Annotation of rpl/lapack/lapack/dsyevr_2stage.f, revision 1.1
1.1 ! bertrand 1: *> \brief <b> DSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices</b>
! 2: *
! 3: * @precisions fortran d -> s
! 4: *
! 5: * =========== DOCUMENTATION ===========
! 6: *
! 7: * Online html documentation available at
! 8: * http://www.netlib.org/lapack/explore-html/
! 9: *
! 10: *> \htmlonly
! 11: *> Download DSYEVR_2STAGE + dependencies
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyevr_2stage.f">
! 13: *> [TGZ]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyevr_2stage.f">
! 15: *> [ZIP]</a>
! 16: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyevr_2stage.f">
! 17: *> [TXT]</a>
! 18: *> \endhtmlonly
! 19: *
! 20: * Definition:
! 21: * ===========
! 22: *
! 23: * SUBROUTINE DSYEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
! 24: * IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK,
! 25: * LWORK, IWORK, LIWORK, INFO )
! 26: *
! 27: * IMPLICIT NONE
! 28: *
! 29: * .. Scalar Arguments ..
! 30: * CHARACTER JOBZ, RANGE, UPLO
! 31: * INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
! 32: * DOUBLE PRECISION ABSTOL, VL, VU
! 33: * ..
! 34: * .. Array Arguments ..
! 35: * INTEGER ISUPPZ( * ), IWORK( * )
! 36: * DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
! 37: * ..
! 38: *
! 39: *
! 40: *> \par Purpose:
! 41: * =============
! 42: *>
! 43: *> \verbatim
! 44: *>
! 45: *> DSYEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors
! 46: *> of a real symmetric matrix A using the 2stage technique for
! 47: *> the reduction to tridiagonal. Eigenvalues and eigenvectors can be
! 48: *> selected by specifying either a range of values or a range of
! 49: *> indices for the desired eigenvalues.
! 50: *>
! 51: *> DSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call
! 52: *> to DSYTRD. Then, whenever possible, DSYEVR_2STAGE calls DSTEMR to compute
! 53: *> the eigenspectrum using Relatively Robust Representations. DSTEMR
! 54: *> computes eigenvalues by the dqds algorithm, while orthogonal
! 55: *> eigenvectors are computed from various "good" L D L^T representations
! 56: *> (also known as Relatively Robust Representations). Gram-Schmidt
! 57: *> orthogonalization is avoided as far as possible. More specifically,
! 58: *> the various steps of the algorithm are as follows.
! 59: *>
! 60: *> For each unreduced block (submatrix) of T,
! 61: *> (a) Compute T - sigma I = L D L^T, so that L and D
! 62: *> define all the wanted eigenvalues to high relative accuracy.
! 63: *> This means that small relative changes in the entries of D and L
! 64: *> cause only small relative changes in the eigenvalues and
! 65: *> eigenvectors. The standard (unfactored) representation of the
! 66: *> tridiagonal matrix T does not have this property in general.
! 67: *> (b) Compute the eigenvalues to suitable accuracy.
! 68: *> If the eigenvectors are desired, the algorithm attains full
! 69: *> accuracy of the computed eigenvalues only right before
! 70: *> the corresponding vectors have to be computed, see steps c) and d).
! 71: *> (c) For each cluster of close eigenvalues, select a new
! 72: *> shift close to the cluster, find a new factorization, and refine
! 73: *> the shifted eigenvalues to suitable accuracy.
! 74: *> (d) For each eigenvalue with a large enough relative separation compute
! 75: *> the corresponding eigenvector by forming a rank revealing twisted
! 76: *> factorization. Go back to (c) for any clusters that remain.
! 77: *>
! 78: *> The desired accuracy of the output can be specified by the input
! 79: *> parameter ABSTOL.
! 80: *>
! 81: *> For more details, see DSTEMR's documentation and:
! 82: *> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
! 83: *> to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
! 84: *> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
! 85: *> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
! 86: *> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
! 87: *> 2004. Also LAPACK Working Note 154.
