Annotation of rpl/lapack/lapack/zheevr_2stage.f, revision 1.1
1.1 ! bertrand 1: *> \brief <b> ZHEEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices</b>
! 2: *
! 3: * @precisions fortran z -> s d c
! 4: *
! 5: * =========== DOCUMENTATION ===========
! 6: *
! 7: * Online html documentation available at
! 8: * http://www.netlib.org/lapack/explore-html/
! 9: *
! 10: *> \htmlonly
! 11: *> Download ZHEEVR_2STAGE + dependencies
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zheevr_2stage.f">
! 13: *> [TGZ]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zheevr_2stage.f">
! 15: *> [ZIP]</a>
! 16: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zheevr_2stage.f">
! 17: *> [TXT]</a>
! 18: *> \endhtmlonly
! 19: *
! 20: * Definition:
! 21: * ===========
! 22: *
! 23: * SUBROUTINE ZHEEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
! 24: * IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ,
! 25: * WORK, LWORK, RWORK, LRWORK, IWORK,
! 26: * LIWORK, INFO )
! 27: *
! 28: * IMPLICIT NONE
! 29: *
! 30: * .. Scalar Arguments ..
! 31: * CHARACTER JOBZ, RANGE, UPLO
! 32: * INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
! 33: * $ M, N
! 34: * DOUBLE PRECISION ABSTOL, VL, VU
! 35: * ..
! 36: * .. Array Arguments ..
! 37: * INTEGER ISUPPZ( * ), IWORK( * )
! 38: * DOUBLE PRECISION RWORK( * ), W( * )
! 39: * COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
! 40: * ..
! 41: *
! 42: *
! 43: *> \par Purpose:
! 44: * =============
! 45: *>
! 46: *> \verbatim
! 47: *>
! 48: *> ZHEEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors
! 49: *> of a complex Hermitian matrix A using the 2stage technique for
! 50: *> the reduction to tridiagonal. Eigenvalues and eigenvectors can
! 51: *> be selected by specifying either a range of values or a range of
! 52: *> indices for the desired eigenvalues.
! 53: *>
! 54: *> ZHEEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call
! 55: *> to ZHETRD. Then, whenever possible, ZHEEVR_2STAGE calls ZSTEMR to compute
! 56: *> eigenspectrum using Relatively Robust Representations. ZSTEMR
! 57: *> computes eigenvalues by the dqds algorithm, while orthogonal
! 58: *> eigenvectors are computed from various "good" L D L^T representations
! 59: *> (also known as Relatively Robust Representations). Gram-Schmidt
! 60: *> orthogonalization is avoided as far as possible. More specifically,
! 61: *> the various steps of the algorithm are as follows.
! 62: *>
! 63: *> For each unreduced block (submatrix) of T,
! 64: *> (a) Compute T - sigma I = L D L^T, so that L and D
! 65: *> define all the wanted eigenvalues to high relative accuracy.
! 66: *> This means that small relative changes in the entries of D and L
! 67: *> cause only small relative changes in the eigenvalues and
! 68: *> eigenvectors. The standard (unfactored) representation of the
! 69: *> tridiagonal matrix T does not have this property in general.
! 70: *> (b) Compute the eigenvalues to suitable accuracy.
! 71: *> If the eigenvectors are desired, the algorithm attains full
! 72: *> accuracy of the computed eigenvalues only right before
! 73: *> the corresponding vectors have to be computed, see steps c) and d).
! 74: *> (c) For each cluster of close eigenvalues, select a new
! 75: *> shift close to the cluster, find a new factorization, and refine
! 76: *> the shifted eigenvalues to suitable accuracy.
! 77: *> (d) For each eigenvalue with a large enough relative separation compute
! 78: *> the corresponding eigenvector by forming a rank revealing twisted
! 79: *> factorization. Go back to (c) for any clusters that remain.
! 80: *>
! 81: *> The desired accuracy of the output can be specified by the input
! 82: *> parameter ABSTOL.
! 83: *>
! 84: *> For more details, see DSTEMR's documentation and:
! 85: *> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
! 86: *> to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
! 87: *> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
! 88: *> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
! 89: *> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
! 90: *> 2004. Also LAPACK Working Note 154.
