LAPACK 3.3.1
Linear Algebra PACKage

zhbgvx.f

Go to the documentation of this file.
00001       SUBROUTINE ZHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
00002      $                   LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
00003      $                   LDZ, WORK, RWORK, IWORK, IFAIL, INFO )
00004 *
00005 *  -- LAPACK driver routine (version 3.2) --
00006 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00007 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00008 *     November 2006
00009 *
00010 *     .. Scalar Arguments ..
00011       CHARACTER          JOBZ, RANGE, UPLO
00012       INTEGER            IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
00013      $                   N
00014       DOUBLE PRECISION   ABSTOL, VL, VU
00015 *     ..
00016 *     .. Array Arguments ..
00017       INTEGER            IFAIL( * ), IWORK( * )
00018       DOUBLE PRECISION   RWORK( * ), W( * )
00019       COMPLEX*16         AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
00020      $                   WORK( * ), Z( LDZ, * )
00021 *     ..
00022 *
00023 *  Purpose
00024 *  =======
00025 *
00026 *  ZHBGVX computes all the eigenvalues, and optionally, the eigenvectors
00027 *  of a complex generalized Hermitian-definite banded eigenproblem, of
00028 *  the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian
00029 *  and banded, and B is also positive definite.  Eigenvalues and
00030 *  eigenvectors can be selected by specifying either all eigenvalues,
00031 *  a range of values or a range of indices for the desired eigenvalues.
00032 *
00033 *  Arguments
00034 *  =========
00035 *
00036 *  JOBZ    (input) CHARACTER*1
00037 *          = 'N':  Compute eigenvalues only;
00038 *          = 'V':  Compute eigenvalues and eigenvectors.
00039 *
00040 *  RANGE   (input) CHARACTER*1
00041 *          = 'A': all eigenvalues will be found;
00042 *          = 'V': all eigenvalues in the half-open interval (VL,VU]
00043 *                 will be found;
00044 *          = 'I': the IL-th through IU-th eigenvalues will be found.
00045 *
00046 *  UPLO    (input) CHARACTER*1
00047 *          = 'U':  Upper triangles of A and B are stored;
00048 *          = 'L':  Lower triangles of A and B are stored.
00049 *
00050 *  N       (input) INTEGER
00051 *          The order of the matrices A and B.  N >= 0.
00052 *
00053 *  KA      (input) INTEGER
00054 *          The number of superdiagonals of the matrix A if UPLO = 'U',
00055 *          or the number of subdiagonals if UPLO = 'L'. KA >= 0.
00056 *
00057 *  KB      (input) INTEGER
00058 *          The number of superdiagonals of the matrix B if UPLO = 'U',
00059 *          or the number of subdiagonals if UPLO = 'L'. KB >= 0.
00060 *
00061 *  AB      (input/output) COMPLEX*16 array, dimension (LDAB, N)
00062 *          On entry, the upper or lower triangle of the Hermitian band
00063 *          matrix A, stored in the first ka+1 rows of the array.  The
00064 *          j-th column of A is stored in the j-th column of the array AB
00065 *          as follows:
00066 *          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
00067 *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
00068 *
00069 *          On exit, the contents of AB are destroyed.
00070 *
00071 *  LDAB    (input) INTEGER
00072 *          The leading dimension of the array AB.  LDAB >= KA+1.
00073 *
00074 *  BB      (input/output) COMPLEX*16 array, dimension (LDBB, N)
00075 *          On entry, the upper or lower triangle of the Hermitian band
00076 *          matrix B, stored in the first kb+1 rows of the array.  The
00077 *          j-th column of B is stored in the j-th column of the array BB
00078 *          as follows:
00079 *          if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
00080 *          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
00081 *
00082 *          On exit, the factor S from the split Cholesky factorization
00083 *          B = S**H*S, as returned by ZPBSTF.
00084 *
00085 *  LDBB    (input) INTEGER
00086 *          The leading dimension of the array BB.  LDBB >= KB+1.