! 88: *> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
! 89: *> tridiagonal eigenvalue/eigenvector problem",
! 90: *> Computer Science Division Technical Report No. UCB/CSD-97-971,
! 91: *> UC Berkeley, May 1997.
! 92: *>
! 93: *>
! 94: *> Note 1 : DSYEVR_2STAGE calls DSTEMR when the full spectrum is requested
! 95: *> on machines which conform to the ieee-754 floating point standard.
! 96: *> DSYEVR_2STAGE calls DSTEBZ and SSTEIN on non-ieee machines and
! 97: *> when partial spectrum requests are made.
! 98: *>
! 99: *> Normal execution of DSTEMR may create NaNs and infinities and
! 100: *> hence may abort due to a floating point exception in environments
! 101: *> which do not handle NaNs and infinities in the ieee standard default
! 102: *> manner.
! 103: *> \endverbatim
! 104: *
! 105: * Arguments:
! 106: * ==========
! 107: *
! 108: *> \param[in] JOBZ
! 109: *> \verbatim
! 110: *> JOBZ is CHARACTER*1
! 111: *> = 'N': Compute eigenvalues only;
! 112: *> = 'V': Compute eigenvalues and eigenvectors.
! 113: *> Not available in this release.
! 114: *> \endverbatim
! 115: *>
! 116: *> \param[in] RANGE
! 117: *> \verbatim
! 118: *> RANGE is CHARACTER*1
! 119: *> = 'A': all eigenvalues will be found.
! 120: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
! 121: *> will be found.
! 122: *> = 'I': the IL-th through IU-th eigenvalues will be found.
! 123: *> For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
! 124: *> DSTEIN are called
! 125: *> \endverbatim
! 126: *>
! 127: *> \param[in] UPLO
! 128: *> \verbatim
! 129: *> UPLO is CHARACTER*1
! 130: *> = 'U': Upper triangle of A is stored;
! 131: *> = 'L': Lower triangle of A is stored.
! 132: *> \endverbatim
! 133: *>
! 134: *> \param[in] N
! 135: *> \verbatim
! 136: *> N is INTEGER
! 137: *> The order of the matrix A. N >= 0.
! 138: *> \endverbatim
! 139: *>
! 140: *> \param[in,out] A
! 141: *> \verbatim
! 142: *> A is DOUBLE PRECISION array, dimension (LDA, N)
! 143: *> On entry, the symmetric matrix A. If UPLO = 'U', the
! 144: *> leading N-by-N upper triangular part of A contains the
! 145: *> upper triangular part of the matrix A. If UPLO = 'L',
! 146: *> the leading N-by-N lower triangular part of A contains
! 147: *> the lower triangular part of the matrix A.
! 148: *> On exit, the lower triangle (if UPLO='L') or the upper
! 149: *> triangle (if UPLO='U') of A, including the diagonal, is
! 150: *> destroyed.
! 151: *> \endverbatim
! 152: *>
! 153: *> \param[in] LDA
! 154: *> \verbatim
! 155: *> LDA is INTEGER
! 156: *> The leading dimension of the array A. LDA >= max(1,N).
! 157: *> \endverbatim
! 158: *>
! 159: *> \param[in] VL
! 160: *> \verbatim
! 161: *> VL is DOUBLE PRECISION
! 162: *> If RANGE='V', the lower bound of the interval to
! 163: *> be searched for eigenvalues. VL < VU.
! 164: *> Not referenced if RANGE = 'A' or 'I'.
! 165: *> \endverbatim
! 166: *>
! 167: *> \param[in] VU
! 168: *> \verbatim
! 169: *> VU is DOUBLE PRECISION
! 170: *> If RANGE='V', the upper bound of the interval to
! 171: *> be searched for eigenvalues. VL < VU.
! 172: *> Not referenced if RANGE = 'A' or 'I'.