! 91: *> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
! 92: *> tridiagonal eigenvalue/eigenvector problem",
! 93: *> Computer Science Division Technical Report No. UCB/CSD-97-971,
! 94: *> UC Berkeley, May 1997.
! 95: *>
! 96: *>
! 97: *> Note 1 : ZHEEVR_2STAGE calls ZSTEMR when the full spectrum is requested
! 98: *> on machines which conform to the ieee-754 floating point standard.
! 99: *> ZHEEVR_2STAGE calls DSTEBZ and ZSTEIN on non-ieee machines and
! 100: *> when partial spectrum requests are made.
! 101: *>
! 102: *> Normal execution of ZSTEMR may create NaNs and infinities and
! 103: *> hence may abort due to a floating point exception in environments
! 104: *> which do not handle NaNs and infinities in the ieee standard default
! 105: *> manner.
! 106: *> \endverbatim
! 107: *
! 108: * Arguments:
! 109: * ==========
! 110: *
! 111: *> \param[in] JOBZ
! 112: *> \verbatim
! 113: *> JOBZ is CHARACTER*1
! 114: *> = 'N': Compute eigenvalues only;
! 115: *> = 'V': Compute eigenvalues and eigenvectors.
! 116: *> Not available in this release.
! 117: *> \endverbatim
! 118: *>
! 119: *> \param[in] RANGE
! 120: *> \verbatim
! 121: *> RANGE is CHARACTER*1
! 122: *> = 'A': all eigenvalues will be found.
! 123: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
! 124: *> will be found.
! 125: *> = 'I': the IL-th through IU-th eigenvalues will be found.
! 126: *> For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
! 127: *> ZSTEIN are called
! 128: *> \endverbatim
! 129: *>
! 130: *> \param[in] UPLO
! 131: *> \verbatim
! 132: *> UPLO is CHARACTER*1
! 133: *> = 'U': Upper triangle of A is stored;
! 134: *> = 'L': Lower triangle of A is stored.
! 135: *> \endverbatim
! 136: *>
! 137: *> \param[in] N
! 138: *> \verbatim
! 139: *> N is INTEGER
! 140: *> The order of the matrix A. N >= 0.
! 141: *> \endverbatim
! 142: *>
! 143: *> \param[in,out] A
! 144: *> \verbatim
! 145: *> A is COMPLEX*16 array, dimension (LDA, N)
! 146: *> On entry, the Hermitian matrix A. If UPLO = 'U', the
! 147: *> leading N-by-N upper triangular part of A contains the
! 148: *> upper triangular part of the matrix A. If UPLO = 'L',
! 149: *> the leading N-by-N lower triangular part of A contains
! 150: *> the lower triangular part of the matrix A.
! 151: *> On exit, the lower triangle (if UPLO='L') or the upper
! 152: *> triangle (if UPLO='U') of A, including the diagonal, is
! 153: *> destroyed.
! 154: *> \endverbatim
! 155: *>
! 156: *> \param[in] LDA
! 157: *> \verbatim
! 158: *> LDA is INTEGER
! 159: *> The leading dimension of the array A. LDA >= max(1,N).
! 160: *> \endverbatim
! 161: *>
! 162: *> \param[in] VL
! 163: *> \verbatim
! 164: *> VL is DOUBLE PRECISION
! 165: *> If RANGE='V', the lower bound of the interval to
! 166: *> be searched for eigenvalues. VL < VU.
! 167: *> Not referenced if RANGE = 'A' or 'I'.
! 168: *> \endverbatim
! 169: *>
! 170: *> \param[in] VU
! 171: *> \verbatim
! 172: *> VU is DOUBLE PRECISION
! 173: *> If RANGE='V', the upper bound of the interval to
! 174: *> be searched for eigenvalues. VL < VU.
! 175: *> Not referenced if RANGE = 'A' or 'I'.
! 176: *> \endverbatim
! 177: *>
! 178: *> \param[in] IL
! 179: *> \verbatim
! 180: *> IL is INTEGER
! 181: *> If RANGE='I', the index of the
! 182: *> smallest eigenvalue to be returned.
! 183: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
! 184: *> Not referenced if RANGE = 'A' or 'V'.
! 185: *> \endverbatim
! 186: *>
! 187: *> \param[in] IU
! 188: *> \verbatim
! 189: *> IU is INTEGER
! 190: *> If RANGE='I', the index of the
! 191: *> largest eigenvalue to be returned.
