1*> \brief <b> CGEEVX 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 CGEEVX + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgeevx.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgeevx.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgeevx.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE CGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, 22* LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, 23* RCONDV, WORK, LWORK, RWORK, INFO ) 24* 25* .. Scalar Arguments .. 26* CHARACTER BALANC, JOBVL, JOBVR, SENSE 27* INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N 28* REAL ABNRM 29* .. 30* .. Array Arguments .. 31* REAL RCONDE( * ), RCONDV( * ), RWORK( * ), 32* $ SCALE( * ) 33* COMPLEX A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), 34* $ W( * ), WORK( * ) 35* .. 36* 37* 38*> \par Purpose: 39* ============= 40*> 41*> \verbatim 42*> 43*> CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the 44*> eigenvalues and, optionally, the left and/or right eigenvectors. 45*> 46*> Optionally also, it computes a balancing transformation to improve 47*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI, 48*> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues 49*> (RCONDE), and reciprocal condition numbers for the right 50*> eigenvectors (RCONDV). 51*> 52*> The right eigenvector v(j) of A satisfies 53*> A * v(j) = lambda(j) * v(j) 54*> where lambda(j) is its eigenvalue. 55*> The left eigenvector u(j) of A satisfies 56*> u(j)**H * A = lambda(j) * u(j)**H 57*> where u(j)**H denotes the conjugate transpose of u(j). 58*> 59*> The computed eigenvectors are normalized to have Euclidean norm 60*> equal to 1 and largest component real. 61*> 62*> Balancing a matrix means permuting the rows and columns to make it 63*> more nearly upper triangular, and applying a diagonal similarity 64*> transformation D * A * D**(-1), where D is a diagonal matrix, to 65*> make its rows and columns closer in norm and the condition numbers 66*> of its eigenvalues and eigenvectors smaller. The computed 67*> reciprocal condition numbers correspond to the balanced matrix. 68*> Permuting rows and columns will not change the condition numbers 69*> (in exact arithmetic) but diagonal scaling will. For further 70*> explanation of balancing, see section 4.10.2 of the LAPACK 71*> Users' Guide. 72*> \endverbatim 73* 74* Arguments: 75* ========== 76* 77*> \param[in] BALANC 78*> \verbatim 79*> BALANC is CHARACTER*1 80*> Indicates how the input matrix should be diagonally scaled 81*> and/or permuted to improve the conditioning of its 82*> eigenvalues. 83*> = 'N': Do not diagonally scale or permute; 84*> = 'P': Perform permutations to make the matrix more nearly 85*> upper triangular. Do not diagonally scale; 86*> = 'S': Diagonally scale the matrix, ie. replace A by 87*> D*A*D**(-1), where D is a diagonal matrix chosen 88*> to make the rows and columns of A more equal in 89*> norm. Do not permute; 90*> = 'B': Both diagonally scale and permute A. 91*> 92*> Computed reciprocal condition numbers will be for the matrix 93*> after balancing and/or permuting. Permuting does not change 94*> condition numbers (in exact arithmetic), but balancing does. 95*> \endverbatim 96*> 97*> \param[in] JOBVL 98*> \verbatim 99*> JOBVL is CHARACTER*1 100*> = 'N': left eigenvectors of A are not computed; 101*> = 'V': left eigenvectors of A are computed. 