! 173: *> \endverbatim
! 174: *>
! 175: *> \param[in] IL
! 176: *> \verbatim
! 177: *> IL is INTEGER
! 178: *> If RANGE='I', the index of the
! 179: *> smallest eigenvalue to be returned.
! 180: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
! 181: *> Not referenced if RANGE = 'A' or 'V'.
! 182: *> \endverbatim
! 183: *>
! 184: *> \param[in] IU
! 185: *> \verbatim
! 186: *> IU is INTEGER
! 187: *> If RANGE='I', the index of the
! 188: *> largest eigenvalue to be returned.
! 189: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
! 190: *> Not referenced if RANGE = 'A' or 'V'.
! 191: *> \endverbatim
! 192: *>
! 193: *> \param[in] ABSTOL
! 194: *> \verbatim
! 195: *> ABSTOL is DOUBLE PRECISION
! 196: *> The absolute error tolerance for the eigenvalues.
! 197: *> An approximate eigenvalue is accepted as converged
! 198: *> when it is determined to lie in an interval [a,b]
! 199: *> of width less than or equal to
! 200: *>
! 201: *> ABSTOL + EPS * max( |a|,|b| ) ,
! 202: *>
! 203: *> where EPS is the machine precision. If ABSTOL is less than
! 204: *> or equal to zero, then EPS*|T| will be used in its place,
! 205: *> where |T| is the 1-norm of the tridiagonal matrix obtained
! 206: *> by reducing A to tridiagonal form.
! 207: *>
! 208: *> See "Computing Small Singular Values of Bidiagonal Matrices
! 209: *> with Guaranteed High Relative Accuracy," by Demmel and
! 210: *> Kahan, LAPACK Working Note #3.
! 211: *>
! 212: *> If high relative accuracy is important, set ABSTOL to
! 213: *> DLAMCH( 'Safe minimum' ). Doing so will guarantee that
! 214: *> eigenvalues are computed to high relative accuracy when
! 215: *> possible in future releases. The current code does not
! 216: *> make any guarantees about high relative accuracy, but
! 217: *> future releases will. See J. Barlow and J. Demmel,
! 218: *> "Computing Accurate Eigensystems of Scaled Diagonally
! 219: *> Dominant Matrices", LAPACK Working Note #7, for a discussion
! 220: *> of which matrices define their eigenvalues to high relative
! 221: *> accuracy.
! 222: *> \endverbatim
! 223: *>
! 224: *> \param[out] M
! 225: *> \verbatim
! 226: *> M is INTEGER
! 227: *> The total number of eigenvalues found. 0 <= M <= N.
! 228: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
! 229: *> \endverbatim
! 230: *>
! 231: *> \param[out] W
! 232: *> \verbatim
! 233: *> W is DOUBLE PRECISION array, dimension (N)
! 234: *> The first M elements contain the selected eigenvalues in
! 235: *> ascending order.
! 236: *> \endverbatim
! 237: *>
! 238: *> \param[out] Z
! 239: *> \verbatim
! 240: *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
! 241: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
! 242: *> contain the orthonormal eigenvectors of the matrix A
! 243: *> corresponding to the selected eigenvalues, with the i-th
! 244: *> column of Z holding the eigenvector associated with W(i).
! 245: *> If JOBZ = 'N', then Z is not referenced.
! 246: *> Note: the user must ensure that at least max(1,M) columns are
! 247: *> supplied in the array Z; if RANGE = 'V', the exact value of M
! 248: *> is not known in advance and an upper bound must be used.
! 249: *> Supplying N columns is always safe.
! 250: *> \endverbatim
! 251: *>
! 252: *> \param[in] LDZ
! 253: *> \verbatim
! 254: *> LDZ is INTEGER
! 255: *> The leading dimension of the array Z. LDZ >= 1, and if
! 256: *> JOBZ = 'V', LDZ >= max(1,N).
