. WANTSB ) THEN 512 MINWRK = 2*N*( N + 1) 513 END IF 514 MAXWRK = MINWRK 515 MAXWRK = MAX( MAXWRK, 516 $ N + N*ILAENV( 1, 'ZGEQRF', ' ', N, 1, N, 0 ) ) 517 MAXWRK = MAX( MAXWRK, 518 $ N + N*ILAENV( 1, 'ZUNMQR', ' ', N, 1, N, 0 ) ) 519 IF( ILVL ) THEN 520 MAXWRK = MAX( MAXWRK, N + 521 $ N*ILAENV( 1, 'ZUNGQR', ' ', N, 1, N, 0 ) ) 522 END IF 523 END IF 524 WORK( 1 ) = MAXWRK 525* 526 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN 527 INFO = -25 528 END IF 529 END IF 530* 531 IF( INFO.NE.0 ) THEN 532 CALL XERBLA( 'ZGGEVX', -INFO ) 533 RETURN 534 ELSE IF( LQUERY ) THEN 535 RETURN 536 END IF 537* 538* Quick return if possible 539* 540 IF( N.EQ.0 ) 541 $ RETURN 542* 543* Get machine constants 544* 545 EPS = DLAMCH( 'P' ) 546 SMLNUM = DLAMCH( 'S' ) 547 BIGNUM = ONE / SMLNUM 548 CALL DLABAD( SMLNUM, BIGNUM ) 549 SMLNUM = SQRT( SMLNUM ) / EPS 550 BIGNUM = ONE / SMLNUM 551* 552* Scale A if max element outside range [SMLNUM,BIGNUM] 553* 554 ANRM = ZLANGE( 'M', N, N, A, LDA, RWORK ) 555 ILASCL = .FALSE. 556 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN 557 ANRMTO = SMLNUM 558 ILASCL = .TRUE. 559 ELSE IF( ANRM.GT.BIGNUM ) THEN 560 ANRMTO = BIGNUM 561 ILASCL = .TRUE. 562 END IF 563 IF( ILASCL ) 564 $ CALL ZLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR ) 565* 566* Scale B if max element outside range [SMLNUM,BIGNUM] 567* 568 BNRM = ZLANGE( 'M', N, N, B, LDB, RWORK ) 569 ILBSCL = .FALSE. 570 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN 571 BNRMTO = SMLNUM 572 ILBSCL = .TRUE. 573 ELSE IF( BNRM.GT.BIGNUM ) THEN 574 BNRMTO = BIGNUM 575 ILBSCL = .TRUE. 576 END IF 577 IF( ILBSCL ) 578 $ CALL ZLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR ) 579* 580* Permute and/or balance the matrix pair (A,B) 581* (Real Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise) 582* 583 CALL ZGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, 584 $ RWORK, IERR ) 585* 586* Compute ABNRM and BBNRM 587* 588 ABNRM = ZLANGE( '1', N, N, A, LDA, RWORK( 1 ) ) 589 IF( ILASCL ) THEN 590 RWORK( 1 ) = ABNRM 591 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, RWORK( 1 ), 1, 592 $ IERR ) 593 ABNRM = RWORK( 1 ) 594 END IF 595* 596 BBNRM = ZLANGE( '1', N, N, B, LDB, RWORK( 1 ) ) 597 IF( ILBSCL ) THEN 598 RWORK( 1 ) = BBNRM 599 CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, RWORK( 1 ), 1, 600 $ IERR ) 601 BBNRM = RWORK( 1 ) 602 END IF 603* 604* Reduce B to triangular form (QR decomposition of B) 605* (Complex Workspace: need N, prefer N*NB ) 606* 607 IROWS = IHI + 1 - ILO 608 IF( ILV .OR. .NOT.WANTSN ) THEN 609 ICOLS = N + 1 - ILO 610 ELSE 611 ICOLS = IROWS 612 END IF 613 ITAU = 1 614 IWRK = ITAU + IROWS 615 CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), 616 $ WORK( IWRK ), LWORK+1-IWRK, IERR ) 617* 618* Apply the unitary transformation to A 619* (Complex Workspace: need N, prefer N*NB) 620* 621 CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, 622 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ), 623 $ LWORK+1-IWRK, IERR ) 624* 625* Initialize VL and/or VR 626* (Workspace: need N, prefer N*NB) 627* 628 IF( ILVL ) THEN 629 CALL ZLASET( 'Full', N, N, CZERO, CONE, VL, LDVL ) 630 IF( IROWS.GT.1 ) THEN 631 CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, 632 $ VL( ILO+1, ILO ), LDVL ) 633 END IF 634 CALL ZUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL, 635 $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) 636 END IF 637* 638 IF( ILVR ) 639 $ CALL ZLASET( 'Full', N, N, CZERO, CONE, VR, LDVR ) 640* 641* Reduce to generalized Hessenberg form 642* (Workspace: none needed) 643* 644 IF( ILV .OR. .NOT.WANTSN ) THEN 645* 646* Eigenvectors requested -- work on whole matrix. 