LAPACK: SRC/dlaed2.f Source File

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00001       SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
00002      $                   Q2, INDX, INDXC, INDXP, COLTYP, 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            INFO, K, LDQ, N, N1
00011       DOUBLE PRECISION   RHO
00012 *     ..
00013 *     .. Array Arguments ..
00014       INTEGER            COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
00015      $                   INDXQ( * )
00016       DOUBLE PRECISION   D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
00017      $                   W( * ), Z( * )
00018 *     ..
00019 *
00020 *  Purpose
00021 *  =======
00022 *
00023 *  DLAED2 merges the two sets of eigenvalues together into a single
00024 *  sorted set.  Then it tries to deflate the size of the problem.
00025 *  There are two ways in which deflation can occur:  when two or more
00026 *  eigenvalues are close together or if there is a tiny entry in the
00027 *  Z vector.  For each such occurrence the order of the related secular
00028 *  equation problem is reduced by one.
00029 *
00030 *  Arguments
00031 *  =========
00032 *
00033 *  K      (output) INTEGER
00034 *         The number of non-deflated eigenvalues, and the order of the
00035 *         related secular equation. 0 <= K <=N.
00036 *
00037 *  N      (input) INTEGER
00038 *         The dimension of the symmetric tridiagonal matrix.  N >= 0.
00039 *
00040 *  N1     (input) INTEGER
00041 *         The location of the last eigenvalue in the leading sub-matrix.
00042 *         min(1,N) <= N1 <= N/2.
00043 *
00044 *  D      (input/output) DOUBLE PRECISION array, dimension (N)
00045 *         On entry, D contains the eigenvalues of the two submatrices to
00046 *         be combined.
00047 *         On exit, D contains the trailing (N-K) updated eigenvalues
00048 *         (those which were deflated) sorted into increasing order.
00049 *
00050 *  Q      (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
00051 *         On entry, Q contains the eigenvectors of two submatrices in
00052 *         the two square blocks with corners at (1,1), (N1,N1)
00053 *         and (N1+1, N1+1), (N,N).
00054 *         On exit, Q contains the trailing (N-K) updated eigenvectors
00055 *         (those which were deflated) in its last N-K columns.
00056 *
00057 *  LDQ    (input) INTEGER
00058 *         The leading dimension of the array Q.  LDQ >= max(1,N).
00059 *
00060 *  INDXQ  (input/output) INTEGER array, dimension (N)
00061 *         The permutation which separately sorts the two sub-problems
00062 *         in D into ascending order.  Note that elements in the second
00063 *         half of this permutation must first have N1 added to their
00064 *         values. Destroyed on exit.
00065 *
00066 *  RHO    (input/output) DOUBLE PRECISION
00067 *         On entry, the off-diagonal element associated with the rank-1
00068 *         cut which originally split the two submatrices which are now
00069 *         being recombined.
00070 *         On exit, RHO has been modified to the value required by
00071 *         DLAED3.
00072 *
00073 *  Z      (input) DOUBLE PRECISION array, dimension (N)
00074 *         On entry, Z contains the updating vector (the last
00075 *         row of the first sub-eigenvector matrix and the first row of
00076 *         the second sub-eigenvector matrix).
00077 *         On exit, the contents of Z have been destroyed by the updating
00078 *         process.
00079 *
00080 *  DLAMDA (output) DOUBLE PRECISION array, dimension (N)
00081 *         A copy of the first K eigenvalues which will be used by
00082 *         DLAED3 to form the secular equation.
00083 *
00084 *  W      (output) DOUBLE PRECISION array, dimension (N)
00085 *         The first k values of the final deflation-altered z-vector
00086 *         which will be passed to DLAED3.
00087 *
00088 *  Q2     (output) DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
00089 *         A copy of the first K eigenvectors which will be used by
00090 *         DLAED3 in a matrix multiply (DGEMM) to solve for the new
00091 *         eigenvectors.
00092 *
00093 *  INDX   (workspace) INTEGER array, dimension (N)
00094 *         The permutation used to sort the contents of DLAMDA into
00095 *         ascending order.
00096 *
00097 *  INDXC  (output) INTEGER array, dimension (N)
00098 *         The permutation used to arrange the columns of the deflated
00099 *         Q matrix into three groups:  the first group contains non-zero
00100 *         elements only at and above N1, the second contains
00101 *         non-zero elements only below N1, and the third is dense.