! 257: *> \endverbatim
! 258: *>
! 259: *> \param[out] ISUPPZ
! 260: *> \verbatim
! 261: *> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
! 262: *> The support of the eigenvectors in Z, i.e., the indices
! 263: *> indicating the nonzero elements in Z. The i-th eigenvector
! 264: *> is nonzero only in elements ISUPPZ( 2*i-1 ) through
! 265: *> ISUPPZ( 2*i ). This is an output of DSTEMR (tridiagonal
! 266: *> matrix). The support of the eigenvectors of A is typically
! 267: *> 1:N because of the orthogonal transformations applied by DORMTR.
! 268: *> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
! 269: *> \endverbatim
! 270: *>
! 271: *> \param[out] WORK
! 272: *> \verbatim
! 273: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
! 274: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 275: *> \endverbatim
! 276: *>
! 277: *> \param[in] LWORK
! 278: *> \verbatim
! 279: *> LWORK is INTEGER
! 280: *> The dimension of the array WORK.
! 281: *> If JOBZ = 'N' and N > 1, LWORK must be queried.
! 282: *> LWORK = MAX(1, 26*N, dimension) where
! 283: *> dimension = max(stage1,stage2) + (KD+1)*N + 5*N
! 284: *> = N*KD + N*max(KD+1,FACTOPTNB)
! 285: *> + max(2*KD*KD, KD*NTHREADS)
! 286: *> + (KD+1)*N + 5*N
! 287: *> where KD is the blocking size of the reduction,
! 288: *> FACTOPTNB is the blocking used by the QR or LQ
! 289: *> algorithm, usually FACTOPTNB=128 is a good choice
! 290: *> NTHREADS is the number of threads used when
! 291: *> openMP compilation is enabled, otherwise =1.
! 292: *> If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available
! 293: *>
! 294: *> If LWORK = -1, then a workspace query is assumed; the routine
! 295: *> only calculates the optimal size of the WORK array, returns
! 296: *> this value as the first entry of the WORK array, and no error
! 297: *> message related to LWORK is issued by XERBLA.
! 298: *> \endverbatim
! 299: *>
! 300: *> \param[out] IWORK
! 301: *> \verbatim
! 302: *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
! 303: *> On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.
! 304: *> \endverbatim
! 305: *>
! 306: *> \param[in] LIWORK
! 307: *> \verbatim
! 308: *> LIWORK is INTEGER
! 309: *> The dimension of the array IWORK. LIWORK >= max(1,10*N).
! 310: *>
! 311: *> If LIWORK = -1, then a workspace query is assumed; the
! 312: *> routine only calculates the optimal size of the IWORK array,
! 313: *> returns this value as the first entry of the IWORK array, and
! 314: *> no error message related to LIWORK is issued by XERBLA.
! 315: *> \endverbatim
! 316: *>
! 317: *> \param[out] INFO
! 318: *> \verbatim
! 319: *> INFO is INTEGER
! 320: *> = 0: successful exit
! 321: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 322: *> > 0: Internal error
! 323: *> \endverbatim
! 324: *
! 325: * Authors:
! 326: * ========
! 327: *
! 328: *> \author Univ. of Tennessee
! 329: *> \author Univ. of California Berkeley
! 330: *> \author Univ. of Colorado Denver
! 331: *> \author NAG Ltd.
! 332: *
! 333: *> \date June 2016
! 334: *
! 335: *> \ingroup doubleSYeigen
! 336: *
! 337: *> \par Contributors:
! 338: * ==================
! 339: *>
! 340: *> Inderjit Dhillon, IBM Almaden, USA \n
! 341: *> Osni Marques, LBNL/NERSC, USA \n
! 342: *> Ken Stanley, Computer Science Division, University of
! 343: *> California at Berkeley, USA \n
! 344: *> Jason Riedy, Computer Science Division, University of
! 345: *> California at Berkeley, USA \n
! 346: *>
! 347: *> \par Further Details:
! 348: * =====================
! 349: *>
! 350: *> \verbatim
! 351: *>
! 352: *> All details about the 2stage techniques are available in:
! 353: *>
! 354: *> Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
! 355: *> Parallel reduction to condensed forms for symmetric eigenvalue problems
! 356: *> using aggregated fine-grained and memory-aware kernels. In Proceedings
! 357: *> of 2011 International Conference for High Performance Computing,
! 358: *> Networking, Storage and Analysis (SC '11), New York, NY, USA,
! 359: *> Article 8 , 11 pages.
