 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ clar1v()

 subroutine clar1v ( integer N, integer B1, integer BN, real LAMBDA, real, dimension( * ) D, real, dimension( * ) L, real, dimension( * ) LD, real, dimension( * ) LLD, real PIVMIN, real GAPTOL, complex, dimension( * ) Z, logical WANTNC, integer NEGCNT, real ZTZ, real MINGMA, integer R, integer, dimension( * ) ISUPPZ, real NRMINV, real RESID, real RQCORR, real, dimension( * ) WORK )

CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

Purpose:
``` CLAR1V computes the (scaled) r-th column of the inverse of
the sumbmatrix in rows B1 through BN of the tridiagonal matrix
L D L**T - sigma I. When sigma is close to an eigenvalue, the
computed vector is an accurate eigenvector. Usually, r corresponds
to the index where the eigenvector is largest in magnitude.
The following steps accomplish this computation :
(a) Stationary qd transform,  L D L**T - sigma I = L(+) D(+) L(+)**T,
(b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,
(c) Computation of the diagonal elements of the inverse of
L D L**T - sigma I by combining the above transforms, and choosing
r as the index where the diagonal of the inverse is (one of the)
largest in magnitude.
(d) Computation of the (scaled) r-th column of the inverse using the
twisted factorization obtained by combining the top part of the
the stationary and the bottom part of the progressive transform.```
Parameters
 [in] N ``` N is INTEGER The order of the matrix L D L**T.``` [in] B1 ``` B1 is INTEGER First index of the submatrix of L D L**T.``` [in] BN ``` BN is INTEGER Last index of the submatrix of L D L**T.``` [in] LAMBDA ``` LAMBDA is REAL The shift. In order to compute an accurate eigenvector, LAMBDA should be a good approximation to an eigenvalue of L D L**T.``` [in] L ``` L is REAL array, dimension (N-1) The (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to N-1.``` [in] D ``` D is REAL array, dimension (N) The n diagonal elements of the diagonal matrix D.``` [in] LD ``` LD is REAL array, dimension (N-1) The n-1 elements L(i)*D(i).``` [in] LLD ``` LLD is REAL array, dimension (N-1) The n-1 elements L(i)*L(i)*D(i).``` [in] PIVMIN ``` PIVMIN is REAL The minimum pivot in the Sturm sequence.``` [in] GAPTOL ``` GAPTOL is REAL Tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual.``` [in,out] Z ``` Z is COMPLEX array, dimension (N) On input, all entries of Z must be set to 0. On output, Z contains the (scaled) r-th column of the inverse. The scaling is such that Z(R) equals 1.``` [in] WANTNC ``` WANTNC is LOGICAL Specifies whether NEGCNT has to be computed.``` [out] NEGCNT ``` NEGCNT is INTEGER If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin in the matrix factorization L D L**T, and NEGCNT = -1 otherwise.``` [out] ZTZ ``` ZTZ is REAL The square of the 2-norm of Z.``` [out] MINGMA ``` MINGMA is REAL The reciprocal of the largest (in magnitude) diagonal element of the inverse of L D L**T - sigma I.``` [in,out] R ``` R is INTEGER The twist index for the twisted factorization used to compute Z. On input, 0 <= R <= N. If R is input as 0, R is set to the index where (L D L**T - sigma I)^{-1} is largest in magnitude. If 1 <= R <= N, R is unchanged. On output, R contains the twist index used to compute Z. Ideally, R designates the position of the maximum entry in the eigenvector.``` [out] ISUPPZ ``` ISUPPZ is INTEGER array, dimension (2) The support of the vector in Z, i.e., the vector Z is nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).``` [out] NRMINV ``` NRMINV is REAL NRMINV = 1/SQRT( ZTZ )``` [out] RESID ``` RESID is REAL The residual of the FP vector. RESID = ABS( MINGMA )/SQRT( ZTZ )``` [out] RQCORR ``` RQCORR is REAL The Rayleigh Quotient correction to LAMBDA. RQCORR = MINGMA*TMP``` [out] WORK ` WORK is REAL array, dimension (4*N)`
Contributors:
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA

Definition at line 227 of file clar1v.f.

230 *
231 * -- LAPACK auxiliary routine --
232 * -- LAPACK is a software package provided by Univ. of Tennessee, --
233 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
234 *
235 * .. Scalar Arguments ..
236  LOGICAL WANTNC
237  INTEGER B1, BN, N, NEGCNT, R
238  REAL GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID,
239  \$ RQCORR, ZTZ
240 * ..
