Annotation of rpl/lapack/lapack/dsyevr.f, revision 1.9
1.9 ! bertrand 1: *> \brief <b> DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY 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 DSYEVR + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyevr.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyevr.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyevr.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
! 22: * ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
! 23: * IWORK, LIWORK, INFO )
! 24: *
! 25: * .. Scalar Arguments ..
! 26: * CHARACTER JOBZ, RANGE, UPLO
! 27: * INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
! 28: * DOUBLE PRECISION ABSTOL, VL, VU
! 29: * ..
! 30: * .. Array Arguments ..
! 31: * INTEGER ISUPPZ( * ), IWORK( * )
! 32: * DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
! 33: * ..
! 34: *
! 35: *
! 36: *> \par Purpose:
! 37: * =============
! 38: *>
! 39: *> \verbatim
! 40: *>
! 41: *> DSYEVR computes selected eigenvalues and, optionally, eigenvectors
! 42: *> of a real symmetric matrix A. Eigenvalues and eigenvectors can be
! 43: *> selected by specifying either a range of values or a range of
! 44: *> indices for the desired eigenvalues.
! 45: *>
! 46: *> DSYEVR first reduces the matrix A to tridiagonal form T with a call
! 47: *> to DSYTRD. Then, whenever possible, DSYEVR calls DSTEMR to compute
! 48: *> the eigenspectrum using Relatively Robust Representations. DSTEMR
! 49: *> computes eigenvalues by the dqds algorithm, while orthogonal
! 50: *> eigenvectors are computed from various "good" L D L^T representations
! 51: *> (also known as Relatively Robust Representations). Gram-Schmidt
! 52: *> orthogonalization is avoided as far as possible. More specifically,
! 53: *> the various steps of the algorithm are as follows.
! 54: *>
! 55: *> For each unreduced block (submatrix) of T,
! 56: *> (a) Compute T - sigma I = L D L^T, so that L and D
! 57: *> define all the wanted eigenvalues to high relative accuracy.
! 58: *> This means that small relative changes in the entries of D and L
! 59: *> cause only small relative changes in the eigenvalues and
! 60: *> eigenvectors. The standard (unfactored) representation of the
! 61: *> tridiagonal matrix T does not have this property in general.
! 62: *> (b) Compute the eigenvalues to suitable accuracy.
! 63: *> If the eigenvectors are desired, the algorithm attains full
! 64: *> accuracy of the computed eigenvalues only right before
! 65: *> the corresponding vectors have to be computed, see steps c) and d).
! 66: *> (c) For each cluster of close eigenvalues, select a new
! 67: *> shift close to the cluster, find a new factorization, and refine
! 68: *> the shifted eigenvalues to suitable accuracy.
! 69: *> (d) For each eigenvalue with a large enough relative separation compute
! 70: *> the corresponding eigenvector by forming a rank revealing twisted
! 71: *> factorization. Go back to (c) for any clusters that remain.
! 72: *>
! 73: *> The desired accuracy of the output can be specified by the input
! 74: *> parameter ABSTOL.
! 75: *>
! 76: *> For more details, see DSTEMR's documentation and:
! 77: *> - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
! 78: *> to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
! 79: *> Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
! 80: *> - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
! 81: *> Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
! 82: *> 2004. Also LAPACK Working Note 154.
! 83: *> - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
! 84: *> tridiagonal eigenvalue/eigenvector problem",
! 85: *> Computer Science Division Technical Report No. UCB/CSD-97-971,
! 86: *> UC Berkeley, May 1997.
! 87: *>
! 88: *>
! 89: *> Note 1 : DSYEVR calls DSTEMR when the full spectrum is requested
! 90: *> on machines which conform to the ieee-754 floating point standard.
! 91: *> DSYEVR calls DSTEBZ and SSTEIN on non-ieee machines and
! 92: *> when partial spectrum requests are made.
! 93: *>
! 94: *> Normal execution of DSTEMR may create NaNs and infinities and
! 95: *> hence may abort due to a floating point exception in environments
! 96: *> which do not handle NaNs and infinities in the ieee standard default
! 97: *> manner.
