Annotation of rpl/lapack/lapack/dgeevx.f, revision 1.9
1.9 ! bertrand 1: *> \brief <b> DGEEVX 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 DGEEVX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeevx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeevx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeevx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
! 22: * VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
! 23: * RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
! 24: *
! 25: * .. Scalar Arguments ..
! 26: * CHARACTER BALANC, JOBVL, JOBVR, SENSE
! 27: * INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
! 28: * DOUBLE PRECISION ABNRM
! 29: * ..
! 30: * .. Array Arguments ..
! 31: * INTEGER IWORK( * )
! 32: * DOUBLE PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ),
! 33: * $ SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
! 34: * $ WI( * ), WORK( * ), WR( * )
! 35: * ..
! 36: *
! 37: *
! 38: *> \par Purpose:
! 39: * =============
! 40: *>
! 41: *> \verbatim
! 42: *>
! 43: *> DGEEVX computes for an N-by-N real 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)**T * A = lambda(j) * u(j)**T
! 57: *> where u(j)**T denotes the 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, i.e. 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 DOUBLE PRECISION 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 real Schur form of the balanced
! 138: *> version of the input 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] WR
! 148: *> \verbatim
! 149: *> WR is DOUBLE PRECISION array, dimension (N)
! 150: *> \endverbatim
! 151: *>
! 152: *> \param[out] WI
! 153: *> \verbatim
! 154: *> WI is DOUBLE PRECISION array, dimension (N)
! 155: *> WR and WI contain the real and imaginary parts,
! 156: *> respectively, of the computed eigenvalues. Complex
! 157: *> conjugate pairs of eigenvalues will appear consecutively
! 158: *> with the eigenvalue having the positive imaginary part
! 159: *> first.
! 160: *> \endverbatim
! 161: *>
! 162: *> \param[out] VL
! 163: *> \verbatim
! 164: *> VL is DOUBLE PRECISION array, dimension (LDVL,N)
! 165: *> If JOBVL = 'V', the left eigenvectors u(j) are stored one
! 166: *> after another in the columns of VL, in the same order
! 167: *> as their eigenvalues.
! 168: *> If JOBVL = 'N', VL is not referenced.
! 169: *> If the j-th eigenvalue is real, then u(j) = VL(:,j),
! 170: *> the j-th column of VL.
! 171: *> If the j-th and (j+1)-st eigenvalues form a complex
! 172: *> conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
! 173: *> u(j+1) = VL(:,j) - i*VL(:,j+1).
! 174: *> \endverbatim
! 175: *>
! 176: *> \param[in] LDVL
! 177: *> \verbatim
! 178: *> LDVL is INTEGER
! 179: *> The leading dimension of the array VL. LDVL >= 1; if
! 180: *> JOBVL = 'V', LDVL >= N.
! 181: *> \endverbatim
! 182: *>
! 183: *> \param[out] VR
! 184: *> \verbatim
! 185: *> VR is DOUBLE PRECISION array, dimension (LDVR,N)
! 186: *> If JOBVR = 'V', the right eigenvectors v(j) are stored one
! 187: *> after another in the columns of VR, in the same order
! 188: *> as their eigenvalues.
! 189: *> If JOBVR = 'N', VR is not referenced.
! 190: *> If the j-th eigenvalue is real, then v(j) = VR(:,j),
! 191: *> the j-th column of VR.
! 192: *> If the j-th and (j+1)-st eigenvalues form a complex
! 193: *> conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
! 194: *> v(j+1) = VR(:,j) - i*VR(:,j+1).
! 195: *> \endverbatim
! 196: *>
! 197: *> \param[in] LDVR
! 198: *> \verbatim
! 199: *> LDVR is INTEGER
! 200: *> The leading dimension of the array VR. LDVR >= 1, and if
! 201: *> JOBVR = 'V', LDVR >= N.
! 202: *> \endverbatim
! 203: *>
! 204: *> \param[out] ILO
! 205: *> \verbatim
! 206: *> ILO is INTEGER
! 207: *> \endverbatim
! 208: *>
! 209: *> \param[out] IHI
! 210: *> \verbatim
! 211: *> IHI is INTEGER
! 212: *> ILO and IHI are integer values determined when A was
! 213: *> balanced. The balanced A(i,j) = 0 if I > J and
! 214: *> J = 1,...,ILO-1 or I = IHI+1,...,N.
