Annotation of rpl/lapack/lapack/dstevx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
! 2: $ M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
! 3: *
! 4: * -- LAPACK driver routine (version 3.2) --
! 5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 7: * November 2006
! 8: *
! 9: * .. Scalar Arguments ..
! 10: CHARACTER JOBZ, RANGE
! 11: INTEGER IL, INFO, IU, LDZ, M, N
! 12: DOUBLE PRECISION ABSTOL, VL, VU
! 13: * ..
! 14: * .. Array Arguments ..
! 15: INTEGER IFAIL( * ), IWORK( * )
! 16: DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
! 17: * ..
! 18: *
! 19: * Purpose
! 20: * =======
! 21: *
! 22: * DSTEVX computes selected eigenvalues and, optionally, eigenvectors
! 23: * of a real symmetric tridiagonal matrix A. Eigenvalues and
! 24: * eigenvectors can be selected by specifying either a range of values
! 25: * or a range of indices for the desired eigenvalues.
! 26: *
! 27: * Arguments
! 28: * =========
! 29: *
! 30: * JOBZ (input) CHARACTER*1
! 31: * = 'N': Compute eigenvalues only;
! 32: * = 'V': Compute eigenvalues and eigenvectors.
! 33: *
! 34: * RANGE (input) CHARACTER*1
! 35: * = 'A': all eigenvalues will be found.
! 36: * = 'V': all eigenvalues in the half-open interval (VL,VU]
! 37: * will be found.
! 38: * = 'I': the IL-th through IU-th eigenvalues will be found.
! 39: *
! 40: * N (input) INTEGER
! 41: * The order of the matrix. N >= 0.
! 42: *
! 43: * D (input/output) DOUBLE PRECISION array, dimension (N)
! 44: * On entry, the n diagonal elements of the tridiagonal matrix
! 45: * A.
! 46: * On exit, D may be multiplied by a constant factor chosen
! 47: * to avoid over/underflow in computing the eigenvalues.
! 48: *
! 49: * E (input/output) DOUBLE PRECISION array, dimension (max(1,N-1))
! 50: * On entry, the (n-1) subdiagonal elements of the tridiagonal
! 51: * matrix A in elements 1 to N-1 of E.
! 52: * On exit, E may be multiplied by a constant factor chosen
! 53: * to avoid over/underflow in computing the eigenvalues.
! 54: *
! 55: * VL (input) DOUBLE PRECISION
! 56: * VU (input) DOUBLE PRECISION
! 57: * If RANGE='V', the lower and upper bounds of the interval to
! 58: * be searched for eigenvalues. VL < VU.
! 59: * Not referenced if RANGE = 'A' or 'I'.
! 60: *
! 61: * IL (input) INTEGER
! 62: * IU (input) INTEGER
! 63: * If RANGE='I', the indices (in ascending order) of the
! 64: * smallest and largest eigenvalues to be returned.
! 65: * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
! 66: * Not referenced if RANGE = 'A' or 'V'.
! 67: *
! 68: * ABSTOL (input) DOUBLE PRECISION
! 69: * The absolute error tolerance for the eigenvalues.
! 70: * An approximate eigenvalue is accepted as converged
! 71: * when it is determined to lie in an interval [a,b]
! 72: * of width less than or equal to
! 73: *
! 74: * ABSTOL + EPS * max( |a|,|b| ) ,
! 75: *
! 76: * where EPS is the machine precision. If ABSTOL is less
! 77: * than or equal to zero, then EPS*|T| will be used in
! 78: * its place, where |T| is the 1-norm of the tridiagonal
! 79: * matrix.
! 80: *
! 81: * Eigenvalues will be computed most accurately when ABSTOL is
! 82: * set to twice the underflow threshold 2*DLAMCH('S'), not zero.
! 83: * If this routine returns with INFO>0, indicating that some
! 84: * eigenvectors did not converge, try setting ABSTOL to
! 85: * 2*DLAMCH('S').
! 86: *
! 87: * See "Computing Small Singular Values of Bidiagonal Matrices
! 88: * with Guaranteed High Relative Accuracy," by Demmel and
! 89: * Kahan, LAPACK Working Note #3.
! 90: *
! 91: * M (output) INTEGER
! 92: * The total number of eigenvalues found. 0 <= M <= N.
! 93: * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
! 94: *
! 95: * W (output) DOUBLE PRECISION array, dimension (N)
! 96: * The first M elements contain the selected eigenvalues in
! 97: * ascending order.
! 98: *
! 99: * Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
! 100: * If JOBZ = 'V', then if INFO = 0, the first M columns of Z
! 101: * contain the orthonormal eigenvectors of the matrix A
! 102: * corresponding to the selected eigenvalues, with the i-th
! 103: * column of Z holding the eigenvector associated with W(i).