! 192: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
! 193: *> Not referenced if RANGE = 'A' or 'V'.
! 194: *> \endverbatim
! 195: *>
! 196: *> \param[in] ABSTOL
! 197: *> \verbatim
! 198: *> ABSTOL is DOUBLE PRECISION
! 199: *> The absolute error tolerance for the eigenvalues.
! 200: *> An approximate eigenvalue is accepted as converged
! 201: *> when it is determined to lie in an interval [a,b]
! 202: *> of width less than or equal to
! 203: *>
! 204: *> ABSTOL + EPS * max( |a|,|b| ) ,
! 205: *>
! 206: *> where EPS is the machine precision. If ABSTOL is less than
! 207: *> or equal to zero, then EPS*|T| will be used in its place,
! 208: *> where |T| is the 1-norm of the tridiagonal matrix obtained
! 209: *> by reducing A to tridiagonal form.
! 210: *>
! 211: *> See "Computing Small Singular Values of Bidiagonal Matrices
! 212: *> with Guaranteed High Relative Accuracy," by Demmel and
! 213: *> Kahan, LAPACK Working Note #3.
! 214: *>
! 215: *> If high relative accuracy is important, set ABSTOL to
! 216: *> DLAMCH( 'Safe minimum' ). Doing so will guarantee that
! 217: *> eigenvalues are computed to high relative accuracy when
! 218: *> possible in future releases. The current code does not
! 219: *> make any guarantees about high relative accuracy, but
! 220: *> furutre releases will. See J. Barlow and J. Demmel,
! 221: *> "Computing Accurate Eigensystems of Scaled Diagonally
! 222: *> Dominant Matrices", LAPACK Working Note #7, for a discussion
! 223: *> of which matrices define their eigenvalues to high relative
! 224: *> accuracy.
! 225: *> \endverbatim
! 226: *>
! 227: *> \param[out] M
! 228: *> \verbatim
! 229: *> M is INTEGER
! 230: *> The total number of eigenvalues found. 0 <= M <= N.
! 231: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
! 232: *> \endverbatim
! 233: *>
! 234: *> \param[out] W
! 235: *> \verbatim
! 236: *> W is DOUBLE PRECISION array, dimension (N)
! 237: *> The first M elements contain the selected eigenvalues in
! 238: *> ascending order.
! 239: *> \endverbatim
! 240: *>
! 241: *> \param[out] Z
! 242: *> \verbatim
! 243: *> Z is COMPLEX*16 array, dimension (LDZ, max(1,M))
! 244: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
! 245: *> contain the orthonormal eigenvectors of the matrix A
! 246: *> corresponding to the selected eigenvalues, with the i-th
! 247: *> column of Z holding the eigenvector associated with W(i).
! 248: *> If JOBZ = 'N', then Z is not referenced.
! 249: *> Note: the user must ensure that at least max(1,M) columns are
! 250: *> supplied in the array Z; if RANGE = 'V', the exact value of M
! 251: *> is not known in advance and an upper bound must be used.
! 252: *> \endverbatim
! 253: *>
! 254: *> \param[in] LDZ
! 255: *> \verbatim
! 256: *> LDZ is INTEGER
! 257: *> The leading dimension of the array Z. LDZ >= 1, and if
! 258: *> JOBZ = 'V', LDZ >= max(1,N).
! 259: *> \endverbatim
! 260: *>
! 261: *> \param[out] ISUPPZ
! 262: *> \verbatim
! 263: *> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
! 264: *> The support of the eigenvectors in Z, i.e., the indices
! 265: *> indicating the nonzero elements in Z. The i-th eigenvector
! 266: *> is nonzero only in elements ISUPPZ( 2*i-1 ) through
! 267: *> ISUPPZ( 2*i ). This is an output of ZSTEMR (tridiagonal
! 268: *> matrix). The support of the eigenvectors of A is typically
! 269: *> 1:N because of the unitary transformations applied by ZUNMTR.
! 270: *> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
! 271: *> \endverbatim
! 272: *>
! 273: *> \param[out] WORK
! 274: *> \verbatim
! 275: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
! 276: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 277: *> \endverbatim
! 278: *>
! 279: *> \param[in] LWORK
! 280: *> \verbatim
! 281: *> LWORK is INTEGER
! 282: *> The dimension of the array WORK.
