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SUBROUTINE SSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
$ M, W, Z, LDZ, ISUPPZ, 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, RANGE
INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
REAL ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER ISUPPZ( * ), IWORK( * )
REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* SSTEVR computes selected eigenvalues and, optionally, eigenvectors
* of a real symmetric tridiagonal matrix T. Eigenvalues and
* eigenvectors can be selected by specifying either a range of values
* or a range of indices for the desired eigenvalues.
*
* Whenever possible, SSTEVR calls SSTEMR to compute the
* eigenspectrum using Relatively Robust Representations. SSTEMR
* computes eigenvalues by the dqds algorithm, while orthogonal
* eigenvectors are computed from various "good" L D L^T representations
* (also known as Relatively Robust Representations). Gram-Schmidt
* orthogonalization is avoided as far as possible. More specifically,
* the various steps of the algorithm are as follows. For the i-th
* unreduced block of T,
* (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
* is a relatively robust representation,
* (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
* relative accuracy by the dqds algorithm,
* (c) If there is a cluster of close eigenvalues, "choose" sigma_i
* close to the cluster, and go to step (a),
* (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
* compute the corresponding eigenvector by forming a
* rank-revealing twisted factorization.
* The desired accuracy of the output can be specified by the input
* parameter ABSTOL.
*
* For more details, see "A new O(n^2) algorithm for the symmetric
* tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,
* Computer Science Division Technical Report No. UCB//CSD-97-971,
* UC Berkeley, May 1997.
*
*
* Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested
* on machines which conform to the ieee-754 floating point standard.
* SSTEVR calls SSTEBZ and SSTEIN on non-ieee machines and
* when partial spectrum requests are made.
*
* Normal execution of SSTEMR may create NaNs and infinities and
* hence may abort due to a floating point exception in environments
* which do not handle NaNs and infinities in the ieee standard default
* manner.
*
* 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.
********** For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and
********** SSTEIN are called
*
* N (input) INTEGER
* The order of the matrix. N >= 0.
*
* D (input/output) REAL array, dimension (N)
* On entry, the n diagonal elements of the tridiagonal matrix
* A.
* On exit, D may be multiplied by a constant factor chosen
* to avoid over/underflow in computing the eigenvalues.
*
* E (input/output) REAL array, dimension (max(1,N-1))
* On entry, the (n-1) subdiagonal elements of the tridiagonal
* matrix A in elements 1 to N-1 of E.
* On exit, E may be multiplied by a constant factor chosen
* to avoid over/underflow in computing the eigenvalues.
*
* 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.
*
* See "Computing Small Singular Values of Bidiagonal Matrices
* with Guaranteed High Relative Accuracy," by Demmel and
* Kahan, LAPACK Working Note #3.
*
* If high relative accuracy is important, set ABSTOL to
* SLAMCH( 'Safe minimum' ). Doing so will guarantee that
* eigenvalues are computed to high relative accuracy when
* possible in future releases. The current code does not
* make any guarantees about high relative accuracy, but
* future releases will. See J. Barlow and J. Demmel,
* "Computing Accurate Eigensystems of Scaled Diagonally
* Dominant Matrices", LAPACK Working Note #7, for a discussion
* of which matrices define their eigenvalues to high relative
* accuracy.
*
* 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)
* The first M elements contain the selected eigenvalues in
* ascending order.
*
* Z (output) REAL array, dimension (LDZ, max(1,M) )
* If JOBZ = 'V', then if INFO = 0, the first M columns of Z
* contain the orthonormal eigenvectors of the matrix A
* corresponding to the selected eigenvalues, with the i-th
* column of Z holding the eigenvector associated with W(i).
* Note: the user must ensure that at least max(1,M) columns are
* supplied in the array Z; if RANGE = 'V', the exact value of M
* is not known in advance and an upper bound must be used.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1, and if
* JOBZ = 'V', LDZ >= max(1,N).
*
* ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
* The support of the eigenvectors in Z, i.e., the indices
* indicating the nonzero elements in Z. The i-th eigenvector
* is nonzero only in elements ISUPPZ( 2*i-1 ) through
* ISUPPZ( 2*i ).
********** Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
*
* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal (and
* minimal) LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= 20*N.
*
* 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 (and
* minimal) LIWORK.