! 360: *> http://doi.acm.org/10.1145/2063384.2063394
! 361: *>
! 362: *> A. Haidar, J. Kurzak, P. Luszczek, 2013.
! 363: *> An improved parallel singular value algorithm and its implementation
! 364: *> for multicore hardware, In Proceedings of 2013 International Conference
! 365: *> for High Performance Computing, Networking, Storage and Analysis (SC '13).
! 366: *> Denver, Colorado, USA, 2013.
! 367: *> Article 90, 12 pages.
! 368: *> http://doi.acm.org/10.1145/2503210.2503292
! 369: *>
! 370: *> A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
! 371: *> A novel hybrid CPU-GPU generalized eigensolver for electronic structure
! 372: *> calculations based on fine-grained memory aware tasks.
! 373: *> International Journal of High Performance Computing Applications.
! 374: *> Volume 28 Issue 2, Pages 196-209, May 2014.
! 375: *> http://hpc.sagepub.com/content/28/2/196
! 376: *>
! 377: *> \endverbatim
! 378: *
! 379: * =====================================================================
! 380: SUBROUTINE DSYEVR_2STAGE( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
! 381: $ IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK,
! 382: $ LWORK, IWORK, LIWORK, INFO )
! 383: *
! 384: IMPLICIT NONE
! 385: *
! 386: * -- LAPACK driver routine (version 3.7.0) --
! 387: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 388: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 389: * June 2016
! 390: *
! 391: * .. Scalar Arguments ..
! 392: CHARACTER JOBZ, RANGE, UPLO
! 393: INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
! 394: DOUBLE PRECISION ABSTOL, VL, VU
! 395: * ..
! 396: * .. Array Arguments ..
! 397: INTEGER ISUPPZ( * ), IWORK( * )
! 398: DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
! 399: * ..
! 400: *
! 401: * =====================================================================
! 402: *
! 403: * .. Parameters ..
! 404: DOUBLE PRECISION ZERO, ONE, TWO
! 405: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
! 406: * ..
! 407: * .. Local Scalars ..
! 408: LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ,
! 409: $ TRYRAC
! 410: CHARACTER ORDER
! 411: INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE,
! 412: $ INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU,
! 413: $ INDWK, INDWKN, ISCALE, J, JJ, LIWMIN,
! 414: $ LLWORK, LLWRKN, LWMIN, NSPLIT,
! 415: $ LHTRD, LWTRD, KD, IB, INDHOUS
! 416: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
! 417: $ SIGMA, SMLNUM, TMP1, VLL, VUU
! 418: * ..
! 419: * .. External Functions ..
! 420: LOGICAL LSAME
! 421: INTEGER ILAENV
! 422: DOUBLE PRECISION DLAMCH, DLANSY
! 423: EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY
! 424: * ..
! 425: * .. External Subroutines ..
! 426: EXTERNAL DCOPY, DORMTR, DSCAL, DSTEBZ, DSTEMR, DSTEIN,
! 427: $ DSTERF, DSWAP, DSYTRD_2STAGE, XERBLA
! 428: * ..
! 429: * .. Intrinsic Functions ..
! 430: INTRINSIC MAX, MIN, SQRT
! 431: * ..
! 432: * .. Executable Statements ..
! 433: *
! 434: * Test the input parameters.
