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author | jason <jason@8a072113-8704-0410-8d35-dd094bca7971> | 2008-10-28 01:38:50 +0000 |
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committer | jason <jason@8a072113-8704-0410-8d35-dd094bca7971> | 2008-10-28 01:38:50 +0000 |
commit | baba851215b44ac3b60b9248eb02bcce7eb76247 (patch) | |
tree | 8c0f5c006875532a30d4409f5e94b0f310ff00a7 /SRC/dsyevr.f | |
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Move LAPACK trunk into position.
Diffstat (limited to 'SRC/dsyevr.f')
-rw-r--r-- | SRC/dsyevr.f | 551 |
1 files changed, 551 insertions, 0 deletions
diff --git a/SRC/dsyevr.f b/SRC/dsyevr.f new file mode 100644 index 00000000..c213c998 --- /dev/null +++ b/SRC/dsyevr.f @@ -0,0 +1,551 @@ + SUBROUTINE DSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, + $ ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, + $ IWORK, LIWORK, 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, LDA, LDZ, LIWORK, LWORK, M, N + DOUBLE PRECISION ABSTOL, VL, VU +* .. +* .. Array Arguments .. + INTEGER ISUPPZ( * ), IWORK( * ) + DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * ) +* .. +* +* Purpose +* ======= +* +* DSYEVR computes selected eigenvalues and, optionally, eigenvectors +* of a real symmetric matrix A. Eigenvalues and eigenvectors can be +* selected by specifying either a range of values or a range of +* indices for the desired eigenvalues. +* +* DSYEVR first reduces the matrix A to tridiagonal form T with a call +* to DSYTRD. Then, whenever possible, DSYEVR calls DSTEMR to compute +* the eigenspectrum using Relatively Robust Representations. DSTEMR +* 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 each unreduced block (submatrix) of T, +* (a) Compute T - sigma I = L D L^T, so that L and D +* define all the wanted eigenvalues to high relative accuracy. +* This means that small relative changes in the entries of D and L +* cause only small relative changes in the eigenvalues and +* eigenvectors. The standard (unfactored) representation of the +* tridiagonal matrix T does not have this property in general. +* (b) Compute the eigenvalues to suitable accuracy. +* If the eigenvectors are desired, the algorithm attains full +* accuracy of the computed eigenvalues only right before +* the corresponding vectors have to be computed, see steps c) and d). +* (c) For each cluster of close eigenvalues, select a new +* shift close to the cluster, find a new factorization, and refine +* the shifted eigenvalues to suitable accuracy. +* (d) For each eigenvalue with a large enough relative separation compute +* the corresponding eigenvector by forming a rank revealing twisted +* factorization. Go back to (c) for any clusters that remain. +* +* The desired accuracy of the output can be specified by the input +* parameter ABSTOL. +* +* For more details, see DSTEMR's documentation and: +* - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations +* to compute orthogonal eigenvectors of symmetric tridiagonal matrices," +* Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. +* - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and +* Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, +* 2004. Also LAPACK Working Note 154. +* - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric +* tridiagonal eigenvalue/eigenvector problem", +* Computer Science Division Technical Report No. UCB/CSD-97-971, +* UC Berkeley, May 1997. +* +* +* Note 1 : DSYEVR calls DSTEMR when the full spectrum is requested +* on machines which conform to the ieee-754 floating point standard. +* DSYEVR calls DSTEBZ and SSTEIN on non-ieee machines and +* when partial spectrum requests are made. +* +* Normal execution of DSTEMR 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, DSTEBZ and +********** DSTEIN are called +* +* 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. +* +* A (input/output) DOUBLE PRECISION array, dimension (LDA, N) +* On entry, the symmetric matrix A. If UPLO = 'U', the +* leading N-by-N upper triangular part of A contains the +* upper triangular part of the matrix A. If UPLO = 'L', +* the leading N-by-N lower triangular part of A contains +* the lower triangular part of the matrix A. +* On exit, the lower triangle (if UPLO='L') or the upper +* triangle (if UPLO='U') of A, including the diagonal, is +* destroyed. +* +* LDA (input) INTEGER +* The leading dimension of the array A. LDA >= 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. +* +* 