! 215: *> \endverbatim
! 216: *>
! 217: *> \param[out] SCALE
! 218: *> \verbatim
! 219: *> SCALE is DOUBLE PRECISION array, dimension (N)
! 220: *> Details of the permutations and scaling factors applied
! 221: *> when balancing A. If P(j) is the index of the row and column
! 222: *> interchanged with row and column j, and D(j) is the scaling
! 223: *> factor applied to row and column j, then
! 224: *> SCALE(J) = P(J), for J = 1,...,ILO-1
! 225: *> = D(J), for J = ILO,...,IHI
! 226: *> = P(J) for J = IHI+1,...,N.
! 227: *> The order in which the interchanges are made is N to IHI+1,
! 228: *> then 1 to ILO-1.
! 229: *> \endverbatim
! 230: *>
! 231: *> \param[out] ABNRM
! 232: *> \verbatim
! 233: *> ABNRM is DOUBLE PRECISION
! 234: *> The one-norm of the balanced matrix (the maximum
! 235: *> of the sum of absolute values of elements of any column).
! 236: *> \endverbatim
! 237: *>
! 238: *> \param[out] RCONDE
! 239: *> \verbatim
! 240: *> RCONDE is DOUBLE PRECISION array, dimension (N)
! 241: *> RCONDE(j) is the reciprocal condition number of the j-th
! 242: *> eigenvalue.
! 243: *> \endverbatim
! 244: *>
! 245: *> \param[out] RCONDV
! 246: *> \verbatim
! 247: *> RCONDV is DOUBLE PRECISION array, dimension (N)
! 248: *> RCONDV(j) is the reciprocal condition number of the j-th
! 249: *> right eigenvector.
! 250: *> \endverbatim
! 251: *>
! 252: *> \param[out] WORK
! 253: *> \verbatim
! 254: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
! 255: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 256: *> \endverbatim
! 257: *>
! 258: *> \param[in] LWORK
! 259: *> \verbatim
! 260: *> LWORK is INTEGER
! 261: *> The dimension of the array WORK. If SENSE = 'N' or 'E',
! 262: *> LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
! 263: *> LWORK >= 3*N. If SENSE = 'V' or 'B', LWORK >= N*(N+6).
! 264: *> For good performance, LWORK must generally be larger.
! 265: *>
! 266: *> If LWORK = -1, then a workspace query is assumed; the routine
! 267: *> only calculates the optimal size of the WORK array, returns
! 268: *> this value as the first entry of the WORK array, and no error
! 269: *> message related to LWORK is issued by XERBLA.
! 270: *> \endverbatim
! 271: *>
! 272: *> \param[out] IWORK
! 273: *> \verbatim
! 274: *> IWORK is INTEGER array, dimension (2*N-2)
! 275: *> If SENSE = 'N' or 'E', not referenced.
! 276: *> \endverbatim
! 277: *>
! 278: *> \param[out] INFO
! 279: *> \verbatim
! 280: *> INFO is INTEGER
! 281: *> = 0: successful exit
! 282: *> < 0: if INFO = -i, the i-th argument had an illegal value.
! 283: *> > 0: if INFO = i, the QR algorithm failed to compute all the
! 284: *> eigenvalues, and no eigenvectors or condition numbers
! 285: *> have been computed; elements 1:ILO-1 and i+1:N of WR
! 286: *> and WI contain eigenvalues which have converged.
! 287: *> \endverbatim
! 288: *
! 289: * Authors:
! 290: * ========
! 291: *
! 292: *> \author Univ. of Tennessee
! 293: *> \author Univ. of California Berkeley
! 294: *> \author Univ. of Colorado Denver
! 295: *> \author NAG Ltd.