00087 *
00088 *  Q       (output) COMPLEX*16 array, dimension (LDQ, N)
00089 *          If JOBZ = 'V', the n-by-n matrix used in the reduction of
00090 *          A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
00091 *          and consequently C to tridiagonal form.
00092 *          If JOBZ = 'N', the array Q is not referenced.
00093 *
00094 *  LDQ     (input) INTEGER
00095 *          The leading dimension of the array Q.  If JOBZ = 'N',
00096 *          LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
00097 *
00098 *  VL      (input) DOUBLE PRECISION
00099 *  VU      (input) DOUBLE PRECISION
00100 *          If RANGE='V', the lower and upper bounds of the interval to
00101 *          be searched for eigenvalues. VL < VU.
00102 *          Not referenced if RANGE = 'A' or 'I'.
00103 *
00104 *  IL      (input) INTEGER
00105 *  IU      (input) INTEGER
00106 *          If RANGE='I', the indices (in ascending order) of the
00107 *          smallest and largest eigenvalues to be returned.
00108 *          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
00109 *          Not referenced if RANGE = 'A' or 'V'.
00110 *
00111 *  ABSTOL  (input) DOUBLE PRECISION
00112 *          The absolute error tolerance for the eigenvalues.
00113 *          An approximate eigenvalue is accepted as converged
00114 *          when it is determined to lie in an interval [a,b]
00115 *          of width less than or equal to
00116 *
00117 *                  ABSTOL + EPS *   max( |a|,|b| ) ,
00118 *
00119 *          where EPS is the machine precision.  If ABSTOL is less than
00120 *          or equal to zero, then  EPS*|T|  will be used in its place,
00121 *          where |T| is the 1-norm of the tridiagonal matrix obtained
00122 *          by reducing AP to tridiagonal form.
00123 *
00124 *          Eigenvalues will be computed most accurately when ABSTOL is
00125 *          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
00126 *          If this routine returns with INFO>0, indicating that some
00127 *          eigenvectors did not converge, try setting ABSTOL to
00128 *          2*DLAMCH('S').
00129 *
00130 *  M       (output) INTEGER
00131 *          The total number of eigenvalues found.  0 <= M <= N.
00132 *          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
00133 *
00134 *  W       (output) DOUBLE PRECISION array, dimension (N)
00135 *          If INFO = 0, the eigenvalues in ascending order.
00136 *
00137 *  Z       (output) COMPLEX*16 array, dimension (LDZ, N)
00138 *          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
00139 *          eigenvectors, with the i-th column of Z holding the
00140 *          eigenvector associated with W(i). The eigenvectors are
00141 *          normalized so that Z**H*B*Z = I.
00142 *          If JOBZ = 'N', then Z is not referenced.
00143 *
00144 *  LDZ     (input) INTEGER
00145 *          The leading dimension of the array Z.  LDZ >= 1, and if
00146 *          JOBZ = 'V', LDZ >= N.
00147 *
00148 *  WORK    (workspace) COMPLEX*16 array, dimension (N)
00149 *
00150 *  RWORK   (workspace) DOUBLE PRECISION array, dimension (7*N)
00151 *
00152 *  IWORK   (workspace) INTEGER array, dimension (5*N)
00153 *
00154 *  IFAIL   (output) INTEGER array, dimension (N)
00155 *          If JOBZ = 'V', then if INFO = 0, the first M elements of
00156 *          IFAIL are zero.  If INFO > 0, then IFAIL contains the
00157 *          indices of the eigenvectors that failed to converge.
00158 *          If JOBZ = 'N', then IFAIL is not referenced.
00159 *
00160 *  INFO    (output) INTEGER
00161 *          = 0:  successful exit
00162 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00163 *          > 0:  if INFO = i, and i is:
00164 *             <= N:  then i eigenvectors failed to converge.  Their
00165 *                    indices are stored in array IFAIL.
00166 *             > N:   if INFO = N + i, for 1 <= i <= N, then ZPBSTF
00167 *                    returned INFO = i: B is not positive definite.
