SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, \$ LWORK, IWORK, LIWORK, INFO ) * * -- LAPACK driver routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, KD, LDAB, LDZ, LIWORK, LWORK, N * .. * .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION AB( LDAB, * ), W( * ), WORK( * ), Z( LDZ, * ) * .. * * Purpose * ======= * * DSBEVD computes all the eigenvalues and, optionally, eigenvectors of * a real symmetric band matrix A. If eigenvectors are desired, it uses * a divide and conquer algorithm. * * The divide and conquer algorithm makes very mild assumptions about * floating point arithmetic. It will work on machines with a guard * digit in add/subtract, or on those binary machines without guard * digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or * Cray-2. It could conceivably fail on hexadecimal or decimal machines * without guard digits, but we know of none. * * Arguments * ========= * * JOBZ (input) CHARACTER*1 * = 'N': Compute eigenvalues only; * = 'V': Compute eigenvalues and eigenvectors. * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * KD (input) INTEGER * The number of superdiagonals of the matrix A if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KD >= 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 KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). * * On exit, AB is overwritten by values generated during the * reduction to tridiagonal form. If UPLO = 'U', the first * superdiagonal and the diagonal of the tridiagonal matrix T * are returned in rows KD and KD+1 of AB, and if UPLO = 'L', * the diagonal and first subdiagonal of T are returned in the * first two rows of AB. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KD + 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 orthonormal * eigenvectors of the matrix A, with the i-th column of Z * holding the eigenvector associated with W(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 (LWORK) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. * IF N <= 1, LWORK must be at least 1. * If JOBZ = 'N' and N > 2, LWORK must be at least 2*N. * If JOBZ = 'V' and N > 2, LWORK must be at least * ( 1 + 5*N + 2*N**2 ). * * If LWORK = -1, then a workspace query is assumed; the routine * only calculates the optimal sizes of the WORK and IWORK * arrays, returns these values as the first entries of the WORK * and IWORK arrays, and no error message related to LWORK or * LIWORK is issued by XERBLA. * * IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. * * LIWORK (input) INTEGER * The dimension of the array LIWORK. * If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. * If JOBZ = 'V' and N > 2, LIWORK must be at least 3 + 5*N. * * If LIWORK = -1, then a workspace query is assumed; the * routine only calculates the optimal sizes of the WORK and * IWORK arrays, returns these values as the first entries of * the WORK and IWORK arrays, and no error message related to * LWORK or LIWORK is issued by XERBLA. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, the algorithm failed to converge; i * off-diagonal elements of an intermediate tridiagonal * form did not converge to zero. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL LOWER, LQUERY, WANTZ INTEGER IINFO, INDE, INDWK2, INDWRK, ISCALE, LIWMIN, \$ LLWRK2, LWMIN DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, \$ SMLNUM * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANSB EXTERNAL LSAME, DLAMCH, DLANSB * .. * .. External Subroutines .. EXTERNAL DGEMM, DLACPY, DLASCL, DSBTRD, DSCAL, DSTEDC, \$ DSTERF, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC SQRT * .. * .. Executable Statements .. * * Test the input parameters. * WANTZ = LSAME( JOBZ, 'V' ) LOWER = LSAME( UPLO, 'L' ) LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) * INFO = 0 IF( N.LE.1 ) THEN LIWMIN = 1 LWMIN = 1 ELSE IF( WANTZ ) THEN LIWMIN = 3 + 5*N LWMIN = 1 + 5*N + 2*N**2 ELSE LIWMIN = 1 LWMIN = 2*N END IF END IF IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KD.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -6 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -9 END IF * IF( INFO.EQ.0 ) THEN WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN * IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -11 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -13 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSBEVD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * IF( N.EQ.1 ) THEN W( 1 ) = AB( 1, 1 ) IF( WANTZ ) \$ Z( 1, 1 ) = ONE RETURN END IF * * Get machine constants. * SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = SQRT( BIGNUM ) * * Scale matrix to allowable range, if necessary. * ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK ) ISCALE = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) THEN IF( LOWER ) THEN CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) ELSE CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO ) END IF END IF * * Call DSBTRD to reduce symmetric band matrix to tridiagonal form. * INDE = 1 INDWRK = INDE + N INDWK2 = INDWRK + N*N LLWRK2 = LWORK - INDWK2 + 1 CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, W, WORK( INDE ), Z, LDZ, \$ WORK( INDWRK ), IINFO ) * * For eigenvalues only, call DSTERF. For eigenvectors, call SSTEDC. * IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, WORK( INDE ), INFO ) ELSE CALL DSTEDC( 'I', N, W, WORK( INDE ), WORK( INDWRK ), N, \$ WORK( INDWK2 ), LLWRK2, IWORK, LIWORK, INFO ) CALL DGEMM( 'N', 'N', N, N, N, ONE, Z, LDZ, WORK( INDWRK ), N, \$ ZERO, WORK( INDWK2 ), N ) CALL DLACPY( 'A', N, N, WORK( INDWK2 ), N, Z, LDZ ) END IF * * If matrix was scaled, then rescale eigenvalues appropriately. * IF( ISCALE.EQ.1 ) \$ CALL DSCAL( N, ONE / SIGMA, W, 1 ) * WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN RETURN * * End of DSBEVD * END