*
* LIWORK (input) INTEGER
* The dimension of the array IWORK. LIWORK >= 10*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: Internal error
*
* Further Details
* ===============
*
* Based on contributions by
* Inderjit Dhillon, IBM Almaden, USA
* Osni Marques, LBNL/NERSC, USA
* Ken Stanley, Computer Science Division, University of
* California at Berkeley, USA
* Jason Riedy, Computer Science Division, University of
* California at Berkeley, USA
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, TEST, LQUERY, VALEIG, WANTZ,
$ TRYRAC
CHARACTER ORDER
INTEGER I, IEEEOK, IMAX, INDIBL, INDIFL, INDISP,
$ INDIWO, ISCALE, J, JJ, LIWMIN, LWMIN, NSPLIT
REAL BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
$ TMP1, TNRM, VLL, VUU
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL SLAMCH, SLANST
EXTERNAL LSAME, ILAENV, SLAMCH, SLANST
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SSCAL, SSTEBZ, SSTEMR, SSTEIN, SSTERF,
$ SSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
*
* Test the input parameters.
*
IEEEOK = ILAENV( 10, 'SSTEVR', 'N', 1, 2, 3, 4 )
*
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
*
LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
LWMIN = MAX( 1, 20*N )
LIWMIN = MAX(1, 10*N )
*
*
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( N.LT.0 ) THEN
INFO = -3
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL )
$ INFO = -7
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -9
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -14
END IF
END IF
*
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -17
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -19
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SSTEVR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( ALLEIG .OR. INDEIG ) THEN
M = 1
W( 1 ) = D( 1 )
ELSE
IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
M = 1
W( 1 ) = D( 1 )
END IF
END IF
IF( WANTZ )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = SLAMCH( 'Safe minimum' )
EPS = SLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
*
* Scale matrix to allowable range, if necessary.
*
ISCALE = 0
VLL = VL
VUU = VU
*
TNRM = SLANST( 'M', N, D, E )
IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / TNRM
ELSE IF( TNRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / TNRM
END IF
IF( ISCALE.EQ.1 ) THEN
CALL SSCAL( N, SIGMA, D, 1 )
CALL SSCAL( N-1, SIGMA, E( 1 ), 1 )
IF( VALEIG ) THEN
VLL = VL*SIGMA
VUU = VU*SIGMA
END IF
END IF
* Initialize indices into workspaces. Note: These indices are used only
* if SSTERF or SSTEMR fail.
* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and
* stores the block indices of each of the M<=N eigenvalues.
INDIBL = 1
* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and
* stores the starting and finishing indices of each block.
INDISP = INDIBL + N
* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
* that corresponding to eigenvectors that fail to converge in
* SSTEIN. This information is discarded; if any fail, the driver
* returns INFO > 0.
INDIFL = INDISP + N
* INDIWO is the offset of the remaining integer workspace.
INDIWO = INDISP + N
*
* If all eigenvalues are desired, then
* call SSTERF or SSTEMR. 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. IEEEOK.EQ.1 ) THEN
CALL SCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
IF( .NOT.WANTZ ) THEN
CALL SCOPY( N, D, 1, W, 1 )
CALL SSTERF( N, W, WORK, INFO )
ELSE
CALL SCOPY( N, D, 1, WORK( N+1 ), 1 )
IF (ABSTOL .LE. TWO*N*EPS) THEN
TRYRAC = .TRUE.
ELSE
TRYRAC = .FALSE.
END IF
CALL SSTEMR( JOBZ, 'A', N, WORK( N+1 ), WORK, VL, VU, IL,
$ IU, M, W, Z, LDZ, N, ISUPPZ, TRYRAC,
$ WORK( 2*N+1 ), LWORK-2*N, IWORK, LIWORK, INFO )
*
END IF
IF( INFO.EQ.0 ) THEN
M = N
GO TO 10
END IF
INFO = 0
END IF
*
* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M,
$ NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK,
$ IWORK( INDIWO ), INFO )
*
IF( WANTZ ) THEN
CALL SSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ),
$ Z, LDZ, WORK, IWORK( INDIWO ), IWORK( INDIFL ),
$ INFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
10 CONTINUE
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = M
ELSE
IMAX = INFO - 1
END IF
CALL SSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
* If eigenvalues are not in order, then sort them, along with
* eigenvectors.
*
IF( WANTZ ) THEN
DO 30 J = 1, M - 1
I = 0
TMP1 = W( J )
DO 20 JJ = J + 1, M
IF( W( JJ ).LT.TMP1 ) THEN
I = JJ
TMP1 = W( JJ )
END IF
20 CONTINUE
*
IF( I.NE.0 ) THEN
W( I ) = W( J )
W( J ) = TMP1
CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
END IF
30 CONTINUE
END IF
*
* Causes problems with tests 19 & 20:
* IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002
*
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
RETURN
*
* End of SSTEVR
*
END
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