! 98: *> \endverbatim
! 99: *
! 100: * Arguments:
! 101: * ==========
! 102: *
! 103: *> \param[in] JOBZ
! 104: *> \verbatim
! 105: *> JOBZ is CHARACTER*1
! 106: *> = 'N': Compute eigenvalues only;
! 107: *> = 'V': Compute eigenvalues and eigenvectors.
! 108: *> \endverbatim
! 109: *>
! 110: *> \param[in] RANGE
! 111: *> \verbatim
! 112: *> RANGE is CHARACTER*1
! 113: *> = 'A': all eigenvalues will be found.
! 114: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
! 115: *> will be found.
! 116: *> = 'I': the IL-th through IU-th eigenvalues will be found.
! 117: *> For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
! 118: *> DSTEIN are called
! 119: *> \endverbatim
! 120: *>
! 121: *> \param[in] UPLO
! 122: *> \verbatim
! 123: *> UPLO is CHARACTER*1
! 124: *> = 'U': Upper triangle of A is stored;
! 125: *> = 'L': Lower triangle of A is stored.
! 126: *> \endverbatim
! 127: *>
! 128: *> \param[in] N
! 129: *> \verbatim
! 130: *> N is INTEGER
! 131: *> The order of the matrix A. N >= 0.
! 132: *> \endverbatim
! 133: *>
! 134: *> \param[in,out] A
! 135: *> \verbatim
! 136: *> A is DOUBLE PRECISION array, dimension (LDA, N)
! 137: *> On entry, the symmetric matrix A. If UPLO = 'U', the
! 138: *> leading N-by-N upper triangular part of A contains the
! 139: *> upper triangular part of the matrix A. If UPLO = 'L',
! 140: *> the leading N-by-N lower triangular part of A contains
! 141: *> the lower triangular part of the matrix A.
! 142: *> On exit, the lower triangle (if UPLO='L') or the upper
! 143: *> triangle (if UPLO='U') of A, including the diagonal, is
! 144: *> destroyed.
! 145: *> \endverbatim
! 146: *>
! 147: *> \param[in] LDA
! 148: *> \verbatim
! 149: *> LDA is INTEGER
! 150: *> The leading dimension of the array A. LDA >= max(1,N).
! 151: *> \endverbatim
! 152: *>
! 153: *> \param[in] VL
! 154: *> \verbatim
! 155: *> VL is DOUBLE PRECISION
! 156: *> \endverbatim
! 157: *>
! 158: *> \param[in] VU
! 159: *> \verbatim
! 160: *> VU is DOUBLE PRECISION
! 161: *> If RANGE='V', the lower and upper bounds of the interval to
! 162: *> be searched for eigenvalues. VL < VU.
! 163: *> Not referenced if RANGE = 'A' or 'I'.
! 164: *> \endverbatim
! 165: *>
! 166: *> \param[in] IL
! 167: *> \verbatim
! 168: *> IL is INTEGER
! 169: *> \endverbatim
! 170: *>
! 171: *> \param[in] IU
! 172: *> \verbatim
! 173: *> IU is INTEGER
! 174: *> If RANGE='I', the indices (in ascending order) of the
! 175: *> smallest and largest eigenvalues to be returned.
! 176: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
! 177: *> Not referenced if RANGE = 'A' or 'V'.
! 178: *> \endverbatim
! 179: *>
! 180: *> \param[in] ABSTOL
! 181: *> \verbatim
! 182: *> ABSTOL is DOUBLE PRECISION
! 183: *> The absolute error tolerance for the eigenvalues.
! 184: *> An approximate eigenvalue is accepted as converged
! 185: *> when it is determined to lie in an interval [a,b]
! 186: *> of width less than or equal to
! 187: *>
! 188: *> ABSTOL + EPS * max( |a|,|b| ) ,
! 189: *>
! 190: *> where EPS is the machine precision. If ABSTOL is less than
! 191: *> or equal to zero, then EPS*|T| will be used in its place,
! 192: *> where |T| is the 1-norm of the tridiagonal matrix obtained
! 193: *> by reducing A to tridiagonal form.
! 194: *>
! 195: *> See "Computing Small Singular Values of Bidiagonal Matrices
! 196: *> with Guaranteed High Relative Accuracy," by Demmel and
! 197: *> Kahan, LAPACK Working Note #3.