241 * .. Array Arguments ..
242  INTEGER ISUPPZ( * )
243  REAL D( * ), L( * ), LD( * ), LLD( * ),
244  \$ WORK( * )
245  COMPLEX Z( * )
246 * ..
247 *
248 * =====================================================================
249 *
250 * .. Parameters ..
251  REAL ZERO, ONE
252  parameter( zero = 0.0e0, one = 1.0e0 )
253  COMPLEX CONE
254  parameter( cone = ( 1.0e0, 0.0e0 ) )
255
256 * ..
257 * .. Local Scalars ..
258  LOGICAL SAWNAN1, SAWNAN2
259  INTEGER I, INDLPL, INDP, INDS, INDUMN, NEG1, NEG2, R1,
260  \$ R2
261  REAL DMINUS, DPLUS, EPS, S, TMP
262 * ..
263 * .. External Functions ..
264  LOGICAL SISNAN
265  REAL SLAMCH
266  EXTERNAL sisnan, slamch
267 * ..
268 * .. Intrinsic Functions ..
269  INTRINSIC abs, real
270 * ..
271 * .. Executable Statements ..
272 *
273  eps = slamch( 'Precision' )
274
275
276  IF( r.EQ.0 ) THEN
277  r1 = b1
278  r2 = bn
279  ELSE
280  r1 = r
281  r2 = r
282  END IF
283
284 * Storage for LPLUS
285  indlpl = 0
286 * Storage for UMINUS
287  indumn = n
288  inds = 2*n + 1
289  indp = 3*n + 1
290
291  IF( b1.EQ.1 ) THEN
292  work( inds ) = zero
293  ELSE
294  work( inds+b1-1 ) = lld( b1-1 )
295  END IF
296
297 *
298 * Compute the stationary transform (using the differential form)
299 * until the index R2.
300 *
301  sawnan1 = .false.
302  neg1 = 0
303  s = work( inds+b1-1 ) - lambda
304  DO 50 i = b1, r1 - 1
305  dplus = d( i ) + s
306  work( indlpl+i ) = ld( i ) / dplus
307  IF(dplus.LT.zero) neg1 = neg1 + 1
308  work( inds+i ) = s*work( indlpl+i )*l( i )
309  s = work( inds+i ) - lambda
310  50 CONTINUE
311  sawnan1 = sisnan( s )
312  IF( sawnan1 ) GOTO 60
313  DO 51 i = r1, r2 - 1
314  dplus = d( i ) + s
315  work( indlpl+i ) = ld( i ) / dplus
316  work( inds+i ) = s*work( indlpl+i )*l( i )
317  s = work( inds+i ) - lambda
318  51 CONTINUE
319  sawnan1 = sisnan( s )
320 *
321  60 CONTINUE
322  IF( sawnan1 ) THEN
323 * Runs a slower version of the above loop if a NaN is detected
324  neg1 = 0
325  s = work( inds+b1-1 ) - lambda
326  DO 70 i = b1, r1 - 1
327  dplus = d( i ) + s
328  IF(abs(dplus).LT.pivmin) dplus = -pivmin
329  work( indlpl+i ) = ld( i ) / dplus
330  IF(dplus.LT.zero) neg1 = neg1 + 1
331  work( inds+i ) = s*work( indlpl+i )*l( i )
332  IF( work( indlpl+i ).EQ.zero )
333  \$ work( inds+i ) = lld( i )
334  s = work( inds+i ) - lambda
335  70 CONTINUE
336  DO 71 i = r1, r2 - 1
337  dplus = d( i ) + s
338  IF(abs(dplus).LT.pivmin) dplus = -pivmin
339  work( indlpl+i ) = ld( i ) / dplus
340  work( inds+i ) = s*work( indlpl+i )*l( i )
341  IF( work( indlpl+i ).EQ.zero )
342  \$ work( inds+i ) = lld( i )
343  s = work( inds+i ) - lambda
344  71 CONTINUE
345  END IF
346 *
347 * Compute the progressive transform (using the differential form)
348 * until the index R1
349 *
350  sawnan2 = .false.
