Annotation of rpl/lapack/lapack/dggevx.f, revision 1.8
1.8 ! bertrand 1: *> \brief <b> DGGEVX 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 DGGEVX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggevx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggevx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggevx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
! 22: * ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
! 23: * IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
! 24: * RCONDV, WORK, LWORK, 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 A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
! 35: * $ B( LDB, * ), BETA( * ), LSCALE( * ),
! 36: * $ RCONDE( * ), RCONDV( * ), RSCALE( * ),
! 37: * $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
! 38: * ..
! 39: *
! 40: *
! 41: *> \par Purpose:
! 42: * =============
! 43: *>
! 44: *> \verbatim
! 45: *>
! 46: *> DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
! 47: *> the generalized eigenvalues, and optionally, the left and/or right
! 48: *> generalized eigenvectors.
! 49: *>
! 50: *> Optionally also, it computes a balancing transformation to improve
! 51: *> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
! 52: *> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
! 53: *> the eigenvalues (RCONDE), and reciprocal condition numbers for the
! 54: *> right eigenvectors (RCONDV).
! 55: *>
! 56: *> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
! 57: *> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
! 58: *> singular. It is usually represented as the pair (alpha,beta), as
! 59: *> there is a reasonable interpretation for beta=0, and even for both
! 60: *> being zero.
! 61: *>
! 62: *> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
! 63: *> of (A,B) satisfies
! 64: *>
! 65: *> A * v(j) = lambda(j) * B * v(j) .
! 66: *>
! 67: *> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
! 68: *> of (A,B) satisfies
! 69: *>
! 70: *> u(j)**H * A = lambda(j) * u(j)**H * B.
! 71: *>
! 72: *> where u(j)**H is the conjugate-transpose of u(j).
! 73: *>
! 74: *> \endverbatim
! 75: *
! 76: * Arguments:
! 77: * ==========
! 78: *
! 79: *> \param[in] BALANC
! 80: *> \verbatim
! 81: *> BALANC is CHARACTER*1
! 82: *> Specifies the balance option to be performed.
! 83: *> = 'N': do not diagonally scale or permute;
! 84: *> = 'P': permute only;
! 85: *> = 'S': scale only;
! 86: *> = 'B': both permute and scale.
! 87: *> Computed reciprocal condition numbers will be for the
! 88: *> matrices after permuting and/or balancing. Permuting does
! 89: *> not change condition numbers (in exact arithmetic), but
! 90: *> balancing does.
! 91: *> \endverbatim
! 92: *>
! 93: *> \param[in] JOBVL
! 94: *> \verbatim
! 95: *> JOBVL is CHARACTER*1
! 96: *> = 'N': do not compute the left generalized eigenvectors;
! 97: *> = 'V': compute the left generalized eigenvectors.
! 98: *> \endverbatim
! 99: *>
! 100: *> \param[in] JOBVR
! 101: *> \verbatim
! 102: *> JOBVR is CHARACTER*1
! 103: *> = 'N': do not compute the right generalized eigenvectors;
! 104: *> = 'V': compute the right generalized eigenvectors.
! 105: *> \endverbatim
! 106: *>
! 107: *> \param[in] SENSE
! 108: *> \verbatim
! 109: *> SENSE is CHARACTER*1
! 110: *> Determines which reciprocal condition numbers are computed.
! 111: *> = 'N': none are computed;
! 112: *> = 'E': computed for eigenvalues only;
! 113: *> = 'V': computed for eigenvectors only;
! 114: *> = 'B': computed for eigenvalues and eigenvectors.
! 115: *> \endverbatim
! 116: *>
! 117: *> \param[in] N
! 118: *> \verbatim
! 119: *> N is INTEGER
! 120: *> The order of the matrices A, B, VL, and VR. N >= 0.
! 121: *> \endverbatim
! 122: *>
! 123: *> \param[in,out] A
! 124: *> \verbatim
! 125: *> A is DOUBLE PRECISION array, dimension (LDA, N)
! 126: *> On entry, the matrix A in the pair (A,B).