! 198: *>
! 199: *> If high relative accuracy is important, set ABSTOL to
! 200: *> DLAMCH( 'Safe minimum' ). Doing so will guarantee that
! 201: *> eigenvalues are computed to high relative accuracy when
! 202: *> possible in future releases. The current code does not
! 203: *> make any guarantees about high relative accuracy, but
! 204: *> future releases will. See J. Barlow and J. Demmel,
! 205: *> "Computing Accurate Eigensystems of Scaled Diagonally
! 206: *> Dominant Matrices", LAPACK Working Note #7, for a discussion
! 207: *> of which matrices define their eigenvalues to high relative
! 208: *> accuracy.
! 209: *> \endverbatim
! 210: *>
! 211: *> \param[out] M
! 212: *> \verbatim
! 213: *> M is INTEGER
! 214: *> The total number of eigenvalues found. 0 <= M <= N.
! 215: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
! 216: *> \endverbatim
! 217: *>
! 218: *> \param[out] W
! 219: *> \verbatim
! 220: *> W is DOUBLE PRECISION array, dimension (N)
! 221: *> The first M elements contain the selected eigenvalues in
! 222: *> ascending order.
! 223: *> \endverbatim
! 224: *>
! 225: *> \param[out] Z
! 226: *> \verbatim
! 227: *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
! 228: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
! 229: *> contain the orthonormal eigenvectors of the matrix A
! 230: *> corresponding to the selected eigenvalues, with the i-th
! 231: *> column of Z holding the eigenvector associated with W(i).
! 232: *> If JOBZ = 'N', then Z is not referenced.
! 233: *> Note: the user must ensure that at least max(1,M) columns are
! 234: *> supplied in the array Z; if RANGE = 'V', the exact value of M
! 235: *> is not known in advance and an upper bound must be used.
! 236: *> Supplying N columns is always safe.
! 237: *> \endverbatim
! 238: *>
! 239: *> \param[in] LDZ
! 240: *> \verbatim
! 241: *> LDZ is INTEGER
! 242: *> The leading dimension of the array Z. LDZ >= 1, and if
! 243: *> JOBZ = 'V', LDZ >= max(1,N).
! 244: *> \endverbatim
! 245: *>
! 246: *> \param[out] ISUPPZ
! 247: *> \verbatim
! 248: *> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
! 249: *> The support of the eigenvectors in Z, i.e., the indices
! 250: *> indicating the nonzero elements in Z. The i-th eigenvector
! 251: *> is nonzero only in elements ISUPPZ( 2*i-1 ) through
! 252: *> ISUPPZ( 2*i ).
! 253: *> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
! 254: *> \endverbatim
! 255: *>
! 256: *> \param[out] WORK
! 257: *> \verbatim
! 258: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
! 259: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 260: *> \endverbatim
! 261: *>
! 262: *> \param[in] LWORK
! 263: *> \verbatim
! 264: *> LWORK is INTEGER
! 265: *> The dimension of the array WORK. LWORK >= max(1,26*N).
! 266: *> For optimal efficiency, LWORK >= (NB+6)*N,
! 267: *> where NB is the max of the blocksize for DSYTRD and DORMTR
! 268: *> returned by ILAENV.
! 269: *>
! 270: *> If LWORK = -1, then a workspace query is assumed; the routine
! 271: *> only calculates the optimal size of the WORK array, returns
! 272: *> this value as the first entry of the WORK array, and no error
! 273: *> message related to LWORK is issued by XERBLA.
! 274: *> \endverbatim
! 275: *>
! 276: *> \param[out] IWORK
! 277: *> \verbatim
! 278: *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
! 279: *> On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.
! 280: *> \endverbatim
! 281: *>
! 282: *> \param[in] LIWORK
! 283: *> \verbatim
! 284: *> LIWORK is INTEGER
! 285: *> The dimension of the array IWORK. LIWORK >= max(1,10*N).
! 286: *>
! 287: *> If LIWORK = -1, then a workspace query is assumed; the
! 288: *> routine only calculates the optimal size of the IWORK array,
! 289: *> returns this value as the first entry of the IWORK array, and
! 290: *> no error message related to LIWORK is issued by XERBLA.
