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