! 296: *
! 297: *> \date November 2011
! 298: *
! 299: *> \ingroup doubleGEeigen
! 300: *
! 301: * =====================================================================
1.1 bertrand 302: SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
303: $ VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
304: $ RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
305: *
1.9 ! bertrand 306: * -- LAPACK driver routine (version 3.4.0) --
1.1 bertrand 307: * -- LAPACK is a software package provided by Univ. of Tennessee, --
308: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 309: * November 2011
1.1 bertrand 310: *
311: * .. Scalar Arguments ..
312: CHARACTER BALANC, JOBVL, JOBVR, SENSE
313: INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
314: DOUBLE PRECISION ABNRM
315: * ..
316: * .. Array Arguments ..
317: INTEGER IWORK( * )
318: DOUBLE PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ),
319: $ SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
320: $ WI( * ), WORK( * ), WR( * )
321: * ..
322: *
323: * =====================================================================
324: *
325: * .. Parameters ..
326: DOUBLE PRECISION ZERO, ONE
327: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
328: * ..
329: * .. Local Scalars ..
330: LOGICAL LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE,
331: $ WNTSNN, WNTSNV
332: CHARACTER JOB, SIDE
333: INTEGER HSWORK, I, ICOND, IERR, ITAU, IWRK, K, MAXWRK,
334: $ MINWRK, NOUT
335: DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
336: $ SN
337: * ..
338: * .. Local Arrays ..
339: LOGICAL SELECT( 1 )
340: DOUBLE PRECISION DUM( 1 )
341: * ..
342: * .. External Subroutines ..
343: EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY,
344: $ DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC,
345: $ DTRSNA, XERBLA
346: * ..
347: * .. External Functions ..
348: LOGICAL LSAME
349: INTEGER IDAMAX, ILAENV
350: DOUBLE PRECISION DLAMCH, DLANGE, DLAPY2, DNRM2
351: EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE, DLAPY2,
352: $ DNRM2
353: * ..
354: * .. Intrinsic Functions ..
355: INTRINSIC MAX, SQRT
356: * ..
357: * .. Executable Statements ..
358: *
359: * Test the input arguments
360: *
361: INFO = 0
362: LQUERY = ( LWORK.EQ.-1 )
363: WANTVL = LSAME( JOBVL, 'V' )
364: WANTVR = LSAME( JOBVR, 'V' )
365: WNTSNN = LSAME( SENSE, 'N' )
366: WNTSNE = LSAME( SENSE, 'E' )
367: WNTSNV = LSAME( SENSE, 'V' )
368: WNTSNB = LSAME( SENSE, 'B' )
369: IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC,
370: $ 'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) )
371: $ THEN
372: INFO = -1
373: ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
374: INFO = -2
375: ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
376: INFO = -3
377: ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR.
378: $ ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND.
379: $ WANTVR ) ) ) THEN
380: INFO = -4
381: ELSE IF( N.LT.0 ) THEN
382: INFO = -5
383: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
384: INFO = -7
385: ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
386: INFO = -11
387: ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
388: INFO = -13
389: END IF
390: *
391: * Compute workspace
392: * (Note: Comments in the code beginning "Workspace:" describe the
393: * minimal amount of workspace needed at that point in the code,
394: * as well as the preferred amount for good performance.
395: * NB refers to the optimal block size for the immediately
396: * following subroutine, as returned by ILAENV.
397: * HSWORK refers to the workspace preferred by DHSEQR, as
398: * calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
399: * the worst case.)
