LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dlaed1.f
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00001 *> \brief \b DLAED1
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download DLAED1 + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed1.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
00022 *                          INFO )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       INTEGER            CUTPNT, INFO, LDQ, N
00026 *       DOUBLE PRECISION   RHO
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       INTEGER            INDXQ( * ), IWORK( * )
00030 *       DOUBLE PRECISION   D( * ), Q( LDQ, * ), WORK( * )
00031 *       ..
00032 *  
00033 *
00034 *> \par Purpose:
00035 *  =============
00036 *>
00037 *> \verbatim
00038 *>
00039 *> DLAED1 computes the updated eigensystem of a diagonal
00040 *> matrix after modification by a rank-one symmetric matrix.  This
00041 *> routine is used only for the eigenproblem which requires all
00042 *> eigenvalues and eigenvectors of a tridiagonal matrix.  DLAED7 handles
00043 *> the case in which eigenvalues only or eigenvalues and eigenvectors
00044 *> of a full symmetric matrix (which was reduced to tridiagonal form)
00045 *> are desired.
00046 *>
00047 *>   T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
00048 *>
00049 *>    where Z = Q**T*u, u is a vector of length N with ones in the
00050 *>    CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
00051 *>
00052 *>    The eigenvectors of the original matrix are stored in Q, and the
00053 *>    eigenvalues are in D.  The algorithm consists of three stages:
00054 *>
00055 *>       The first stage consists of deflating the size of the problem
00056 *>       when there are multiple eigenvalues or if there is a zero in
00057 *>       the Z vector.  For each such occurence the dimension of the
00058 *>       secular equation problem is reduced by one.  This stage is
00059 *>       performed by the routine DLAED2.
00060 *>
00061 *>       The second stage consists of calculating the updated
00062 *>       eigenvalues. This is done by finding the roots of the secular
00063 *>       equation via the routine DLAED4 (as called by DLAED3).
00064 *>       This routine also calculates the eigenvectors of the current
00065 *>       problem.
00066 *>
00067 *>       The final stage consists of computing the updated eigenvectors
00068 *>       directly using the updated eigenvalues.  The eigenvectors for
00069 *>       the current problem are multiplied with the eigenvectors from
00070 *>       the overall problem.
00071 *> \endverbatim
00072 *
00073 *  Arguments:
00074 *  ==========
00075 *
00076 *> \param[in] N
00077 *> \verbatim
00078 *>          N is INTEGER
00079 *>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
00080 *> \endverbatim
00081 *>
00082 *> \param[in,out] D
00083 *> \verbatim
00084 *>          D is DOUBLE PRECISION array, dimension (N)
00085 *>         On entry, the eigenvalues of the rank-1-perturbed matrix.
00086 *>         On exit, the eigenvalues of the repaired matrix.
00087 *> \endverbatim
00088 *>
00089 *> \param[in,out] Q
00090 *> \verbatim
00091 *>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
00092 *>         On entry, the eigenvectors of the rank-1-perturbed matrix.
00093 *>         On exit, the eigenvectors of the repaired tridiagonal matrix.
00094 *> \endverbatim
00095 *>
00096 *> \param[in] LDQ
00097 *> \verbatim
00098 *>          LDQ is INTEGER
00099 *>         The leading dimension of the array Q.  LDQ >= max(1,N).
00100 *> \endverbatim
00101 *>
00102 *> \param[in,out] INDXQ
00103 *> \verbatim
00104 *>          INDXQ is INTEGER array, dimension (N)
00105 *>         On entry, the permutation which separately sorts the two
00106 *>         subproblems in D into ascending order.
00107 *>         On exit, the permutation which will reintegrate the
00108 *>         subproblems back into sorted order,
00109 *>         i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
00110 *> \endverbatim
00111 *>
00112 *> \param[in] RHO
00113 *> \verbatim
00114 *>          RHO is DOUBLE PRECISION
00115 *>         The subdiagonal entry used to create the rank-1 modification.
00116 *> \endverbatim
00117 *>
00118 *> \param[in] CUTPNT
00119 *> \verbatim
00120 *>          CUTPNT is INTEGER
00121 *>         The location of the last eigenvalue in the leading sub-matrix.
00122 *>         min(1,N) <= CUTPNT <= N/2.
00123 *> \endverbatim
00124 *>
00125 *> \param[out] WORK
00126 *> \verbatim
00127 *>          WORK is DOUBLE PRECISION array, dimension (4*N + N**2)
00128 *> \endverbatim
00129 *>
00130 *> \param[out] IWORK
00131 *> \verbatim
00132 *>          IWORK is INTEGER array, dimension (4*N)
00133 *> \endverbatim
00134 *>
00135 *> \param[out] INFO
00136 *> \verbatim
00137 *>          INFO is INTEGER
00138 *>          = 0:  successful exit.
00139 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
00140 *>          > 0:  if INFO = 1, an eigenvalue did not converge
00141 *> \endverbatim
00142 *
00143 *  Authors:
00144 *  ========
00145 *
00146 *> \author Univ. of Tennessee 
00147 *> \author Univ. of California Berkeley 
00148 *> \author Univ. of Colorado Denver 
00149 *> \author NAG Ltd. 
