Annotation of rpl/lapack/lapack/dsyevx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
! 2: $ ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
! 3: $ IFAIL, INFO )
! 4: *
! 5: * -- LAPACK driver routine (version 3.2) --
! 6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 8: * November 2006
! 9: *
! 10: * .. Scalar Arguments ..
! 11: CHARACTER JOBZ, RANGE, UPLO
! 12: INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
! 13: DOUBLE PRECISION ABSTOL, VL, VU
! 14: * ..
! 15: * .. Array Arguments ..
! 16: INTEGER IFAIL( * ), IWORK( * )
! 17: DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
! 18: * ..
! 19: *
! 20: * Purpose
! 21: * =======
! 22: *
! 23: * DSYEVX computes selected eigenvalues and, optionally, eigenvectors
! 24: * of a real symmetric matrix A. Eigenvalues and eigenvectors can be
! 25: * selected by specifying either a range of values or a range of indices
! 26: * for the desired eigenvalues.
! 27: *
! 28: * Arguments
! 29: * =========
! 30: *
! 31: * JOBZ (input) CHARACTER*1
! 32: * = 'N': Compute eigenvalues only;
! 33: * = 'V': Compute eigenvalues and eigenvectors.
! 34: *
! 35: * RANGE (input) CHARACTER*1
! 36: * = 'A': all eigenvalues will be found.
! 37: * = 'V': all eigenvalues in the half-open interval (VL,VU]
! 38: * will be found.
! 39: * = 'I': the IL-th through IU-th eigenvalues will be found.
! 40: *
! 41: * UPLO (input) CHARACTER*1
! 42: * = 'U': Upper triangle of A is stored;
! 43: * = 'L': Lower triangle of A is stored.
! 44: *
! 45: * N (input) INTEGER
! 46: * The order of the matrix A. N >= 0.
! 47: *
! 48: * A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
! 49: * On entry, the symmetric matrix A. If UPLO = 'U', the
! 50: * leading N-by-N upper triangular part of A contains the
! 51: * upper triangular part of the matrix A. If UPLO = 'L',
! 52: * the leading N-by-N lower triangular part of A contains
! 53: * the lower triangular part of the matrix A.
! 54: * On exit, the lower triangle (if UPLO='L') or the upper
! 55: * triangle (if UPLO='U') of A, including the diagonal, is
! 56: * destroyed.
! 57: *
! 58: * LDA (input) INTEGER
! 59: * The leading dimension of the array A. LDA >= max(1,N).
! 60: *
! 61: * VL (input) DOUBLE PRECISION
! 62: * VU (input) DOUBLE PRECISION
! 63: * If RANGE='V', the lower and upper bounds of the interval to
! 64: * be searched for eigenvalues. VL < VU.
! 65: * Not referenced if RANGE = 'A' or 'I'.
! 66: *
! 67: * IL (input) INTEGER
! 68: * IU (input) INTEGER
! 69: * If RANGE='I', the indices (in ascending order) of the
! 70: * smallest and largest eigenvalues to be returned.
! 71: * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
! 72: * Not referenced if RANGE = 'A' or 'V'.
! 73: *
! 74: * ABSTOL (input) DOUBLE PRECISION
! 75: * The absolute error tolerance for the eigenvalues.
! 76: * An approximate eigenvalue is accepted as converged
! 77: * when it is determined to lie in an interval [a,b]
! 78: * of width less than or equal to
! 79: *
! 80: * ABSTOL + EPS * max( |a|,|b| ) ,
! 81: *
! 82: * where EPS is the machine precision. If ABSTOL is less than
! 83: * or equal to zero, then EPS*|T| will be used in its place,
! 84: * where |T| is the 1-norm of the tridiagonal matrix obtained
! 85: * by reducing A to tridiagonal form.
! 86: *
! 87: * Eigenvalues will be computed most accurately when ABSTOL is
! 88: * set to twice the underflow threshold 2*DLAMCH('S'), not zero.
! 89: * If this routine returns with INFO>0, indicating that some
! 90: * eigenvectors did not converge, try setting ABSTOL to
! 91: * 2*DLAMCH('S').
! 92: *
! 93: * See "Computing Small Singular Values of Bidiagonal Matrices
! 94: * with Guaranteed High Relative Accuracy," by Demmel and
! 95: * Kahan, LAPACK Working Note #3.