! 127: *> On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
! 128: *> or both, then A contains the first part of the real Schur
! 129: *> form of the "balanced" versions of the input A and B.
! 130: *> \endverbatim
! 131: *>
! 132: *> \param[in] LDA
! 133: *> \verbatim
! 134: *> LDA is INTEGER
! 135: *> The leading dimension of A. LDA >= max(1,N).
! 136: *> \endverbatim
! 137: *>
! 138: *> \param[in,out] B
! 139: *> \verbatim
! 140: *> B is DOUBLE PRECISION array, dimension (LDB, N)
! 141: *> On entry, the matrix B in the pair (A,B).
! 142: *> On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
! 143: *> or both, then B contains the second part of the real Schur
! 144: *> form of the "balanced" versions of the input A and B.
! 145: *> \endverbatim
! 146: *>
! 147: *> \param[in] LDB
! 148: *> \verbatim
! 149: *> LDB is INTEGER
! 150: *> The leading dimension of B. LDB >= max(1,N).
! 151: *> \endverbatim
! 152: *>
! 153: *> \param[out] ALPHAR
! 154: *> \verbatim
! 155: *> ALPHAR is DOUBLE PRECISION array, dimension (N)
! 156: *> \endverbatim
! 157: *>
! 158: *> \param[out] ALPHAI
! 159: *> \verbatim
! 160: *> ALPHAI is DOUBLE PRECISION array, dimension (N)
! 161: *> \endverbatim
! 162: *>
! 163: *> \param[out] BETA
! 164: *> \verbatim
! 165: *> BETA is DOUBLE PRECISION array, dimension (N)
! 166: *> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
! 167: *> be the generalized eigenvalues. If ALPHAI(j) is zero, then
! 168: *> the j-th eigenvalue is real; if positive, then the j-th and
! 169: *> (j+1)-st eigenvalues are a complex conjugate pair, with
! 170: *> ALPHAI(j+1) negative.
! 171: *>
! 172: *> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
! 173: *> may easily over- or underflow, and BETA(j) may even be zero.
! 174: *> Thus, the user should avoid naively computing the ratio
! 175: *> ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
! 176: *> than and usually comparable with norm(A) in magnitude, and
! 177: *> BETA always less than and usually comparable with norm(B).
! 178: *> \endverbatim
! 179: *>
! 180: *> \param[out] VL
! 181: *> \verbatim
! 182: *> VL is DOUBLE PRECISION array, dimension (LDVL,N)
! 183: *> If JOBVL = 'V', the left eigenvectors u(j) are stored one
! 184: *> after another in the columns of VL, in the same order as
! 185: *> their eigenvalues. If the j-th eigenvalue is real, then
! 186: *> u(j) = VL(:,j), the j-th column of VL. If the j-th and
! 187: *> (j+1)-th eigenvalues form a complex conjugate pair, then
! 188: *> u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
! 189: *> Each eigenvector will be scaled so the largest component have
! 190: *> abs(real part) + abs(imag. part) = 1.
! 191: *> Not referenced if JOBVL = 'N'.
! 192: *> \endverbatim
! 193: *>
! 194: *> \param[in] LDVL
! 195: *> \verbatim
! 196: *> LDVL is INTEGER
! 197: *> The leading dimension of the matrix VL. LDVL >= 1, and
! 198: *> if JOBVL = 'V', LDVL >= N.
! 199: *> \endverbatim
! 200: *>
! 201: *> \param[out] VR
! 202: *> \verbatim
! 203: *> VR is DOUBLE PRECISION array, dimension (LDVR,N)
! 204: *> If JOBVR = 'V', the right eigenvectors v(j) are stored one
! 205: *> after another in the columns of VR, in the same order as
! 206: *> their eigenvalues. If the j-th eigenvalue is real, then
! 207: *> v(j) = VR(:,j), the j-th column of VR. If the j-th and
! 208: *> (j+1)-th eigenvalues form a complex conjugate pair, then
! 209: *> v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
! 210: *> Each eigenvector will be scaled so the largest component have
! 211: *> abs(real part) + abs(imag. part) = 1.
