 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ slaed1()

 subroutine slaed1 ( integer N, real, dimension( * ) D, real, dimension( ldq, * ) Q, integer LDQ, integer, dimension( * ) INDXQ, real RHO, integer CUTPNT, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

SLAED1 used by SSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.

Purpose:
``` SLAED1 computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix.  This
routine is used only for the eigenproblem which requires all
eigenvalues and eigenvectors of a tridiagonal matrix.  SLAED7 handles
the case in which eigenvalues only or eigenvalues and eigenvectors
of a full symmetric matrix (which was reduced to tridiagonal form)
are desired.

T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)

where Z = Q**T*u, u is a vector of length N with ones in the
CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

The eigenvectors of the original matrix are stored in Q, and the
eigenvalues are in D.  The algorithm consists of three stages:

The first stage consists of deflating the size of the problem
when there are multiple eigenvalues or if there is a zero in
the Z vector.  For each such occurrence the dimension of the
secular equation problem is reduced by one.  This stage is
performed by the routine SLAED2.

The second stage consists of calculating the updated
eigenvalues. This is done by finding the roots of the secular
equation via the routine SLAED4 (as called by SLAED3).
This routine also calculates the eigenvectors of the current
problem.

The final stage consists of computing the updated eigenvectors
directly using the updated eigenvalues.  The eigenvectors for
the current problem are multiplied with the eigenvectors from
the overall problem.```
Parameters
 [in] N ``` N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.``` [in,out] D ``` D is REAL array, dimension (N) On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.``` [in,out] Q ``` Q is REAL array, dimension (LDQ,N) On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.``` [in] LDQ ``` LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).``` [in,out] INDXQ ``` INDXQ is INTEGER array, dimension (N) On entry, the permutation which separately sorts the two subproblems in D into ascending order. On exit, the permutation which will reintegrate the subproblems back into sorted order, i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.``` [in] RHO ``` RHO is REAL The subdiagonal entry used to create the rank-1 modification.``` [in] CUTPNT ``` CUTPNT is INTEGER The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N/2.``` [out] WORK ` WORK is REAL array, dimension (4*N + N**2)` [out] IWORK ` IWORK is INTEGER array, dimension (4*N)` [out] INFO ``` INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge```
Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee

Definition at line 161 of file slaed1.f.

163 *
164 * -- LAPACK computational routine --
165 * -- LAPACK is a software package provided by Univ. of Tennessee, --
166 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
167 *
168 * .. Scalar Arguments ..
169  INTEGER CUTPNT, INFO, LDQ, N
170  REAL RHO
171 * ..
172 * .. Array Arguments ..
173  INTEGER INDXQ( * ), IWORK( * )
174  REAL D( * ), Q( LDQ, * ), WORK( * )
175 * ..
176 *
177 * =====================================================================
178 *
179 * .. Local Scalars ..
180  INTEGER COLTYP, CPP1, I, IDLMDA, INDX, INDXC, INDXP,
181  \$ IQ2, IS, IW, IZ, K, N1, N2
182 * ..
183 * .. External Subroutines ..
184  EXTERNAL scopy, slaed2, slaed3, slamrg, xerbla
185 * ..
186 * .. Intrinsic Functions ..
187  INTRINSIC max, min
188 * ..
189 * .. Executable Statements ..
190 *
191 * Test the input parameters.
192 *
193  info = 0
194 *
195  IF( n.LT.0 ) THEN
196  info = -1
197  ELSE IF( ldq.LT.max( 1, n ) ) THEN
198  info = -4
199  ELSE IF( min( 1, n / 2 ).GT.cutpnt .OR. ( n / 2 ).LT.cutpnt ) THEN
200  info = -7
201  END IF
202  IF( info.NE.0 ) THEN
203  CALL xerbla( 'SLAED1', -info )
204  RETURN
205  END IF
206 *
207 * Quick return if possible
208 *
209  IF( n.EQ.0 )
210  \$ RETURN
211 *
212 * The following values are integer pointers which indicate
213 * the portion of the workspace
214 * used by a particular array in SLAED2 and SLAED3.
215 *
216  iz = 1
217  idlmda = iz + n
218  iw = idlmda + n
219  iq2 = iw + n
220 *
221  indx = 1
222  indxc = indx + n
223  coltyp = indxc + n
224  indxp = coltyp + n
225 *
226 *
227 * Form the z-vector which consists of the last row of Q_1 and the
228 * first row of Q_2.
229 *
230  CALL scopy( cutpnt, q( cutpnt, 1 ), ldq, work( iz ), 1 )
231  cpp1 = cutpnt + 1
232  CALL scopy( n-cutpnt, q( cpp1, cpp1 ), ldq, work( iz+cutpnt ), 1 )
233 *
234 * Deflate eigenvalues.
235 *
236  CALL slaed2( k, n, cutpnt, d, q, ldq, indxq, rho, work( iz ),
237  \$ work( idlmda ), work( iw ), work( iq2 ),
238  \$ iwork( indx ), iwork( indxc ), iwork( indxp ),
239  \$ iwork( coltyp ), info )
240 *
241  IF( info.NE.0 )
242  \$ GO TO 20
243 *
244 * Solve Secular Equation.
245 *
246  IF( k.NE.0 ) THEN
247  is = ( iwork( coltyp )+iwork( coltyp+1 ) )*cutpnt +
248  \$ ( iwork( coltyp+1 )+iwork( coltyp+2 ) )*( n-cutpnt ) + iq2
249  CALL slaed3( k, n, cutpnt, d, q, ldq, rho, work( idlmda ),
250  \$ work( iq2 ), iwork( indxc ), iwork( coltyp ),
251  \$ work( iw ), work( is ), info )
252  IF( info.NE.0 )
253  \$ GO TO 20
254 *
255 * Prepare the INDXQ sorting permutation.
256 *
257  n1 = k
258  n2 = n - k
259  CALL slamrg( n1, n2, d, 1, -1, indxq )
260  ELSE
261  DO 10 i = 1, n
262  indxq( i ) = i
263  10 CONTINUE
264  END IF
265 *
266  20 CONTINUE
267  RETURN
268 *
269 * End of SLAED1
270 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slamrg(N1, N2, A, STRD1, STRD2, INDEX)
SLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single...
Definition: slamrg.f:99
subroutine slaed3(K, N, N1, D, Q, LDQ, RHO, DLAMDA, Q2, INDX, CTOT, W, S, INFO)
SLAED3 used by SSTEDC. Finds the roots of the secular equation and updates the eigenvectors....
Definition: slaed3.f:185
subroutine slaed2(K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W, Q2, INDX, INDXC, INDXP, COLTYP, INFO)
SLAED2 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matri...
Definition: slaed2.f:212
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
Here is the call graph for this function:
Here is the caller graph for this function: