ssteqr2.f

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      SUBROUTINE ssteqr2( COMPZ, N, D, E, Z, LDZ, NR, WORK, INFO )
*
*  -- LAPACK routine (version 2.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      CHARACTER          COMPZ
      INTEGER            INFO, LDZ, N, NR
*     ..
*     .. Array Arguments ..
      REAL               D( * ), E( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  SSTEQR2 is a modified version of LAPACK routine SSTEQR.
*  SSTEQR2 computes all eigenvalues and, optionally, eigenvectors of a
*  symmetric tridiagonal matrix using the implicit QL or QR method.
*  running SSTEQR2 to perform updates on a distributed matrix Q.
*  Proper usage of SSTEQR2 can be gleaned from examination of ScaLAPACK's
*  PSSYEV.
*
*  Arguments
*  =========
*
*  COMPZ   (input) CHARACTER*1
*          = 'N':  Compute eigenvalues only.
*          = 'I':  Compute eigenvalues and eigenvectors of the
*                  tridiagonal matrix.  Z must be initialized to the
*                  identity matrix by PDLASET or DLASET prior to entering
*                  this subroutine.
*
*  N       (input) INTEGER
*          The order of the 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 (n-1) subdiagonal elements of the tridiagonal
*          matrix.
*          On exit, E has been destroyed.
*
*  Z       (local input/local output) REAL array, global
*          dimension (N, N), local dimension (LDZ, NR).
*          On entry, if  COMPZ = 'V', then Z contains the orthogonal
*          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 symmetric 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, and if
*          eigenvectors are desired, then  LDZ >= max(1,N).
*
*  NR      (input) INTEGER
*          NR = MAX(1, NUMROC( N, NB, MYPROW, 0, NPROCS ) ).
*          If COMPZ = 'N', then NR is not referenced.
*
*  WORK    (workspace) REAL array, dimension (max(1,2*N-2))
*          If COMPZ = 'N', then WORK is not referenced.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  the algorithm has failed to find all the eigenvalues in
*                a total of 30*N iterations; if INFO = i, then i
*                elements of E have not converged to zero; on exit, D
*                and E contain the elements of a symmetric tridiagonal
*                matrix which is orthogonally similar to the original
*                matrix.
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE, TWO, THREE
      parameter( zero = 0.0e0, one = 1.0e0, two = 2.0e0,
     $                   three = 3.0e0 )
      INTEGER            MAXIT
      parameter( maxit = 30 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ICOMPZ, II, ISCALE, J, JTOT, K, L, L1, LEND,
     $                   LENDM1, LENDP1, LENDSV, LM1, LSV, M, MM, MM1,
     $                   NM1, NMAXIT
      REAL               ANORM, B, C, EPS, EPS2, F, G, P, R, RT1, RT2,
     $                   S, SAFMAX, SAFMIN, SSFMAX, SSFMIN, TST
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      REAL               SLAMCH, SLANST, SLAPY2
      EXTERNAL           lsame, slamch, slanst, slapy2
*     ..
*     .. External Subroutines ..
      EXTERNAL           slae2, slaev2, slartg, slascl, slasr,
     $                   slasrt, sswap, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, sign, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
*
      IF( lsame( compz, 'N' ) ) THEN
         icompz = 0
      ELSE IF( lsame( compz, 'I' ) ) THEN
         icompz = 1
      ELSE
         icompz = -1
      END IF
      IF( icompz.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( icompz.GT.0 .AND. ldz.LT.max( 1, nr ) ) THEN
         info = -6
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SSTEQR2', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   RETURN
*
      IF( n.EQ.1 ) THEN
         IF( icompz.EQ.1 )
     $      z( 1, 1 ) = one
         RETURN
      END IF
*
*     Determine the unit roundoff and over/underflow thresholds.
*
      eps = slamch( 'E' )
      eps2 = eps**2
      safmin = slamch( 'S' )
      safmax = one / safmin
      ssfmax = sqrt( safmax ) / three
      ssfmin = sqrt( safmin ) / eps2
*
*     Compute the eigenvalues and eigenvectors of the tridiagonal
*     matrix.
*
      nmaxit = n*maxit
      jtot = 0
*
*     Determine where the matrix splits and choose QL or QR iteration
*     for each block, according to whether top or bottom diagonal
*     element is smaller.
*
      l1 = 1
      nm1 = n - 1
*
   10 CONTINUE
      IF( l1.GT.n )
     $   GO TO 160
      IF( l1.GT.1 )
     $   e( l1-1 ) = zero
      IF( l1.LE.nm1 ) THEN
         DO 20 m = l1, nm1
            tst = abs( e( m ) )
            IF( tst.EQ.zero )
     $         GO TO 30
            IF( tst.LE.( sqrt( abs( d( m ) ) )*sqrt( abs( d( m+
     $          1 ) ) ) )*eps ) THEN
               e( m ) = zero
               GO TO 30
            END IF
   20    CONTINUE
      END IF
      m = n
*
   30 CONTINUE
      l = l1
      lsv = l
      lend = m
      lendsv = lend
      l1 = m + 1
      IF( lend.EQ.l )
     $   GO TO 10
*
*     Scale submatrix in rows and columns L to LEND
*
      anorm = slanst( 'I', lend-l+1, d( l ), e( l ) )
      iscale = 0
      IF( anorm.EQ.zero )
     $   GO TO 10
      IF( anorm.GT.ssfmax ) THEN
         iscale = 1
         CALL slascl( 'G', 0, 0, anorm, ssfmax, lend-l+1, 1, d( l ), n,
     $                info )
         CALL slascl( 'G', 0, 0, anorm, ssfmax, lend-l, 1, e( l ), n,
     $                info )
      ELSE IF( anorm.LT.ssfmin ) THEN
         iscale = 2
         CALL slascl( 'G', 0, 0, anorm, ssfmin, lend-l+1, 1, d( l ), n,
     $                info )
         CALL slascl( 'G', 0, 0, anorm, ssfmin, lend-l, 1, e( l ), n,
     $                info )
      END IF
*
*     Choose between QL and QR iteration
*
      IF( abs( d( lend ) ).LT.abs( d( l ) ) ) THEN
         lend = lsv
         l = lendsv
      END IF
*
      IF( lend.GT.l ) THEN
*
*        QL Iteration
*
*        Look for small subdiagonal element.
*
   40    CONTINUE
         IF( l.NE.lend ) THEN
            lendm1 = lend - 1
            DO 50 m = l, lendm1
               tst = abs( e( m ) )**2
               IF( tst.LE.( eps2*abs( d( m ) ) )*abs( d( m+1 ) )+
     $             safmin )GO TO 60
   50       CONTINUE
         END IF
*
         m = lend
*
   60    CONTINUE
         IF( m.LT.lend )
     $      e( m ) = zero
         p = d( l )
         IF( m.EQ.l )
     $      GO TO 80
*
*        If remaining matrix is 2-by-2, use SLAE2 or SLAEV2
*        to compute its eigensystem.
*
         IF( m.EQ.l+1 ) THEN
            IF( icompz.GT.0 ) THEN
               CALL slaev2( d( l ), e( l ), d( l+1 ), rt1, rt2, c, s )
               work( l ) = c
               work( n-1+l ) = s
               CALL slasr( 'R', 'V', 'B', nr, 2, work( l ),
     $                     work( n-1+l ), z( 1, l ), ldz )
            ELSE
               CALL slae2( d( l ), e( l ), d( l+1 ), rt1, rt2 )
            END IF
            d( l ) = rt1
            d( l+1 ) = rt2
            e( l ) = zero
            l = l + 2
            IF( l.LE.lend )
     $         GO TO 40
            GO TO 140
         END IF
*
         IF( jtot.EQ.nmaxit )
     $      GO TO 140
         jtot = jtot + 1
*
*        Form shift.
*
         g = ( d( l+1 )-p ) / ( two*e( l ) )
         r = slapy2( g, one )
         g = d( m ) - p + ( e( l ) / ( g+sign( r, g ) ) )
*
         s = one
         c = one
         p = zero
*
*        Inner loop
*
         mm1 = m - 1
         DO 70 i = mm1, l, -1
            f = s*e( i )
            b = c*e( i )
            CALL slartg( g, f, c, s, r )
            IF( i.NE.m-1 )
     $         e( i+1 ) = r
            g = d( i+1 ) - p
            r = ( d( i )-g )*s + two*c*b
            p = s*r
            d( i+1 ) = g + p
            g = c*r - b
*
*           If eigenvectors are desired, then save rotations.
*
            IF( icompz.GT.0 ) THEN
               work( i ) = c
               work( n-1+i ) = -s
            END IF
*
   70    CONTINUE
*
*        If eigenvectors are desired, then apply saved rotations.
*
         IF( icompz.GT.0 ) THEN
            mm = m - l + 1
            CALL slasr( 'R', 'V', 'B', nr, mm, work( l ), work( n-1+l ),
     $                  z( 1, l ), ldz )
         END IF
*
         d( l ) = d( l ) - p
         e( l ) = g
         GO TO 40
*
*        Eigenvalue found.
*
   80    CONTINUE
         d( l ) = p
*
         l = l + 1
         IF( l.LE.lend )
     $      GO TO 40
         GO TO 140
*
      ELSE
*
*        QR Iteration
*
*        Look for small superdiagonal element.