647* 648 CALL ZGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL, 649 $ LDVL, VR, LDVR, IERR ) 650 ELSE 651 CALL ZGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA, 652 $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR ) 653 END IF 654* 655* Perform QZ algorithm (Compute eigenvalues, and optionally, the 656* Schur forms and Schur vectors) 657* (Complex Workspace: need N) 658* (Real Workspace: need N) 659* 660 IWRK = ITAU 661 IF( ILV .OR. .NOT.WANTSN ) THEN 662 CHTEMP = 'S' 663 ELSE 664 CHTEMP = 'E' 665 END IF 666* 667 CALL ZHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, 668 $ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWRK ), 669 $ LWORK+1-IWRK, RWORK, IERR ) 670 IF( IERR.NE.0 ) THEN 671 IF( IERR.GT.0 .AND. IERR.LE.N ) THEN 672 INFO = IERR 673 ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN 674 INFO = IERR - N 675 ELSE 676 INFO = N + 1 677 END IF 678 GO TO 90 679 END IF 680* 681* Compute Eigenvectors and estimate condition numbers if desired 682* ZTGEVC: (Complex Workspace: need 2*N ) 683* (Real Workspace: need 2*N ) 684* ZTGSNA: (Complex Workspace: need 2*N*N if SENSE='V' or 'B') 685* (Integer Workspace: need N+2 ) 686* 687 IF( ILV .OR. .NOT.WANTSN ) THEN 688 IF( ILV ) THEN 689 IF( ILVL ) THEN 690 IF( ILVR ) THEN 691 CHTEMP = 'B' 692 ELSE 693 CHTEMP = 'L' 694 END IF 695 ELSE 696 CHTEMP = 'R' 697 END IF 698* 699 CALL ZTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, 700 $ LDVL, VR, LDVR, N, IN, WORK( IWRK ), RWORK, 701 $ IERR ) 702 IF( IERR.NE.0 ) THEN 703 INFO = N + 2 704 GO TO 90 705 END IF 706 END IF 707* 708 IF( .NOT.WANTSN ) THEN 709* 710* compute eigenvectors (ZTGEVC) and estimate condition 711* numbers (ZTGSNA). Note that the definition of the condition 712* number is not invariant under transformation (u,v) to 713* (Q*u, Z*v), where (u,v) are eigenvectors of the generalized 714* Schur form (S,T), Q and Z are orthogonal matrices. In order 715* to avoid using extra 2*N*N workspace, we have to 716* re-calculate eigenvectors and estimate the condition numbers 717* one at a time. 718* 719 DO 20 I = 1, N 720* 721 DO 10 J = 1, N 722 BWORK( J ) = .FALSE. 723 10 CONTINUE 724 BWORK( I ) = .TRUE. 725* 726 IWRK = N + 1 727 IWRK1 = IWRK + N 728* 729 IF( WANTSE .OR. WANTSB ) THEN 730 CALL ZTGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB, 731 $ WORK( 1 ), N, WORK( IWRK ), N, 1, M, 732 $ WORK( IWRK1 ), RWORK, IERR ) 733 IF( IERR.NE.0 ) THEN 734 INFO = N + 2 735 GO TO 90 736 END IF 737 END IF 738* 739 CALL ZTGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB, 740 $ WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ), 741 $ RCONDV( I ), 1, M, WORK( IWRK1 ), 742 $ LWORK-IWRK1+1, IWORK, IERR ) 743* 744 20 CONTINUE 745 END IF 746 END IF 747* 748* Undo balancing on VL and VR and normalization 749* (Workspace: none needed) 750* 751 IF( ILVL ) THEN 752 CALL ZGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N, VL, 753 $ LDVL, IERR ) 754* 755 DO 50 JC = 1, N 756 TEMP = ZERO 757 DO 30 JR = 1, N 758 TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) ) 759 30 CONTINUE 760 IF( TEMP.LT.SMLNUM ) 761 $ GO TO 50 762 TEMP = ONE / TEMP 763 DO 40 JR = 1, N 764 VL( JR, JC ) = VL( JR, JC )*TEMP 765 40 CONTINUE 766 50 CONTINUE 767 END IF 768* 769 IF( ILVR ) THEN 770 CALL ZGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N, VR, 771 $ LDVR, IERR ) 772 DO 80 JC = 1, N 773 TEMP = ZERO 774 DO 60 JR = 1, N 775 TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) ) 776 60 CONTINUE 777 IF( TEMP.LT.SMLNUM ) 778 $ GO TO 80 779 TEMP = ONE / TEMP 780 DO 70 JR = 1, N 781 VR( JR, JC ) = VR( JR, JC )*TEMP 782 70 CONTINUE 783 80 CONTINUE 784 END IF 785* 786* Undo scaling if necessary 787* 788 90 CONTINUE 789* 790 IF( ILASCL ) 791 $ CALL ZLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR ) 792* 793 IF( ILBSCL ) 794 $ CALL ZLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) 795* 796 WORK( 1 ) = MAXWRK 797 RETURN 798* 799* End of ZGGEVX 800* 801 END 802