! 291: *> \endverbatim
! 292: *>
! 293: *> \param[out] INFO
! 294: *> \verbatim
! 295: *> INFO is INTEGER
! 296: *> = 0: successful exit
! 297: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 298: *> > 0: Internal error
! 299: *> \endverbatim
! 300: *
! 301: * Authors:
! 302: * ========
! 303: *
! 304: *> \author Univ. of Tennessee
! 305: *> \author Univ. of California Berkeley
! 306: *> \author Univ. of Colorado Denver
! 307: *> \author NAG Ltd.
! 308: *
! 309: *> \date November 2011
! 310: *
! 311: *> \ingroup doubleSYeigen
! 312: *
! 313: *> \par Contributors:
! 314: * ==================
! 315: *>
! 316: *> Inderjit Dhillon, IBM Almaden, USA \n
! 317: *> Osni Marques, LBNL/NERSC, USA \n
! 318: *> Ken Stanley, Computer Science Division, University of
! 319: *> California at Berkeley, USA \n
! 320: *> Jason Riedy, Computer Science Division, University of
! 321: *> California at Berkeley, USA \n
! 322: *>
! 323: * =====================================================================
1.1 bertrand 324: SUBROUTINE DSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
325: $ ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
326: $ IWORK, LIWORK, INFO )
327: *
1.9 ! bertrand 328: * -- LAPACK driver routine (version 3.4.0) --
1.1 bertrand 329: * -- LAPACK is a software package provided by Univ. of Tennessee, --
330: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 331: * November 2011
1.1 bertrand 332: *
333: * .. Scalar Arguments ..
334: CHARACTER JOBZ, RANGE, UPLO
335: INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
336: DOUBLE PRECISION ABSTOL, VL, VU
337: * ..
338: * .. Array Arguments ..
339: INTEGER ISUPPZ( * ), IWORK( * )
340: DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
341: * ..
342: *
343: * =====================================================================
344: *
345: * .. Parameters ..
346: DOUBLE PRECISION ZERO, ONE, TWO
347: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
348: * ..
349: * .. Local Scalars ..
350: LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ,
351: $ TRYRAC
352: CHARACTER ORDER
353: INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE,
354: $ INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU,
355: $ INDWK, INDWKN, ISCALE, J, JJ, LIWMIN,
356: $ LLWORK, LLWRKN, LWKOPT, LWMIN, NB, NSPLIT
357: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
358: $ SIGMA, SMLNUM, TMP1, VLL, VUU
359: * ..
360: * .. External Functions ..
361: LOGICAL LSAME
362: INTEGER ILAENV
363: DOUBLE PRECISION DLAMCH, DLANSY
364: EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY
365: * ..
366: * .. External Subroutines ..
367: EXTERNAL DCOPY, DORMTR, DSCAL, DSTEBZ, DSTEMR, DSTEIN,
368: $ DSTERF, DSWAP, DSYTRD, XERBLA
369: * ..
370: * .. Intrinsic Functions ..
371: INTRINSIC MAX, MIN, SQRT
372: * ..
373: * .. Executable Statements ..
374: *
375: * Test the input parameters.