! 283: *> If JOBZ = 'N' and N > 1, LWORK must be queried.
! 284: *> LWORK = MAX(1, 26*N, dimension) where
! 285: *> dimension = max(stage1,stage2) + (KD+1)*N + N
! 286: *> = N*KD + N*max(KD+1,FACTOPTNB)
! 287: *> + max(2*KD*KD, KD*NTHREADS)
! 288: *> + (KD+1)*N + N
! 289: *> where KD is the blocking size of the reduction,
! 290: *> FACTOPTNB is the blocking used by the QR or LQ
! 291: *> algorithm, usually FACTOPTNB=128 is a good choice
! 292: *> NTHREADS is the number of threads used when
! 293: *> openMP compilation is enabled, otherwise =1.
! 294: *> If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available
! 295: *>
! 296: *> If LWORK = -1, then a workspace query is assumed; the routine
! 297: *> only calculates the optimal sizes of the WORK, RWORK and
! 298: *> IWORK arrays, returns these values as the first entries of
! 299: *> the WORK, RWORK and IWORK arrays, and no error message
! 300: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
! 301: *> \endverbatim
! 302: *>
! 303: *> \param[out] RWORK
! 304: *> \verbatim
! 305: *> RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
! 306: *> On exit, if INFO = 0, RWORK(1) returns the optimal
! 307: *> (and minimal) LRWORK.
! 308: *> \endverbatim
! 309: *>
! 310: *> \param[in] LRWORK
! 311: *> \verbatim
! 312: *> LRWORK is INTEGER
! 313: *> The length of the array RWORK. LRWORK >= max(1,24*N).
! 314: *>
! 315: *> If LRWORK = -1, then a workspace query is assumed; the
! 316: *> routine only calculates the optimal sizes of the WORK, RWORK
! 317: *> and IWORK arrays, returns these values as the first entries
! 318: *> of the WORK, RWORK and IWORK arrays, and no error message
! 319: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
! 320: *> \endverbatim
! 321: *>
! 322: *> \param[out] IWORK
! 323: *> \verbatim
! 324: *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
! 325: *> On exit, if INFO = 0, IWORK(1) returns the optimal
! 326: *> (and minimal) LIWORK.
! 327: *> \endverbatim
! 328: *>
! 329: *> \param[in] LIWORK
! 330: *> \verbatim
! 331: *> LIWORK is INTEGER
! 332: *> The dimension of the array IWORK. LIWORK >= max(1,10*N).
! 333: *>
! 334: *> If LIWORK = -1, then a workspace query is assumed; the
! 335: *> routine only calculates the optimal sizes of the WORK, RWORK
! 336: *> and IWORK arrays, returns these values as the first entries
! 337: *> of the WORK, RWORK and IWORK arrays, and no error message
! 338: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
! 339: *> \endverbatim
! 340: *>
! 341: *> \param[out] INFO
! 342: *> \verbatim
! 343: *> INFO is INTEGER
! 344: *> = 0: successful exit
! 345: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 346: *> > 0: Internal error
! 347: *> \endverbatim
! 348: *
! 349: * Authors:
! 350: * ========
! 351: *
! 352: *> \author Univ. of Tennessee
! 353: *> \author Univ. of California Berkeley
! 354: *> \author Univ. of Colorado Denver
! 355: *> \author NAG Ltd.
! 356: *
! 357: *> \date June 2016
! 358: *
! 359: *> \ingroup complex16HEeigen
! 360: *
! 361: *> \par Contributors:
! 362: * ==================
! 363: *>
! 364: *> Inderjit Dhillon, IBM Almaden, USA \n
! 365: *> Osni Marques, LBNL/NERSC, USA \n
! 366: *> Ken Stanley, Computer Science Division, University of
! 367: *> California at Berkeley, USA \n
! 368: *> Jason Riedy, Computer Science Division, University of
! 369: *> California at Berkeley, USA \n
! 370: *>
! 371: *> \par Further Details:
! 372: * =====================
! 373: *>
! 374: *> \verbatim
! 375: *>
! 376: *> All details about the 2stage techniques are available in:
! 377: *>
! 378: *> Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
! 379: *> Parallel reduction to condensed forms for symmetric eigenvalue problems
! 380: *> using aggregated fine-grained and memory-aware kernels. In Proceedings
! 381: *> of 2011 International Conference for High Performance Computing,
! 382: *> Networking, Storage and Analysis (SC '11), New York, NY, USA,
! 383: *> Article 8 , 11 pages.