! 96: *
! 97: * M (output) INTEGER
! 98: * The total number of eigenvalues found. 0 <= M <= N.
! 99: * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
! 100: *
! 101: * W (output) DOUBLE PRECISION array, dimension (N)
! 102: * On normal exit, the first M elements contain the selected
! 103: * eigenvalues in ascending order.
! 104: *
! 105: * Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
! 106: * If JOBZ = 'V', then if INFO = 0, the first M columns of Z
! 107: * contain the orthonormal eigenvectors of the matrix A
! 108: * corresponding to the selected eigenvalues, with the i-th
! 109: * column of Z holding the eigenvector associated with W(i).
! 110: * If an eigenvector fails to converge, then that column of Z
! 111: * contains the latest approximation to the eigenvector, and the
! 112: * index of the eigenvector is returned in IFAIL.
! 113: * If JOBZ = 'N', then Z is not referenced.
! 114: * Note: the user must ensure that at least max(1,M) columns are
! 115: * supplied in the array Z; if RANGE = 'V', the exact value of M
! 116: * is not known in advance and an upper bound must be used.
! 117: *
! 118: * LDZ (input) INTEGER
! 119: * The leading dimension of the array Z. LDZ >= 1, and if
! 120: * JOBZ = 'V', LDZ >= max(1,N).
! 121: *
! 122: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
! 123: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 124: *
! 125: * LWORK (input) INTEGER
! 126: * The length of the array WORK. LWORK >= 1, when N <= 1;
! 127: * otherwise 8*N.
! 128: * For optimal efficiency, LWORK >= (NB+3)*N,
! 129: * where NB is the max of the blocksize for DSYTRD and DORMTR
! 130: * returned by ILAENV.
! 131: *
! 132: * If LWORK = -1, then a workspace query is assumed; the routine
! 133: * only calculates the optimal size of the WORK array, returns
! 134: * this value as the first entry of the WORK array, and no error
! 135: * message related to LWORK is issued by XERBLA.
! 136: *
! 137: * IWORK (workspace) INTEGER array, dimension (5*N)
! 138: *
! 139: * IFAIL (output) INTEGER array, dimension (N)
! 140: * If JOBZ = 'V', then if INFO = 0, the first M elements of
! 141: * IFAIL are zero. If INFO > 0, then IFAIL contains the
! 142: * indices of the eigenvectors that failed to converge.
! 143: * If JOBZ = 'N', then IFAIL is not referenced.
! 144: *
! 145: * INFO (output) INTEGER
! 146: * = 0: successful exit
! 147: * < 0: if INFO = -i, the i-th argument had an illegal value
! 148: * > 0: if INFO = i, then i eigenvectors failed to converge.
! 149: * Their indices are stored in array IFAIL.
! 150: *
! 151: * =====================================================================
! 152: *
! 153: * .. Parameters ..
! 154: DOUBLE PRECISION ZERO, ONE
! 155: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 156: * ..
! 157: * .. Local Scalars ..
! 158: LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
! 159: $ WANTZ
! 160: CHARACTER ORDER
! 161: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
! 162: $ INDISP, INDIWO, INDTAU, INDWKN, INDWRK, ISCALE,
! 163: $ ITMP1, J, JJ, LLWORK, LLWRKN, LWKMIN,
! 164: $ LWKOPT, NB, NSPLIT
! 165: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
! 166: $ SIGMA, SMLNUM, TMP1, VLL, VUU
! 167: * ..
! 168: * .. External Functions ..
! 169: LOGICAL LSAME
! 170: INTEGER ILAENV
! 171: DOUBLE PRECISION DLAMCH, DLANSY
! 172: EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY
! 173: * ..
! 174: * .. External Subroutines ..
! 175: EXTERNAL DCOPY, DLACPY, DORGTR, DORMTR, DSCAL, DSTEBZ,
! 176: $ DSTEIN, DSTEQR, DSTERF, DSWAP, DSYTRD, XERBLA
! 177: * ..
! 178: * .. Intrinsic Functions ..
! 179: INTRINSIC MAX, MIN, SQRT
! 180: * ..
! 181: * .. Executable Statements ..
! 182: *
! 183: * Test the input parameters.