! 212: *> Not referenced if JOBVR = 'N'.
! 213: *> \endverbatim
! 214: *>
! 215: *> \param[in] LDVR
! 216: *> \verbatim
! 217: *> LDVR is INTEGER
! 218: *> The leading dimension of the matrix VR. LDVR >= 1, and
! 219: *> if JOBVR = 'V', LDVR >= N.
! 220: *> \endverbatim
! 221: *>
! 222: *> \param[out] ILO
! 223: *> \verbatim
! 224: *> ILO is INTEGER
! 225: *> \endverbatim
! 226: *>
! 227: *> \param[out] IHI
! 228: *> \verbatim
! 229: *> IHI is INTEGER
! 230: *> ILO and IHI are integer values such that on exit
! 231: *> A(i,j) = 0 and B(i,j) = 0 if i > j and
! 232: *> j = 1,...,ILO-1 or i = IHI+1,...,N.
! 233: *> If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
! 234: *> \endverbatim
! 235: *>
! 236: *> \param[out] LSCALE
! 237: *> \verbatim
! 238: *> LSCALE is DOUBLE PRECISION array, dimension (N)
! 239: *> Details of the permutations and scaling factors applied
! 240: *> to the left side of A and B. If PL(j) is the index of the
! 241: *> row interchanged with row j, and DL(j) is the scaling
! 242: *> factor applied to row j, then
! 243: *> LSCALE(j) = PL(j) for j = 1,...,ILO-1
! 244: *> = DL(j) for j = ILO,...,IHI
! 245: *> = PL(j) for j = IHI+1,...,N.
! 246: *> The order in which the interchanges are made is N to IHI+1,
! 247: *> then 1 to ILO-1.
! 248: *> \endverbatim
! 249: *>
! 250: *> \param[out] RSCALE
! 251: *> \verbatim
! 252: *> RSCALE is DOUBLE PRECISION array, dimension (N)
! 253: *> Details of the permutations and scaling factors applied
! 254: *> to the right side of A and B. If PR(j) is the index of the
! 255: *> column interchanged with column j, and DR(j) is the scaling
! 256: *> factor applied to column j, then
! 257: *> RSCALE(j) = PR(j) for j = 1,...,ILO-1
! 258: *> = DR(j) for j = ILO,...,IHI
! 259: *> = PR(j) for j = IHI+1,...,N
! 260: *> The order in which the interchanges are made is N to IHI+1,
! 261: *> then 1 to ILO-1.
! 262: *> \endverbatim
! 263: *>
! 264: *> \param[out] ABNRM
! 265: *> \verbatim
! 266: *> ABNRM is DOUBLE PRECISION
! 267: *> The one-norm of the balanced matrix A.
! 268: *> \endverbatim
! 269: *>
! 270: *> \param[out] BBNRM
! 271: *> \verbatim
! 272: *> BBNRM is DOUBLE PRECISION
! 273: *> The one-norm of the balanced matrix B.
! 274: *> \endverbatim
! 275: *>
! 276: *> \param[out] RCONDE
! 277: *> \verbatim
! 278: *> RCONDE is DOUBLE PRECISION array, dimension (N)
! 279: *> If SENSE = 'E' or 'B', the reciprocal condition numbers of
! 280: *> the eigenvalues, stored in consecutive elements of the array.
! 281: *> For a complex conjugate pair of eigenvalues two consecutive
! 282: *> elements of RCONDE are set to the same value. Thus RCONDE(j),
! 283: *> RCONDV(j), and the j-th columns of VL and VR all correspond
! 284: *> to the j-th eigenpair.
! 285: *> If SENSE = 'N or 'V', RCONDE is not referenced.
