```      SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
\$                   LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
\$                   LDZ, WORK, IWORK, IFAIL, INFO )
*
*  -- LAPACK driver routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
CHARACTER          JOBZ, RANGE, UPLO
INTEGER            IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
\$                   N
DOUBLE PRECISION   ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
INTEGER            IFAIL( * ), IWORK( * )
DOUBLE PRECISION   AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
\$                   W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  DSBGVX computes selected eigenvalues, and optionally, eigenvectors
*  of a real generalized symmetric-definite banded eigenproblem, of
*  the form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric
*  and banded, and B is also positive definite.  Eigenvalues and
*  eigenvectors can be selected by specifying either all eigenvalues,
*  a range of values or a range of indices for the desired eigenvalues.
*
*  Arguments
*  =========
*
*  JOBZ    (input) CHARACTER*1
*          = 'N':  Compute eigenvalues only;
*          = 'V':  Compute eigenvalues and eigenvectors.
*
*  RANGE   (input) CHARACTER*1
*          = 'A': all eigenvalues will be found.
*          = 'V': all eigenvalues in the half-open interval (VL,VU]
*                 will be found.
*          = 'I': the IL-th through IU-th eigenvalues will be found.
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangles of A and B are stored;
*          = 'L':  Lower triangles of A and B are stored.
*
*  N       (input) INTEGER
*          The order of the matrices A and B.  N >= 0.
*
*  KA      (input) INTEGER
*          The number of superdiagonals of the matrix A if UPLO = 'U',
*          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.
*
*  KB      (input) INTEGER
*          The number of superdiagonals of the matrix B if UPLO = 'U',
*          or the number of subdiagonals if UPLO = 'L'.  KB >= 0.
*
*  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB, N)
*          On entry, the upper or lower triangle of the symmetric band
*          matrix A, stored in the first ka+1 rows of the array.  The
*          j-th column of A is stored in the j-th column of the array AB
*          as follows:
*          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
*          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
*
*          On exit, the contents of AB are destroyed.
*
*  LDAB    (input) INTEGER
*          The leading dimension of the array AB.  LDAB >= KA+1.
*
*  BB      (input/output) DOUBLE PRECISION array, dimension (LDBB, N)
*          On entry, the upper or lower triangle of the symmetric band
*          matrix B, stored in the first kb+1 rows of the array.  The
*          j-th column of B is stored in the j-th column of the array BB
*          as follows:
*          if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
*          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
*
*          On exit, the factor S from the split Cholesky factorization
*          B = S**T*S, as returned by DPBSTF.
*
*  LDBB    (input) INTEGER
*          The leading dimension of the array BB.  LDBB >= KB+1.
*
*  Q       (output) DOUBLE PRECISION array, dimension (LDQ, N)
*          If JOBZ = 'V', the n-by-n matrix used in the reduction of
*          A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
*          and consequently C to tridiagonal form.
*          If JOBZ = 'N', the array Q is not referenced.
*
*  LDQ     (input) INTEGER
*          The leading dimension of the array Q.  If JOBZ = 'N',
*          LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
*
*  VL      (input) DOUBLE PRECISION
*  VU      (input) DOUBLE PRECISION
*          If RANGE='V', the lower and upper bounds of the interval to
*          be searched for eigenvalues. VL < VU.
*          Not referenced if RANGE = 'A' or 'I'.
*
*  IL      (input) INTEGER
*  IU      (input) INTEGER
*          If RANGE='I', the indices (in ascending order) of the
*          smallest and largest eigenvalues to be returned.
*          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*          Not referenced if RANGE = 'A' or 'V'.
*
*  ABSTOL  (input) DOUBLE PRECISION
*          The absolute error tolerance for the eigenvalues.
*          An approximate eigenvalue is accepted as converged
*          when it is determined to lie in an interval [a,b]
*          of width less than or equal to
*
*                  ABSTOL + EPS *   max( |a|,|b| ) ,
*
*          where EPS is the machine precision.  If ABSTOL is less than
*          or equal to zero, then  EPS*|T|  will be used in its place,
*          where |T| is the 1-norm of the tridiagonal matrix obtained
*          by reducing A to tridiagonal form.
*
*          Eigenvalues will be computed most accurately when ABSTOL is
*          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*          If this routine returns with INFO>0, indicating that some
*          eigenvectors did not converge, try setting ABSTOL to
*          2*DLAMCH('S').
*
*  M       (output) INTEGER
*          The total number of eigenvalues found.  0 <= M <= N.
*          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*
*  W       (output) DOUBLE PRECISION array, dimension (N)
*          If INFO = 0, the eigenvalues in ascending order.
*
*  Z       (output) DOUBLE PRECISION array, dimension (LDZ, N)
*          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
*          eigenvectors, with the i-th column of Z holding the
*          eigenvector associated with W(i).  The eigenvectors are
*          normalized so Z**T*B*Z = I.
*          If JOBZ = 'N', then Z is not referenced.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.  LDZ >= 1, and if
*          JOBZ = 'V', LDZ >= max(1,N).