376: *
377: IEEEOK = ILAENV( 10, 'DSYEVR', 'N', 1, 2, 3, 4 )
378: *
379: LOWER = LSAME( UPLO, 'L' )
380: WANTZ = LSAME( JOBZ, 'V' )
381: ALLEIG = LSAME( RANGE, 'A' )
382: VALEIG = LSAME( RANGE, 'V' )
383: INDEIG = LSAME( RANGE, 'I' )
384: *
385: LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
386: *
387: LWMIN = MAX( 1, 26*N )
388: LIWMIN = MAX( 1, 10*N )
389: *
390: INFO = 0
391: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
392: INFO = -1
393: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
394: INFO = -2
395: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
396: INFO = -3
397: ELSE IF( N.LT.0 ) THEN
398: INFO = -4
399: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
400: INFO = -6
401: ELSE
402: IF( VALEIG ) THEN
403: IF( N.GT.0 .AND. VU.LE.VL )
404: $ INFO = -8
405: ELSE IF( INDEIG ) THEN
406: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
407: INFO = -9
408: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
409: INFO = -10
410: END IF
411: END IF
412: END IF
413: IF( INFO.EQ.0 ) THEN
414: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
415: INFO = -15
416: ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
417: INFO = -18
418: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
419: INFO = -20
420: END IF
421: END IF
422: *
423: IF( INFO.EQ.0 ) THEN
424: NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
425: NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) )
426: LWKOPT = MAX( ( NB+1 )*N, LWMIN )
427: WORK( 1 ) = LWKOPT
428: IWORK( 1 ) = LIWMIN
429: END IF
430: *
431: IF( INFO.NE.0 ) THEN
432: CALL XERBLA( 'DSYEVR', -INFO )
433: RETURN
434: ELSE IF( LQUERY ) THEN
435: RETURN
436: END IF
437: *
438: * Quick return if possible
439: *
440: M = 0
441: IF( N.EQ.0 ) THEN
442: WORK( 1 ) = 1
443: RETURN
444: END IF
445: *
446: IF( N.EQ.1 ) THEN
447: WORK( 1 ) = 7
448: IF( ALLEIG .OR. INDEIG ) THEN
449: M = 1
450: W( 1 ) = A( 1, 1 )
451: ELSE
452: IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN
453: M = 1
454: W( 1 ) = A( 1, 1 )
455: END IF
456: END IF
1.5 bertrand 457: IF( WANTZ ) THEN
458: Z( 1, 1 ) = ONE
459: ISUPPZ( 1 ) = 1
460: ISUPPZ( 2 ) = 1
461: END IF
1.1 bertrand 462: RETURN
463: END IF
464: *
465: * Get machine constants.
466: *
467: SAFMIN = DLAMCH( 'Safe minimum' )
468: EPS = DLAMCH( 'Precision' )
469: SMLNUM = SAFMIN / EPS
470: BIGNUM = ONE / SMLNUM
471: RMIN = SQRT( SMLNUM )
472: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
473: *
474: * Scale matrix to allowable range, if necessary.
475: *
476: ISCALE = 0
477: ABSTLL = ABSTOL
1.5 bertrand 478: IF (VALEIG) THEN
479: VLL = VL
480: VUU = VU
481: END IF
1.1 bertrand 482: ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
483: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
484: ISCALE = 1
485: SIGMA = RMIN / ANRM
486: ELSE IF( ANRM.GT.RMAX ) THEN
487: ISCALE = 1
488: SIGMA = RMAX / ANRM
489: END IF
490: IF( ISCALE.EQ.1 ) THEN
491: IF( LOWER ) THEN
492: DO 10 J = 1, N
493: CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 )
494: 10 CONTINUE
495: ELSE
496: DO 20 J = 1, N
497: CALL DSCAL( J, SIGMA, A( 1, J ), 1 )
498: 20 CONTINUE
499: END IF
500: IF( ABSTOL.GT.0 )
501: $ ABSTLL = ABSTOL*SIGMA
502: IF( VALEIG ) THEN
503: VLL = VL*SIGMA
504: VUU = VU*SIGMA
505: END IF
506: END IF
507:
508: * Initialize indices into workspaces. Note: The IWORK indices are
509: * used only if DSTERF or DSTEMR fail.
510:
511: * WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the
512: * elementary reflectors used in DSYTRD.
513: INDTAU = 1
514: * WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries.
515: INDD = INDTAU + N
516: * WORK(INDE:INDE+N-1) stores the off-diagonal entries of the
517: * tridiagonal matrix from DSYTRD.
518: INDE = INDD + N
519: * WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over
520: * -written by DSTEMR (the DSTERF path copies the diagonal to W).
521: INDDD = INDE + N
522: * WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over
523: * -written while computing the eigenvalues in DSTERF and DSTEMR.
524: INDEE = INDDD + N
525: * INDWK is the starting offset of the left-over workspace, and
526: * LLWORK is the remaining workspace size.
527: INDWK = INDEE + N
528: LLWORK = LWORK - INDWK + 1
529:
530: * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
531: * stores the block indices of each of the M<=N eigenvalues.
532: INDIBL = 1
533: * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
534: * stores the starting and finishing indices of each block.
535: INDISP = INDIBL + N
536: * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
537: * that corresponding to eigenvectors that fail to converge in
538: * DSTEIN. This information is discarded; if any fail, the driver
539: * returns INFO > 0.
540: INDIFL = INDISP + N
541: * INDIWO is the offset of the remaining integer workspace.
