LAPACK  3.5.0
LAPACK: Linear Algebra PACKage
slaed2.f File Reference

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Functions/Subroutines

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 matrix is tridiagonal. More...
 

Function/Subroutine Documentation

subroutine slaed2 ( integer  K,
integer  N,
integer  N1,
real, dimension( * )  D,
real, dimension( ldq, * )  Q,
integer  LDQ,
integer, dimension( * )  INDXQ,
real  RHO,
real, dimension( * )  Z,
real, dimension( * )  DLAMDA,
real, dimension( * )  W,
real, dimension( * )  Q2,
integer, dimension( * )  INDX,
integer, dimension( * )  INDXC,
integer, dimension( * )  INDXP,
integer, dimension( * )  COLTYP,
integer  INFO 
)

SLAED2 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.

Download SLAED2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 SLAED2 merges the two sets of eigenvalues together into a single
 sorted set.  Then it tries to deflate the size of the problem.
 There are two ways in which deflation can occur:  when two or more
 eigenvalues are close together or if there is a tiny entry in the
 Z vector.  For each such occurrence the order of the related secular
 equation problem is reduced by one.
Parameters
[out]K
          K is INTEGER
         The number of non-deflated eigenvalues, and the order of the
         related secular equation. 0 <= K <=N.
[in]N
          N is INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.
[in]N1
          N1 is INTEGER
         The location of the last eigenvalue in the leading sub-matrix.
         min(1,N) <= N1 <= N/2.
[in,out]D
          D is REAL array, dimension (N)
         On entry, D contains the eigenvalues of the two submatrices to
         be combined.
         On exit, D contains the trailing (N-K) updated eigenvalues
         (those which were deflated) sorted into increasing order.
[in,out]Q
          Q is REAL array, dimension (LDQ, N)
         On entry, Q contains the eigenvectors of two submatrices in
         the two square blocks with corners at (1,1), (N1,N1)
         and (N1+1, N1+1), (N,N).
         On exit, Q contains the trailing (N-K) updated eigenvectors
         (those which were deflated) in its last N-K columns.
[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)
         The permutation which separately sorts the two sub-problems
         in D into ascending order.  Note that elements in the second
         half of this permutation must first have N1 added to their
         values. Destroyed on exit.
[in,out]RHO
          RHO is REAL
         On entry, the off-diagonal element associated with the rank-1
         cut which originally split the two submatrices which are now
         being recombined.
         On exit, RHO has been modified to the value required by
         SLAED3.
[in]Z
          Z is REAL array, dimension (N)
         On entry, Z contains the updating vector (the last
         row of the first sub-eigenvector matrix and the first row of
         the second sub-eigenvector matrix).
         On exit, the contents of Z have been destroyed by the updating
         process.
[out]DLAMDA
          DLAMDA is REAL array, dimension (N)
         A copy of the first K eigenvalues which will be used by
         SLAED3 to form the secular equation.
[out]W
          W is REAL array, dimension (N)
         The first k values of the final deflation-altered z-vector
         which will be passed to SLAED3.
[out]Q2
          Q2 is REAL array, dimension (N1**2+(N-N1)**2)
         A copy of the first K eigenvectors which will be used by
         SLAED3 in a matrix multiply (SGEMM) to solve for the new
         eigenvectors.
[out]INDX
          INDX is INTEGER array, dimension (N)
         The permutation used to sort the contents of DLAMDA into
         ascending order.
[out]INDXC
          INDXC is INTEGER array, dimension (N)
         The permutation used to arrange the columns of the deflated
         Q matrix into three groups:  the first group contains non-zero
         elements only at and above N1, the second contains
         non-zero elements only below N1, and the third is dense.
[out]INDXP
          INDXP is INTEGER array, dimension (N)
         The permutation used to place deflated values of D at the end
         of the array.  INDXP(1:K) points to the nondeflated D-values
         and INDXP(K+1:N) points to the deflated eigenvalues.
[out]COLTYP
          COLTYP is INTEGER array, dimension (N)
         During execution, a label which will indicate which of the
         following types a column in the Q2 matrix is:
         1 : non-zero in the upper half only;
         2 : dense;
         3 : non-zero in the lower half only;
         4 : deflated.
         On exit, COLTYP(i) is the number of columns of type i,
         for i=1 to 4 only.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
September 2012
Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee

Definition at line 214 of file slaed2.f.

