LAPACK 3.3.1
Linear Algebra PACKage

dsbevd.f

Go to the documentation of this file.
00001       SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
00002      $                   LWORK, IWORK, LIWORK, INFO )
00003 *
00004 *  -- LAPACK driver routine (version 3.2) --
00005 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00006 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00007 *     November 2006
00008 *
00009 *     .. Scalar Arguments ..
00010       CHARACTER          JOBZ, UPLO
00011       INTEGER            INFO, KD, LDAB, LDZ, LIWORK, LWORK, N
00012 *     ..
00013 *     .. Array Arguments ..
00014       INTEGER            IWORK( * )
00015       DOUBLE PRECISION   AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * )
00016 *     ..
00017 *
00018 *  Purpose
00019 *  =======
00020 *
00021 *  DSBEVD computes all the eigenvalues and, optionally, eigenvectors of
00022 *  a real symmetric band matrix A. If eigenvectors are desired, it uses
00023 *  a divide and conquer algorithm.
00024 *
00025 *  The divide and conquer algorithm makes very mild assumptions about
00026 *  floating point arithmetic. It will work on machines with a guard
00027 *  digit in add/subtract, or on those binary machines without guard
00028 *  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
00029 *  Cray-2. It could conceivably fail on hexadecimal or decimal machines
00030 *  without guard digits, but we know of none.
00031 *
00032 *  Arguments
00033 *  =========
00034 *
00035 *  JOBZ    (input) CHARACTER*1
00036 *          = 'N':  Compute eigenvalues only;
00037 *          = 'V':  Compute eigenvalues and eigenvectors.
00038 *
00039 *  UPLO    (input) CHARACTER*1
00040 *          = 'U':  Upper triangle of A is stored;
00041 *          = 'L':  Lower triangle of A is stored.
00042 *
00043 *  N       (input) INTEGER
00044 *          The order of the matrix A.  N >= 0.
00045 *
00046 *  KD      (input) INTEGER
00047 *          The number of superdiagonals of the matrix A if UPLO = 'U',
00048 *          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
00049 *
00050 *  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
00051 *          On entry, the upper or lower triangle of the symmetric band
00052 *          matrix A, stored in the first KD+1 rows of the array.  The
00053 *          j-th column of A is stored in the j-th column of the array AB
00054 *          as follows:
00055 *          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
00056 *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
00057 *
00058 *          On exit, AB is overwritten by values generated during the
00059 *          reduction to tridiagonal form.  If UPLO = 'U', the first
00060 *          superdiagonal and the diagonal of the tridiagonal matrix T
00061 *          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
00062 *          the diagonal and first subdiagonal of T are returned in the
00063 *          first two rows of AB.
00064 *
00065 *  LDAB    (input) INTEGER
00066 *          The leading dimension of the array AB.  LDAB >= KD + 1.
00067 *
00068 *  W       (output) DOUBLE PRECISION array, dimension (N)
00069 *          If INFO = 0, the eigenvalues in ascending order.
00070 *
00071 *  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
00072 *          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
00073 *          eigenvectors of the matrix A, with the i-th column of Z
00074 *          holding the eigenvector associated with W(i).
00075 *          If JOBZ = 'N', then Z is not referenced.
00076 *
00077 *  LDZ     (input) INTEGER
00078 *          The leading dimension of the array Z.  LDZ >= 1, and if
00079 *          JOBZ = 'V', LDZ >= max(1,N).
00080 *
00081 *  WORK    (workspace/output) DOUBLE PRECISION array,
00082 *                                         dimension (LWORK)
00083 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00084 *
00085 *  LWORK   (input) INTEGER
00086 *          The dimension of the array WORK.
00087 *          IF N <= 1,                LWORK must be at least 1.
00088 *          If JOBZ  = 'N' and N > 2, LWORK must be at least 2*N.
00089 *          If JOBZ  = 'V' and N > 2, LWORK must be at least
00090 *                         ( 1 + 5*N + 2*N**2 ).
00091 *
00092 *          If LWORK = -1, then a workspace query is assumed; the routine
00093 *          only calculates the optimal sizes of the WORK and IWORK
00094 *          arrays, returns these values as the first entries of the WORK
00095 *          and IWORK arrays, and no error message related to LWORK or
00096 *          LIWORK is issued by XERBLA.
00097 *
00098 *  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
00099 *          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
00100 *
00101 *  LIWORK  (input) INTEGER
00102 *          The dimension of the array LIWORK.
00103 *          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
00104 *          If JOBZ  = 'V' and N > 2, LIWORK must be at least 3 + 5*N.
00105 *
00106 *          If LIWORK = -1, then a workspace query is assumed; the
00107 *          routine only calculates the optimal sizes of the WORK and
00108 *          IWORK arrays, returns these values as the first entries of
00109 *          the WORK and IWORK arrays, and no error message related to
00110 *          LWORK or LIWORK is issued by XERBLA.
00111 *
00112 *  INFO    (output) INTEGER
00113 *          = 0:  successful exit
00114 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00115 *          > 0:  if INFO = i, the algorithm failed to converge; i
00116 *                off-diagonal elements of an intermediate tridiagonal
00117 *                form did not converge to zero.