! 286: *> \endverbatim
! 287: *>
! 288: *> \param[out] RCONDV
! 289: *> \verbatim
! 290: *> RCONDV is DOUBLE PRECISION array, dimension (N)
! 291: *> If SENSE = 'V' or 'B', the estimated reciprocal condition
! 292: *> numbers of the eigenvectors, stored in consecutive elements
! 293: *> of the array. For a complex eigenvector two consecutive
! 294: *> elements of RCONDV are set to the same value. If the
! 295: *> eigenvalues cannot be reordered to compute RCONDV(j),
! 296: *> RCONDV(j) is set to 0; this can only occur when the true
! 297: *> value would be very small anyway.
! 298: *> If SENSE = 'N' or 'E', RCONDV is not referenced.
! 299: *> \endverbatim
! 300: *>
! 301: *> \param[out] WORK
! 302: *> \verbatim
! 303: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
! 304: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 305: *> \endverbatim
! 306: *>
! 307: *> \param[in] LWORK
! 308: *> \verbatim
! 309: *> LWORK is INTEGER
! 310: *> The dimension of the array WORK. LWORK >= max(1,2*N).
! 311: *> If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
! 312: *> LWORK >= max(1,6*N).
! 313: *> If SENSE = 'E' or 'B', LWORK >= max(1,10*N).
! 314: *> If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
! 315: *>
! 316: *> If LWORK = -1, then a workspace query is assumed; the routine
! 317: *> only calculates the optimal size of the WORK array, returns
! 318: *> this value as the first entry of the WORK array, and no error
! 319: *> message related to LWORK is issued by XERBLA.
! 320: *> \endverbatim
! 321: *>
! 322: *> \param[out] IWORK
! 323: *> \verbatim
! 324: *> IWORK is INTEGER array, dimension (N+6)
! 325: *> If SENSE = 'E', IWORK is not referenced.
! 326: *> \endverbatim
! 327: *>
! 328: *> \param[out] BWORK
! 329: *> \verbatim
! 330: *> BWORK is LOGICAL array, dimension (N)
! 331: *> If SENSE = 'N', BWORK is not referenced.
! 332: *> \endverbatim
! 333: *>
! 334: *> \param[out] INFO
! 335: *> \verbatim
! 336: *> INFO is INTEGER
! 337: *> = 0: successful exit
! 338: *> < 0: if INFO = -i, the i-th argument had an illegal value.
! 339: *> = 1,...,N:
! 340: *> The QZ iteration failed. No eigenvectors have been
! 341: *> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
! 342: *> should be correct for j=INFO+1,...,N.
! 343: *> > N: =N+1: other than QZ iteration failed in DHGEQZ.
! 344: *> =N+2: error return from DTGEVC.
! 345: *> \endverbatim
! 346: *
! 347: * Authors:
! 348: * ========
! 349: *
! 350: *> \author Univ. of Tennessee
! 351: *> \author Univ. of California Berkeley
! 352: *> \author Univ. of Colorado Denver
! 353: *> \author NAG Ltd.
! 354: *
! 355: *> \date November 2011
! 356: *
! 357: *> \ingroup doubleGEeigen
! 358: *
! 359: *> \par Further Details:
! 360: * =====================
! 361: *>
! 362: *> \verbatim
! 363: *>
! 364: *> Balancing a matrix pair (A,B) includes, first, permuting rows and
! 365: *> columns to isolate eigenvalues, second, applying diagonal similarity
! 366: *> transformation to the rows and columns to make the rows and columns
! 367: *> as close in norm as possible. The computed reciprocal condition
! 368: *> numbers correspond to the balanced matrix. Permuting rows and columns
! 369: *> will not change the condition numbers (in exact arithmetic) but
! 370: *> diagonal scaling will. For further explanation of balancing, see
! 371: *> section 4.11.1.2 of LAPACK Users' Guide.
! 372: *>
! 373: *> An approximate error bound on the chordal distance between the i-th
! 374: *> computed generalized eigenvalue w and the corresponding exact
! 375: *> eigenvalue lambda is
! 376: *>
! 377: *> chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
! 378: *>
! 379: *> An approximate error bound for the angle between the i-th computed
! 380: *> eigenvector VL(i) or VR(i) is given by
! 381: *>
! 382: *> EPS * norm(ABNRM, BBNRM) / DIF(i).