00168 *                    The factorization of B could not be completed and
00169 *                    no eigenvalues or eigenvectors were computed.
00170 *
00171 *  Further Details
00172 *  ===============
00173 *
00174 *  Based on contributions by
00175 *     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
00176 *
00177 *  =====================================================================
00178 *
00179 *     .. Parameters ..
00180       DOUBLE PRECISION   ZERO
00181       PARAMETER          ( ZERO = 0.0D+0 )
00182       COMPLEX*16         CZERO, CONE
00183       PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
00184      $                   CONE = ( 1.0D+0, 0.0D+0 ) )
00185 *     ..
00186 *     .. Local Scalars ..
00187       LOGICAL            ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ
00188       CHARACTER          ORDER, VECT
00189       INTEGER            I, IINFO, INDD, INDE, INDEE, INDIBL, INDISP,
00190      $                   INDIWK, INDRWK, INDWRK, ITMP1, J, JJ, NSPLIT
00191       DOUBLE PRECISION   TMP1
00192 *     ..
00193 *     .. External Functions ..
00194       LOGICAL            LSAME
00195       EXTERNAL           LSAME
00196 *     ..
00197 *     .. External Subroutines ..
00198       EXTERNAL           DCOPY, DSTEBZ, DSTERF, XERBLA, ZCOPY, ZGEMV,
00199      $                   ZHBGST, ZHBTRD, ZLACPY, ZPBSTF, ZSTEIN, ZSTEQR,
00200      $                   ZSWAP
00201 *     ..
00202 *     .. Intrinsic Functions ..
00203       INTRINSIC          MIN
00204 *     ..
00205 *     .. Executable Statements ..
00206 *
00207 *     Test the input parameters.
00208 *
00209       WANTZ = LSAME( JOBZ, 'V' )
00210       UPPER = LSAME( UPLO, 'U' )
00211       ALLEIG = LSAME( RANGE, 'A' )
00212       VALEIG = LSAME( RANGE, 'V' )
00213       INDEIG = LSAME( RANGE, 'I' )
00214 *
00215       INFO = 0
00216       IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
00217          INFO = -1
00218       ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
00219          INFO = -2
00220       ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
00221          INFO = -3
00222       ELSE IF( N.LT.0 ) THEN
00223          INFO = -4
00224       ELSE IF( KA.LT.0 ) THEN
00225          INFO = -5
00226       ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
00227          INFO = -6
00228       ELSE IF( LDAB.LT.KA+1 ) THEN
00229          INFO = -8
00230       ELSE IF( LDBB.LT.KB+1 ) THEN
00231          INFO = -10
00232       ELSE IF( LDQ.LT.1 .OR. ( WANTZ .AND. LDQ.LT.N ) ) THEN
00233          INFO = -12
00234       ELSE
00235          IF( VALEIG ) THEN
00236             IF( N.GT.0 .AND. VU.LE.VL )
00237      $         INFO = -14
00238          ELSE IF( INDEIG ) THEN
00239             IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
00240                INFO = -15
00241             ELSE IF ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
00242                INFO = -16
00243             END IF
00244          END IF
00245       END IF
00246       IF( INFO.EQ.0) THEN
00247          IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
00248             INFO = -21
00249          END IF
00250       END IF
00251 *
00252       IF( INFO.NE.0 ) THEN
00253          CALL XERBLA( 'ZHBGVX', -INFO )
00254          RETURN
00255       END IF
00256 *
00257 *     Quick return if possible
00258 *
00259       M = 0
00260       IF( N.EQ.0 )
00261      $   RETURN
00262 *
00263 *     Form a split Cholesky factorization of B.
00264 *
00265       CALL ZPBSTF( UPLO, N, KB, BB, LDBB, INFO )
00266       IF( INFO.NE.0 ) THEN
00267          INFO = N + INFO
00268          RETURN
00269       END IF
00270 *
00271 *     Transform problem to standard eigenvalue problem.
00272 *
00273       CALL ZHBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ,
00274      $             WORK, RWORK, IINFO )
00275 *
00276 *     Solve the standard eigenvalue problem.