00150 *
00151 *> \date November 2011
00152 *
00153 *> \ingroup auxOTHERcomputational
00154 *
00155 *> \par Contributors:
00156 *  ==================
00157 *>
00158 *> Jeff Rutter, Computer Science Division, University of California
00159 *> at Berkeley, USA \n
00160 *>  Modified by Francoise Tisseur, University of Tennessee
00161 *>
00162 *  =====================================================================
00163       SUBROUTINE DLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
00164      $                   INFO )
00165 *
00166 *  -- LAPACK computational routine (version 3.4.0) --
00167 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00168 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00169 *     November 2011
00170 *
00171 *     .. Scalar Arguments ..
00172       INTEGER            CUTPNT, INFO, LDQ, N
00173       DOUBLE PRECISION   RHO
00174 *     ..
00175 *     .. Array Arguments ..
00176       INTEGER            INDXQ( * ), IWORK( * )
00177       DOUBLE PRECISION   D( * ), Q( LDQ, * ), WORK( * )
00178 *     ..
00179 *
00180 *  =====================================================================
00181 *
00182 *     .. Local Scalars ..
00183       INTEGER            COLTYP, I, IDLMDA, INDX, INDXC, INDXP, IQ2, IS,
00184      $                   IW, IZ, K, N1, N2, ZPP1
00185 *     ..
00186 *     .. External Subroutines ..
00187       EXTERNAL           DCOPY, DLAED2, DLAED3, DLAMRG, XERBLA
00188 *     ..
00189 *     .. Intrinsic Functions ..
00190       INTRINSIC          MAX, MIN
00191 *     ..
00192 *     .. Executable Statements ..
00193 *
00194 *     Test the input parameters.
00195 *
00196       INFO = 0
00197 *
00198       IF( N.LT.0 ) THEN
00199          INFO = -1
00200       ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
00201          INFO = -4
00202       ELSE IF( MIN( 1, N / 2 ).GT.CUTPNT .OR. ( N / 2 ).LT.CUTPNT ) THEN
00203          INFO = -7
00204       END IF
00205       IF( INFO.NE.0 ) THEN
00206          CALL XERBLA( 'DLAED1', -INFO )
00207          RETURN
00208       END IF
00209 *
00210 *     Quick return if possible
00211 *
00212       IF( N.EQ.0 )
00213      $   RETURN
00214 *
00215 *     The following values are integer pointers which indicate
00216 *     the portion of the workspace
00217 *     used by a particular array in DLAED2 and DLAED3.
00218 *
00219       IZ = 1
00220       IDLMDA = IZ + N
00221       IW = IDLMDA + N
00222       IQ2 = IW + N
00223 *
00224       INDX = 1
00225       INDXC = INDX + N
00226       COLTYP = INDXC + N
00227       INDXP = COLTYP + N
00228 *
00229 *
00230 *     Form the z-vector which consists of the last row of Q_1 and the
00231 *     first row of Q_2.
00232 *
00233       CALL DCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 )
00234       ZPP1 = CUTPNT + 1
00235       CALL DCOPY( N-CUTPNT, Q( ZPP1, ZPP1 ), LDQ, WORK( IZ+CUTPNT ), 1 )
00236 *
00237 *     Deflate eigenvalues.
00238 *
00239       CALL DLAED2( K, N, CUTPNT, D, Q, LDQ, INDXQ, RHO, WORK( IZ ),
00240      $             WORK( IDLMDA ), WORK( IW ), WORK( IQ2 ),
00241      $             IWORK( INDX ), IWORK( INDXC ), IWORK( INDXP ),
00242      $             IWORK( COLTYP ), INFO )
00243 *
00244       IF( INFO.NE.0 )
00245      $   GO TO 20
00246 *
00247 *     Solve Secular Equation.
00248 *
00249       IF( K.NE.0 ) THEN
00250          IS = ( IWORK( COLTYP )+IWORK( COLTYP+1 ) )*CUTPNT +
00251      $        ( IWORK( COLTYP+1 )+IWORK( COLTYP+2 ) )*( N-CUTPNT ) + IQ2
00252          CALL DLAED3( K, N, CUTPNT, D, Q, LDQ, RHO, WORK( IDLMDA ),
00253      $                WORK( IQ2 ), IWORK( INDXC ), IWORK( COLTYP ),
00254      $                WORK( IW ), WORK( IS ), INFO )
00255          IF( INFO.NE.0 )
00256      $      GO TO 20
00257 *
00258 *     Prepare the INDXQ sorting permutation.
00259 *
00260          N1 = K
00261          N2 = N - K
00262          CALL DLAMRG( N1, N2, D, 1, -1, INDXQ )
00263       ELSE
00264          DO 10 I = 1, N
00265             INDXQ( I ) = I
00266    10    CONTINUE
00267       END IF
00268 *
00269    20 CONTINUE
00270       RETURN
00271 *
00272 *     End of DLAED1
00273 *
00274       END
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