! 383: *>
! 384: *> For further explanation of the reciprocal condition numbers RCONDE
! 385: *> and RCONDV, see section 4.11 of LAPACK User's Guide.
! 386: *> \endverbatim
! 387: *>
! 388: * =====================================================================
1.1 bertrand 389: SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
390: $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
391: $ IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
392: $ RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
393: *
1.8 ! bertrand 394: * -- LAPACK driver routine (version 3.4.0) --
1.1 bertrand 395: * -- LAPACK is a software package provided by Univ. of Tennessee, --
396: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 ! bertrand 397: * November 2011
1.1 bertrand 398: *
399: * .. Scalar Arguments ..
400: CHARACTER BALANC, JOBVL, JOBVR, SENSE
401: INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
402: DOUBLE PRECISION ABNRM, BBNRM
403: * ..
404: * .. Array Arguments ..
405: LOGICAL BWORK( * )
406: INTEGER IWORK( * )
407: DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
408: $ B( LDB, * ), BETA( * ), LSCALE( * ),
409: $ RCONDE( * ), RCONDV( * ), RSCALE( * ),
410: $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
411: * ..
412: *
413: * =====================================================================
414: *
415: * .. Parameters ..
416: DOUBLE PRECISION ZERO, ONE
417: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
418: * ..
419: * .. Local Scalars ..
420: LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL,
421: $ PAIR, WANTSB, WANTSE, WANTSN, WANTSV
422: CHARACTER CHTEMP
423: INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS,
424: $ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK,
425: $ MINWRK, MM
426: DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
427: $ SMLNUM, TEMP
428: * ..
429: * .. Local Arrays ..
430: LOGICAL LDUMMA( 1 )
431: * ..
432: * .. External Subroutines ..
433: EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD,
434: $ DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGEVC,
435: $ DTGSNA, XERBLA
436: * ..
437: * .. External Functions ..
438: LOGICAL LSAME
439: INTEGER ILAENV
440: DOUBLE PRECISION DLAMCH, DLANGE
441: EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
442: * ..
443: * .. Intrinsic Functions ..
444: INTRINSIC ABS, MAX, SQRT
445: * ..
446: * .. Executable Statements ..
447: *
448: * Decode the input arguments
449: *
450: IF( LSAME( JOBVL, 'N' ) ) THEN
451: IJOBVL = 1
452: ILVL = .FALSE.
453: ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
454: IJOBVL = 2
455: ILVL = .TRUE.
456: ELSE
457: IJOBVL = -1
458: ILVL = .FALSE.
459: END IF
460: *
461: IF( LSAME( JOBVR, 'N' ) ) THEN
462: IJOBVR = 1
463: ILVR = .FALSE.
464: ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
465: IJOBVR = 2
466: ILVR = .TRUE.
467: ELSE
468: IJOBVR = -1
469: ILVR = .FALSE.
470: END IF
471: ILV = ILVL .OR. ILVR
472: *
473: NOSCL = LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'P' )
474: WANTSN = LSAME( SENSE, 'N' )
475: WANTSE = LSAME( SENSE, 'E' )
476: WANTSV = LSAME( SENSE, 'V' )
477: WANTSB = LSAME( SENSE, 'B' )
478: *
479: * Test the input arguments
480: *
481: INFO = 0
482: LQUERY = ( LWORK.EQ.-1 )
483: IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC,
484: $ 'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) )
485: $ THEN
486: INFO = -1
487: ELSE IF( IJOBVL.LE.0 ) THEN
488: INFO = -2
489: ELSE IF( IJOBVR.LE.0 ) THEN
490: INFO = -3
491: ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) )
492: $ THEN
493: INFO = -4
494: ELSE IF( N.LT.0 ) THEN
495: INFO = -5
496: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
497: INFO = -7
498: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
499: INFO = -9
500: ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
501: INFO = -14
502: ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
503: INFO = -16
504: END IF
505: *
506: * Compute workspace
507: * (Note: Comments in the code beginning "Workspace:" describe the
508: * minimal amount of workspace needed at that point in the code,
509: * as well as the preferred amount for good performance.
