ssbgvx.f

          SUBROUTINE SSBGVX( 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
          REAL               ABSTOL, VL, VU
    *     ..
    *     .. Array Arguments ..
          INTEGER            IFAIL( * ), IWORK( * )
          REAL               AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
         $                   W( * ), WORK( * ), Z( LDZ, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SSBGVX 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) REAL 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) REAL 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 SPBSTF.
    *
    *  LDBB    (input) INTEGER
    *          The leading dimension of the array BB.  LDBB >= KB+1.
    *
    *  Q       (output) REAL 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) REAL
    *  VU      (input) REAL
    *          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) REAL
    *          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*SLAMCH('S'), not zero.
    *          If this routine returns with INFO>0, indicating that some
    *          eigenvectors did not converge, try setting ABSTOL to
    *          2*SLAMCH('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) REAL array, dimension (N)
    *          If INFO = 0, the eigenvalues in ascending order.
    *
    *  Z       (output) REAL 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) REAL array, dimension (7N)
    *
    *  IWORK   (workspace/output) INTEGER array, dimension (5N)
    *
    *  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 : SPBSTF 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 ..
          REAL               ZERO, ONE
          PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+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
          REAL               TMP1
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          EXTERNAL           LSAME
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SCOPY, SGEMV, SLACPY, SPBSTF, SSBGST, SSBTRD,
         $                   SSTEBZ, SSTEIN, SSTEQR, SSTERF, SSWAP, 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( 'SSBGVX', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          M = 0
          IF( N.EQ.0 )
         $   RETURN
    *
    *     Form a split Cholesky factorization of B.
    *
          CALL SPBSTF( UPLO, N, KB, BB, LDBB, INFO )
          IF( INFO.NE.0 ) THEN
             INFO = N + INFO
             RETURN
          END IF
    *
    *     Transform problem to standard eigenvalue problem.
    *
          CALL SSBGST( 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 SSBTRD( 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 SSTERF or SSTEQR.  If this fails for some
    *     eigenvalue, then try SSTEBZ.
    *
          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 SCOPY( N, WORK( INDD ), 1, W, 1 )
             INDEE = INDWRK + 2*N
             CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
             IF( .NOT.WANTZ ) THEN
                CALL SSTERF( N, W, WORK( INDEE ), INFO )
             ELSE
                CALL SLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
                CALL SSTEQR( 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 SSTEBZ and, if eigenvectors are desired,
    *     call SSTEIN.
    *
          IF( WANTZ ) THEN
             ORDER = 'B'
          ELSE
             ORDER = 'E'
          END IF
          INDIBL = 1
          INDISP = INDIBL + N
          INDIWO = INDISP + N
          CALL SSTEBZ( 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 SSTEIN( 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 SSTEIN.
    *
             DO 20 J = 1, M
                CALL SCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
                CALL SGEMV( '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 SSWAP( 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 SSBGVX
    *
          END