102*> If SENSE = 'E' or 'B', JOBVL must = 'V'. 103*> \endverbatim 104*> 105*> \param[in] JOBVR 106*> \verbatim 107*> JOBVR is CHARACTER*1 108*> = 'N': right eigenvectors of A are not computed; 109*> = 'V': right eigenvectors of A are computed. 110*> If SENSE = 'E' or 'B', JOBVR must = 'V'. 111*> \endverbatim 112*> 113*> \param[in] SENSE 114*> \verbatim 115*> SENSE is CHARACTER*1 116*> Determines which reciprocal condition numbers are computed. 117*> = 'N': None are computed; 118*> = 'E': Computed for eigenvalues only; 119*> = 'V': Computed for right eigenvectors only; 120*> = 'B': Computed for eigenvalues and right eigenvectors. 121*> 122*> If SENSE = 'E' or 'B', both left and right eigenvectors 123*> must also be computed (JOBVL = 'V' and JOBVR = 'V'). 124*> \endverbatim 125*> 126*> \param[in] N 127*> \verbatim 128*> N is INTEGER 129*> The order of the matrix A. N >= 0. 130*> \endverbatim 131*> 132*> \param[in,out] A 133*> \verbatim 134*> A is COMPLEX array, dimension (LDA,N) 135*> On entry, the N-by-N matrix A. 136*> On exit, A has been overwritten. If JOBVL = 'V' or 137*> JOBVR = 'V', A contains the Schur form of the balanced 138*> version of the matrix A. 139*> \endverbatim 140*> 141*> \param[in] LDA 142*> \verbatim 143*> LDA is INTEGER 144*> The leading dimension of the array A. LDA >= max(1,N). 145*> \endverbatim 146*> 147*> \param[out] W 148*> \verbatim 149*> W is COMPLEX array, dimension (N) 150*> W contains the computed eigenvalues. 151*> \endverbatim 152*> 153*> \param[out] VL 154*> \verbatim 155*> VL is COMPLEX array, dimension (LDVL,N) 156*> If JOBVL = 'V', the left eigenvectors u(j) are stored one 157*> after another in the columns of VL, in the same order 158*> as their eigenvalues. 159*> If JOBVL = 'N', VL is not referenced. 160*> u(j) = VL(:,j), the j-th column of VL. 161*> \endverbatim 162*> 163*> \param[in] LDVL 164*> \verbatim 165*> LDVL is INTEGER 166*> The leading dimension of the array VL. LDVL >= 1; if 167*> JOBVL = 'V', LDVL >= N. 168*> \endverbatim 169*> 170*> \param[out] VR 171*> \verbatim 172*> VR is COMPLEX array, dimension (LDVR,N) 173*> If JOBVR = 'V', the right eigenvectors v(j) are stored one 174*> after another in the columns of VR, in the same order 175*> as their eigenvalues. 176*> If JOBVR = 'N', VR is not referenced. 177*> v(j) = VR(:,j), the j-th column of VR. 178*> \endverbatim 179*> 180*> \param[in] LDVR 181*> \verbatim 182*> LDVR is INTEGER 183*> The leading dimension of the array VR. LDVR >= 1; if 184*> JOBVR = 'V', LDVR >= N. 185*> \endverbatim 186*> 187*> \param[out] ILO 188*> \verbatim 189*> ILO is INTEGER 190*> \endverbatim 191*> 192*> \param[out] IHI 193*> \verbatim 194*> IHI is INTEGER 195*> ILO and IHI are integer values determined when A was 196*> balanced. The balanced A(i,j) = 0 if I > J and 197*> J = 1,...,ILO-1 or I = IHI+1,...,N. 198*> \endverbatim 199*> 200*> \param[out] SCALE 201*> \verbatim 202*> SCALE is REAL array, dimension (N) 203*> Details of the permutations and scaling factors applied 204*> when balancing A. If P(j) is the index of the row and column 205*> interchanged with row and column j, and D(j) is the scaling 206*> factor applied to row and column j, then 207*> SCALE(J) = P(J), for J = 1,...,ILO-1 208*> = D(J), for J = ILO,...,IHI 209*> = P(J) for J = IHI+1,...,N. 210*> The order in which the interchanges are made is N to IHI+1, 211*> then 1 to ILO-1. 