400: *
401: IF( INFO.EQ.0 ) THEN
402: IF( N.EQ.0 ) THEN
403: MINWRK = 1
404: MAXWRK = 1
405: ELSE
406: MAXWRK = N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
407: *
408: IF( WANTVL ) THEN
409: CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL,
410: $ WORK, -1, INFO )
411: ELSE IF( WANTVR ) THEN
412: CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR,
413: $ WORK, -1, INFO )
414: ELSE
415: IF( WNTSNN ) THEN
416: CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR,
417: $ LDVR, WORK, -1, INFO )
418: ELSE
419: CALL DHSEQR( 'S', 'N', N, 1, N, A, LDA, WR, WI, VR,
420: $ LDVR, WORK, -1, INFO )
421: END IF
422: END IF
423: HSWORK = WORK( 1 )
424: *
425: IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
426: MINWRK = 2*N
427: IF( .NOT.WNTSNN )
428: $ MINWRK = MAX( MINWRK, N*N+6*N )
429: MAXWRK = MAX( MAXWRK, HSWORK )
430: IF( .NOT.WNTSNN )
431: $ MAXWRK = MAX( MAXWRK, N*N + 6*N )
432: ELSE
433: MINWRK = 3*N
434: IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
435: $ MINWRK = MAX( MINWRK, N*N + 6*N )
436: MAXWRK = MAX( MAXWRK, HSWORK )
437: MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'DORGHR',
438: $ ' ', N, 1, N, -1 ) )
439: IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
440: $ MAXWRK = MAX( MAXWRK, N*N + 6*N )
441: MAXWRK = MAX( MAXWRK, 3*N )
442: END IF
443: MAXWRK = MAX( MAXWRK, MINWRK )
444: END IF
445: WORK( 1 ) = MAXWRK
446: *
447: IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
448: INFO = -21
449: END IF
450: END IF
451: *
452: IF( INFO.NE.0 ) THEN
453: CALL XERBLA( 'DGEEVX', -INFO )
454: RETURN
455: ELSE IF( LQUERY ) THEN
456: RETURN
457: END IF
458: *
459: * Quick return if possible
460: *
461: IF( N.EQ.0 )
462: $ RETURN
463: *
464: * Get machine constants
465: *
466: EPS = DLAMCH( 'P' )
467: SMLNUM = DLAMCH( 'S' )
468: BIGNUM = ONE / SMLNUM
469: CALL DLABAD( SMLNUM, BIGNUM )
470: SMLNUM = SQRT( SMLNUM ) / EPS
471: BIGNUM = ONE / SMLNUM
472: *
473: * Scale A if max element outside range [SMLNUM,BIGNUM]
474: *
475: ICOND = 0
476: ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
477: SCALEA = .FALSE.
478: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
479: SCALEA = .TRUE.
480: CSCALE = SMLNUM
481: ELSE IF( ANRM.GT.BIGNUM ) THEN
482: SCALEA = .TRUE.
483: CSCALE = BIGNUM
484: END IF
485: IF( SCALEA )
486: $ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
487: *
488: * Balance the matrix and compute ABNRM
489: *
490: CALL DGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR )
491: ABNRM = DLANGE( '1', N, N, A, LDA, DUM )
492: IF( SCALEA ) THEN
493: DUM( 1 ) = ABNRM
494: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
495: ABNRM = DUM( 1 )
496: END IF
497: *
498: * Reduce to upper Hessenberg form
499: * (Workspace: need 2*N, prefer N+N*NB)
500: *
501: ITAU = 1
502: IWRK = ITAU + N
503: CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
504: $ LWORK-IWRK+1, IERR )
505: *
506: IF( WANTVL ) THEN
507: *
508: * Want left eigenvectors
509: * Copy Householder vectors to VL
510: *
511: SIDE = 'L'
512: CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL )
513: *
514: * Generate orthogonal matrix in VL
515: * (Workspace: need 2*N-1, prefer N+(N-1)*NB)
516: *
517: CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
518: $ LWORK-IWRK+1, IERR )
519: *
520: * Perform QR iteration, accumulating Schur vectors in VL
521: * (Workspace: need 1, prefer HSWORK (see comments) )
522: *
523: IWRK = ITAU
524: CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL,
525: $ WORK( IWRK ), LWORK-IWRK+1, INFO )
526: *
527: IF( WANTVR ) THEN
528: *
529: * Want left and right eigenvectors
530: * Copy Schur vectors to VR
531: *
532: SIDE = 'B'
533: CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
534: END IF
535: *
536: ELSE IF( WANTVR ) THEN
537: *
538: * Want right eigenvectors
539: * Copy Householder vectors to VR
540: *
541: SIDE = 'R'
542: CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR )
543: *
544: * Generate orthogonal matrix in VR
545: * (Workspace: need 2*N-1, prefer N+(N-1)*NB)