00277 *     Reduce Hermitian band matrix to tridiagonal form.
00278 *
00279       INDD = 1
00280       INDE = INDD + N
00281       INDRWK = INDE + N
00282       INDWRK = 1
00283       IF( WANTZ ) THEN
00284          VECT = 'U'
00285       ELSE
00286          VECT = 'N'
00287       END IF
00288       CALL ZHBTRD( VECT, UPLO, N, KA, AB, LDAB, RWORK( INDD ),
00289      $             RWORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
00290 *
00291 *     If all eigenvalues are desired and ABSTOL is less than or equal
00292 *     to zero, then call DSTERF or ZSTEQR.  If this fails for some
00293 *     eigenvalue, then try DSTEBZ.
00294 *
00295       TEST = .FALSE.
00296       IF( INDEIG ) THEN
00297          IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
00298             TEST = .TRUE.
00299          END IF
00300       END IF
00301       IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
00302          CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
00303          INDEE = INDRWK + 2*N
00304          CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
00305          IF( .NOT.WANTZ ) THEN
00306             CALL DSTERF( N, W, RWORK( INDEE ), INFO )
00307          ELSE
00308             CALL ZLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
00309             CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
00310      $                   RWORK( INDRWK ), INFO )
00311             IF( INFO.EQ.0 ) THEN
00312                DO 10 I = 1, N
00313                   IFAIL( I ) = 0
00314    10          CONTINUE
00315             END IF
00316          END IF
00317          IF( INFO.EQ.0 ) THEN
00318             M = N
00319             GO TO 30
00320          END IF
00321          INFO = 0
00322       END IF
00323 *
00324 *     Otherwise, call DSTEBZ and, if eigenvectors are desired,
00325 *     call ZSTEIN.
00326 *
00327       IF( WANTZ ) THEN
00328          ORDER = 'B'
00329       ELSE
00330          ORDER = 'E'
00331       END IF
00332       INDIBL = 1
00333       INDISP = INDIBL + N
00334       INDIWK = INDISP + N
00335       CALL DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL,
00336      $             RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
00337      $             IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
00338      $             IWORK( INDIWK ), INFO )
00339 *
00340       IF( WANTZ ) THEN
00341          CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
00342      $                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
00343      $                RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
00344 *
00345 *        Apply unitary matrix used in reduction to tridiagonal
00346 *        form to eigenvectors returned by ZSTEIN.
00347 *
00348          DO 20 J = 1, M
00349             CALL ZCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
00350             CALL ZGEMV( 'N', N, N, CONE, Q, LDQ, WORK, 1, CZERO,
00351      $                  Z( 1, J ), 1 )
00352    20    CONTINUE
00353       END IF
00354 *
00355    30 CONTINUE
00356 *
00357 *     If eigenvalues are not in order, then sort them, along with
00358 *     eigenvectors.
00359 *
00360       IF( WANTZ ) THEN
00361          DO 50 J = 1, M - 1
00362             I = 0
00363             TMP1 = W( J )
00364             DO 40 JJ = J + 1, M
00365                IF( W( JJ ).LT.TMP1 ) THEN
00366                   I = JJ
00367                   TMP1 = W( JJ )
00368                END IF
00369    40       CONTINUE
00370 *
00371             IF( I.NE.0 ) THEN
00372                ITMP1 = IWORK( INDIBL+I-1 )
00373                W( I ) = W( J )
00374                IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
00375                W( J ) = TMP1
00376                IWORK( INDIBL+J-1 ) = ITMP1
00377                CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
00378                IF( INFO.NE.0 ) THEN
00379                   ITMP1 = IFAIL( I )
00380                   IFAIL( I ) = IFAIL( J )
00381                   IFAIL( J ) = ITMP1
00382                END IF
00383             END IF
00384    50    CONTINUE
00385       END IF
00386 *
00387       RETURN
00388 *
00389 *     End of ZHBGVX
00390 *
00391       END
 All Files Functions