! 384: *> http://doi.acm.org/10.1145/2063384.2063394
! 385: *>
! 386: *> A. Haidar, J. Kurzak, P. Luszczek, 2013.
! 387: *> An improved parallel singular value algorithm and its implementation
! 388: *> for multicore hardware, In Proceedings of 2013 International Conference
! 389: *> for High Performance Computing, Networking, Storage and Analysis (SC '13).
! 390: *> Denver, Colorado, USA, 2013.
! 391: *> Article 90, 12 pages.
! 392: *> http://doi.acm.org/10.1145/2503210.2503292
! 393: *>
! 394: *> A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
! 395: *> A novel hybrid CPU-GPU generalized eigensolver for electronic structure
! 396: *> calculations based on fine-grained memory aware tasks.
! 397: *> International Journal of High Performance Computing Applications.
! 398: *> Volume 28 Issue 2, Pages 196-209, May 2014.
! 399: *> http://hpc.sagepub.com/content/28/2/196
! 400: *>
! 401: *> \endverbatim
! 402: *
! 403: * =====================================================================
! 404: SUBROUTINE ZHEEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
! 405: $ IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ,
! 406: $ WORK, LWORK, RWORK, LRWORK, IWORK,
! 407: $ LIWORK, INFO )
! 408: *
! 409: IMPLICIT NONE
! 410: *
! 411: * -- LAPACK driver routine (version 3.7.0) --
! 412: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 413: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 414: * June 2016
! 415: *
! 416: * .. Scalar Arguments ..
! 417: CHARACTER JOBZ, RANGE, UPLO
! 418: INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK,
! 419: $ M, N
! 420: DOUBLE PRECISION ABSTOL, VL, VU
! 421: * ..
! 422: * .. Array Arguments ..
! 423: INTEGER ISUPPZ( * ), IWORK( * )
! 424: DOUBLE PRECISION RWORK( * ), W( * )
! 425: COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
! 426: * ..
! 427: *
! 428: * =====================================================================
! 429: *
! 430: * .. Parameters ..
! 431: DOUBLE PRECISION ZERO, ONE, TWO
! 432: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
! 433: * ..
! 434: * .. Local Scalars ..
! 435: LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
! 436: $ WANTZ, TRYRAC
! 437: CHARACTER ORDER
! 438: INTEGER I, IEEEOK, IINFO, IMAX, INDIBL, INDIFL, INDISP,
! 439: $ INDIWO, INDRD, INDRDD, INDRE, INDREE, INDRWK,
! 440: $ INDTAU, INDWK, INDWKN, ISCALE, ITMP1, J, JJ,
! 441: $ LIWMIN, LLWORK, LLRWORK, LLWRKN, LRWMIN,
! 442: $ LWMIN, NSPLIT, LHTRD, LWTRD, KD, IB, INDHOUS
! 443: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
! 444: $ SIGMA, SMLNUM, TMP1, VLL, VUU
! 445: * ..
! 446: * .. External Functions ..
! 447: LOGICAL LSAME
! 448: INTEGER ILAENV
! 449: DOUBLE PRECISION DLAMCH, ZLANSY
! 450: EXTERNAL LSAME, ILAENV, DLAMCH, ZLANSY
! 451: * ..
! 452: * .. External Subroutines ..
! 453: EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
! 454: $ ZHETRD_2STAGE, ZSTEMR, ZSTEIN, ZSWAP, ZUNMTR
! 455: * ..
! 456: * .. Intrinsic Functions ..
! 457: INTRINSIC DBLE, MAX, MIN, SQRT
! 458: * ..
! 459: * .. Executable Statements ..
! 460: *
! 461: * Test the input parameters.