212*> \endverbatim 213*> 214*> \param[out] ABNRM 215*> \verbatim 216*> ABNRM is REAL 217*> The one-norm of the balanced matrix (the maximum 218*> of the sum of absolute values of elements of any column). 219*> \endverbatim 220*> 221*> \param[out] RCONDE 222*> \verbatim 223*> RCONDE is REAL array, dimension (N) 224*> RCONDE(j) is the reciprocal condition number of the j-th 225*> eigenvalue. 226*> \endverbatim 227*> 228*> \param[out] RCONDV 229*> \verbatim 230*> RCONDV is REAL array, dimension (N) 231*> RCONDV(j) is the reciprocal condition number of the j-th 232*> right eigenvector. 233*> \endverbatim 234*> 235*> \param[out] WORK 236*> \verbatim 237*> WORK is COMPLEX array, dimension (MAX(1,LWORK)) 238*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 239*> \endverbatim 240*> 241*> \param[in] LWORK 242*> \verbatim 243*> LWORK is INTEGER 244*> The dimension of the array WORK. If SENSE = 'N' or 'E', 245*> LWORK >= max(1,2*N), and if SENSE = 'V' or 'B', 246*> LWORK >= N*N+2*N. 247*> For good performance, LWORK must generally be larger. 248*> 249*> If LWORK = -1, then a workspace query is assumed; the routine 250*> only calculates the optimal size of the WORK array, returns 251*> this value as the first entry of the WORK array, and no error 252*> message related to LWORK is issued by XERBLA. 253*> \endverbatim 254*> 255*> \param[out] RWORK 256*> \verbatim 257*> RWORK is REAL array, dimension (2*N) 258*> \endverbatim 259*> 260*> \param[out] INFO 261*> \verbatim 262*> INFO is INTEGER 263*> = 0: successful exit 264*> < 0: if INFO = -i, the i-th argument had an illegal value. 265*> > 0: if INFO = i, the QR algorithm failed to compute all the 266*> eigenvalues, and no eigenvectors or condition numbers 267*> have been computed; elements 1:ILO-1 and i+1:N of W 268*> contain eigenvalues which have converged. 269*> \endverbatim 270* 271* Authors: 272* ======== 273* 274*> \author Univ. of Tennessee 275*> \author Univ. of California Berkeley 276*> \author Univ. of Colorado Denver 277*> \author NAG Ltd. 278* 279*> \date November 2011 280* 281*> \ingroup complexGEeigen 282* 283* ===================================================================== 284 SUBROUTINE CGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, 285 $ LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, 286 $ RCONDV, WORK, LWORK, RWORK, INFO ) 287* 288* -- LAPACK driver routine (version 3.4.0) -- 289* -- LAPACK is a software package provided by Univ. of Tennessee, -- 290* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 291* November 2011 292* 293* .. Scalar Arguments .. 294 CHARACTER BALANC, JOBVL, JOBVR, SENSE 295 INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N 296 REAL ABNRM 297* .. 298* .. Array Arguments .. 299 REAL RCONDE( * ), RCONDV( * ), RWORK( * ), 300 $ SCALE( * ) 301 COMPLEX A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), 302 $ W( * ), WORK( * ) 303* .. 304* 305* ===================================================================== 306* 307* .. Parameters .. 