LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ slaed8()

subroutine slaed8 ( integer  ICOMPQ,
integer  K,
integer  N,
integer  QSIZ,
real, dimension( * )  D,
real, dimension( ldq, * )  Q,
integer  LDQ,
integer, dimension( * )  INDXQ,
real  RHO,
integer  CUTPNT,
real, dimension( * )  Z,
real, dimension( * )  DLAMDA,
real, dimension( ldq2, * )  Q2,
integer  LDQ2,
real, dimension( * )  W,
integer, dimension( * )  PERM,
integer  GIVPTR,
integer, dimension( 2, * )  GIVCOL,
real, dimension( 2, * )  GIVNUM,
integer, dimension( * )  INDXP,
integer, dimension( * )  INDX,
integer  INFO 
)

SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Download SLAED8 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 SLAED8 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 element in the
 Z vector.  For each such occurrence the order of the related secular
 equation problem is reduced by one.
Parameters
[in]ICOMPQ
          ICOMPQ is INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used
                to reduce the original matrix to tridiagonal form.
[out]K
          K is INTEGER
         The number of non-deflated eigenvalues, and the order of the
         related secular equation.
[in]N
          N is INTEGER
         The dimension of the symmetric tridiagonal matrix.  N >= 0.
[in]QSIZ
          QSIZ is INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.
[in,out]D
          D is REAL array, dimension (N)
         On entry, the eigenvalues of the two submatrices to be
         combined.  On exit, the trailing (N-K) updated eigenvalues
         (those which were deflated) sorted into increasing order.
[in,out]Q
          Q is REAL array, dimension (LDQ,N)
         If ICOMPQ = 0, Q is not referenced.  Otherwise,
         on entry, Q contains the eigenvectors of the partially solved
         system which has been previously updated in matrix
         multiplies with other partially solved eigensystems.
         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]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 CUTPNT added to
         their values in order to be accurate.
[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]CUTPNT
          CUTPNT is INTEGER
         The location of the last eigenvalue in the leading
         sub-matrix.  min(1,N) <= CUTPNT <= N.
[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 are 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]Q2
          Q2 is REAL array, dimension (LDQ2,N)
         If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
         a copy of the first K eigenvectors which will be used by
         SLAED7 in a matrix multiply (SGEMM) to update the new
         eigenvectors.
[in]LDQ2
          LDQ2 is INTEGER
         The leading dimension of the array Q2.  LDQ2 >= max(1,N).
[out]W
          W is REAL array, dimension (N)
         The first k values of the final deflation-altered z-vector and
         will be passed to SLAED3.
[out]PERM
          PERM is INTEGER array, dimension (N)
         The permutations (from deflation and sorting) to be applied
         to each eigenblock.
[out]GIVPTR
          GIVPTR is INTEGER
         The number of Givens rotations which took place in this
         subproblem.
[out]GIVCOL
          GIVCOL is INTEGER array, dimension (2, N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.
[out]GIVNUM
          GIVNUM is REAL array, dimension (2, N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.
[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]INDX
          INDX is INTEGER array, dimension (N)
         The permutation used to sort the contents of D into ascending
         order.
[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.
Contributors:
Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 240 of file slaed8.f.

243 *
244 * -- LAPACK computational routine --
245 * -- LAPACK is a software package provided by Univ. of Tennessee, --
246 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
247 *
248 * .. Scalar Arguments ..
249  INTEGER CUTPNT, GIVPTR, ICOMPQ, INFO, K, LDQ, LDQ2, N,
250  $ QSIZ
251  REAL RHO
252 * ..
253 * .. Array Arguments ..
254  INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ),
255  $ INDXQ( * ), PERM( * )
256  REAL D( * ), DLAMDA( * ), GIVNUM( 2, * ),
257  $ Q( LDQ, * ), Q2( LDQ2, * ), W( * ), Z( * )
258 * ..
259 *
260 * =====================================================================
261 *
262 * .. Parameters ..
263  REAL MONE, ZERO, ONE, TWO, EIGHT
264  parameter( mone = -1.0e0, zero = 0.0e0, one = 1.0e0,
265  $ two = 2.0e0, eight = 8.0e0 )
266 * ..
267 * .. Local Scalars ..
268 *
269  INTEGER I, IMAX, J, JLAM, JMAX, JP, K2, N1, N1P1, N2
270  REAL C, EPS, S, T, TAU, TOL
271 * ..
272 * .. External Functions ..
273  INTEGER ISAMAX
274  REAL SLAMCH, SLAPY2
275  EXTERNAL isamax, slamch, slapy2
276 * ..
