Annotation of rpl/lapack/lapack/dsbevx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
! 2: $ VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, 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, KD, LDAB, LDQ, LDZ, M, N
! 13: DOUBLE PRECISION ABSTOL, VL, VU
! 14: * ..
! 15: * .. Array Arguments ..
! 16: INTEGER IFAIL( * ), IWORK( * )
! 17: DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
! 18: $ Z( LDZ, * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * DSBEVX computes selected eigenvalues and, optionally, eigenvectors
! 25: * of a real symmetric band matrix A. Eigenvalues and eigenvectors can
! 26: * be selected by specifying either a range of values or a range of
! 27: * indices for the desired eigenvalues.
! 28: *
! 29: * Arguments
! 30: * =========
! 31: *
! 32: * JOBZ (input) CHARACTER*1
! 33: * = 'N': Compute eigenvalues only;
! 34: * = 'V': Compute eigenvalues and eigenvectors.
! 35: *
! 36: * RANGE (input) CHARACTER*1
! 37: * = 'A': all eigenvalues will be found;
! 38: * = 'V': all eigenvalues in the half-open interval (VL,VU]
! 39: * will be found;
! 40: * = 'I': the IL-th through IU-th eigenvalues will be found.
! 41: *
! 42: * UPLO (input) CHARACTER*1
! 43: * = 'U': Upper triangle of A is stored;
! 44: * = 'L': Lower triangle of A is stored.
! 45: *
! 46: * N (input) INTEGER
! 47: * The order of the matrix A. N >= 0.
! 48: *
! 49: * KD (input) INTEGER
! 50: * The number of superdiagonals of the matrix A if UPLO = 'U',
! 51: * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
! 52: *
! 53: * AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
! 54: * On entry, the upper or lower triangle of the symmetric band
! 55: * matrix A, stored in the first KD+1 rows of the array. The
! 56: * j-th column of A is stored in the j-th column of the array AB
! 57: * as follows:
! 58: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
! 59: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
! 60: *
! 61: * On exit, AB is overwritten by values generated during the
! 62: * reduction to tridiagonal form. If UPLO = 'U', the first
! 63: * superdiagonal and the diagonal of the tridiagonal matrix T
! 64: * are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
! 65: * the diagonal and first subdiagonal of T are returned in the
! 66: * first two rows of AB.
! 67: *
! 68: * LDAB (input) INTEGER
! 69: * The leading dimension of the array AB. LDAB >= KD + 1.
! 70: *
! 71: * Q (output) DOUBLE PRECISION array, dimension (LDQ, N)
! 72: * If JOBZ = 'V', the N-by-N orthogonal matrix used in the
! 73: * reduction to tridiagonal form.
! 74: * If JOBZ = 'N', the array Q is not referenced.
! 75: *
! 76: * LDQ (input) INTEGER
! 77: * The leading dimension of the array Q. If JOBZ = 'V', then
! 78: * LDQ >= max(1,N).
! 79: *
! 80: * VL (input) DOUBLE PRECISION
! 81: * VU (input) DOUBLE PRECISION
! 82: * If RANGE='V', the lower and upper bounds of the interval to
! 83: * be searched for eigenvalues. VL < VU.
! 84: * Not referenced if RANGE = 'A' or 'I'.
! 85: *
! 86: * IL (input) INTEGER
! 87: * IU (input) INTEGER
! 88: * If RANGE='I', the indices (in ascending order) of the
! 89: * smallest and largest eigenvalues to be returned.
! 90: * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
! 91: * Not referenced if RANGE = 'A' or 'V'.
! 92: *
! 93: * ABSTOL (input) DOUBLE PRECISION
! 94: * The absolute error tolerance for the eigenvalues.
! 95: * An approximate eigenvalue is accepted as converged
! 96: * when it is determined to lie in an interval [a,b]
! 97: * of width less than or equal to
! 98: *
! 99: * ABSTOL + EPS * max( |a|,|b| ) ,
! 100: *
! 101: * where EPS is the machine precision. If ABSTOL is less than
! 102: * or equal to zero, then EPS*|T| will be used in its place,
! 103: * where |T| is the 1-norm of the tridiagonal matrix obtained
! 104: * by reducing AB to tridiagonal form.