00102 *
00103 *  INDXP  (workspace) INTEGER array, dimension (N)
00104 *         The permutation used to place deflated values of D at the end
00105 *         of the array.  INDXP(1:K) points to the nondeflated D-values
00106 *         and INDXP(K+1:N) points to the deflated eigenvalues.
00107 *
00108 *  COLTYP (workspace/output) INTEGER array, dimension (N)
00109 *         During execution, a label which will indicate which of the
00110 *         following types a column in the Q2 matrix is:
00111 *         1 : non-zero in the upper half only;
00112 *         2 : dense;
00113 *         3 : non-zero in the lower half only;
00114 *         4 : deflated.
00115 *         On exit, COLTYP(i) is the number of columns of type i,
00116 *         for i=1 to 4 only.
00117 *
00118 *  INFO   (output) INTEGER
00119 *          = 0:  successful exit.
00120 *          < 0:  if INFO = -i, the i-th argument had an illegal value.
00121 *
00122 *  Further Details
00123 *  ===============
00124 *
00125 *  Based on contributions by
00126 *     Jeff Rutter, Computer Science Division, University of California
00127 *     at Berkeley, USA
00128 *  Modified by Francoise Tisseur, University of Tennessee.
00129 *
00130 *  =====================================================================
00131 *
00132 *     .. Parameters ..
00133       DOUBLE PRECISION   MONE, ZERO, ONE, TWO, EIGHT
00134       PARAMETER          ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
00135      $                   TWO = 2.0D0, EIGHT = 8.0D0 )
00136 *     ..
00137 *     .. Local Arrays ..
00138       INTEGER            CTOT( 4 ), PSM( 4 )
00139 *     ..
00140 *     .. Local Scalars ..
00141       INTEGER            CT, I, IMAX, IQ1, IQ2, J, JMAX, JS, K2, N1P1,
00142      $                   N2, NJ, PJ
00143       DOUBLE PRECISION   C, EPS, S, T, TAU, TOL
00144 *     ..
00145 *     .. External Functions ..
00146       INTEGER            IDAMAX
00147       DOUBLE PRECISION   DLAMCH, DLAPY2
00148       EXTERNAL           IDAMAX, DLAMCH, DLAPY2
00149 *     ..
00150 *     .. External Subroutines ..
00151       EXTERNAL           DCOPY, DLACPY, DLAMRG, DROT, DSCAL, XERBLA
00152 *     ..
00153 *     .. Intrinsic Functions ..
00154       INTRINSIC          ABS, MAX, MIN, SQRT
00155 *     ..
00156 *     .. Executable Statements ..
00157 *
00158 *     Test the input parameters.
00159 *
00160       INFO = 0
00161 *
00162       IF( N.LT.0 ) THEN
00163          INFO = -2
00164       ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
00165          INFO = -6
00166       ELSE IF( MIN( 1, ( N / 2 ) ).GT.N1 .OR. ( N / 2 ).LT.N1 ) THEN
00167          INFO = -3
00168       END IF
00169       IF( INFO.NE.0 ) THEN
00170          CALL XERBLA( 'DLAED2', -INFO )
00171          RETURN
00172       END IF
00173 *
00174 *     Quick return if possible
00175 *
00176       IF( N.EQ.0 )
00177      $   RETURN
00178 *
00179       N2 = N - N1
00180       N1P1 = N1 + 1
00181 *
00182       IF( RHO.LT.ZERO ) THEN
00183          CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
00184       END IF
00185 *
00186 *     Normalize z so that norm(z) = 1.  Since z is the concatenation of
00187 *     two normalized vectors, norm2(z) = sqrt(2).
00188 *
00189       T = ONE / SQRT( TWO )
00190       CALL DSCAL( N, T, Z, 1 )
00191 *
00192 *     RHO = ABS( norm(z)**2 * RHO )
00193 *
00194       RHO = ABS( TWO*RHO )
00195 *
00196 *     Sort the eigenvalues into increasing order
00197 *
00198       DO 10 I = N1P1, N
00199          INDXQ( I ) = INDXQ( I ) + N1
00200    10 CONTINUE
00201 *
00202 *     re-integrate the deflated parts from the last pass
00203 *
00204       DO 20 I = 1, N
00205          DLAMDA( I ) = D( INDXQ( I ) )
00206    20 CONTINUE
00207       CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDXC )
00208       DO 30 I = 1, N
00209          INDX( I ) = INDXQ( INDXC( I ) )
00210    30 CONTINUE
00211 *
00212 *     Calculate the allowable deflation tolerance
00213 *
00214       IMAX = IDAMAX( N, Z, 1 )
00215       JMAX = IDAMAX( N, D, 1 )
00216       EPS = DLAMCH( 'Epsilon' )
00217       TOL = EIGHT*EPS*MAX( ABS( D( JMAX ) ), ABS( Z( IMAX ) ) )
00218 *
00219 *     If the rank-1 modifier is small enough, no more needs to be done
00220 *     except to reorganize Q so that its columns correspond with the
00221 *     elements in D.