! 184: *
! 185: LOWER = LSAME( UPLO, 'L' )
! 186: WANTZ = LSAME( JOBZ, 'V' )
! 187: ALLEIG = LSAME( RANGE, 'A' )
! 188: VALEIG = LSAME( RANGE, 'V' )
! 189: INDEIG = LSAME( RANGE, 'I' )
! 190: LQUERY = ( LWORK.EQ.-1 )
! 191: *
! 192: INFO = 0
! 193: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
! 194: INFO = -1
! 195: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
! 196: INFO = -2
! 197: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
! 198: INFO = -3
! 199: ELSE IF( N.LT.0 ) THEN
! 200: INFO = -4
! 201: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 202: INFO = -6
! 203: ELSE
! 204: IF( VALEIG ) THEN
! 205: IF( N.GT.0 .AND. VU.LE.VL )
! 206: $ INFO = -8
! 207: ELSE IF( INDEIG ) THEN
! 208: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
! 209: INFO = -9
! 210: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
! 211: INFO = -10
! 212: END IF
! 213: END IF
! 214: END IF
! 215: IF( INFO.EQ.0 ) THEN
! 216: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
! 217: INFO = -15
! 218: END IF
! 219: END IF
! 220: *
! 221: IF( INFO.EQ.0 ) THEN
! 222: IF( N.LE.1 ) THEN
! 223: LWKMIN = 1
! 224: WORK( 1 ) = LWKMIN
! 225: ELSE
! 226: LWKMIN = 8*N
! 227: NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
! 228: NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) )
! 229: LWKOPT = MAX( LWKMIN, ( NB + 3 )*N )
! 230: WORK( 1 ) = LWKOPT
! 231: END IF
! 232: *
! 233: IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
! 234: $ INFO = -17
! 235: END IF
! 236: *
! 237: IF( INFO.NE.0 ) THEN
! 238: CALL XERBLA( 'DSYEVX', -INFO )
! 239: RETURN
! 240: ELSE IF( LQUERY ) THEN
! 241: RETURN
! 242: END IF
! 243: *
! 244: * Quick return if possible
! 245: *
! 246: M = 0
! 247: IF( N.EQ.0 ) THEN
! 248: RETURN
! 249: END IF
! 250: *
! 251: IF( N.EQ.1 ) THEN
! 252: IF( ALLEIG .OR. INDEIG ) THEN
! 253: M = 1
! 254: W( 1 ) = A( 1, 1 )
! 255: ELSE
! 256: IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN
! 257: M = 1
! 258: W( 1 ) = A( 1, 1 )
! 259: END IF
! 260: END IF
! 261: IF( WANTZ )
! 262: $ Z( 1, 1 ) = ONE
! 263: RETURN
! 264: END IF
! 265: *
! 266: * Get machine constants.
! 267: *
! 268: SAFMIN = DLAMCH( 'Safe minimum' )
! 269: EPS = DLAMCH( 'Precision' )
! 270: SMLNUM = SAFMIN / EPS
! 271: BIGNUM = ONE / SMLNUM
! 272: RMIN = SQRT( SMLNUM )
! 273: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
! 274: *
! 275: * Scale matrix to allowable range, if necessary.
! 276: *
! 277: ISCALE = 0
! 278: ABSTLL = ABSTOL
! 279: IF( VALEIG ) THEN
! 280: VLL = VL
! 281: VUU = VU
! 282: END IF
! 283: ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
! 284: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
! 285: ISCALE = 1
! 286: SIGMA = RMIN / ANRM
! 287: ELSE IF( ANRM.GT.RMAX ) THEN
! 288: ISCALE = 1
! 289: SIGMA = RMAX / ANRM
! 290: END IF
! 291: IF( ISCALE.EQ.1 ) THEN
! 292: IF( LOWER ) THEN
! 293: DO 10 J = 1, N
! 294: CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 )
! 295: 10 CONTINUE
! 296: ELSE
! 297: DO 20 J = 1, N
! 298: CALL DSCAL( J, SIGMA, A( 1, J ), 1 )
! 299: 20 CONTINUE
! 300: END IF
! 301: IF( ABSTOL.GT.0 )
! 302: $ ABSTLL = ABSTOL*SIGMA
! 303: IF( VALEIG ) THEN
! 304: VLL = VL*SIGMA
! 305: VUU = VU*SIGMA
! 306: END IF
! 307: END IF
! 308: *
! 309: * Call DSYTRD to reduce symmetric matrix to tridiagonal form.