542: INDIWO = INDISP + N
543:
544: *
545: * Call DSYTRD to reduce symmetric matrix to tridiagonal form.
546: *
547: CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ),
548: $ WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO )
549: *
550: * If all eigenvalues are desired
551: * then call DSTERF or DSTEMR and DORMTR.
552: *
553: IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND.
554: $ IEEEOK.EQ.1 ) THEN
555: IF( .NOT.WANTZ ) THEN
556: CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
557: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
558: CALL DSTERF( N, W, WORK( INDEE ), INFO )
559: ELSE
560: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
561: CALL DCOPY( N, WORK( INDD ), 1, WORK( INDDD ), 1 )
562: *
563: IF (ABSTOL .LE. TWO*N*EPS) THEN
564: TRYRAC = .TRUE.
565: ELSE
566: TRYRAC = .FALSE.
567: END IF
568: CALL DSTEMR( JOBZ, 'A', N, WORK( INDDD ), WORK( INDEE ),
569: $ VL, VU, IL, IU, M, W, Z, LDZ, N, ISUPPZ,
570: $ TRYRAC, WORK( INDWK ), LWORK, IWORK, LIWORK,
571: $ INFO )
572: *
573: *
574: *
575: * Apply orthogonal matrix used in reduction to tridiagonal
576: * form to eigenvectors returned by DSTEIN.
577: *
578: IF( WANTZ .AND. INFO.EQ.0 ) THEN
579: INDWKN = INDE
580: LLWRKN = LWORK - INDWKN + 1
581: CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA,
582: $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ),
583: $ LLWRKN, IINFO )
584: END IF
585: END IF
586: *
587: *
588: IF( INFO.EQ.0 ) THEN
589: * Everything worked. Skip DSTEBZ/DSTEIN. IWORK(:) are
590: * undefined.
591: M = N
592: GO TO 30
593: END IF
594: INFO = 0
595: END IF
596: *
597: * Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN.
598: * Also call DSTEBZ and DSTEIN if DSTEMR fails.
599: *
600: IF( WANTZ ) THEN
601: ORDER = 'B'
602: ELSE
603: ORDER = 'E'
604: END IF
605:
606: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
607: $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
608: $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWK ),
609: $ IWORK( INDIWO ), INFO )
610: *
611: IF( WANTZ ) THEN
612: CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
613: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
614: $ WORK( INDWK ), IWORK( INDIWO ), IWORK( INDIFL ),
615: $ INFO )
616: *
617: * Apply orthogonal matrix used in reduction to tridiagonal
618: * form to eigenvectors returned by DSTEIN.
619: *
620: INDWKN = INDE
621: LLWRKN = LWORK - INDWKN + 1
622: CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
623: $ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
624: END IF
625: *
626: * If matrix was scaled, then rescale eigenvalues appropriately.
627: *
628: * Jump here if DSTEMR/DSTEIN succeeded.
629: 30 CONTINUE
630: IF( ISCALE.EQ.1 ) THEN
631: IF( INFO.EQ.0 ) THEN
632: IMAX = M
633: ELSE
634: IMAX = INFO - 1
635: END IF
636: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
637: END IF
638: *
639: * If eigenvalues are not in order, then sort them, along with
640: * eigenvectors. Note: We do not sort the IFAIL portion of IWORK.
641: * It may not be initialized (if DSTEMR/DSTEIN succeeded), and we do
642: * not return this detailed information to the user.
643: *
644: IF( WANTZ ) THEN
645: DO 50 J = 1, M - 1
646: I = 0
647: TMP1 = W( J )
648: DO 40 JJ = J + 1, M
649: IF( W( JJ ).LT.TMP1 ) THEN
650: I = JJ
651: TMP1 = W( JJ )
652: END IF
653: 40 CONTINUE
654: *
655: IF( I.NE.0 ) THEN
656: W( I ) = W( J )
657: W( J ) = TMP1
658: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
659: END IF
660: 50 CONTINUE
661: END IF
662: *
663: * Set WORK(1) to optimal workspace size.
664: *
665: WORK( 1 ) = LWKOPT
666: IWORK( 1 ) = LIWMIN
667: *
668: RETURN
669: *
670: * End of DSYEVR
671: *
672: END
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