! 104: * If an eigenvector fails to converge (INFO > 0), then that
! 105: * column of Z contains the latest approximation to the
! 106: * eigenvector, and the index of the eigenvector is returned
! 107: * in IFAIL. If JOBZ = 'N', then Z is not referenced.
! 108: * Note: the user must ensure that at least max(1,M) columns are
! 109: * supplied in the array Z; if RANGE = 'V', the exact value of M
! 110: * is not known in advance and an upper bound must be used.
! 111: *
! 112: * LDZ (input) INTEGER
! 113: * The leading dimension of the array Z. LDZ >= 1, and if
! 114: * JOBZ = 'V', LDZ >= max(1,N).
! 115: *
! 116: * WORK (workspace) DOUBLE PRECISION array, dimension (5*N)
! 117: *
! 118: * IWORK (workspace) INTEGER array, dimension (5*N)
! 119: *
! 120: * IFAIL (output) INTEGER array, dimension (N)
! 121: * If JOBZ = 'V', then if INFO = 0, the first M elements of
! 122: * IFAIL are zero. If INFO > 0, then IFAIL contains the
! 123: * indices of the eigenvectors that failed to converge.
! 124: * If JOBZ = 'N', then IFAIL is not referenced.
! 125: *
! 126: * INFO (output) INTEGER
! 127: * = 0: successful exit
! 128: * < 0: if INFO = -i, the i-th argument had an illegal value
! 129: * > 0: if INFO = i, then i eigenvectors failed to converge.
! 130: * Their indices are stored in array IFAIL.
! 131: *
! 132: * =====================================================================
! 133: *
! 134: * .. Parameters ..
! 135: DOUBLE PRECISION ZERO, ONE
! 136: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
! 137: * ..
! 138: * .. Local Scalars ..
! 139: LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
! 140: CHARACTER ORDER
! 141: INTEGER I, IMAX, INDIBL, INDISP, INDIWO, INDWRK,
! 142: $ ISCALE, ITMP1, J, JJ, NSPLIT
! 143: DOUBLE PRECISION BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
! 144: $ TMP1, TNRM, VLL, VUU
! 145: * ..
! 146: * .. External Functions ..
! 147: LOGICAL LSAME
! 148: DOUBLE PRECISION DLAMCH, DLANST
! 149: EXTERNAL LSAME, DLAMCH, DLANST
! 150: * ..
! 151: * .. External Subroutines ..
! 152: EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTEIN, DSTEQR, DSTERF,
! 153: $ DSWAP, XERBLA
! 154: * ..
! 155: * .. Intrinsic Functions ..
! 156: INTRINSIC MAX, MIN, SQRT
! 157: * ..
! 158: * .. Executable Statements ..
! 159: *
! 160: * Test the input parameters.
! 161: *
! 162: WANTZ = LSAME( JOBZ, 'V' )
! 163: ALLEIG = LSAME( RANGE, 'A' )
! 164: VALEIG = LSAME( RANGE, 'V' )
! 165: INDEIG = LSAME( RANGE, 'I' )
! 166: *
! 167: INFO = 0
! 168: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
! 169: INFO = -1
! 170: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
! 171: INFO = -2
! 172: ELSE IF( N.LT.0 ) THEN
! 173: INFO = -3
! 174: ELSE
! 175: IF( VALEIG ) THEN
! 176: IF( N.GT.0 .AND. VU.LE.VL )
! 177: $ INFO = -7
! 178: ELSE IF( INDEIG ) THEN
! 179: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
! 180: INFO = -8
! 181: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
! 182: INFO = -9
! 183: END IF
! 184: END IF
! 185: END IF
! 186: IF( INFO.EQ.0 ) THEN
! 187: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
! 188: $ INFO = -14
! 189: END IF
! 190: *
! 191: IF( INFO.NE.0 ) THEN
! 192: CALL XERBLA( 'DSTEVX', -INFO )
! 193: RETURN
! 194: END IF
! 195: *
! 196: * Quick return if possible
! 197: *
! 198: M = 0
! 199: IF( N.EQ.0 )
! 200: $ RETURN
! 201: *
! 202: IF( N.EQ.1 ) THEN
! 203: IF( ALLEIG .OR. INDEIG ) THEN
! 204: M = 1
! 205: W( 1 ) = D( 1 )
! 206: ELSE
! 207: IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
! 208: M = 1
! 209: W( 1 ) = D( 1 )
! 210: END IF
! 211: END IF
! 212: IF( WANTZ )
! 213: $ Z( 1, 1 ) = ONE
! 214: RETURN
! 215: END IF
! 216: *
! 217: * Get machine constants.