! 462: *
! 463: IEEEOK = ILAENV( 10, 'ZHEEVR', 'N', 1, 2, 3, 4 )
! 464: *
! 465: LOWER = LSAME( UPLO, 'L' )
! 466: WANTZ = LSAME( JOBZ, 'V' )
! 467: ALLEIG = LSAME( RANGE, 'A' )
! 468: VALEIG = LSAME( RANGE, 'V' )
! 469: INDEIG = LSAME( RANGE, 'I' )
! 470: *
! 471: LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LRWORK.EQ.-1 ) .OR.
! 472: $ ( LIWORK.EQ.-1 ) )
! 473: *
! 474: KD = ILAENV( 17, 'DSYTRD_2STAGE', JOBZ, N, -1, -1, -1 )
! 475: IB = ILAENV( 18, 'DSYTRD_2STAGE', JOBZ, N, KD, -1, -1 )
! 476: LHTRD = ILAENV( 19, 'DSYTRD_2STAGE', JOBZ, N, KD, IB, -1 )
! 477: LWTRD = ILAENV( 20, 'DSYTRD_2STAGE', JOBZ, N, KD, IB, -1 )
! 478: LWMIN = N + LHTRD + LWTRD
! 479: LRWMIN = MAX( 1, 24*N )
! 480: LIWMIN = MAX( 1, 10*N )
! 481: *
! 482: INFO = 0
! 483: IF( .NOT.( LSAME( JOBZ, 'N' ) ) ) THEN
! 484: INFO = -1
! 485: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
! 486: INFO = -2
! 487: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
! 488: INFO = -3
! 489: ELSE IF( N.LT.0 ) THEN
! 490: INFO = -4
! 491: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 492: INFO = -6
! 493: ELSE
! 494: IF( VALEIG ) THEN
! 495: IF( N.GT.0 .AND. VU.LE.VL )
! 496: $ INFO = -8
! 497: ELSE IF( INDEIG ) THEN
! 498: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
! 499: INFO = -9
! 500: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
! 501: INFO = -10
! 502: END IF
! 503: END IF
! 504: END IF
! 505: IF( INFO.EQ.0 ) THEN
! 506: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
! 507: INFO = -15
! 508: END IF
! 509: END IF
! 510: *
! 511: IF( INFO.EQ.0 ) THEN
! 512: WORK( 1 ) = LWMIN
! 513: RWORK( 1 ) = LRWMIN
! 514: IWORK( 1 ) = LIWMIN
! 515: *
! 516: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
! 517: INFO = -18
! 518: ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
! 519: INFO = -20
! 520: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
! 521: INFO = -22
! 522: END IF
! 523: END IF
! 524: *
! 525: IF( INFO.NE.0 ) THEN
! 526: CALL XERBLA( 'ZHEEVR_2STAGE', -INFO )
! 527: RETURN
! 528: ELSE IF( LQUERY ) THEN
! 529: RETURN
! 530: END IF
! 531: *
! 532: * Quick return if possible
! 533: *
! 534: M = 0
! 535: IF( N.EQ.0 ) THEN
! 536: WORK( 1 ) = 1
! 537: RETURN
! 538: END IF
! 539: *
! 540: IF( N.EQ.1 ) THEN
! 541: WORK( 1 ) = 2
! 542: IF( ALLEIG .OR. INDEIG ) THEN
! 543: M = 1
! 544: W( 1 ) = DBLE( A( 1, 1 ) )
! 545: ELSE
! 546: IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) )
! 547: $ THEN
! 548: M = 1
! 549: W( 1 ) = DBLE( A( 1, 1 ) )
! 550: END IF
! 551: END IF
! 552: IF( WANTZ ) THEN
! 553: Z( 1, 1 ) = ONE
! 554: ISUPPZ( 1 ) = 1
! 555: ISUPPZ( 2 ) = 1
! 556: END IF
! 557: RETURN
! 558: END IF
! 559: *
! 560: * Get machine constants.
! 561: *
! 562: SAFMIN = DLAMCH( 'Safe minimum' )
! 563: EPS = DLAMCH( 'Precision' )
! 564: SMLNUM = SAFMIN / EPS
! 565: BIGNUM = ONE / SMLNUM
! 566: RMIN = SQRT( SMLNUM )
! 567: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
! 568: *
! 569: * Scale matrix to allowable range, if necessary.