510: * NB refers to the optimal block size for the immediately
511: * following subroutine, as returned by ILAENV. The workspace is
512: * computed assuming ILO = 1 and IHI = N, the worst case.)
513: *
514: IF( INFO.EQ.0 ) THEN
515: IF( N.EQ.0 ) THEN
516: MINWRK = 1
517: MAXWRK = 1
518: ELSE
519: IF( NOSCL .AND. .NOT.ILV ) THEN
520: MINWRK = 2*N
521: ELSE
522: MINWRK = 6*N
523: END IF
524: IF( WANTSE .OR. WANTSB ) THEN
525: MINWRK = 10*N
526: END IF
527: IF( WANTSV .OR. WANTSB ) THEN
528: MINWRK = MAX( MINWRK, 2*N*( N + 4 ) + 16 )
529: END IF
530: MAXWRK = MINWRK
531: MAXWRK = MAX( MAXWRK,
532: $ N + N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) )
533: MAXWRK = MAX( MAXWRK,
534: $ N + N*ILAENV( 1, 'DORMQR', ' ', N, 1, N, 0 ) )
535: IF( ILVL ) THEN
536: MAXWRK = MAX( MAXWRK, N +
537: $ N*ILAENV( 1, 'DORGQR', ' ', N, 1, N, 0 ) )
538: END IF
539: END IF
540: WORK( 1 ) = MAXWRK
541: *
542: IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
543: INFO = -26
544: END IF
545: END IF
546: *
547: IF( INFO.NE.0 ) THEN
548: CALL XERBLA( 'DGGEVX', -INFO )
549: RETURN
550: ELSE IF( LQUERY ) THEN
551: RETURN
552: END IF
553: *
554: * Quick return if possible
555: *
556: IF( N.EQ.0 )
557: $ RETURN
558: *
559: *
560: * Get machine constants
561: *
562: EPS = DLAMCH( 'P' )
563: SMLNUM = DLAMCH( 'S' )
564: BIGNUM = ONE / SMLNUM
565: CALL DLABAD( SMLNUM, BIGNUM )
566: SMLNUM = SQRT( SMLNUM ) / EPS
567: BIGNUM = ONE / SMLNUM
568: *
569: * Scale A if max element outside range [SMLNUM,BIGNUM]
570: *
571: ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
572: ILASCL = .FALSE.
573: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
574: ANRMTO = SMLNUM
575: ILASCL = .TRUE.
576: ELSE IF( ANRM.GT.BIGNUM ) THEN
577: ANRMTO = BIGNUM
578: ILASCL = .TRUE.
579: END IF
580: IF( ILASCL )
581: $ CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
582: *
583: * Scale B if max element outside range [SMLNUM,BIGNUM]
584: *
585: BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
586: ILBSCL = .FALSE.
587: IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
588: BNRMTO = SMLNUM
589: ILBSCL = .TRUE.
590: ELSE IF( BNRM.GT.BIGNUM ) THEN
591: BNRMTO = BIGNUM
592: ILBSCL = .TRUE.