00118 *
00119 *  =====================================================================
00120 *
00121 *     .. Parameters ..
00122       DOUBLE PRECISION   ZERO, ONE
00123       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00124 *     ..
00125 *     .. Local Scalars ..
00126       LOGICAL            LOWER, LQUERY, WANTZ
00127       INTEGER            IINFO, INDE, INDWK2, INDWRK, ISCALE, LIWMIN,
00128      $                   LLWRK2, LWMIN
00129       DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
00130      $                   SMLNUM
00131 *     ..
00132 *     .. External Functions ..
00133       LOGICAL            LSAME
00134       DOUBLE PRECISION   DLAMCH, DLANSB
00135       EXTERNAL           LSAME, DLAMCH, DLANSB
00136 *     ..
00137 *     .. External Subroutines ..
00138       EXTERNAL           DGEMM, DLACPY, DLASCL, DSBTRD, DSCAL, DSTEDC,
00139      $                   DSTERF, XERBLA
00140 *     ..
00141 *     .. Intrinsic Functions ..
00142       INTRINSIC          SQRT
00143 *     ..
00144 *     .. Executable Statements ..
00145 *
00146 *     Test the input parameters.
00147 *
00148       WANTZ = LSAME( JOBZ, 'V' )
00149       LOWER = LSAME( UPLO, 'L' )
00150       LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
00151 *
00152       INFO = 0
00153       IF( N.LE.1 ) THEN
00154          LIWMIN = 1
00155          LWMIN = 1
00156       ELSE
00157          IF( WANTZ ) THEN
00158             LIWMIN = 3 + 5*N
00159             LWMIN = 1 + 5*N + 2*N**2
00160          ELSE
00161             LIWMIN = 1
00162             LWMIN = 2*N
00163          END IF
00164       END IF
00165       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00166          INFO = -1
00167       ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
00168          INFO = -2
00169       ELSE IF( N.LT.0 ) THEN
00170          INFO = -3
00171       ELSE IF( KD.LT.0 ) THEN
00172          INFO = -4
00173       ELSE IF( LDAB.LT.KD+1 ) THEN
00174          INFO = -6
00175       ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
00176          INFO = -9
00177       END IF
00178 *
00179       IF( INFO.EQ.0 ) THEN
00180          WORK( 1 ) = LWMIN
00181          IWORK( 1 ) = LIWMIN
00182 *
00183          IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
00184             INFO = -11
00185          ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
00186             INFO = -13
00187          END IF
00188       END IF
00189 *
00190       IF( INFO.NE.0 ) THEN
00191          CALL XERBLA( 'DSBEVD', -INFO )
00192          RETURN
00193       ELSE IF( LQUERY ) THEN
00194          RETURN
00195       END IF
00196 *
00197 *     Quick return if possible
00198 *
00199       IF( N.EQ.0 )
00200      $   RETURN
00201 *
00202       IF( N.EQ.1 ) THEN
00203          W( 1 ) = AB( 1, 1 )
00204          IF( WANTZ )
00205      $      Z( 1, 1 ) = ONE
00206          RETURN
00207       END IF
00208 *
00209 *     Get machine constants.
00210 *
00211       SAFMIN = DLAMCH( 'Safe minimum' )
00212       EPS = DLAMCH( 'Precision' )
00213       SMLNUM = SAFMIN / EPS
00214       BIGNUM = ONE / SMLNUM
00215       RMIN = SQRT( SMLNUM )
00216       RMAX = SQRT( BIGNUM )
00217 *
00218 *     Scale matrix to allowable range, if necessary.
00219 *
00220       ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
00221       ISCALE = 0
00222       IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
00223          ISCALE = 1
00224          SIGMA = RMIN / ANRM
00225       ELSE IF( ANRM.GT.RMAX ) THEN
00226          ISCALE = 1
00227          SIGMA = RMAX / ANRM
00228       END IF
00229       IF( ISCALE.EQ.1 ) THEN
00230          IF( LOWER ) THEN
00231             CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
00232          ELSE
00233             CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
00234          END IF
00235       END IF
00236 *
00237 *     Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
00238 *
00239       INDE = 1
00240       INDWRK = INDE + N
00241       INDWK2 = INDWRK + N*N
00242       LLWRK2 = LWORK - INDWK2 + 1
00243       CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, WORK( INDE ), Z, LDZ,
00244      $             WORK( INDWRK ), IINFO )
00245 *
00246 *     For eigenvalues only, call DSTERF.  For eigenvectors, call SSTEDC.
00247 *
00248       IF( .NOT.WANTZ ) THEN
00249          CALL DSTERF( N, W, WORK( INDE ), INFO )
00250       ELSE
00251          CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N,
00252      $                WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO )
00253          CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ), N,
00254      $               ZERO, WORK( INDWK2 ), N )
00255          CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ )
00256       END IF
00257 *
00258 *     If matrix was scaled, then rescale eigenvalues appropriately.
00259 *
00260       IF( ISCALE.EQ.1 )
00261      $   CALL DSCAL( N, ONE / SIGMA, W, 1 )
00262 *
00263       WORK( 1 ) = LWMIN
00264       IWORK( 1 ) = LIWMIN
00265       RETURN
00266 *
00267 *     End of DSBEVD
00268 *
00269       END
 All Files Functions