! 570: *
! 571: ISCALE = 0
! 572: ABSTLL = ABSTOL
! 573: IF (VALEIG) THEN
! 574: VLL = VL
! 575: VUU = VU
! 576: END IF
! 577: ANRM = ZLANSY( 'M', UPLO, N, A, LDA, RWORK )
! 578: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
! 579: ISCALE = 1
! 580: SIGMA = RMIN / ANRM
! 581: ELSE IF( ANRM.GT.RMAX ) THEN
! 582: ISCALE = 1
! 583: SIGMA = RMAX / ANRM
! 584: END IF
! 585: IF( ISCALE.EQ.1 ) THEN
! 586: IF( LOWER ) THEN
! 587: DO 10 J = 1, N
! 588: CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 )
! 589: 10 CONTINUE
! 590: ELSE
! 591: DO 20 J = 1, N
! 592: CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 )
! 593: 20 CONTINUE
! 594: END IF
! 595: IF( ABSTOL.GT.0 )
! 596: $ ABSTLL = ABSTOL*SIGMA
! 597: IF( VALEIG ) THEN
! 598: VLL = VL*SIGMA
! 599: VUU = VU*SIGMA
! 600: END IF
! 601: END IF
! 602:
! 603: * Initialize indices into workspaces. Note: The IWORK indices are
! 604: * used only if DSTERF or ZSTEMR fail.
! 605:
! 606: * WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the
! 607: * elementary reflectors used in ZHETRD.
! 608: INDTAU = 1
! 609: * INDWK is the starting offset of the remaining complex workspace,
! 610: * and LLWORK is the remaining complex workspace size.
! 611: INDHOUS = INDTAU + N
! 612: INDWK = INDHOUS + LHTRD
! 613: LLWORK = LWORK - INDWK + 1
! 614:
! 615: * RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal
! 616: * entries.
! 617: INDRD = 1
! 618: * RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the
! 619: * tridiagonal matrix from ZHETRD.
! 620: INDRE = INDRD + N
! 621: * RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over
! 622: * -written by ZSTEMR (the DSTERF path copies the diagonal to W).
! 623: INDRDD = INDRE + N
! 624: * RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over
! 625: * -written while computing the eigenvalues in DSTERF and ZSTEMR.
! 626: INDREE = INDRDD + N
! 627: * INDRWK is the starting offset of the left-over real workspace, and
! 628: * LLRWORK is the remaining workspace size.
! 629: INDRWK = INDREE + N
! 630: LLRWORK = LRWORK - INDRWK + 1
! 631:
! 632: * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
! 633: * stores the block indices of each of the M<=N eigenvalues.
! 634: INDIBL = 1
! 635: * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
! 636: * stores the starting and finishing indices of each block.
! 637: INDISP = INDIBL + N
! 638: * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
! 639: * that corresponding to eigenvectors that fail to converge in
! 640: * ZSTEIN. This information is discarded; if any fail, the driver
! 641: * returns INFO > 0.
! 642: INDIFL = INDISP + N
! 643: * INDIWO is the offset of the remaining integer workspace.
! 644: INDIWO = INDIFL + N
! 645:
! 646: *
! 647: * Call ZHETRD_2STAGE to reduce Hermitian matrix to tridiagonal form.
! 648: *
! 649: CALL ZHETRD_2STAGE( JOBZ, UPLO, N, A, LDA, RWORK( INDRD ),
! 650: $ RWORK( INDRE ), WORK( INDTAU ),
! 651: $ WORK( INDHOUS ), LHTRD,
! 652: $ WORK( INDWK ), LLWORK, IINFO )
! 653: *
! 654: * If all eigenvalues are desired
! 655: * then call DSTERF or ZSTEMR and ZUNMTR.
! 656: *
! 657: TEST = .FALSE.
! 658: IF( INDEIG ) THEN
! 659: IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
! 660: TEST = .TRUE.
! 661: END IF
! 662: END IF
! 663: IF( ( ALLEIG.OR.TEST ) .AND. ( IEEEOK.EQ.1 ) ) THEN
! 664: IF( .NOT.WANTZ ) THEN
! 665: CALL DCOPY( N, RWORK( INDRD ), 1, W, 1 )
! 666: CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
! 667: CALL DSTERF( N, W, RWORK( INDREE ), INFO )
! 668: ELSE
! 669: CALL DCOPY( N-1, RWORK( INDRE ), 1, RWORK( INDREE ), 1 )
! 670: CALL DCOPY( N, RWORK( INDRD ), 1, RWORK( INDRDD ), 1 )
! 671: *
! 672: IF (ABSTOL .LE. TWO*N*EPS) THEN
! 673: TRYRAC = .TRUE.