! 435: *
! 436: IEEEOK = ILAENV( 10, 'DSYEVR', 'N', 1, 2, 3, 4 )
! 437: *
! 438: LOWER = LSAME( UPLO, 'L' )
! 439: WANTZ = LSAME( JOBZ, 'V' )
! 440: ALLEIG = LSAME( RANGE, 'A' )
! 441: VALEIG = LSAME( RANGE, 'V' )
! 442: INDEIG = LSAME( RANGE, 'I' )
! 443: *
! 444: LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
! 445: *
! 446: KD = ILAENV( 17, 'DSYTRD_2STAGE', JOBZ, N, -1, -1, -1 )
! 447: IB = ILAENV( 18, 'DSYTRD_2STAGE', JOBZ, N, KD, -1, -1 )
! 448: LHTRD = ILAENV( 19, 'DSYTRD_2STAGE', JOBZ, N, KD, IB, -1 )
! 449: LWTRD = ILAENV( 20, 'DSYTRD_2STAGE', JOBZ, N, KD, IB, -1 )
! 450: LWMIN = MAX( 26*N, 5*N + LHTRD + LWTRD )
! 451: LIWMIN = MAX( 1, 10*N )
! 452: *
! 453: INFO = 0
! 454: IF( .NOT.( LSAME( JOBZ, 'N' ) ) ) THEN
! 455: INFO = -1
! 456: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
! 457: INFO = -2
! 458: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
! 459: INFO = -3
! 460: ELSE IF( N.LT.0 ) THEN
! 461: INFO = -4
! 462: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 463: INFO = -6
! 464: ELSE
! 465: IF( VALEIG ) THEN
! 466: IF( N.GT.0 .AND. VU.LE.VL )
! 467: $ INFO = -8
! 468: ELSE IF( INDEIG ) THEN
! 469: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
! 470: INFO = -9
! 471: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
! 472: INFO = -10
! 473: END IF
! 474: END IF
! 475: END IF
! 476: IF( INFO.EQ.0 ) THEN
! 477: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
! 478: INFO = -15
! 479: ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
! 480: INFO = -18
! 481: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
! 482: INFO = -20
! 483: END IF
! 484: END IF
! 485: *
! 486: IF( INFO.EQ.0 ) THEN
! 487: * NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
! 488: * NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) )
! 489: * LWKOPT = MAX( ( NB+1 )*N, LWMIN )
! 490: WORK( 1 ) = LWMIN
! 491: IWORK( 1 ) = LIWMIN
! 492: END IF
! 493: *
! 494: IF( INFO.NE.0 ) THEN
! 495: CALL XERBLA( 'DSYEVR_2STAGE', -INFO )
! 496: RETURN
! 497: ELSE IF( LQUERY ) THEN
! 498: RETURN
! 499: END IF
! 500: *
! 501: * Quick return if possible
! 502: *
! 503: M = 0
! 504: IF( N.EQ.0 ) THEN
! 505: WORK( 1 ) = 1
! 506: RETURN
! 507: END IF
! 508: *
! 509: IF( N.EQ.1 ) THEN
! 510: WORK( 1 ) = 7
! 511: IF( ALLEIG .OR. INDEIG ) THEN
! 512: M = 1
! 513: W( 1 ) = A( 1, 1 )
! 514: ELSE
! 515: IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN
! 516: M = 1
! 517: W( 1 ) = A( 1, 1 )
! 518: END IF
! 519: END IF
! 520: IF( WANTZ ) THEN
! 521: Z( 1, 1 ) = ONE
! 522: ISUPPZ( 1 ) = 1
! 523: ISUPPZ( 2 ) = 1
! 524: END IF
! 525: RETURN
! 526: END IF
! 527: *
! 528: * Get machine constants.
! 529: *
! 530: SAFMIN = DLAMCH( 'Safe minimum' )
! 531: EPS = DLAMCH( 'Precision' )
! 532: SMLNUM = SAFMIN / EPS
! 533: BIGNUM = ONE / SMLNUM
! 534: RMIN = SQRT( SMLNUM )
! 535: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
! 536: *
! 537: * Scale matrix to allowable range, if necessary.