! 105: *
! 106: * Eigenvalues will be computed most accurately when ABSTOL is
! 107: * set to twice the underflow threshold 2*DLAMCH('S'), not zero.
! 108: * If this routine returns with INFO>0, indicating that some
! 109: * eigenvectors did not converge, try setting ABSTOL to
! 110: * 2*DLAMCH('S').
! 111: *
! 112: * See "Computing Small Singular Values of Bidiagonal Matrices
! 113: * with Guaranteed High Relative Accuracy," by Demmel and
! 114: * Kahan, LAPACK Working Note #3.
! 115: *
! 116: * M (output) INTEGER
! 117: * The total number of eigenvalues found. 0 <= M <= N.
! 118: * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
! 119: *
! 120: * W (output) DOUBLE PRECISION array, dimension (N)
! 121: * The first M elements contain the selected eigenvalues in
! 122: * ascending order.
! 123: *
! 124: * Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
! 125: * If JOBZ = 'V', then if INFO = 0, the first M columns of Z
! 126: * contain the orthonormal eigenvectors of the matrix A
! 127: * corresponding to the selected eigenvalues, with the i-th
! 128: * column of Z holding the eigenvector associated with W(i).
! 129: * If an eigenvector fails to converge, then that column of Z
! 130: * contains the latest approximation to the eigenvector, and the
! 131: * index of the eigenvector is returned in IFAIL.
! 132: * If JOBZ = 'N', then Z is not referenced.
! 133: * Note: the user must ensure that at least max(1,M) columns are
! 134: * supplied in the array Z; if RANGE = 'V', the exact value of M
! 135: * is not known in advance and an upper bound must be used.
! 136: *
! 137: * LDZ (input) INTEGER
! 138: * The leading dimension of the array Z. LDZ >= 1, and if
! 139: * JOBZ = 'V', LDZ >= max(1,N).
! 140: *
! 141: * WORK (workspace) DOUBLE PRECISION array, dimension (7*N)
! 142: *
! 143: * IWORK (workspace) INTEGER array, dimension (5*N)
! 144: *
! 145: * IFAIL (output) INTEGER array, dimension (N)
! 146: * If JOBZ = 'V', then if INFO = 0, the first M elements of
! 147: * IFAIL are zero. If INFO > 0, then IFAIL contains the
! 148: * indices of the eigenvectors that failed to converge.
! 149: * If JOBZ = 'N', then IFAIL is not referenced.
! 150: *
! 151: * INFO (output) INTEGER
! 152: * = 0: successful exit.
! 153: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 154: * > 0: if INFO = i, then i eigenvectors failed to converge.
! 155: * Their indices are stored in array IFAIL.
! 156: *
! 157: * =====================================================================
! 158: *
! 159: * .. Parameters ..
! 160: DOUBLE PRECISION ZERO, ONE
! 161: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
! 162: * ..
! 163: * .. Local Scalars ..
! 164: LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ
! 165: CHARACTER ORDER
! 166: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
! 167: $ INDISP, INDIWO, INDWRK, ISCALE, ITMP1, J, JJ,
! 168: $ NSPLIT
! 169: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
! 170: $ SIGMA, SMLNUM, TMP1, VLL, VUU
! 171: * ..
! 172: * .. External Functions ..
! 173: LOGICAL LSAME
! 174: DOUBLE PRECISION DLAMCH, DLANSB
! 175: EXTERNAL LSAME, DLAMCH, DLANSB
! 176: * ..
! 177: * .. External Subroutines ..
! 178: EXTERNAL DCOPY, DGEMV, DLACPY, DLASCL, DSBTRD, DSCAL,
! 179: $ DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
! 180: * ..
! 181: * .. Intrinsic Functions ..
! 182: INTRINSIC MAX, MIN, SQRT
! 183: * ..