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 +* DLAMCH( '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) DOUBLE PRECISION array, dimension (N) +* The first M elements contain the selected eigenvalues in +* ascending order. +* +* Z (output) DOUBLE PRECISION 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). +* If JOBZ = 'N', then Z is not referenced. +* 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. +* Supplying N columns is always safe. +* +* 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) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) +* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. +* +* LWORK (input) INTEGER +* The dimension of the array WORK. LWORK >= max(1,26*N). +* For optimal efficiency, LWORK >= (NB+6)*N, +* where NB is the max of the blocksize for DSYTRD and DORMTR +* returned by ILAENV. +* +* If LWORK = -1, then a workspace query is assumed; the routine +* only calculates the optimal size of the WORK array, returns +* this value as the first entry of the WORK array, and no error +* message related to LWORK is issued by XERBLA. +* +* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) +* On exit, if INFO = 0, IWORK(1) returns the optimal LWORK. +* +* LIWORK (input) INTEGER +* The dimension of the array IWORK. LIWORK >= max(1,10*N). +* +* If LIWORK = -1, then a workspace query is assumed; the +* routine only calculates the optimal size of the IWORK array, +* returns this value as the first entry of the IWORK array, and +* no error message related to 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 .. + DOUBLE PRECISION ZERO, ONE, TWO + PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 ) +* .. +* .. Local Scalars .. + LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, VALEIG, WANTZ, + $ TRYRAC + CHARACTER ORDER + INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE, + $ INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU, + $ INDWK, INDWKN, ISCALE, J, JJ, LIWMIN, + $ LLWORK, LLWRKN, LWKOPT, LWMIN, NB, NSPLIT + DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, + $ SIGMA, SMLNUM, TMP1, VLL, VUU +* .. +* .. External Functions .. + LOGICAL LSAME + INTEGER ILAENV + DOUBLE PRECISION DLAMCH, DLANSY + EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY +* .. +* .. External Subroutines .. + EXTERNAL DCOPY, DORMTR, DSCAL, DSTEBZ, DSTEMR, DSTEIN, + $ DSTERF, DSWAP, DSYTRD, XERBLA +* .. +* .. Intrinsic Functions .. + INTRINSIC MAX, MIN, SQRT +* .. +* .. Executable Statements .. +* +* Test the input parameters. +* + IEEEOK = ILAENV( 10, 'DSYEVR', 'N', 1, 2, 3, 4 ) +* + LOWER = LSAME( UPLO, 'L' ) + 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, 26*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( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN + INFO = -3 + ELSE IF( N.LT.0 ) THEN + INFO = -4 + ELSE IF( LDA.LT.MAX( 1, N ) ) THEN + INFO = -6 + ELSE + IF( VALEIG ) THEN + IF( N.GT.0 .AND. VU.LE.VL ) + $ INFO = -8 + ELSE IF( INDEIG ) THEN + IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN + INFO = -9 + ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN + INFO = -10 + END IF + END IF + END IF + IF( INFO.EQ.0 ) THEN + IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN + INFO = -15 + ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN + INFO = -18 + ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN + INFO = -20 + END IF + END IF +* + IF( INFO.EQ.0 ) THEN + NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 ) + NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) ) + LWKOPT = MAX( ( NB+1 )*N, LWMIN ) + WORK( 1 ) = LWKOPT + IWORK( 1 ) = LIWMIN + END IF +* + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'DSYEVR', -INFO ) + RETURN + ELSE IF( LQUERY ) THEN + RETURN + END IF +* +* Quick return if possible +* + M = 0 + IF( N.EQ.0 ) THEN + WORK( 1 ) = 1 + RETURN + END IF +* + IF( N.EQ.1 ) THEN + WORK( 1 ) = 7 + IF( ALLEIG .OR. INDEIG ) THEN + M = 1 + W( 1 ) = A( 1, 1 ) + ELSE + IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN + M = 1 + W( 1 ) = A( 1, 1 ) + END IF + END IF + 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 = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) ) +* +* Scale matrix to allowable range, if necessary. +* + ISCALE = 0 + ABSTLL = ABSTOL + VLL = VL + VUU = VU + ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK ) + 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 + DO 10 J = 1, N + CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 ) + 10 CONTINUE + ELSE + DO 20 J = 1, N + CALL DSCAL( J, SIGMA, A( 1, J ), 1 ) + 20 CONTINUE + END IF + IF( ABSTOL.GT.0 ) + $ ABSTLL = ABSTOL*SIGMA + IF( VALEIG ) THEN + VLL = VL*SIGMA + VUU = VU*SIGMA + END IF + END IF + +* Initialize indices into workspaces. Note: The IWORK indices are +* used only if DSTERF or DSTEMR fail. + +* WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the +* elementary reflectors used in DSYTRD. + INDTAU = 1 +* WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries. + INDD = INDTAU + N +* WORK(INDE:INDE+N-1) stores the off-diagonal entries of the +* tridiagonal matrix from DSYTRD. + INDE = INDD + N +* WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over +* -written by DSTEMR (the DSTERF path copies the diagonal to W). + INDDD = INDE + N +* WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over +* -written while computing the eigenvalues in DSTERF and DSTEMR. + INDEE = INDDD + N +* INDWK is the starting offset of the left-over workspace, and +* LLWORK is the remaining workspace size. + INDWK = INDEE + N + LLWORK = LWORK - INDWK + 1 + +* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and +* stores the block indices of each of the M<=N eigenvalues. + INDIBL = 1 +* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ 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 +* DSTEIN. 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 + +* +* Call DSYTRD to reduce symmetric matrix to tridiagonal form. +* + CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ), + $ WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO ) +* +* If all eigenvalues are desired +* then call DSTERF or DSTEMR and DORMTR. +* + IF( ( ALLEIG .OR. ( INDEIG .AND. IL.EQ.1 .AND. IU.EQ.N ) ) .AND. + $ IEEEOK.EQ.1 ) THEN + IF( .NOT.WANTZ ) THEN + CALL DCOPY( N, WORK( INDD ), 1, W, 1 ) + CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) + CALL DSTERF( N, W, WORK( INDEE ), INFO ) + ELSE + CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 ) + CALL DCOPY( N, WORK( INDD ), 1, WORK( INDDD ), 1 ) +* + IF (ABSTOL .LE. TWO*N*EPS) THEN + TRYRAC = .TRUE. + ELSE + TRYRAC = .FALSE. + END IF + CALL DSTEMR( JOBZ, 'A', N, WORK( INDDD ), WORK( INDEE ), + $ VL, VU, IL, IU, M, W, Z, LDZ, N, ISUPPZ, + $ TRYRAC, WORK( INDWK ), LWORK, IWORK, LIWORK, + $ INFO ) +* +* +* +* Apply orthogonal matrix used in reduction to tridiagonal +* form to eigenvectors returned by DSTEIN. +* + IF( WANTZ .AND. INFO.EQ.0 ) THEN + INDWKN = INDE + LLWRKN = LWORK - INDWKN + 1 + CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, + $ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ), + $ LLWRKN, IINFO ) + END IF + END IF +* +* + IF( INFO.EQ.0 ) THEN +* Everything worked. Skip DSTEBZ/DSTEIN. IWORK(:) are +* undefined. + M = N + GO TO 30 + END IF + INFO = 0 + END IF +* +* Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN. +* Also call DSTEBZ and DSTEIN if DSTEMR fails. +* + IF( WANTZ ) THEN + ORDER = 'B' + ELSE + ORDER = 'E' + END IF + + CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL, + $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W, + $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWK ), + $ IWORK( INDIWO ), INFO ) +* + IF( WANTZ ) THEN + CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W, + $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ, + $ WORK( INDWK ), IWORK( INDIWO ), IWORK( INDIFL ), + $ INFO ) +* +* Apply orthogonal matrix used in reduction to tridiagonal +* form to eigenvectors returned by DSTEIN. +* + INDWKN = INDE + LLWRKN = LWORK - INDWKN + 1 + CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z, + $ LDZ, WORK( INDWKN ), LLWRKN, IINFO ) + END IF +* +* If matrix was scaled, then rescale eigenvalues appropriately. +* +* Jump here if DSTEMR/DSTEIN succeeded. + 30 CONTINUE + IF( ISCALE.EQ.1 ) THEN + IF( INFO.EQ.0 ) THEN + IMAX = M + ELSE + IMAX = INFO - 1 + END IF + CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) + END IF +* +* If eigenvalues are not in order, then sort them, along with +* eigenvectors. Note: We do not sort the IFAIL portion of IWORK. +* It may not be initialized (if DSTEMR/DSTEIN succeeded), and we do +* not return this detailed information to the user. +* + 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 + W( I ) = W( J ) + W( J ) = TMP1 + CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 ) + END IF + 50 CONTINUE + END IF +* +* Set WORK(1) to optimal workspace size. +* + WORK( 1 ) = LWKOPT + IWORK( 1 ) = LIWMIN +* + RETURN +* +* End of DSYEVR +* + END |