546: *
547: CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
548: $ LWORK-IWRK+1, IERR )
549: *
550: * Perform QR iteration, accumulating Schur vectors in VR
551: * (Workspace: need 1, prefer HSWORK (see comments) )
552: *
553: IWRK = ITAU
554: CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
555: $ WORK( IWRK ), LWORK-IWRK+1, INFO )
556: *
557: ELSE
558: *
559: * Compute eigenvalues only
560: * If condition numbers desired, compute Schur form
561: *
562: IF( WNTSNN ) THEN
563: JOB = 'E'
564: ELSE
565: JOB = 'S'
566: END IF
567: *
568: * (Workspace: need 1, prefer HSWORK (see comments) )
569: *
570: IWRK = ITAU
571: CALL DHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
572: $ WORK( IWRK ), LWORK-IWRK+1, INFO )
573: END IF
574: *
575: * If INFO > 0 from DHSEQR, then quit
576: *
577: IF( INFO.GT.0 )
578: $ GO TO 50
579: *
580: IF( WANTVL .OR. WANTVR ) THEN
581: *
582: * Compute left and/or right eigenvectors
583: * (Workspace: need 3*N)
584: *
585: CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
586: $ N, NOUT, WORK( IWRK ), IERR )
587: END IF
588: *
589: * Compute condition numbers if desired
590: * (Workspace: need N*N+6*N unless SENSE = 'E')
591: *
592: IF( .NOT.WNTSNN ) THEN
593: CALL DTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
594: $ RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, IWORK,
595: $ ICOND )
596: END IF
597: *
598: IF( WANTVL ) THEN
599: *
600: * Undo balancing of left eigenvectors
601: *
602: CALL DGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL,
603: $ IERR )
604: *
605: * Normalize left eigenvectors and make largest component real
606: *
607: DO 20 I = 1, N
608: IF( WI( I ).EQ.ZERO ) THEN
609: SCL = ONE / DNRM2( N, VL( 1, I ), 1 )
610: CALL DSCAL( N, SCL, VL( 1, I ), 1 )
611: ELSE IF( WI( I ).GT.ZERO ) THEN
612: SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ),
613: $ DNRM2( N, VL( 1, I+1 ), 1 ) )
614: CALL DSCAL( N, SCL, VL( 1, I ), 1 )
615: CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 )
616: DO 10 K = 1, N
617: WORK( K ) = VL( K, I )**2 + VL( K, I+1 )**2
618: 10 CONTINUE
619: K = IDAMAX( N, WORK, 1 )
620: CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
621: CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
622: VL( K, I+1 ) = ZERO
623: END IF
624: 20 CONTINUE
625: END IF
626: *
627: IF( WANTVR ) THEN
628: *
629: * Undo balancing of right eigenvectors
630: *
631: CALL DGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR,
632: $ IERR )
633: *
634: * Normalize right eigenvectors and make largest component real
635: *
636: DO 40 I = 1, N
637: IF( WI( I ).EQ.ZERO ) THEN
638: SCL = ONE / DNRM2( N, VR( 1, I ), 1 )
639: CALL DSCAL( N, SCL, VR( 1, I ), 1 )
640: ELSE IF( WI( I ).GT.ZERO ) THEN
641: SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ),
642: $ DNRM2( N, VR( 1, I+1 ), 1 ) )
643: CALL DSCAL( N, SCL, VR( 1, I ), 1 )
644: CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 )
645: DO 30 K = 1, N
646: WORK( K ) = VR( K, I )**2 + VR( K, I+1 )**2
647: 30 CONTINUE
648: K = IDAMAX( N, WORK, 1 )
649: CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
650: CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
651: VR( K, I+1 ) = ZERO
652: END IF
653: 40 CONTINUE
654: END IF
655: *
656: * Undo scaling if necessary
657: *
658: 50 CONTINUE
659: IF( SCALEA ) THEN
660: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ),
661: $ MAX( N-INFO, 1 ), IERR )
662: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ),
663: $ MAX( N-INFO, 1 ), IERR )
664: IF( INFO.EQ.0 ) THEN
665: IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 )
666: $ CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N,
667: $ IERR )
668: ELSE
669: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
670: $ IERR )
671: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
672: $ IERR )
673: END IF
674: END IF
675: *
676: WORK( 1 ) = MAXWRK
677: RETURN
678: *
679: * End of DGEEVX
680: *
681: END
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