! 184: * .. Executable Statements ..
! 185: *
! 186: * Test the input parameters.
! 187: *
! 188: WANTZ = LSAME( JOBZ, 'V' )
! 189: ALLEIG = LSAME( RANGE, 'A' )
! 190: VALEIG = LSAME( RANGE, 'V' )
! 191: INDEIG = LSAME( RANGE, 'I' )
! 192: LOWER = LSAME( UPLO, 'L' )
! 193: *
! 194: INFO = 0
! 195: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
! 196: INFO = -1
! 197: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
! 198: INFO = -2
! 199: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
! 200: INFO = -3
! 201: ELSE IF( N.LT.0 ) THEN
! 202: INFO = -4
! 203: ELSE IF( KD.LT.0 ) THEN
! 204: INFO = -5
! 205: ELSE IF( LDAB.LT.KD+1 ) THEN
! 206: INFO = -7
! 207: ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN
! 208: INFO = -9
! 209: ELSE
! 210: IF( VALEIG ) THEN
! 211: IF( N.GT.0 .AND. VU.LE.VL )
! 212: $ INFO = -11
! 213: ELSE IF( INDEIG ) THEN
! 214: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
! 215: INFO = -12
! 216: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
! 217: INFO = -13
! 218: END IF
! 219: END IF
! 220: END IF
! 221: IF( INFO.EQ.0 ) THEN
! 222: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
! 223: $ INFO = -18
! 224: END IF
! 225: *
! 226: IF( INFO.NE.0 ) THEN
! 227: CALL XERBLA( 'DSBEVX', -INFO )
! 228: RETURN
! 229: END IF
! 230: *
! 231: * Quick return if possible
! 232: *
! 233: M = 0
! 234: IF( N.EQ.0 )
! 235: $ RETURN
! 236: *
! 237: IF( N.EQ.1 ) THEN
! 238: M = 1
! 239: IF( LOWER ) THEN
! 240: TMP1 = AB( 1, 1 )
! 241: ELSE
! 242: TMP1 = AB( KD+1, 1 )
! 243: END IF
! 244: IF( VALEIG ) THEN
! 245: IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) )
! 246: $ M = 0
! 247: END IF
! 248: IF( M.EQ.1 ) THEN
! 249: W( 1 ) = TMP1
! 250: IF( WANTZ )
! 251: $ Z( 1, 1 ) = ONE
! 252: END IF
! 253: RETURN
! 254: END IF
! 255: *
! 256: * Get machine constants.
! 257: *
! 258: SAFMIN = DLAMCH( 'Safe minimum' )
! 259: EPS = DLAMCH( 'Precision' )
! 260: SMLNUM = SAFMIN / EPS
! 261: BIGNUM = ONE / SMLNUM
! 262: RMIN = SQRT( SMLNUM )
! 263: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
! 264: *
! 265: * Scale matrix to allowable range, if necessary.
! 266: *
! 267: ISCALE = 0
! 268: ABSTLL = ABSTOL
! 269: IF( VALEIG ) THEN
! 270: VLL = VL
! 271: VUU = VU
! 272: ELSE
! 273: VLL = ZERO
! 274: VUU = ZERO
! 275: END IF
! 276: ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
! 277: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
! 278: ISCALE = 1
! 279: SIGMA = RMIN / ANRM
! 280: ELSE IF( ANRM.GT.RMAX ) THEN
! 281: ISCALE = 1
! 282: SIGMA = RMAX / ANRM
! 283: END IF
! 284: IF( ISCALE.EQ.1 ) THEN
! 285: IF( LOWER ) THEN
! 286: CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
! 287: ELSE
! 288: CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
! 289: END IF
! 290: IF( ABSTOL.GT.0 )
! 291: $ ABSTLL = ABSTOL*SIGMA
! 292: IF( VALEIG ) THEN
! 293: VLL = VL*SIGMA
! 294: VUU = VU*SIGMA
! 295: END IF
! 296: END IF
! 297: *
! 298: * Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
! 299: *
! 300: INDD = 1
! 301: INDE = INDD + N
! 302: INDWRK = INDE + N
! 303: CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, WORK( INDD ),
! 304: $ WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
! 305: *
! 306: * If all eigenvalues are desired and ABSTOL is less than or equal
! 307: * to zero, then call DSTERF or SSTEQR. If this fails for some
! 308: * eigenvalue, then try DSTEBZ.