! 538: *
! 539: ISCALE = 0
! 540: ABSTLL = ABSTOL
! 541: IF (VALEIG) THEN
! 542: VLL = VL
! 543: VUU = VU
! 544: END IF
! 545: ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
! 546: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
! 547: ISCALE = 1
! 548: SIGMA = RMIN / ANRM
! 549: ELSE IF( ANRM.GT.RMAX ) THEN
! 550: ISCALE = 1
! 551: SIGMA = RMAX / ANRM
! 552: END IF
! 553: IF( ISCALE.EQ.1 ) THEN
! 554: IF( LOWER ) THEN
! 555: DO 10 J = 1, N
! 556: CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 )
! 557: 10 CONTINUE
! 558: ELSE
! 559: DO 20 J = 1, N
! 560: CALL DSCAL( J, SIGMA, A( 1, J ), 1 )
! 561: 20 CONTINUE
! 562: END IF
! 563: IF( ABSTOL.GT.0 )
! 564: $ ABSTLL = ABSTOL*SIGMA
! 565: IF( VALEIG ) THEN
! 566: VLL = VL*SIGMA
! 567: VUU = VU*SIGMA
! 568: END IF
! 569: END IF
! 570:
! 571: * Initialize indices into workspaces. Note: The IWORK indices are
! 572: * used only if DSTERF or DSTEMR fail.
! 573:
! 574: * WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the
! 575: * elementary reflectors used in DSYTRD.
! 576: INDTAU = 1
! 577: * WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries.
! 578: INDD = INDTAU + N
! 579: * WORK(INDE:INDE+N-1) stores the off-diagonal entries of the
! 580: * tridiagonal matrix from DSYTRD.
! 581: INDE = INDD + N
! 582: * WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over
! 583: * -written by DSTEMR (the DSTERF path copies the diagonal to W).
! 584: INDDD = INDE + N
! 585: * WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over
! 586: * -written while computing the eigenvalues in DSTERF and DSTEMR.
! 587: INDEE = INDDD + N
! 588: * INDHOUS is the starting offset Householder storage of stage 2
! 589: INDHOUS = INDEE + N
! 590: * INDWK is the starting offset of the left-over workspace, and
! 591: * LLWORK is the remaining workspace size.
! 592: INDWK = INDHOUS + LHTRD
! 593: LLWORK = LWORK - INDWK + 1
! 594:
! 595:
! 596: * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
! 597: * stores the block indices of each of the M<=N eigenvalues.
! 598: INDIBL = 1
! 599: * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
! 600: * stores the starting and finishing indices of each block.
! 601: INDISP = INDIBL + N
! 602: * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
! 603: * that corresponding to eigenvectors that fail to converge in
! 604: * DSTEIN. This information is discarded; if any fail, the driver
! 605: * returns INFO > 0.
! 606: INDIFL = INDISP + N
! 607: * INDIWO is the offset of the remaining integer workspace.
! 608: INDIWO = INDIFL + N
! 609:
! 610: *
! 611: * Call DSYTRD_2STAGE to reduce symmetric matrix to tridiagonal form.
! 612: *
! 613: *
! 614: CALL DSYTRD_2STAGE( JOBZ, UPLO, N, A, LDA, WORK( INDD ),
! 615: $ WORK( INDE ), WORK( INDTAU ), WORK( INDHOUS ),
! 616: $ LHTRD, WORK( INDWK ), LLWORK, IINFO )
! 617: *
! 618: * If all eigenvalues are desired
! 619: * then call DSTERF or DSTEMR and DORMTR.
! 620: *
! 621: IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND.
! 622: $ IEEEOK.EQ.1 ) THEN
! 623: IF( .NOT.WANTZ ) THEN
! 624: CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
! 625: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
! 626: CALL DSTERF( N, W, WORK( INDEE ), INFO )
! 627: ELSE
! 628: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
! 629: CALL DCOPY( N, WORK( INDD ), 1, WORK( INDDD ), 1 )
! 630: *
! 631: IF (ABSTOL .LE. TWO*N*EPS) THEN
! 632: TRYRAC = .TRUE.