593: END IF
594: IF( ILBSCL )
595: $ CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
596: *
597: * Permute and/or balance the matrix pair (A,B)
598: * (Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
599: *
600: CALL DGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE,
601: $ WORK, IERR )
602: *
603: * Compute ABNRM and BBNRM
604: *
605: ABNRM = DLANGE( '1', N, N, A, LDA, WORK( 1 ) )
606: IF( ILASCL ) THEN
607: WORK( 1 ) = ABNRM
608: CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, WORK( 1 ), 1,
609: $ IERR )
610: ABNRM = WORK( 1 )
611: END IF
612: *
613: BBNRM = DLANGE( '1', N, N, B, LDB, WORK( 1 ) )
614: IF( ILBSCL ) THEN
615: WORK( 1 ) = BBNRM
616: CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, WORK( 1 ), 1,
617: $ IERR )
618: BBNRM = WORK( 1 )
619: END IF
620: *
621: * Reduce B to triangular form (QR decomposition of B)
622: * (Workspace: need N, prefer N*NB )
623: *
624: IROWS = IHI + 1 - ILO
625: IF( ILV .OR. .NOT.WANTSN ) THEN
626: ICOLS = N + 1 - ILO
627: ELSE
628: ICOLS = IROWS
629: END IF
630: ITAU = 1
631: IWRK = ITAU + IROWS
632: CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
633: $ WORK( IWRK ), LWORK+1-IWRK, IERR )
634: *
635: * Apply the orthogonal transformation to A
636: * (Workspace: need N, prefer N*NB)
637: *
638: CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
639: $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
640: $ LWORK+1-IWRK, IERR )
641: *
642: * Initialize VL and/or VR
643: * (Workspace: need N, prefer N*NB)
644: *
645: IF( ILVL ) THEN
646: CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
647: IF( IROWS.GT.1 ) THEN
648: CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
649: $ VL( ILO+1, ILO ), LDVL )
650: END IF
651: CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
652: $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
653: END IF
654: *
655: IF( ILVR )
656: $ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
657: *
658: * Reduce to generalized Hessenberg form
659: * (Workspace: none needed)
660: *
661: IF( ILV .OR. .NOT.WANTSN ) THEN
662: *
663: * Eigenvectors requested -- work on whole matrix.
664: *
665: CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
666: $ LDVL, VR, LDVR, IERR )
667: ELSE
668: CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
669: $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
670: END IF
671: *
672: * Perform QZ algorithm (Compute eigenvalues, and optionally, the
673: * Schur forms and Schur vectors)
674: * (Workspace: need N)
675: *
676: IF( ILV .OR. .NOT.WANTSN ) THEN
677: CHTEMP = 'S'
678: ELSE
679: CHTEMP = 'E'
680: END IF
681: *
682: CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
683: $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK,
684: $ LWORK, IERR )
685: IF( IERR.NE.0 ) THEN
686: IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
687: INFO = IERR
688: ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
689: INFO = IERR - N
690: ELSE
691: INFO = N + 1
692: END IF
693: GO TO 130
694: END IF
695: *
696: * Compute Eigenvectors and estimate condition numbers if desired
697: * (Workspace: DTGEVC: need 6*N
698: * DTGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B',
699: * need N otherwise )
700: *
701: IF( ILV .OR. .NOT.WANTSN ) THEN
702: IF( ILV ) THEN
703: IF( ILVL ) THEN
704: IF( ILVR ) THEN
705: CHTEMP = 'B'
706: ELSE
707: CHTEMP = 'L'
708: END IF
709: ELSE
710: CHTEMP = 'R'
711: END IF
712: *
713: CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL,
714: $ LDVL, VR, LDVR, N, IN, WORK, IERR )
715: IF( IERR.NE.0 ) THEN
716: INFO = N + 2
717: GO TO 130
718: END IF
719: END IF
720: *
721: IF( .NOT.WANTSN ) THEN
722: *
723: * compute eigenvectors (DTGEVC) and estimate condition
724: * numbers (DTGSNA). Note that the definition of the condition
725: * number is not invariant under transformation (u,v) to
726: * (Q*u, Z*v), where (u,v) are eigenvectors of the generalized
727: * Schur form (S,T), Q and Z are orthogonal matrices. In order
728: * to avoid using extra 2*N*N workspace, we have to recalculate
729: * eigenvectors and estimate one condition numbers at a time.
730: *
731: PAIR = .FALSE.
732: DO 20 I = 1, N
733: *
734: IF( PAIR ) THEN
735: PAIR = .FALSE.
736: GO TO 20
737: END IF
738: MM = 1
739: IF( I.LT.N ) THEN
740: IF( A( I+1, I ).NE.ZERO ) THEN
741: PAIR = .TRUE.
742: MM = 2
743: END IF
744: END IF
745: *
746: DO 10 J = 1, N
747: BWORK( J ) = .FALSE.