214 *
215 * -- LAPACK computational routine (version 3.4.2) --
216 * -- LAPACK is a software package provided by Univ. of Tennessee, --
217 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
218 * September 2012
219 *
220 * .. Scalar Arguments ..
221  INTEGER info, k, ldq, n, n1
222  REAL rho
223 * ..
224 * .. Array Arguments ..
225  INTEGER coltyp( * ), indx( * ), indxc( * ), indxp( * ),
226  $ indxq( * )
227  REAL d( * ), dlamda( * ), q( ldq, * ), q2( * ),
228  $ w( * ), z( * )
229 * ..
230 *
231 * =====================================================================
232 *
233 * .. Parameters ..
234  REAL mone, zero, one, two, eight
235  parameter( mone = -1.0e0, zero = 0.0e0, one = 1.0e0,
236  $ two = 2.0e0, eight = 8.0e0 )
237 * ..
238 * .. Local Arrays ..
239  INTEGER ctot( 4 ), psm( 4 )
240 * ..
241 * .. Local Scalars ..
242  INTEGER ct, i, imax, iq1, iq2, j, jmax, js, k2, n1p1,
243  $ n2, nj, pj
244  REAL c, eps, s, t, tau, tol
245 * ..
246 * .. External Functions ..
247  INTEGER isamax
248  REAL slamch, slapy2
249  EXTERNAL isamax, slamch, slapy2
250 * ..
251 * .. External Subroutines ..
252  EXTERNAL scopy, slacpy, slamrg, srot, sscal, xerbla
253 * ..
254 * .. Intrinsic Functions ..
255  INTRINSIC abs, max, min, sqrt
256 * ..
257 * .. Executable Statements ..
258 *
259 * Test the input parameters.
260 *
261  info = 0
262 *
263  IF( n.LT.0 ) THEN
264  info = -2
265  ELSE IF( ldq.LT.max( 1, n ) ) THEN
266  info = -6
267  ELSE IF( min( 1, ( n / 2 ) ).GT.n1 .OR. ( n / 2 ).LT.n1 ) THEN
268  info = -3
269  END IF
270  IF( info.NE.0 ) THEN
271  CALL xerbla( 'SLAED2', -info )
272  RETURN
273  END IF
274 *
275 * Quick return if possible
276 *
277  IF( n.EQ.0 )
278  $ RETURN
279 *
280  n2 = n - n1
281  n1p1 = n1 + 1
282 *
283  IF( rho.LT.zero ) THEN
284  CALL sscal( n2, mone, z( n1p1 ), 1 )
285  END IF
286 *
287 * Normalize z so that norm(z) = 1. Since z is the concatenation of
288 * two normalized vectors, norm2(z) = sqrt(2).
289 *
290  t = one / sqrt( two )
291  CALL sscal( n, t, z, 1 )
292 *
293 * RHO = ABS( norm(z)**2 * RHO )
294 *
295  rho = abs( two*rho )
296 *
297 * Sort the eigenvalues into increasing order
298 *
299  DO 10 i = n1p1, n
300  indxq( i ) = indxq( i ) + n1
301  10 CONTINUE
302 *
303 * re-integrate the deflated parts from the last pass
304 *
305  DO 20 i = 1, n
306  dlamda( i ) = d( indxq( i ) )
307  20 CONTINUE
308  CALL slamrg( n1, n2, dlamda, 1, 1, indxc )
309  DO 30 i = 1, n
310  indx( i ) = indxq( indxc( i ) )
311  30 CONTINUE
312 *
313 * Calculate the allowable deflation tolerance
314 *
315  imax = isamax( n, z, 1 )
316  jmax = isamax( n, d, 1 )
317  eps = slamch( 'Epsilon' )
318  tol = eight*eps*max( abs( d( jmax ) ), abs( z( imax ) ) )
319 *
320 * If the rank-1 modifier is small enough, no more needs to be done
321 * except to reorganize Q so that its columns correspond with the
322 * elements in D.
323 *
324  IF( rho*abs( z( imax ) ).LE.tol ) THEN
325  k = 0
326  iq2 = 1
327  DO 40 j = 1, n
328  i = indx( j )
329  CALL scopy( n, q( 1, i ), 1, q2( iq2 ), 1 )
330  dlamda( j ) = d( i )
331  iq2 = iq2 + n
332  40 CONTINUE
333  CALL slacpy( 'A', n, n, q2, n, q, ldq )
334  CALL scopy( n, dlamda, 1, d, 1 )
335  GO TO 190
336  END IF
337 *
338 * If there are multiple eigenvalues then the problem deflates. Here
339 * the number of equal eigenvalues are found. As each equal
340 * eigenvalue is found, an elementary reflector is computed to rotate
341 * the corresponding eigensubspace so that the corresponding
342 * components of Z are zero in this new basis.
343 *
344  DO 50 i = 1, n1
345  coltyp( i ) = 1
346  50 CONTINUE
347  DO 60 i = n1p1, n
348  coltyp( i ) = 3
349  60 CONTINUE
350 *
351 *
352  k = 0
353  k2 = n + 1
354  DO 70 j = 1, n
355  nj = indx( j )
356  IF( rho*abs( z( nj ) ).LE.tol ) THEN
357 *
358 * Deflate due to small z component.
359 *
360  k2 = k2 - 1
361  coltyp( nj ) = 4
362  indxp( k2 ) = nj
363  IF( j.EQ.n )
364  $ GO TO 100
365  ELSE
366  pj = nj
367  GO TO 80
368  END IF
369  70 CONTINUE
370  80 CONTINUE
371  j = j + 1
372  nj = indx( j )
373  IF( j.GT.n )
374  $ GO TO 100
375  IF( rho*abs( z( nj ) ).LE.tol ) THEN
376 *
377 * Deflate due to small z component.
378 *
379  k2 = k2 - 1
380  coltyp( nj ) = 4
381  indxp( k2 ) = nj
382  ELSE
383 *
384 * Check if eigenvalues are close enough to allow deflation.
385 *
386  s = z( pj )
387  c = z( nj )
388 *
389 * Find sqrt(a**2+b**2) without overflow or
390 * destructive underflow.
391 *
392  tau = slapy2( c, s )
393  t = d( nj ) - d( pj )
394  c = c / tau
395  s = -s / tau
396  IF( abs( t*c*s ).LE.tol ) THEN
397 *
398 * Deflation is possible.
399 *
400  z( nj ) = tau
401  z( pj ) = zero
402  IF( coltyp( nj ).NE.coltyp( pj ) )
403  $ coltyp( nj ) = 2
404  coltyp( pj ) = 4
405  CALL srot( n, q( 1, pj ), 1, q( 1, nj ), 1, c, s )
406  t = d( pj )*c**2 + d( nj )*s**2
407  d( nj ) = d( pj )*s**2 + d( nj )*c**2
408  d( pj ) = t
409  k2 = k2 - 1
410  i = 1
411  90 CONTINUE
412  IF( k2+i.LE.n ) THEN
413  IF( d( pj ).LT.d( indxp( k2+i ) ) ) THEN
414  indxp( k2+i-1 ) = indxp( k2+i )
415  indxp( k2+i ) = pj
416  i = i + 1
417  GO TO 90
418  ELSE
419  indxp( k2+i-1 ) = pj
420  END IF
421  ELSE
422  indxp( k2+i-1 ) = pj
423  END IF
424  pj = nj
425  ELSE
426  k = k + 1
427  dlamda( k ) = d( pj )
428  w( k ) = z( pj )
429  indxp( k ) = pj
430  pj = nj
431  END IF
432  END IF
433  GO TO 80
434  100 CONTINUE
435 *
436 * Record the last eigenvalue.
437 *
438  k = k + 1
439  dlamda( k ) = d( pj )
440  w( k ) = z( pj )
441  indxp( k ) = pj
442 *
443 * Count up the total number of the various types of columns, then
444 * form a permutation which positions the four column types into
445 * four uniform groups (although one or more of these groups may be
