LAPACK 3.3.0

slaed1.f

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00001       SUBROUTINE SLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
00002      $                   INFO )
00003 *
00004 *  -- LAPACK routine (version 3.2) --
00005 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00006 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00007 *     November 2006
00008 *
00009 *     .. Scalar Arguments ..
00010       INTEGER            CUTPNT, INFO, LDQ, N
00011       REAL               RHO
00012 *     ..
00013 *     .. Array Arguments ..
00014       INTEGER            INDXQ( * ), IWORK( * )
00015       REAL               D( * ), Q( LDQ, * ), WORK( * )
00016 *     ..
00017 *
00018 *  Purpose
00019 *  =======
00020 *
00021 *  SLAED1 computes the updated eigensystem of a diagonal
00022 *  matrix after modification by a rank-one symmetric matrix.  This
00023 *  routine is used only for the eigenproblem which requires all
00024 *  eigenvalues and eigenvectors of a tridiagonal matrix.  SLAED7 handles
00025 *  the case in which eigenvalues only or eigenvalues and eigenvectors
00026 *  of a full symmetric matrix (which was reduced to tridiagonal form)
00027 *  are desired.
00028 *
00029 *    T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out)
00030 *
00031 *     where Z = Q'u, u is a vector of length N with ones in the
00032 *     CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
00033 *
00034 *     The eigenvectors of the original matrix are stored in Q, and the
00035 *     eigenvalues are in D.  The algorithm consists of three stages:
00036 *
00037 *        The first stage consists of deflating the size of the problem
00038 *        when there are multiple eigenvalues or if there is a zero in
00039 *        the Z vector.  For each such occurence the dimension of the
00040 *        secular equation problem is reduced by one.  This stage is
00041 *        performed by the routine SLAED2.
00042 *
00043 *        The second stage consists of calculating the updated
00044 *        eigenvalues. This is done by finding the roots of the secular
00045 *        equation via the routine SLAED4 (as called by SLAED3).
00046 *        This routine also calculates the eigenvectors of the current
00047 *        problem.
00048 *
00049 *        The final stage consists of computing the updated eigenvectors
00050 *        directly using the updated eigenvalues.  The eigenvectors for
00051 *        the current problem are multiplied with the eigenvectors from
00052 *        the overall problem.
00053 *
00054 *  Arguments
00055 *  =========
00056 *
00057 *  N      (input) INTEGER
00058 *         The dimension of the symmetric tridiagonal matrix.  N >= 0.
00059 *
00060 *  D      (input/output) REAL array, dimension (N)
00061 *         On entry, the eigenvalues of the rank-1-perturbed matrix.
00062 *         On exit, the eigenvalues of the repaired matrix.
00063 *
00064 *  Q      (input/output) REAL array, dimension (LDQ,N)
00065 *         On entry, the eigenvectors of the rank-1-perturbed matrix.
00066 *         On exit, the eigenvectors of the repaired tridiagonal matrix.
00067 *
00068 *  LDQ    (input) INTEGER
00069 *         The leading dimension of the array Q.  LDQ >= max(1,N).
00070 *
00071 *  INDXQ  (input/output) INTEGER array, dimension (N)
00072 *         On entry, the permutation which separately sorts the two
00073 *         subproblems in D into ascending order.
00074 *         On exit, the permutation which will reintegrate the
00075 *         subproblems back into sorted order,
00076 *         i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
00077 *
00078 *  RHO    (input) REAL
00079 *         The subdiagonal entry used to create the rank-1 modification.
00080 *
00081 *  CUTPNT (input) INTEGER
00082 *         The location of the last eigenvalue in the leading sub-matrix.
00083 *         min(1,N) <= CUTPNT <= N/2.
00084 *
00085 *  WORK   (workspace) REAL array, dimension (4*N + N**2)
00086 *
00087 *  IWORK  (workspace) INTEGER array, dimension (4*N)
00088 *
00089 *  INFO   (output) INTEGER
00090 *          = 0:  successful exit.
00091 *          < 0:  if INFO = -i, the i-th argument had an illegal value.
00092 *          > 0:  if INFO = 1, an eigenvalue did not converge
00093 *
00094 *  Further Details
00095 *  ===============
00096 *
00097 *  Based on contributions by
00098 *     Jeff Rutter, Computer Science Division, University of California
00099 *     at Berkeley, USA
00100 *  Modified by Francoise Tisseur, University of Tennessee.
00101 *
00102 *  =====================================================================
00103 *
00104 *     .. Local Scalars ..
00105       INTEGER            COLTYP, CPP1, I, IDLMDA, INDX, INDXC, INDXP,
00106      $                   IQ2, IS, IW, IZ, K, N1, N2
00107 *     ..
00108 *     .. External Subroutines ..
00109       EXTERNAL           SCOPY, SLAED2, SLAED3, SLAMRG, XERBLA
00110 *     ..
00111 *     .. Intrinsic Functions ..
00112       INTRINSIC          MAX, MIN
00113 *     ..
00114 *     .. Executable Statements ..
00115 *
00116 *     Test the input parameters.
00117 *
00118       INFO = 0
00119 *
00120       IF( N.LT.0 ) THEN
00121          INFO = -1
00122       ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
00123          INFO = -4
00124       ELSE IF( MIN( 1, N / 2 ).GT.CUTPNT .OR. ( N / 2 ).LT.CUTPNT ) THEN
00125          INFO = -7
00126       END IF
00127       IF( INFO.NE.0 ) THEN
00128          CALL XERBLA( 'SLAED1', -INFO )
00129          RETURN
00130       END IF
00131 *
00132 *     Quick return if possible
00133 *
00134       IF( N.EQ.0 )
00135      $   RETURN
00136 *
00137 *     The following values are integer pointers which indicate
00138 *     the portion of the workspace
00139 *     used by a particular array in SLAED2 and SLAED3.
00140 *
00141       IZ = 1
00142       IDLMDA = IZ + N
00143       IW = IDLMDA + N
00144       IQ2 = IW + N
00145 *
00146       INDX = 1
00147       INDXC = INDX + N
00148       COLTYP = INDXC + N
00149       INDXP = COLTYP + N
00150 *
00151 *
00152 *     Form the z-vector which consists of the last row of Q_1 and the
00153 *     first row of Q_2.
00154 *
00155       CALL SCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 )
00156       CPP1 = CUTPNT + 1
00157       CALL SCOPY( N-CUTPNT, Q( CPP1, CPP1 ), LDQ, WORK( IZ+CUTPNT ), 1 )
00158 *
00159 *     Deflate eigenvalues.
00160 *
00161       CALL SLAED2( K, N, CUTPNT, D, Q, LDQ, INDXQ, RHO, WORK( IZ ),
00162      $             WORK( IDLMDA ), WORK( IW ), WORK( IQ2 ),
00163      $             IWORK( INDX ), IWORK( INDXC ), IWORK( INDXP ),
00164      $             IWORK( COLTYP ), INFO )
00165 *
00166       IF( INFO.NE.0 )
00167      $   GO TO 20
00168 *
00169 *     Solve Secular Equation.
00170 *
00171       IF( K.NE.0 ) THEN
00172          IS = ( IWORK( COLTYP )+IWORK( COLTYP+1 ) )*CUTPNT +
00173      $        ( IWORK( COLTYP+1 )+IWORK( COLTYP+2 ) )*( N-CUTPNT ) + IQ2
00174          CALL SLAED3( K, N, CUTPNT, D, Q, LDQ, RHO, WORK( IDLMDA ),
00175      $                WORK( IQ2 ), IWORK( INDXC ), IWORK( COLTYP ),
00176      $                WORK( IW ), WORK( IS ), INFO )
00177          IF( INFO.NE.0 )
00178      $      GO TO 20
00179 *
00180 *     Prepare the INDXQ sorting permutation.
00181 *
00182          N1 = K
00183          N2 = N - K
00184          CALL SLAMRG( N1, N2, D, 1, -1, INDXQ )
00185       ELSE
00186          DO 10 I = 1, N
00187             INDXQ( I ) = I
00188    10    CONTINUE
00189       END IF
00190 *
00191    20 CONTINUE
00192       RETURN
00193 *
00194 *     End of SLAED1
00195 *
00196       END
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