*
*  WORK    (workspace/output) DOUBLE PRECISION array, dimension (7*N)
*
*  IWORK   (workspace/output) INTEGER array, dimension (5*N)
*
*  IFAIL   (output) INTEGER array, dimension (M)
*          If JOBZ = 'V', then if INFO = 0, the first M elements of
*          IFAIL are zero.  If INFO > 0, then IFAIL contains the
*          indices of the eigenvalues that failed to converge.
*          If JOBZ = 'N', then IFAIL is not referenced.
*
*  INFO    (output) INTEGER
*          = 0 : successful exit
*          < 0 : if INFO = -i, the i-th argument had an illegal value
*          <= N: if INFO = i, then i eigenvectors failed to converge.
*                  Their indices are stored in IFAIL.
*          > N : DPBSTF returned an error code; i.e.,
*                if INFO = N + i, for 1 <= i <= N, then the leading
*                minor of order i of B is not positive definite.
*                The factorization of B could not be completed and
*                no eigenvalues or eigenvectors were computed.
*
*  Further Details
*  ===============
*
*  Based on contributions by
*     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
*  =====================================================================
*
*     .. Parameters ..
DOUBLE PRECISION   ZERO, ONE
PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
LOGICAL            ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ
CHARACTER          ORDER, VECT
INTEGER            I, IINFO, INDD, INDE, INDEE, INDIBL, INDISP,
\$                   INDIWO, INDWRK, ITMP1, J, JJ, NSPLIT
DOUBLE PRECISION   TMP1
*     ..
*     .. External Functions ..
LOGICAL            LSAME
EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
EXTERNAL           DCOPY, DGEMV, DLACPY, DPBSTF, DSBGST, DSBTRD,
\$                   DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
UPPER = LSAME( UPLO, 'U' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -2
ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( KA.LT.0 ) THEN
INFO = -5
ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
INFO = -6
ELSE IF( LDAB.LT.KA+1 ) THEN
INFO = -8
ELSE IF( LDBB.LT.KB+1 ) THEN
INFO = -10
ELSE IF( LDQ.LT.1 .OR. ( WANTZ .AND. LDQ.LT.N ) ) THEN
INFO = -12
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL )
\$         INFO = -14
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -15
ELSE IF ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -16
END IF
END IF
END IF
IF( INFO.EQ.0) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -21
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSBGVX', -INFO )
RETURN
END IF
*
*     Quick return if possible
*
M = 0
IF( N.EQ.0 )
\$   RETURN
*
*     Form a split Cholesky factorization of B.
*
CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO )
IF( INFO.NE.0 ) THEN
INFO = N + INFO
RETURN
END IF
*
*     Transform problem to standard eigenvalue problem.
*
CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ,
\$             WORK, IINFO )
*
*     Reduce symmetric band matrix to tridiagonal form.
*
INDD = 1
INDE = INDD + N
INDWRK = INDE + N
IF( WANTZ ) THEN
VECT = 'U'
ELSE
VECT = 'N'
END IF
CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, WORK( INDD ),
\$             WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
*
*     If all eigenvalues are desired and ABSTOL is less than or equal
*     to zero, then call DSTERF or SSTEQR.  If this fails for some
*     eigenvalue, then try DSTEBZ.
*
TEST = .FALSE.
IF( INDEIG ) THEN
IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
TEST = .TRUE.
END IF
END IF
IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
INDEE = INDWRK + 2*N
CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
IF( .NOT.WANTZ ) THEN
CALL DSTERF( N, W, WORK( INDEE ), INFO )
ELSE
CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
\$                   WORK( INDWRK ), INFO )
IF( INFO.EQ.0 ) THEN
DO 10 I = 1, N
IFAIL( I ) = 0
10          CONTINUE
END IF
END IF
IF( INFO.EQ.0 ) THEN
M = N
GO TO 30
END IF
INFO = 0
END IF
*
*     Otherwise, call DSTEBZ and, if eigenvectors are desired,
*     call DSTEIN.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
INDIBL = 1
INDISP = INDIBL + N
INDIWO = INDISP + N
CALL DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL,
\$             WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
\$             IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
\$             IWORK( INDIWO ), INFO )
*
IF( WANTZ ) THEN
CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
\$                IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
\$                WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
*
*        Apply transformation matrix used in reduction to tridiagonal
*        form to eigenvectors returned by DSTEIN.
*
DO 20 J = 1, M
CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO,
\$                  Z( 1, J ), 1 )
20    CONTINUE
END IF
*
30 CONTINUE
*
*     If eigenvalues are not in order, then sort them, along with
*     eigenvectors.
*
IF( WANTZ ) THEN
DO 50 J = 1, M - 1
I = 0
TMP1 = W( J )
DO 40 JJ = J + 1, M
IF( W( JJ ).LT.TMP1 ) THEN
I = JJ
TMP1 = W( JJ )
END IF
40       CONTINUE
*
IF( I.NE.0 ) THEN
ITMP1 = IWORK( INDIBL+I-1 )
W( I ) = W( J )
IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
W( J ) = TMP1
IWORK( INDIBL+J-1 ) = ITMP1
CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
IF( INFO.NE.0 ) THEN
ITMP1 = IFAIL( I )
IFAIL( I ) = IFAIL( J )
IFAIL( J ) = ITMP1
END IF
END IF
50    CONTINUE
END IF
*
RETURN
*
*     End of DSBGVX
*
END

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