446 * empty).
447 *
448  DO 110 j = 1, 4
449  ctot( j ) = 0
450  110 CONTINUE
451  DO 120 j = 1, n
452  ct = coltyp( j )
453  ctot( ct ) = ctot( ct ) + 1
454  120 CONTINUE
455 *
456 * PSM(*) = Position in SubMatrix (of types 1 through 4)
457 *
458  psm( 1 ) = 1
459  psm( 2 ) = 1 + ctot( 1 )
460  psm( 3 ) = psm( 2 ) + ctot( 2 )
461  psm( 4 ) = psm( 3 ) + ctot( 3 )
462  k = n - ctot( 4 )
463 *
464 * Fill out the INDXC array so that the permutation which it induces
465 * will place all type-1 columns first, all type-2 columns next,
466 * then all type-3's, and finally all type-4's.
467 *
468  DO 130 j = 1, n
469  js = indxp( j )
470  ct = coltyp( js )
471  indx( psm( ct ) ) = js
472  indxc( psm( ct ) ) = j
473  psm( ct ) = psm( ct ) + 1
474  130 CONTINUE
475 *
476 * Sort the eigenvalues and corresponding eigenvectors into DLAMDA
477 * and Q2 respectively. The eigenvalues/vectors which were not
478 * deflated go into the first K slots of DLAMDA and Q2 respectively,
479 * while those which were deflated go into the last N - K slots.
480 *
481  i = 1
482  iq1 = 1
483  iq2 = 1 + ( ctot( 1 )+ctot( 2 ) )*n1
484  DO 140 j = 1, ctot( 1 )
485  js = indx( i )
486  CALL scopy( n1, q( 1, js ), 1, q2( iq1 ), 1 )
487  z( i ) = d( js )
488  i = i + 1
489  iq1 = iq1 + n1
490  140 CONTINUE
491 *
492  DO 150 j = 1, ctot( 2 )
493  js = indx( i )
494  CALL scopy( n1, q( 1, js ), 1, q2( iq1 ), 1 )
495  CALL scopy( n2, q( n1+1, js ), 1, q2( iq2 ), 1 )
496  z( i ) = d( js )
497  i = i + 1
498  iq1 = iq1 + n1
499  iq2 = iq2 + n2
500  150 CONTINUE
501 *
502  DO 160 j = 1, ctot( 3 )
503  js = indx( i )
504  CALL scopy( n2, q( n1+1, js ), 1, q2( iq2 ), 1 )
505  z( i ) = d( js )
506  i = i + 1
507  iq2 = iq2 + n2
508  160 CONTINUE
509 *
510  iq1 = iq2
511  DO 170 j = 1, ctot( 4 )
512  js = indx( i )
513  CALL scopy( n, q( 1, js ), 1, q2( iq2 ), 1 )
514  iq2 = iq2 + n
515  z( i ) = d( js )
516  i = i + 1
517  170 CONTINUE
518 *
519 * The deflated eigenvalues and their corresponding vectors go back
520 * into the last N - K slots of D and Q respectively.
521 *
522  IF( k.LT.n ) THEN
523  CALL slacpy( 'A', n, ctot( 4 ), q2( iq1 ), n,
524  $ q( 1, k+1 ), ldq )
525  CALL scopy( n-k, z( k+1 ), 1, d( k+1 ), 1 )
526  END IF
527 *
528 * Copy CTOT into COLTYP for referencing in SLAED3.
529 *
530  DO 180 j = 1, 4
531  coltyp( j ) = ctot( j )
532  180 CONTINUE
533 *
534  190 CONTINUE
535  RETURN
536 *
537 * End of SLAED2
538 *
integer function isamax(N, SX, INCX)
ISAMAX
Definition: isamax.f:53
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:101
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
real function slapy2(X, Y)
SLAPY2 returns sqrt(x2+y2).
Definition: slapy2.f:65
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:55
subroutine srot(N, SX, INCX, SY, INCY, C, S)
SROT
Definition: srot.f:53

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