! 674: ELSE
! 675: TRYRAC = .FALSE.
! 676: END IF
! 677: CALL ZSTEMR( JOBZ, 'A', N, RWORK( INDRDD ),
! 678: $ RWORK( INDREE ), VL, VU, IL, IU, M, W,
! 679: $ Z, LDZ, N, ISUPPZ, TRYRAC,
! 680: $ RWORK( INDRWK ), LLRWORK,
! 681: $ IWORK, LIWORK, INFO )
! 682: *
! 683: * Apply unitary matrix used in reduction to tridiagonal
! 684: * form to eigenvectors returned by ZSTEMR.
! 685: *
! 686: IF( WANTZ .AND. INFO.EQ.0 ) THEN
! 687: INDWKN = INDWK
! 688: LLWRKN = LWORK - INDWKN + 1
! 689: CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA,
! 690: $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ),
! 691: $ LLWRKN, IINFO )
! 692: END IF
! 693: END IF
! 694: *
! 695: *
! 696: IF( INFO.EQ.0 ) THEN
! 697: M = N
! 698: GO TO 30
! 699: END IF
! 700: INFO = 0
! 701: END IF
! 702: *
! 703: * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
! 704: * Also call DSTEBZ and ZSTEIN if ZSTEMR fails.
! 705: *
! 706: IF( WANTZ ) THEN
! 707: ORDER = 'B'
! 708: ELSE
! 709: ORDER = 'E'
! 710: END IF
! 711:
! 712: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
! 713: $ RWORK( INDRD ), RWORK( INDRE ), M, NSPLIT, W,
! 714: $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
! 715: $ IWORK( INDIWO ), INFO )
! 716: *
! 717: IF( WANTZ ) THEN
! 718: CALL ZSTEIN( N, RWORK( INDRD ), RWORK( INDRE ), M, W,
! 719: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
! 720: $ RWORK( INDRWK ), IWORK( INDIWO ), IWORK( INDIFL ),
! 721: $ INFO )
! 722: *
! 723: * Apply unitary matrix used in reduction to tridiagonal
! 724: * form to eigenvectors returned by ZSTEIN.
! 725: *
! 726: INDWKN = INDWK
! 727: LLWRKN = LWORK - INDWKN + 1
! 728: CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
! 729: $ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
! 730: END IF
! 731: *
! 732: * If matrix was scaled, then rescale eigenvalues appropriately.
! 733: *
! 734: 30 CONTINUE
! 735: IF( ISCALE.EQ.1 ) THEN
! 736: IF( INFO.EQ.0 ) THEN
! 737: IMAX = M
! 738: ELSE
! 739: IMAX = INFO - 1
! 740: END IF
! 741: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
! 742: END IF
! 743: *
! 744: * If eigenvalues are not in order, then sort them, along with
! 745: * eigenvectors.
! 746: *
! 747: IF( WANTZ ) THEN
! 748: DO 50 J = 1, M - 1
! 749: I = 0
! 750: TMP1 = W( J )
! 751: DO 40 JJ = J + 1, M
! 752: IF( W( JJ ).LT.TMP1 ) THEN
! 753: I = JJ
! 754: TMP1 = W( JJ )
! 755: END IF
! 756: 40 CONTINUE
! 757: *
! 758: IF( I.NE.0 ) THEN
! 759: ITMP1 = IWORK( INDIBL+I-1 )
! 760: W( I ) = W( J )
! 761: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
! 762: W( J ) = TMP1
! 763: IWORK( INDIBL+J-1 ) = ITMP1
! 764: CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
! 765: END IF
! 766: 50 CONTINUE
! 767: END IF
! 768: *
! 769: * Set WORK(1) to optimal workspace size.
! 770: *
! 771: WORK( 1 ) = LWMIN
! 772: RWORK( 1 ) = LRWMIN
! 773: IWORK( 1 ) = LIWMIN
! 774: *
! 775: RETURN
! 776: *
! 777: * End of ZHEEVR_2STAGE
! 778: *
! 779: END
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