277 * .. External Subroutines ..
278  EXTERNAL scopy, slacpy, slamrg, srot, sscal, xerbla
279 * ..
280 * .. Intrinsic Functions ..
281  INTRINSIC abs, max, min, sqrt
282 * ..
283 * .. Executable Statements ..
284 *
285 * Test the input parameters.
286 *
287  info = 0
288 *
289  IF( icompq.LT.0 .OR. icompq.GT.1 ) THEN
290  info = -1
291  ELSE IF( n.LT.0 ) THEN
292  info = -3
293  ELSE IF( icompq.EQ.1 .AND. qsiz.LT.n ) THEN
294  info = -4
295  ELSE IF( ldq.LT.max( 1, n ) ) THEN
296  info = -7
297  ELSE IF( cutpnt.LT.min( 1, n ) .OR. cutpnt.GT.n ) THEN
298  info = -10
299  ELSE IF( ldq2.LT.max( 1, n ) ) THEN
300  info = -14
301  END IF
302  IF( info.NE.0 ) THEN
303  CALL xerbla( 'SLAED8', -info )
304  RETURN
305  END IF
306 *
307 * Need to initialize GIVPTR to O here in case of quick exit
308 * to prevent an unspecified code behavior (usually sigfault)
309 * when IWORK array on entry to *stedc is not zeroed
310 * (or at least some IWORK entries which used in *laed7 for GIVPTR).
311 *
312  givptr = 0
313 *
314 * Quick return if possible
315 *
316  IF( n.EQ.0 )
317  $ RETURN
318 *
319  n1 = cutpnt
320  n2 = n - n1
321  n1p1 = n1 + 1
322 *
323  IF( rho.LT.zero ) THEN
324  CALL sscal( n2, mone, z( n1p1 ), 1 )
325  END IF
326 *
327 * Normalize z so that norm(z) = 1
328 *
329  t = one / sqrt( two )
330  DO 10 j = 1, n
331  indx( j ) = j
332  10 CONTINUE
333  CALL sscal( n, t, z, 1 )
334  rho = abs( two*rho )
335 *
336 * Sort the eigenvalues into increasing order
337 *
338  DO 20 i = cutpnt + 1, n
339  indxq( i ) = indxq( i ) + cutpnt
340  20 CONTINUE
341  DO 30 i = 1, n
342  dlamda( i ) = d( indxq( i ) )
343  w( i ) = z( indxq( i ) )
344  30 CONTINUE
345  i = 1
346  j = cutpnt + 1
347  CALL slamrg( n1, n2, dlamda, 1, 1, indx )
348  DO 40 i = 1, n
349  d( i ) = dlamda( indx( i ) )
350  z( i ) = w( indx( i ) )
351  40 CONTINUE
352 *
353 * Calculate the allowable deflation tolerance
354 *
355  imax = isamax( n, z, 1 )
356  jmax = isamax( n, d, 1 )
357  eps = slamch( 'Epsilon' )
358  tol = eight*eps*abs( d( jmax ) )
359 *
360 * If the rank-1 modifier is small enough, no more needs to be done
361 * except to reorganize Q so that its columns correspond with the
362 * elements in D.
363 *
364  IF( rho*abs( z( imax ) ).LE.tol ) THEN
365  k = 0
366  IF( icompq.EQ.0 ) THEN
367  DO 50 j = 1, n
368  perm( j ) = indxq( indx( j ) )
369  50 CONTINUE
370  ELSE
371  DO 60 j = 1, n
372  perm( j ) = indxq( indx( j ) )
373  CALL scopy( qsiz, q( 1, perm( j ) ), 1, q2( 1, j ), 1 )
374  60 CONTINUE
375  CALL slacpy( 'A', qsiz, n, q2( 1, 1 ), ldq2, q( 1, 1 ),
376  $ ldq )
377  END IF
378  RETURN
379  END IF
380 *
381 * If there are multiple eigenvalues then the problem deflates. Here
382 * the number of equal eigenvalues are found. As each equal
383 * eigenvalue is found, an elementary reflector is computed to rotate
384 * the corresponding eigensubspace so that the corresponding
385 * components of Z are zero in this new basis.
386 *
387  k = 0
388  k2 = n + 1
389  DO 70 j = 1, n
390  IF( rho*abs( z( j ) ).LE.tol ) THEN
391 *
392 * Deflate due to small z component.