308 REAL ZERO, ONE 309 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) 310* .. 311* .. Local Scalars .. 312 LOGICAL LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE, 313 $ WNTSNN, WNTSNV 314 CHARACTER JOB, SIDE 315 INTEGER HSWORK, I, ICOND, IERR, ITAU, IWRK, K, MAXWRK, 316 $ MINWRK, NOUT 317 REAL ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM 318 COMPLEX TMP 319* .. 320* .. Local Arrays .. 321 LOGICAL SELECT( 1 ) 322 REAL DUM( 1 ) 323* .. 324* .. External Subroutines .. 325 EXTERNAL CGEBAK, CGEBAL, CGEHRD, CHSEQR, CLACPY, CLASCL, 326 $ CSCAL, CSSCAL, CTREVC, CTRSNA, CUNGHR, SLABAD, 327 $ SLASCL, XERBLA 328* .. 329* .. External Functions .. 330 LOGICAL LSAME 331 INTEGER ILAENV, ISAMAX 332 REAL CLANGE, SCNRM2, SLAMCH 333 EXTERNAL LSAME, ILAENV, ISAMAX, CLANGE, SCNRM2, SLAMCH 334* .. 335* .. Intrinsic Functions .. 336 INTRINSIC AIMAG, CMPLX, CONJG, MAX, REAL, SQRT 337* .. 338* .. Executable Statements .. 339* 340* Test the input arguments 341* 342 INFO = 0 343 LQUERY = ( LWORK.EQ.-1 ) 344 WANTVL = LSAME( JOBVL, 'V' ) 345 WANTVR = LSAME( JOBVR, 'V' ) 346 WNTSNN = LSAME( SENSE, 'N' ) 347 WNTSNE = LSAME( SENSE, 'E' ) 348 WNTSNV = LSAME( SENSE, 'V' ) 349 WNTSNB = LSAME( SENSE, 'B' ) 350 IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'S' ) .OR. 351 $ LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) ) THEN 352 INFO = -1 353 ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN 354 INFO = -2 355 ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN 356 INFO = -3 357 ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR. 358 $ ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND. 359 $ WANTVR ) ) ) THEN 360 INFO = -4 361 ELSE IF( N.LT.0 ) THEN 362 INFO = -5 363 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 364 INFO = -7 365 ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN 366 INFO = -10 367 ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN 368 INFO = -12 369 END IF 370* 371* Compute workspace 372* (Note: Comments in the code beginning "Workspace:" describe the 373* minimal amount of workspace needed at that point in the code, 374* as well as the preferred amount for good performance. 375* CWorkspace refers to complex workspace, and RWorkspace to real 376* workspace. NB refers to the optimal block size for the 377* immediately following subroutine, as returned by ILAENV. 378* HSWORK refers to the workspace preferred by CHSEQR, as 379* calculated below. HSWORK is computed assuming ILO=1 and IHI=N, 380* the worst case.) 381* 382 IF( INFO.EQ.0 ) THEN 383 IF( N.EQ.0 ) THEN 384 MINWRK = 1 385 MAXWRK = 1 386 ELSE 387 MAXWRK = N + N*ILAENV( 1, 'CGEHRD', ' ', N, 1, N, 0 ) 388* 389 IF( WANTVL ) THEN 390 CALL CHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VL, LDVL, 391 $ WORK, -1, INFO ) 392 ELSE IF( WANTVR ) THEN 393 CALL CHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VR, LDVR, 394 $ WORK, -1, INFO ) 395 ELSE 396 IF( WNTSNN ) THEN 397 CALL CHSEQR( 'E', 'N', N, 1, N, A, LDA, W, VR, LDVR, 398 $ WORK, -1, INFO ) 399 ELSE 400 CALL CHSEQR( 'S', 'N', N, 1, N, A, LDA, W, VR, LDVR, 401 $ WORK, -1, INFO ) 402 END IF 403 END IF 404 HSWORK = WORK( 1 ) 405* 406 IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN 407 MINWRK = 2*N 408 IF( .NOT.