! 218: *
! 219: SAFMIN = DLAMCH( 'Safe minimum' )
! 220: EPS = DLAMCH( 'Precision' )
! 221: SMLNUM = SAFMIN / EPS
! 222: BIGNUM = ONE / SMLNUM
! 223: RMIN = SQRT( SMLNUM )
! 224: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
! 225: *
! 226: * Scale matrix to allowable range, if necessary.
! 227: *
! 228: ISCALE = 0
! 229: IF( VALEIG ) THEN
! 230: VLL = VL
! 231: VUU = VU
! 232: ELSE
! 233: VLL = ZERO
! 234: VUU = ZERO
! 235: END IF
! 236: TNRM = DLANST( 'M', N, D, E )
! 237: IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
! 238: ISCALE = 1
! 239: SIGMA = RMIN / TNRM
! 240: ELSE IF( TNRM.GT.RMAX ) THEN
! 241: ISCALE = 1
! 242: SIGMA = RMAX / TNRM
! 243: END IF
! 244: IF( ISCALE.EQ.1 ) THEN
! 245: CALL DSCAL( N, SIGMA, D, 1 )
! 246: CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
! 247: IF( VALEIG ) THEN
! 248: VLL = VL*SIGMA
! 249: VUU = VU*SIGMA
! 250: END IF
! 251: END IF
! 252: *
! 253: * If all eigenvalues are desired and ABSTOL is less than zero, then
! 254: * call DSTERF or SSTEQR. If this fails for some eigenvalue, then
! 255: * try DSTEBZ.
! 256: *
! 257: TEST = .FALSE.
! 258: IF( INDEIG ) THEN
! 259: IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
! 260: TEST = .TRUE.
! 261: END IF
! 262: END IF
! 263: IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
! 264: CALL DCOPY( N, D, 1, W, 1 )
! 265: CALL DCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
! 266: INDWRK = N + 1
! 267: IF( .NOT.WANTZ ) THEN
! 268: CALL DSTERF( N, W, WORK, INFO )
! 269: ELSE
! 270: CALL DSTEQR( 'I', N, W, WORK, Z, LDZ, WORK( INDWRK ), INFO )
! 271: IF( INFO.EQ.0 ) THEN
! 272: DO 10 I = 1, N
! 273: IFAIL( I ) = 0
! 274: 10 CONTINUE
! 275: END IF
! 276: END IF
! 277: IF( INFO.EQ.0 ) THEN
! 278: M = N
! 279: GO TO 20
! 280: END IF
! 281: INFO = 0
! 282: END IF
! 283: *
! 284: * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
! 285: *
! 286: IF( WANTZ ) THEN
! 287: ORDER = 'B'
! 288: ELSE
! 289: ORDER = 'E'
! 290: END IF
! 291: INDWRK = 1
! 292: INDIBL = 1
! 293: INDISP = INDIBL + N
! 294: INDIWO = INDISP + N
! 295: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M,
! 296: $ NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ),
! 297: $ WORK( INDWRK ), IWORK( INDIWO ), INFO )
! 298: *
! 299: IF( WANTZ ) THEN
! 300: CALL DSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ),
! 301: $ Z, LDZ, WORK( INDWRK ), IWORK( INDIWO ), IFAIL,
! 302: $ INFO )
! 303: END IF
! 304: *
! 305: * If matrix was scaled, then rescale eigenvalues appropriately.
! 306: *
! 307: 20 CONTINUE
! 308: IF( ISCALE.EQ.1 ) THEN
! 309: IF( INFO.EQ.0 ) THEN
! 310: IMAX = M
! 311: ELSE
! 312: IMAX = INFO - 1
! 313: END IF
! 314: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
! 315: END IF
! 316: *
! 317: * If eigenvalues are not in order, then sort them, along with
! 318: * eigenvectors.
! 319: *
! 320: IF( WANTZ ) THEN
! 321: DO 40 J = 1, M - 1
! 322: I = 0
! 323: TMP1 = W( J )
! 324: DO 30 JJ = J + 1, M
! 325: IF( W( JJ ).LT.TMP1 ) THEN
! 326: I = JJ
! 327: TMP1 = W( JJ )
! 328: END IF
! 329: 30 CONTINUE
! 330: *
! 331: IF( I.NE.0 ) THEN
! 332: ITMP1 = IWORK( INDIBL+I-1 )
! 333: W( I ) = W( J )
! 334: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
! 335: W( J ) = TMP1
! 336: IWORK( INDIBL+J-1 ) = ITMP1
! 337: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
! 338: IF( INFO.NE.0 ) THEN
! 339: ITMP1 = IFAIL( I )
! 340: IFAIL( I ) = IFAIL( J )
! 341: IFAIL( J ) = ITMP1
! 342: END IF
! 343: END IF
! 344: 40 CONTINUE
! 345: END IF
! 346: *
! 347: RETURN
! 348: *
! 349: * End of DSTEVX
! 350: *
! 351: END
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