748: 10 CONTINUE
749: IF( MM.EQ.1 ) THEN
750: BWORK( I ) = .TRUE.
751: ELSE IF( MM.EQ.2 ) THEN
752: BWORK( I ) = .TRUE.
753: BWORK( I+1 ) = .TRUE.
754: END IF
755: *
756: IWRK = MM*N + 1
757: IWRK1 = IWRK + MM*N
758: *
759: * Compute a pair of left and right eigenvectors.
760: * (compute workspace: need up to 4*N + 6*N)
761: *
762: IF( WANTSE .OR. WANTSB ) THEN
763: CALL DTGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB,
764: $ WORK( 1 ), N, WORK( IWRK ), N, MM, M,
765: $ WORK( IWRK1 ), IERR )
766: IF( IERR.NE.0 ) THEN
767: INFO = N + 2
768: GO TO 130
769: END IF
770: END IF
771: *
772: CALL DTGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB,
773: $ WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ),
774: $ RCONDV( I ), MM, M, WORK( IWRK1 ),
775: $ LWORK-IWRK1+1, IWORK, IERR )
776: *
777: 20 CONTINUE
778: END IF
779: END IF
780: *
781: * Undo balancing on VL and VR and normalization
782: * (Workspace: none needed)
783: *
784: IF( ILVL ) THEN
785: CALL DGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N, VL,
786: $ LDVL, IERR )
787: *
788: DO 70 JC = 1, N
789: IF( ALPHAI( JC ).LT.ZERO )
790: $ GO TO 70
791: TEMP = ZERO
792: IF( ALPHAI( JC ).EQ.ZERO ) THEN
793: DO 30 JR = 1, N
794: TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
795: 30 CONTINUE
796: ELSE
797: DO 40 JR = 1, N
798: TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
799: $ ABS( VL( JR, JC+1 ) ) )
800: 40 CONTINUE
801: END IF
802: IF( TEMP.LT.SMLNUM )
803: $ GO TO 70
804: TEMP = ONE / TEMP
805: IF( ALPHAI( JC ).EQ.ZERO ) THEN
806: DO 50 JR = 1, N
807: VL( JR, JC ) = VL( JR, JC )*TEMP
808: 50 CONTINUE
809: ELSE
810: DO 60 JR = 1, N
811: VL( JR, JC ) = VL( JR, JC )*TEMP
812: VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
813: 60 CONTINUE
814: END IF
815: 70 CONTINUE
816: END IF
817: IF( ILVR ) THEN
818: CALL DGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N, VR,
819: $ LDVR, IERR )
820: DO 120 JC = 1, N
821: IF( ALPHAI( JC ).LT.ZERO )
822: $ GO TO 120
823: TEMP = ZERO
824: IF( ALPHAI( JC ).EQ.ZERO ) THEN
825: DO 80 JR = 1, N
826: TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
827: 80 CONTINUE
828: ELSE
829: DO 90 JR = 1, N
830: TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
831: $ ABS( VR( JR, JC+1 ) ) )
832: 90 CONTINUE
833: END IF
834: IF( TEMP.LT.SMLNUM )
835: $ GO TO 120
836: TEMP = ONE / TEMP
837: IF( ALPHAI( JC ).EQ.ZERO ) THEN
838: DO 100 JR = 1, N
839: VR( JR, JC ) = VR( JR, JC )*TEMP
840: 100 CONTINUE
841: ELSE
842: DO 110 JR = 1, N
843: VR( JR, JC ) = VR( JR, JC )*TEMP
844: VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
845: 110 CONTINUE
846: END IF
847: 120 CONTINUE
848: END IF
849: *
850: * Undo scaling if necessary
851: *
852: IF( ILASCL ) THEN
853: CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
854: CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
855: END IF
856: *
857: IF( ILBSCL ) THEN
858: CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
859: END IF
860: *
861: 130 CONTINUE
862: WORK( 1 ) = MAXWRK
863: *
864: RETURN
865: *
866: * End of DGGEVX
867: *
868: END
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