( WNTSNN .OR. WNTSNE ) ) 409 $ MINWRK = MAX( MINWRK, N*N + 2*N ) 410 MAXWRK = MAX( MAXWRK, HSWORK ) 411 IF( .NOT.( WNTSNN .OR. WNTSNE ) ) 412 $ MAXWRK = MAX( MAXWRK, N*N + 2*N ) 413 ELSE 414 MINWRK = 2*N 415 IF( .NOT.( WNTSNN .OR. WNTSNE ) ) 416 $ MINWRK = MAX( MINWRK, N*N + 2*N ) 417 MAXWRK = MAX( MAXWRK, HSWORK ) 418 MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'CUNGHR', 419 $ ' ', N, 1, N, -1 ) ) 420 IF( .NOT.( WNTSNN .OR. WNTSNE ) ) 421 $ MAXWRK = MAX( MAXWRK, N*N + 2*N ) 422 MAXWRK = MAX( MAXWRK, 2*N ) 423 END IF 424 MAXWRK = MAX( MAXWRK, MINWRK ) 425 END IF 426 WORK( 1 ) = MAXWRK 427* 428 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN 429 INFO = -20 430 END IF 431 END IF 432* 433 IF( INFO.NE.0 ) THEN 434 CALL XERBLA( 'CGEEVX', -INFO ) 435 RETURN 436 ELSE IF( LQUERY ) THEN 437 RETURN 438 END IF 439* 440* Quick return if possible 441* 442 IF( N.EQ.0 ) 443 $ RETURN 444* 445* Get machine constants 446* 447 EPS = SLAMCH( 'P' ) 448 SMLNUM = SLAMCH( 'S' ) 449 BIGNUM = ONE / SMLNUM 450 CALL SLABAD( SMLNUM, BIGNUM ) 451 SMLNUM = SQRT( SMLNUM ) / EPS 452 BIGNUM = ONE / SMLNUM 453* 454* Scale A if max element outside range [SMLNUM,BIGNUM] 455* 456 ICOND = 0 457 ANRM = CLANGE( 'M', N, N, A, LDA, DUM ) 458 SCALEA = .FALSE. 459 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN 460 SCALEA = .TRUE. 461 CSCALE = SMLNUM 462 ELSE IF( ANRM.GT.BIGNUM ) THEN 463 SCALEA = .TRUE. 464 CSCALE = BIGNUM 465 END IF 466 IF( SCALEA ) 467 $ CALL CLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) 468* 469* Balance the matrix and compute ABNRM 470* 471 CALL CGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR ) 472 ABNRM = CLANGE( '1', N, N, A, LDA, DUM ) 473 IF( SCALEA ) THEN 474 DUM( 1 ) = ABNRM 475 CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR ) 476 ABNRM = DUM( 1 ) 477 END IF 478* 479* Reduce to upper Hessenberg form 480* (CWorkspace: need 2*N, prefer N+N*NB) 481* (RWorkspace: none) 482* 483 ITAU = 1 484 IWRK = ITAU + N 485 CALL CGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), 486 $ LWORK-IWRK+1, IERR ) 487* 488 IF( WANTVL ) THEN 489* 490* Want left eigenvectors 491* Copy Householder vectors to VL 492* 493 SIDE = 'L' 494 CALL CLACPY( 'L', N, N, A, LDA, VL, LDVL ) 495* 496* Generate unitary matrix in VL 497* (CWorkspace: need 2*N-1, prefer N+(N-1)*NB) 498* (RWorkspace: none) 499* 500 CALL CUNGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ), 501 $ LWORK-IWRK+1, IERR ) 502* 503* Perform QR iteration, accumulating Schur vectors in VL 504* (CWorkspace: need 1, prefer HSWORK (see comments) ) 505* (RWorkspace: none) 506* 507 IWRK = ITAU 508 CALL CHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VL, LDVL, 509 $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 510* 511 IF( WANTVR ) THEN 512* 513* Want left and right eigenvectors 514* Copy Schur vectors to VR 515* 516 SIDE = 'B' 517 CALL CLACPY( 'F', N, N, VL, LDVL, VR, LDVR ) 518 END IF 519* 520 ELSE IF( WANTVR ) THEN 521* 522* Want right eigenvectors 523* Copy Householder vectors to VR 524* 525 SIDE = 'R' 526 CALL CLACPY( 'L', N, N, A, LDA, VR, LDVR ) 527* 528* Generate unitary matrix in VR 529* (CWorkspace: need 2*N-1, prefer N+(N-1)*NB) 530* (RWorkspace: none) 531* 532 CALL CUNGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ), 