! 310: *
! 311: INDTAU = 1
! 312: INDE = INDTAU + N
! 313: INDD = INDE + N
! 314: INDWRK = INDD + N
! 315: LLWORK = LWORK - INDWRK + 1
! 316: CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ),
! 317: $ WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
! 318: *
! 319: * If all eigenvalues are desired and ABSTOL is less than or equal to
! 320: * zero, then call DSTERF or DORGTR and SSTEQR. If this fails for
! 321: * some eigenvalue, then try DSTEBZ.
! 322: *
! 323: TEST = .FALSE.
! 324: IF( INDEIG ) THEN
! 325: IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
! 326: TEST = .TRUE.
! 327: END IF
! 328: END IF
! 329: IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
! 330: CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
! 331: INDEE = INDWRK + 2*N
! 332: IF( .NOT.WANTZ ) THEN
! 333: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
! 334: CALL DSTERF( N, W, WORK( INDEE ), INFO )
! 335: ELSE
! 336: CALL DLACPY( 'A', N, N, A, LDA, Z, LDZ )
! 337: CALL DORGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
! 338: $ WORK( INDWRK ), LLWORK, IINFO )
! 339: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
! 340: CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
! 341: $ WORK( INDWRK ), INFO )
! 342: IF( INFO.EQ.0 ) THEN
! 343: DO 30 I = 1, N
! 344: IFAIL( I ) = 0
! 345: 30 CONTINUE
! 346: END IF
! 347: END IF
! 348: IF( INFO.EQ.0 ) THEN
! 349: M = N
! 350: GO TO 40
! 351: END IF
! 352: INFO = 0
! 353: END IF
! 354: *
! 355: * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
! 356: *
! 357: IF( WANTZ ) THEN
! 358: ORDER = 'B'
! 359: ELSE
! 360: ORDER = 'E'
! 361: END IF
! 362: INDIBL = 1
! 363: INDISP = INDIBL + N
! 364: INDIWO = INDISP + N
! 365: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
! 366: $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
! 367: $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
! 368: $ IWORK( INDIWO ), INFO )
! 369: *
! 370: IF( WANTZ ) THEN
! 371: CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
! 372: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
! 373: $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
! 374: *
! 375: * Apply orthogonal matrix used in reduction to tridiagonal
! 376: * form to eigenvectors returned by DSTEIN.
! 377: *
! 378: INDWKN = INDE
! 379: LLWRKN = LWORK - INDWKN + 1
! 380: CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
! 381: $ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
! 382: END IF
! 383: *
! 384: * If matrix was scaled, then rescale eigenvalues appropriately.
! 385: *
! 386: 40 CONTINUE
! 387: IF( ISCALE.EQ.1 ) THEN
! 388: IF( INFO.EQ.0 ) THEN
! 389: IMAX = M
! 390: ELSE
! 391: IMAX = INFO - 1
! 392: END IF
! 393: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
! 394: END IF
! 395: *
! 396: * If eigenvalues are not in order, then sort them, along with
! 397: * eigenvectors.
! 398: *
! 399: IF( WANTZ ) THEN
! 400: DO 60 J = 1, M - 1
! 401: I = 0
! 402: TMP1 = W( J )
! 403: DO 50 JJ = J + 1, M
! 404: IF( W( JJ ).LT.TMP1 ) THEN
! 405: I = JJ
! 406: TMP1 = W( JJ )
! 407: END IF
! 408: 50 CONTINUE
! 409: *
! 410: IF( I.NE.0 ) THEN
! 411: ITMP1 = IWORK( INDIBL+I-1 )
! 412: W( I ) = W( J )
! 413: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
! 414: W( J ) = TMP1
! 415: IWORK( INDIBL+J-1 ) = ITMP1
! 416: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
! 417: IF( INFO.NE.0 ) THEN
! 418: ITMP1 = IFAIL( I )
! 419: IFAIL( I ) = IFAIL( J )
! 420: IFAIL( J ) = ITMP1
! 421: END IF
! 422: END IF
! 423: 60 CONTINUE
! 424: END IF
! 425: *
! 426: * Set WORK(1) to optimal workspace size.
! 427: *
! 428: WORK( 1 ) = LWKOPT
! 429: *
! 430: RETURN
! 431: *
! 432: * End of DSYEVX
! 433: *
! 434: END
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