00222 *
00223       IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
00224          K = 0
00225          IQ2 = 1
00226          DO 40 J = 1, N
00227             I = INDX( J )
00228             CALL DCOPY( N, Q( 1, I ), 1, Q2( IQ2 ), 1 )
00229             DLAMDA( J ) = D( I )
00230             IQ2 = IQ2 + N
00231    40    CONTINUE
00232          CALL DLACPY( 'A', N, N, Q2, N, Q, LDQ )
00233          CALL DCOPY( N, DLAMDA, 1, D, 1 )
00234          GO TO 190
00235       END IF
00236 *
00237 *     If there are multiple eigenvalues then the problem deflates.  Here
00238 *     the number of equal eigenvalues are found.  As each equal
00239 *     eigenvalue is found, an elementary reflector is computed to rotate
00240 *     the corresponding eigensubspace so that the corresponding
00241 *     components of Z are zero in this new basis.
00242 *
00243       DO 50 I = 1, N1
00244          COLTYP( I ) = 1
00245    50 CONTINUE
00246       DO 60 I = N1P1, N
00247          COLTYP( I ) = 3
00248    60 CONTINUE
00249 *
00250 *
00251       K = 0
00252       K2 = N + 1
00253       DO 70 J = 1, N
00254          NJ = INDX( J )
00255          IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
00256 *
00257 *           Deflate due to small z component.
00258 *
00259             K2 = K2 - 1
00260             COLTYP( NJ ) = 4
00261             INDXP( K2 ) = NJ
00262             IF( J.EQ.N )
00263      $         GO TO 100
00264          ELSE
00265             PJ = NJ
00266             GO TO 80
00267          END IF
00268    70 CONTINUE
00269    80 CONTINUE
00270       J = J + 1
00271       NJ = INDX( J )
00272       IF( J.GT.N )
00273      $   GO TO 100
00274       IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
00275 *
00276 *        Deflate due to small z component.
00277 *
00278          K2 = K2 - 1
00279          COLTYP( NJ ) = 4
00280          INDXP( K2 ) = NJ
00281       ELSE
00282 *
00283 *        Check if eigenvalues are close enough to allow deflation.
00284 *
00285          S = Z( PJ )
00286          C = Z( NJ )
00287 *
00288 *        Find sqrt(a**2+b**2) without overflow or
00289 *        destructive underflow.
00290 *
00291          TAU = DLAPY2( C, S )
00292          T = D( NJ ) - D( PJ )
00293          C = C / TAU
00294          S = -S / TAU
00295          IF( ABS( T*C*S ).LE.TOL ) THEN
00296 *
00297 *           Deflation is possible.
00298 *
00299             Z( NJ ) = TAU
00300             Z( PJ ) = ZERO
00301             IF( COLTYP( NJ ).NE.COLTYP( PJ ) )
00302      $         COLTYP( NJ ) = 2
00303             COLTYP( PJ ) = 4
00304             CALL DROT( N, Q( 1, PJ ), 1, Q( 1, NJ ), 1, C, S )
00305             T = D( PJ )*C**2 + D( NJ )*S**2
00306             D( NJ ) = D( PJ )*S**2 + D( NJ )*C**2
00307             D( PJ ) = T
00308             K2 = K2 - 1
00309             I = 1
00310    90       CONTINUE
00311             IF( K2+I.LE.N ) THEN
00312                IF( D( PJ ).LT.D( INDXP( K2+I ) ) ) THEN
00313                   INDXP( K2+I-1 ) = INDXP( K2+I )
00314                   INDXP( K2+I ) = PJ
00315                   I = I + 1
00316                   GO TO 90
00317                ELSE
00318                   INDXP( K2+I-1 ) = PJ
00319                END IF
00320             ELSE
00321                INDXP( K2+I-1 ) = PJ
00322             END IF
00323             PJ = NJ
00324          ELSE
00325             K = K + 1
00326             DLAMDA( K ) = D( PJ )
00327             W( K ) = Z( PJ )
00328             INDXP( K ) = PJ
00329             PJ = NJ
00330          END IF
00331       END IF
00332       GO TO 80
00333   100 CONTINUE
00334 *
00335 *     Record the last eigenvalue.
00336 *
00337       K = K + 1
00338       DLAMDA( K ) = D( PJ )
00339       W( K ) = Z( PJ )
00340       INDXP( K ) = PJ
00341 *
00342 *     Count up the total number of the various types of columns, then
00343 *     form a permutation which positions the four column types into
00344 *     four uniform groups (although one or more of these groups may be
00345 *     empty).
00346 *
00347       DO 110 J = 1, 4
00348          CTOT( J ) = 0
00349   110 CONTINUE
00350       DO 120 J = 1, N
00351          CT = COLTYP( J )
00352          CTOT( CT ) = CTOT( CT ) + 1
00353   120 CONTINUE
00354 *
00355 *     PSM(*) = Position in SubMatrix (of types 1 through 4)
00356 *
00357       PSM( 1 ) = 1
00358       PSM( 2 ) = 1 + CTOT( 1 )
00359       PSM( 3 ) = PSM( 2 ) + CTOT( 2 )
00360       PSM( 4 ) = PSM( 3 ) + CTOT( 3 )
00361       K = N - CTOT( 4 )
00362 *
00363 *     Fill out the INDXC array so that the permutation which it induces
00364 *     will place all type-1 columns first, all type-2 columns next,
00365 *     then all type-3's, and finally all type-4's.
00366 *
00367       DO 130 J = 1, N
00368          JS = INDXP( J )
00369          CT = COLTYP( JS )
00370          INDX( PSM( CT ) ) = JS
00371          INDXC( PSM( CT ) ) = J
00372          PSM( CT ) = PSM( CT ) + 1
00373   130 CONTINUE
00374 *
00375 *     Sort the eigenvalues and corresponding eigenvectors into DLAMDA
00376 *     and Q2 respectively.  The eigenvalues/vectors which were not
00377 *     deflated go into the first K slots of DLAMDA and Q2 respectively,
00378 *     while those which were deflated go into the last N - K slots.
00379 *
00380       I = 1
00381       IQ1 = 1
00382       IQ2 = 1 + ( CTOT( 1 )+CTOT( 2 ) )*N1
00383       DO 140 J = 1, CTOT( 1 )
00384          JS = INDX( I )
00385          CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
00386          Z( I ) = D( JS )
00387          I = I + 1
00388          IQ1 = IQ1 + N1
00389   140 CONTINUE
00390 *
00391       DO 150 J = 1, CTOT( 2 )
00392          JS = INDX( I )
00393          CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
00394          CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
00395          Z( I ) = D( JS )
00396          I = I + 1
00397          IQ1 = IQ1 + N1
00398          IQ2 = IQ2 + N2
00399   150 CONTINUE
00400 *
00401       DO 160 J = 1, CTOT( 3 )
00402          JS = INDX( I )
00403          CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
00404          Z( I ) = D( JS )
00405          I = I + 1
00406          IQ2 = IQ2 + N2
00407   160 CONTINUE
00408 *
00409       IQ1 = IQ2
00410       DO 170 J = 1, CTOT( 4 )
00411          JS = INDX( I )
00412          CALL DCOPY( N, Q( 1, JS ), 1, Q2( IQ2 ), 1 )
00413          IQ2 = IQ2 + N
00414          Z( I ) = D( JS )
00415          I = I + 1
00416   170 CONTINUE
00417 *
00418 *     The deflated eigenvalues and their corresponding vectors go back
00419 *     into the last N - K slots of D and Q respectively.
00420 *
00421       CALL DLACPY( 'A', N, CTOT( 4 ), Q2( IQ1 ), N, Q( 1, K+1 ), LDQ )
00422       CALL DCOPY( N-K, Z( K+1 ), 1, D( K+1 ), 1 )
00423 *
00424 *     Copy CTOT into COLTYP for referencing in DLAED3.
00425 *
00426       DO 180 J = 1, 4
00427          COLTYP( J ) = CTOT( J )
00428   180 CONTINUE
00429 *
00430   190 CONTINUE
00431       RETURN
00432 *
00433 *     End of DLAED2
00434 *
00435       END