533 $ LWORK-IWRK+1, IERR ) 534* 535* Perform QR iteration, accumulating Schur vectors in VR 536* (CWorkspace: need 1, prefer HSWORK (see comments) ) 537* (RWorkspace: none) 538* 539 IWRK = ITAU 540 CALL CHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VR, LDVR, 541 $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 542* 543 ELSE 544* 545* Compute eigenvalues only 546* If condition numbers desired, compute Schur form 547* 548 IF( WNTSNN ) THEN 549 JOB = 'E' 550 ELSE 551 JOB = 'S' 552 END IF 553* 554* (CWorkspace: need 1, prefer HSWORK (see comments) ) 555* (RWorkspace: none) 556* 557 IWRK = ITAU 558 CALL CHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, W, VR, LDVR, 559 $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 560 END IF 561* 562* If INFO > 0 from CHSEQR, then quit 563* 564 IF( INFO.GT.0 ) 565 $ GO TO 50 566* 567 IF( WANTVL .OR. WANTVR ) THEN 568* 569* Compute left and/or right eigenvectors 570* (CWorkspace: need 2*N) 571* (RWorkspace: need N) 572* 573 CALL CTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, 574 $ N, NOUT, WORK( IWRK ), RWORK, IERR ) 575 END IF 576* 577* Compute condition numbers if desired 578* (CWorkspace: need N*N+2*N unless SENSE = 'E') 579* (RWorkspace: need 2*N unless SENSE = 'E') 580* 581 IF( .NOT.WNTSNN ) THEN 582 CALL CTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, 583 $ RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, RWORK, 584 $ ICOND ) 585 END IF 586* 587 IF( WANTVL ) THEN 588* 589* Undo balancing of left eigenvectors 590* 591 CALL CGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL, 592 $ IERR ) 593* 594* Normalize left eigenvectors and make largest component real 595* 596 DO 20 I = 1, N 597 SCL = ONE / SCNRM2( N, VL( 1, I ), 1 ) 598 CALL CSSCAL( N, SCL, VL( 1, I ), 1 ) 599 DO 10 K = 1, N 600 RWORK( K ) = REAL( VL( K, I ) )**2 + 601 $ AIMAG( VL( K, I ) )**2 602 10 CONTINUE 603 K = ISAMAX( N, RWORK, 1 ) 604 TMP = CONJG( VL( K, I ) ) / SQRT( RWORK( K ) ) 605 CALL CSCAL( N, TMP, VL( 1, I ), 1 ) 606 VL( K, I ) = CMPLX( REAL( VL( K, I ) ), ZERO ) 607 20 CONTINUE 608 END IF 609* 610 IF( WANTVR ) THEN 611* 612* Undo balancing of right eigenvectors 613* 614 CALL CGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR, 615 $ IERR ) 616* 617* Normalize right eigenvectors and make largest component real 618* 619 DO 40 I = 1, N 620 SCL = ONE / SCNRM2( N, VR( 1, I ), 1 ) 621 CALL CSSCAL( N, SCL, VR( 1, I ), 1 ) 622 DO 30 K = 1, N 623 RWORK( K ) = REAL( VR( K, I ) )**2 + 624 $ AIMAG( VR( K, I ) )**2 625 30 CONTINUE 626 K = ISAMAX( N, RWORK, 1 ) 627 TMP = CONJG( VR( K, I ) ) / SQRT( RWORK( K ) ) 628 CALL CSCAL( N, TMP, VR( 1, I ), 1 ) 629 VR( K, I ) = CMPLX( REAL( VR( K, I ) ), ZERO ) 630 40 CONTINUE 631 END IF 632* 633* Undo scaling if necessary 634* 635 50 CONTINUE 636 IF( SCALEA ) THEN 637 CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, W( INFO+1 ), 638 $ MAX( N-INFO, 1 ), IERR ) 639 IF( INFO.EQ.0 ) THEN 640 IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 ) 641 $ CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N, 642 $ IERR ) 643 ELSE 644 CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR ) 645 END IF 646 END IF 647* 648 WORK( 1 ) = MAXWRK 649 RETURN 650* 651* End of CGEEVX 652* 653 END 654