393 *
394  k2 = k2 - 1
395  indxp( k2 ) = j
396  IF( j.EQ.n )
397  $ GO TO 110
398  ELSE
399  jlam = j
400  GO TO 80
401  END IF
402  70 CONTINUE
403  80 CONTINUE
404  j = j + 1
405  IF( j.GT.n )
406  $ GO TO 100
407  IF( rho*abs( z( j ) ).LE.tol ) THEN
408 *
409 * Deflate due to small z component.
410 *
411  k2 = k2 - 1
412  indxp( k2 ) = j
413  ELSE
414 *
415 * Check if eigenvalues are close enough to allow deflation.
416 *
417  s = z( jlam )
418  c = z( j )
419 *
420 * Find sqrt(a**2+b**2) without overflow or
421 * destructive underflow.
422 *
423  tau = slapy2( c, s )
424  t = d( j ) - d( jlam )
425  c = c / tau
426  s = -s / tau
427  IF( abs( t*c*s ).LE.tol ) THEN
428 *
429 * Deflation is possible.
430 *
431  z( j ) = tau
432  z( jlam ) = zero
433 *
434 * Record the appropriate Givens rotation
435 *
436  givptr = givptr + 1
437  givcol( 1, givptr ) = indxq( indx( jlam ) )
438  givcol( 2, givptr ) = indxq( indx( j ) )
439  givnum( 1, givptr ) = c
440  givnum( 2, givptr ) = s
441  IF( icompq.EQ.1 ) THEN
442  CALL srot( qsiz, q( 1, indxq( indx( jlam ) ) ), 1,
443  $ q( 1, indxq( indx( j ) ) ), 1, c, s )
444  END IF
445  t = d( jlam )*c*c + d( j )*s*s
446  d( j ) = d( jlam )*s*s + d( j )*c*c
447  d( jlam ) = t
448  k2 = k2 - 1
449  i = 1
450  90 CONTINUE
451  IF( k2+i.LE.n ) THEN
452  IF( d( jlam ).LT.d( indxp( k2+i ) ) ) THEN
453  indxp( k2+i-1 ) = indxp( k2+i )
454  indxp( k2+i ) = jlam
455  i = i + 1
456  GO TO 90
457  ELSE
458  indxp( k2+i-1 ) = jlam
459  END IF
460  ELSE
461  indxp( k2+i-1 ) = jlam
462  END IF
463  jlam = j
464  ELSE
465  k = k + 1
466  w( k ) = z( jlam )
467  dlamda( k ) = d( jlam )
468  indxp( k ) = jlam
469  jlam = j
470  END IF
471  END IF
472  GO TO 80
473  100 CONTINUE
474 *
475 * Record the last eigenvalue.
476 *
477  k = k + 1
478  w( k ) = z( jlam )
479  dlamda( k ) = d( jlam )
480  indxp( k ) = jlam
481 *
482  110 CONTINUE
483 *
484 * Sort the eigenvalues and corresponding eigenvectors into DLAMDA
485 * and Q2 respectively. The eigenvalues/vectors which were not
486 * deflated go into the first K slots of DLAMDA and Q2 respectively,
487 * while those which were deflated go into the last N - K slots.
488 *
489  IF( icompq.EQ.0 ) THEN
490  DO 120 j = 1, n
491  jp = indxp( j )
492  dlamda( j ) = d( jp )
493  perm( j ) = indxq( indx( jp ) )
494  120 CONTINUE
495  ELSE
496  DO 130 j = 1, n
497  jp = indxp( j )
498  dlamda( j ) = d( jp )
499  perm( j ) = indxq( indx( jp ) )
500  CALL scopy( qsiz, q( 1, perm( j ) ), 1, q2( 1, j ), 1 )
501  130 CONTINUE
502  END IF
503 *
504 * The deflated eigenvalues and their corresponding vectors go back
505 * into the last N - K slots of D and Q respectively.
506 *
507  IF( k.LT.n ) THEN
508  IF( icompq.EQ.0 ) THEN
509  CALL scopy( n-k, dlamda( k+1 ), 1, d( k+1 ), 1 )
510  ELSE
511  CALL scopy( n-k, dlamda( k+1 ), 1, d( k+1 ), 1 )
512  CALL slacpy( 'A', qsiz, n-k, q2( 1, k+1 ), ldq2,
513  $ q( 1, k+1 ), ldq )
514  END IF
515  END IF
516 *
517  RETURN
518 *
519 * End of SLAED8
520 *
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
real function slapy2(X, Y)
SLAPY2 returns sqrt(x2+y2).
Definition: slapy2.f:63
integer function isamax(N, SX, INCX)
ISAMAX
Definition: isamax.f:71
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 srot(N, SX, INCX, SY, INCY, C, S)
SROT
Definition: srot.f:92
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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