SUBROUTINE CSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK, \$ LRWORK, IWORK, LIWORK, INFO ) * * -- LAPACK 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 COMPZ INTEGER INFO, LDZ, LIWORK, LRWORK, LWORK, N * .. * .. Array Arguments .. INTEGER IWORK( * ) REAL D( * ), E( * ), RWORK( * ) COMPLEX WORK( * ), Z( LDZ, * ) * .. * * Purpose * ======= * * CSTEDC computes all eigenvalues and, optionally, eigenvectors of a * symmetric tridiagonal matrix using the divide and conquer method. * The eigenvectors of a full or band complex Hermitian matrix can also * be found if CHETRD or CHPTRD or CHBTRD has been used to reduce this * matrix to tridiagonal form. * * This code 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. See SLAED3 for details. * * Arguments * ========= * * COMPZ (input) CHARACTER*1 * = 'N': Compute eigenvalues only. * = 'I': Compute eigenvectors of tridiagonal matrix also. * = 'V': Compute eigenvectors of original Hermitian matrix * also. On entry, Z contains the unitary matrix used * to reduce the original matrix to tridiagonal form. * * N (input) INTEGER * The dimension of the symmetric tridiagonal matrix. N >= 0. * * D (input/output) REAL array, dimension (N) * On entry, the diagonal elements of the tridiagonal matrix. * On exit, if INFO = 0, the eigenvalues in ascending order. * * E (input/output) REAL array, dimension (N-1) * On entry, the subdiagonal elements of the tridiagonal matrix. * On exit, E has been destroyed. * * Z (input/output) COMPLEX array, dimension (LDZ,N) * On entry, if COMPZ = 'V', then Z contains the unitary * matrix used in the reduction to tridiagonal form. * On exit, if INFO = 0, then if COMPZ = 'V', Z contains the * orthonormal eigenvectors of the original Hermitian matrix, * and if COMPZ = 'I', Z contains the orthonormal eigenvectors * of the symmetric tridiagonal matrix. * If COMPZ = 'N', then Z is not referenced. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1. * If eigenvectors are desired, then LDZ >= max(1,N). * * WORK (workspace/output) COMPLEX 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. * If COMPZ = 'N' or 'I', or N <= 1, LWORK must be at least 1. * If COMPZ = 'V' and N > 1, LWORK must be at least N*N. * Note that for COMPZ = 'V', then if N is less than or * equal to the minimum divide size, usually 25, then LWORK need * only be 1. * * If LWORK = -1, then a workspace query is assumed; the routine * only calculates the optimal sizes of the WORK, RWORK and * IWORK arrays, returns these values as the first entries of * the WORK, RWORK and IWORK arrays, and no error message * related to LWORK or LRWORK or LIWORK is issued by XERBLA. * * RWORK (workspace/output) REAL array, dimension (MAX(1,LRWORK)) * On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK. * * LRWORK (input) INTEGER * The dimension of the array RWORK. * If COMPZ = 'N' or N <= 1, LRWORK must be at least 1. * If COMPZ = 'V' and N > 1, LRWORK must be at least * 1 + 3*N + 2*N*lg N + 3*N**2 , * where lg( N ) = smallest integer k such * that 2**k >= N. * If COMPZ = 'I' and N > 1, LRWORK must be at least * 1 + 4*N + 2*N**2 . * Note that for COMPZ = 'I' or 'V', then if N is less than or * equal to the minimum divide size, usually 25, then LRWORK * need only be max(1,2*(N-1)). * * If LRWORK = -1, then a workspace query is assumed; the * routine only calculates