*
   90    CONTINUE
         IF( l.NE.lend ) THEN
            lendp1 = lend + 1
            DO 100 m = l, lendp1, -1
               tst = abs( e( m-1 ) )**2
               IF( tst.LE.( eps2*abs( d( m ) ) )*abs( d( m-1 ) )+
     $             safmin )GO TO 110
  100       CONTINUE
         END IF
*
         m = lend
*
  110    CONTINUE
         IF( m.GT.lend )
     $      e( m-1 ) = zero
         p = d( l )
         IF( m.EQ.l )
     $      GO TO 130
*
*        If remaining matrix is 2-by-2, use SLAE2 or SLAEV2
*        to compute its eigensystem.
*
         IF( m.EQ.l-1 ) THEN
            IF( icompz.GT.0 ) THEN
               CALL slaev2( d( l-1 ), e( l-1 ), d( l ), rt1, rt2, c, s )
               work( m ) = c
               work( n-1+m ) = s
               CALL slasr( 'r', 'v', 'f', NR, 2, WORK( M ),
     $                     WORK( N-1+M ), Z( 1, L-1 ), LDZ )
            ELSE
               CALL SLAE2( D( L-1 ), E( L-1 ), D( L ), RT1, RT2 )
            END IF
            D( L-1 ) = RT1
            D( L ) = RT2
            E( L-1 ) = ZERO
            L = L - 2
.GE.            IF( LLEND )
     $         GO TO 90
            GO TO 140
         END IF
*
.EQ.         IF( JTOTNMAXIT )
     $      GO TO 140
         JTOT = JTOT + 1
*
*        Form shift.
*
         G = ( D( L-1 )-P ) / ( TWO*E( L-1 ) )
         R = SLAPY2( G, ONE )
         G = D( M ) - P + ( E( L-1 ) / ( G+SIGN( R, G ) ) )
*
         S = ONE
         C = ONE
         P = ZERO
*
*        Inner loop
*
         LM1 = L - 1
         DO 120 I = M, LM1
            F = S*E( I )
            B = C*E( I )
            CALL SLARTG( G, F, C, S, R )
.NE.            IF( IM )
     $         E( I-1 ) = R
            G = D( I ) - P
            R = ( D( I+1 )-G )*S + TWO*C*B
            P = S*R
            D( I ) = G + P
            G = C*R - B
*
*           If eigenvectors are desired, then save rotations.
*
.GT.            IF( ICOMPZ0 ) THEN
               WORK( I ) = C
               WORK( N-1+I ) = S
            END IF
*
  120    CONTINUE
*
*        If eigenvectors are desired, then apply saved rotations.
*
.GT.         IF( ICOMPZ0 ) THEN
            MM = L - M + 1
            CALL SLASR( 'r', 'v', 'f', NR, MM, WORK( M ), WORK( N-1+M ),
     $                  Z( 1, M ), LDZ )
         END IF
*
         D( L ) = D( L ) - P
         E( LM1 ) = G
         GO TO 90
*
*        Eigenvalue found.
*
  130    CONTINUE
         D( L ) = P
*
         L = L - 1
.GE.         IF( LLEND )
     $      GO TO 90
         GO TO 140
*
      END IF
*
*     Undo scaling if necessary
*
  140 CONTINUE
.EQ.      IF( ISCALE1 ) THEN
         CALL SLASCL( 'g', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1,
     $                D( LSV ), N, INFO )
         CALL SLASCL( 'g', 0, 0, SSFMAX, ANORM, LENDSV-LSV, 1, E( LSV ),
     $                N, INFO )
.EQ.      ELSE IF( ISCALE2 ) THEN
         CALL SLASCL( 'g', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1,
     $                D( LSV ), N, INFO )
         CALL SLASCL( 'g', 0, 0, SSFMIN, ANORM, LENDSV-LSV, 1, E( LSV ),
     $                N, INFO )
      END IF
*
*     Check for no convergence to an eigenvalue after a total
*     of N*MAXIT iterations.
*
.LT.      IF( JTOTNMAXIT )
     $   GO TO 10
      DO 150 I = 1, N - 1
.NE.         IF( E( I )ZERO )
     $      INFO = INFO + 1
  150 CONTINUE
      GO TO 190
*
*     Order eigenvalues and eigenvectors.
*
  160 CONTINUE
.EQ.      IF( ICOMPZ0 ) THEN
*
*        Use Quick Sort
*
         CALL SLASRT( 'i', N, D, INFO )
*
      ELSE
*
*        Use Selection Sort to minimize swaps of eigenvectors
*
         DO 180 II = 2, N
            I = II - 1
            K = I
            P = D( I )
            DO 170 J = II, N
.LT.               IF( D( J )P ) THEN
                  K = J
                  P = D( J )
               END IF
  170       CONTINUE
.NE.            IF( KI ) THEN
               D( K ) = D( I )
               D( I ) = P
               CALL SSWAP( NR, Z( 1, I ), 1, Z( 1, K ), 1 )
            END IF
  180    CONTINUE
      END IF
*
  190 CONTINUE
      RETURN
*
*     End of SSTEQR2
*
      END