! 633: ELSE
! 634: TRYRAC = .FALSE.
! 635: END IF
! 636: CALL DSTEMR( JOBZ, 'A', N, WORK( INDDD ), WORK( INDEE ),
! 637: $ VL, VU, IL, IU, M, W, Z, LDZ, N, ISUPPZ,
! 638: $ TRYRAC, WORK( INDWK ), LWORK, IWORK, LIWORK,
! 639: $ INFO )
! 640: *
! 641: *
! 642: *
! 643: * Apply orthogonal matrix used in reduction to tridiagonal
! 644: * form to eigenvectors returned by DSTEMR.
! 645: *
! 646: IF( WANTZ .AND. INFO.EQ.0 ) THEN
! 647: INDWKN = INDE
! 648: LLWRKN = LWORK - INDWKN + 1
! 649: CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA,
! 650: $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ),
! 651: $ LLWRKN, IINFO )
! 652: END IF
! 653: END IF
! 654: *
! 655: *
! 656: IF( INFO.EQ.0 ) THEN
! 657: * Everything worked. Skip DSTEBZ/DSTEIN. IWORK(:) are
! 658: * undefined.
! 659: M = N
! 660: GO TO 30
! 661: END IF
! 662: INFO = 0
! 663: END IF
! 664: *
! 665: * Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN.
! 666: * Also call DSTEBZ and DSTEIN if DSTEMR fails.
! 667: *
! 668: IF( WANTZ ) THEN
! 669: ORDER = 'B'
! 670: ELSE
! 671: ORDER = 'E'
! 672: END IF
! 673:
! 674: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
! 675: $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
! 676: $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWK ),
! 677: $ IWORK( INDIWO ), INFO )
! 678: *
! 679: IF( WANTZ ) THEN
! 680: CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
! 681: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
! 682: $ WORK( INDWK ), IWORK( INDIWO ), IWORK( INDIFL ),
! 683: $ INFO )
! 684: *
! 685: * Apply orthogonal matrix used in reduction to tridiagonal
! 686: * form to eigenvectors returned by DSTEIN.
! 687: *
! 688: INDWKN = INDE
! 689: LLWRKN = LWORK - INDWKN + 1
! 690: CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
! 691: $ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
! 692: END IF
! 693: *
! 694: * If matrix was scaled, then rescale eigenvalues appropriately.
! 695: *
! 696: * Jump here if DSTEMR/DSTEIN succeeded.
! 697: 30 CONTINUE
! 698: IF( ISCALE.EQ.1 ) THEN
! 699: IF( INFO.EQ.0 ) THEN
! 700: IMAX = M
! 701: ELSE
! 702: IMAX = INFO - 1
! 703: END IF
! 704: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
! 705: END IF
! 706: *
! 707: * If eigenvalues are not in order, then sort them, along with
! 708: * eigenvectors. Note: We do not sort the IFAIL portion of IWORK.
! 709: * It may not be initialized (if DSTEMR/DSTEIN succeeded), and we do
! 710: * not return this detailed information to the user.
! 711: *
! 712: IF( WANTZ ) THEN
! 713: DO 50 J = 1, M - 1
! 714: I = 0
! 715: TMP1 = W( J )
! 716: DO 40 JJ = J + 1, M
! 717: IF( W( JJ ).LT.TMP1 ) THEN
! 718: I = JJ
! 719: TMP1 = W( JJ )
! 720: END IF
! 721: 40 CONTINUE
! 722: *
! 723: IF( I.NE.0 ) THEN
! 724: W( I ) = W( J )
! 725: W( J ) = TMP1
! 726: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
! 727: END IF
! 728: 50 CONTINUE
! 729: END IF
! 730: *
! 731: * Set WORK(1) to optimal workspace size.
! 732: *
! 733: WORK( 1 ) = LWMIN
! 734: IWORK( 1 ) = LIWMIN
! 735: *
! 736: RETURN
! 737: *
! 738: * End of DSYEVR_2STAGE
! 739: *
! 740: END
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