! 309: *
! 310: TEST = .FALSE.
! 311: IF (INDEIG) THEN
! 312: IF (IL.EQ.1 .AND. IU.EQ.N) THEN
! 313: TEST = .TRUE.
! 314: END IF
! 315: END IF
! 316: IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
! 317: CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
! 318: INDEE = INDWRK + 2*N
! 319: IF( .NOT.WANTZ ) THEN
! 320: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
! 321: CALL DSTERF( N, W, WORK( INDEE ), INFO )
! 322: ELSE
! 323: CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
! 324: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
! 325: CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
! 326: $ WORK( INDWRK ), INFO )
! 327: IF( INFO.EQ.0 ) THEN
! 328: DO 10 I = 1, N
! 329: IFAIL( I ) = 0
! 330: 10 CONTINUE
! 331: END IF
! 332: END IF
! 333: IF( INFO.EQ.0 ) THEN
! 334: M = N
! 335: GO TO 30
! 336: END IF
! 337: INFO = 0
! 338: END IF
! 339: *
! 340: * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
! 341: *
! 342: IF( WANTZ ) THEN
! 343: ORDER = 'B'
! 344: ELSE
! 345: ORDER = 'E'
! 346: END IF
! 347: INDIBL = 1
! 348: INDISP = INDIBL + N
! 349: INDIWO = INDISP + N
! 350: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
! 351: $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
! 352: $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
! 353: $ IWORK( INDIWO ), INFO )
! 354: *
! 355: IF( WANTZ ) THEN
! 356: CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
! 357: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
! 358: $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
! 359: *
! 360: * Apply orthogonal matrix used in reduction to tridiagonal
! 361: * form to eigenvectors returned by DSTEIN.
! 362: *
! 363: DO 20 J = 1, M
! 364: CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
! 365: CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO,
! 366: $ Z( 1, J ), 1 )
! 367: 20 CONTINUE
! 368: END IF
! 369: *
! 370: * If matrix was scaled, then rescale eigenvalues appropriately.
! 371: *
! 372: 30 CONTINUE
! 373: IF( ISCALE.EQ.1 ) THEN
! 374: IF( INFO.EQ.0 ) THEN
! 375: IMAX = M
! 376: ELSE
! 377: IMAX = INFO - 1
! 378: END IF
! 379: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
! 380: END IF
! 381: *
! 382: * If eigenvalues are not in order, then sort them, along with
! 383: * eigenvectors.
! 384: *
! 385: IF( WANTZ ) THEN
! 386: DO 50 J = 1, M - 1
! 387: I = 0
! 388: TMP1 = W( J )
! 389: DO 40 JJ = J + 1, M
! 390: IF( W( JJ ).LT.TMP1 ) THEN
! 391: I = JJ
! 392: TMP1 = W( JJ )
! 393: END IF
! 394: 40 CONTINUE
! 395: *
! 396: IF( I.NE.0 ) THEN
! 397: ITMP1 = IWORK( INDIBL+I-1 )
! 398: W( I ) = W( J )
! 399: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
! 400: W( J ) = TMP1
! 401: IWORK( INDIBL+J-1 ) = ITMP1
! 402: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
! 403: IF( INFO.NE.0 ) THEN
! 404: ITMP1 = IFAIL( I )
! 405: IFAIL( I ) = IFAIL( J )
! 406: IFAIL( J ) = ITMP1
! 407: END IF
! 408: END IF
! 409: 50 CONTINUE
! 410: END IF
! 411: *
! 412: RETURN
! 413: *
! 414: * End of DSBEVX
! 415: *
! 416: END
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