the optimal sizes of the WORK, RWORK * and IWORK arrays, returns these values as the first entries * of the WORK, RWORK and IWORK arrays, and no error message * related to LWORK or LRWORK 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 IWORK. * If COMPZ = 'N' or N <= 1, LIWORK must be at least 1. * If COMPZ = 'V' or N > 1, LIWORK must be at least * 6 + 6*N + 5*N*lg N. * If COMPZ = 'I' or N > 1, LIWORK must be at least * 3 + 5*N . * Note that for COMPZ = 'I' or 'V', then if N is less than or * equal to the minimum divide size, usually 25, then LIWORK * need only be 1. * * If LIWORK = -1, then a workspace query is assumed; the * routine only calculates the optimal sizes of the WORK, RWORK * and IWORK arrays, returns these values as the first entries * of the WORK, RWORK and IWORK arrays, and no error message * related to LWORK or LRWORK 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: The algorithm failed to compute an eigenvalue while * working on the submatrix lying in rows and columns * INFO/(N+1) through mod(INFO,N+1). * * Further Details * =============== * * Based on contributions by * Jeff Rutter, Computer Science Division, University of California * at Berkeley, USA * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, TWO PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0 ) * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER FINISH, I, ICOMPZ, II, J, K, LGN, LIWMIN, LL, \$ LRWMIN, LWMIN, M, SMLSIZ, START REAL EPS, ORGNRM, P, TINY * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV REAL SLAMCH, SLANST EXTERNAL ILAENV, LSAME, SLAMCH, SLANST * .. * .. External Subroutines .. EXTERNAL XERBLA, CLACPY, CLACRM, CLAED0, CSTEQR, CSWAP, \$ SLASCL, SLASET, SSTEDC, SSTEQR, SSTERF * .. * .. Intrinsic Functions .. INTRINSIC ABS, INT, LOG, MAX, MOD, REAL, SQRT * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) * IF( LSAME( COMPZ, 'N' ) ) THEN ICOMPZ = 0 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN ICOMPZ = 1 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN ICOMPZ = 2 ELSE ICOMPZ = -1 END IF IF( ICOMPZ.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( ( LDZ.LT.1 ) .OR. \$ ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, N ) ) ) THEN INFO = -6 END IF * IF( INFO.EQ.0 ) THEN * * Compute the workspace requirements * SMLSIZ = ILAENV( 9, 'CSTEDC', ' ', 0, 0, 0, 0 ) IF( N.LE.1 .OR. ICOMPZ.EQ.0 ) THEN LWMIN = 1 LIWMIN = 1 LRWMIN = 1 ELSE IF( N.LE.SMLSIZ ) THEN LWMIN = 1 LIWMIN = 1 LRWMIN = 2*( N - 1 ) ELSE IF( ICOMPZ.EQ.1 ) THEN LGN = INT( LOG( REAL( N ) ) / LOG( TWO ) ) IF( 2**LGN.LT.N ) \$ LGN = LGN + 1 IF( 2**LGN.LT.N ) \$ LGN = LGN + 1 LWMIN = N*N LRWMIN = 1 + 3*N + 2*N*LGN + 3*N**2 LIWMIN = 6 + 6*N + 5*N*LGN ELSE IF( ICOMPZ.EQ.2 ) THEN LWMIN = 1 LRWMIN = 1 + 4*N + 2*N**2 LIWMIN = 3 + 5*N END IF WORK( 1 ) = LWMIN RWORK( 1 ) = LRWMIN IWORK( 1 ) = LIWMIN * IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -8 ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN INFO = -10 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -12 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'CSTEDC', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN IF( N.EQ.1 ) THEN IF( ICOMPZ.NE.0 ) \$ Z( 1, 1 ) = ONE RETURN END IF * * If the following conditional clause is removed, then the routine * will use the Divide and Conquer routine to compute only the * eigenvalues, which requires (3N + 3N**2) real workspace and * (2 + 5N + 2N lg(N)) integer workspace. * Since on many architectures SSTERF is much faster than any other * algorithm for finding eigenvalues only, it is used here * as the default. If the conditional clause is removed, then * information on the size of workspace needs to be changed. * * If COMPZ = 'N', use SSTERF to compute the eigenvalues. * IF( ICOMPZ.EQ.0 ) THEN CALL SSTERF( N, D, E, INFO ) GO TO 70 END IF * * If N is smaller than the minimum divide size (SMLSIZ+1), then * solve the problem with another solver. * IF( N.LE.SMLSIZ ) THEN * CALL CSTEQR( COMPZ, N, D, E, Z, LDZ, RWORK, INFO ) * ELSE * * If COMPZ = 'I', we simply call SSTEDC instead. * IF( ICOMPZ.EQ.2 ) THEN CALL SLASET( 'Full', N, N, ZERO, ONE, RWORK, N ) LL = N*N + 1 CALL SSTEDC( 'I', N, D, E, RWORK, N, \$ RWORK( LL ), LRWORK-LL+1, IWORK, LIWORK, INFO ) DO 20 J = 1, N DO 10 I = 1, N Z( I, J ) = RWORK( ( J-1 )*N+I ) 10 CONTINUE 20 CONTINUE GO TO 70 END IF * * From now on, only option left to be handled is COMPZ = 'V', * i.e. ICOMPZ = 1. * * Scale. * ORGNRM = SLANST( 'M', N, D, E ) IF( ORGNRM.EQ.ZERO ) \$ GO TO 70 * EPS = SLAMCH( 'Epsilon' ) * START = 1 * * while ( START <= N ) * 30 CONTINUE IF( START.LE.N ) THEN * * Let FINISH be the position of the next subdiagonal entry * such that E( FINISH ) <= TINY or FINISH = N if no such * subdiagonal exists. The matrix identified by the elements * between START and FINISH constitutes an independent * sub-problem. * FINISH = START 40 CONTINUE IF( FINISH.LT.N ) THEN TINY = EPS*SQRT( ABS( D( FINISH ) ) )* \$ SQRT( ABS( D( FINISH+1 ) ) ) IF( ABS( E( FINISH ) ).GT.TINY ) THEN FINISH = FINISH + 1 GO TO 40 END IF END IF * * (Sub) Problem determined. Compute its size and solve it. * M = FINISH - START + 1 IF( M.GT.SMLSIZ ) THEN * * Scale. * ORGNRM = SLANST( 'M', M, D( START ), E( START ) ) CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, M, 1, D( START ), M, \$ INFO ) CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, M-1, 1, E( START ), \$ M-1, INFO ) * CALL CLAED0( N, M, D( START ), E( START ), Z( 1, START ), \$ LDZ, WORK, N, RWORK, IWORK, INFO ) IF( INFO.GT.0 ) THEN INFO = ( INFO / ( M+1 )+START-1 )*( N+1 ) + \$ MOD( INFO, ( M+1 ) ) + START - 1 GO TO 70 END IF * * Scale back. * CALL SLASCL( 'G', 0, 0, ONE, ORGNRM, M, 1, D( START ), M, \$ INFO ) * ELSE CALL SSTEQR( 'I', M, D( START ), E( START ), RWORK, M, \$ RWORK( M*M+1 ), INFO ) CALL CLACRM( N, M, Z( 1, START ), LDZ, RWORK, M, WORK, N, \$ RWORK( M*M+1 ) ) CALL CLACPY( 'A', N, M, WORK, N, Z( 1, START ), LDZ ) IF( INFO.GT.0 ) THEN INFO = START*( N+1 ) + FINISH GO TO 70 END IF END IF * START = FINISH + 1 GO TO 30 END IF * * endwhile * * If the problem split any number of times, then the eigenvalues * will not be properly ordered. Here we permute the eigenvalues * (and the associated eigenvectors) into ascending order. * IF( M.NE.N ) THEN * * Use Selection Sort to minimize swaps of eigenvectors * DO 60 II = 2, N I = II - 1 K = I P = D( I ) DO 50 J = II, N IF( D( J ).LT.P ) THEN K = J P = D( J ) END IF 50 CONTINUE IF( K.NE.I ) THEN D( K ) = D( I ) D( I ) = P CALL CSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 ) END IF 60 CONTINUE END IF END IF * 70 CONTINUE WORK( 1 ) = LWMIN RWORK( 1 ) = LRWMIN IWORK( 1 ) = LIWMIN * RETURN * * End of CSTEDC * END