351  neg2 = 0
352  work( indp+bn-1 ) = d( bn ) - lambda
353  DO 80 i = bn - 1, r1, -1
354  dminus = lld( i ) + work( indp+i )
355  tmp = d( i ) / dminus
356  IF(dminus.LT.zero) neg2 = neg2 + 1
357  work( indumn+i ) = l( i )*tmp
358  work( indp+i-1 ) = work( indp+i )*tmp - lambda
359  80 CONTINUE
360  tmp = work( indp+r1-1 )
361  sawnan2 = sisnan( tmp )
362
363  IF( sawnan2 ) THEN
364 * Runs a slower version of the above loop if a NaN is detected
365  neg2 = 0
366  DO 100 i = bn-1, r1, -1
367  dminus = lld( i ) + work( indp+i )
368  IF(abs(dminus).LT.pivmin) dminus = -pivmin
369  tmp = d( i ) / dminus
370  IF(dminus.LT.zero) neg2 = neg2 + 1
371  work( indumn+i ) = l( i )*tmp
372  work( indp+i-1 ) = work( indp+i )*tmp - lambda
373  IF( tmp.EQ.zero )
374  \$ work( indp+i-1 ) = d( i ) - lambda
375  100 CONTINUE
376  END IF
377 *
378 * Find the index (from R1 to R2) of the largest (in magnitude)
379 * diagonal element of the inverse
380 *
381  mingma = work( inds+r1-1 ) + work( indp+r1-1 )
382  IF( mingma.LT.zero ) neg1 = neg1 + 1
383  IF( wantnc ) THEN
384  negcnt = neg1 + neg2
385  ELSE
386  negcnt = -1
387  ENDIF
388  IF( abs(mingma).EQ.zero )
389  \$ mingma = eps*work( inds+r1-1 )
390  r = r1
391  DO 110 i = r1, r2 - 1
392  tmp = work( inds+i ) + work( indp+i )
393  IF( tmp.EQ.zero )
394  \$ tmp = eps*work( inds+i )
395  IF( abs( tmp ).LE.abs( mingma ) ) THEN
396  mingma = tmp
397  r = i + 1
398  END IF
399  110 CONTINUE
400 *
401 * Compute the FP vector: solve N^T v = e_r
402 *
403  isuppz( 1 ) = b1
404  isuppz( 2 ) = bn
405  z( r ) = cone
406  ztz = one
407 *
408 * Compute the FP vector upwards from R
409 *
410  IF( .NOT.sawnan1 .AND. .NOT.sawnan2 ) THEN
411  DO 210 i = r-1, b1, -1
412  z( i ) = -( work( indlpl+i )*z( i+1 ) )
413  IF( (abs(z(i))+abs(z(i+1)))* abs(ld(i)).LT.gaptol )
414  \$ THEN
415  z( i ) = zero
416  isuppz( 1 ) = i + 1
417  GOTO 220
418  ENDIF
419  ztz = ztz + real( z( i )*z( i ) )
420  210 CONTINUE
421  220 CONTINUE
422  ELSE
423 * Run slower loop if NaN occurred.
424  DO 230 i = r - 1, b1, -1
425  IF( z( i+1 ).EQ.zero ) THEN
426  z( i ) = -( ld( i+1 ) / ld( i ) )*z( i+2 )
427  ELSE
428  z( i ) = -( work( indlpl+i )*z( i+1 ) )
429  END IF
430  IF( (abs(z(i))+abs(z(i+1)))* abs(ld(i)).LT.gaptol )
431  \$ THEN
432  z( i ) = zero
433  isuppz( 1 ) = i + 1
434  GO TO 240
435  END IF
436  ztz = ztz + real( z( i )*z( i ) )
437  230 CONTINUE
438  240 CONTINUE
439  ENDIF
440
441 * Compute the FP vector downwards from R in blocks of size BLKSIZ
442  IF( .NOT.sawnan1 .AND. .NOT.sawnan2 ) THEN
443  DO 250 i = r, bn-1
444  z( i+1 ) = -( work( indumn+i )*z( i ) )
445  IF( (abs(z(i))+abs(z(i+1)))* abs(ld(i)).LT.gaptol )
446  \$ THEN
447  z( i+1 ) = zero
448  isuppz( 2 ) = i
449  GO TO 260
450  END IF
451  ztz = ztz + real( z( i+1 )*z( i+1 ) )
452  250 CONTINUE
453  260 CONTINUE
454  ELSE
455 * Run slower loop if NaN occurred.
456  DO 270 i = r, bn - 1
457  IF( z( i ).EQ.zero ) THEN
458  z( i+1 ) = -( ld( i-1 ) / ld( i ) )*z( i-1 )
459  ELSE
460  z( i+1 ) = -( work( indumn+i )*z( i ) )
461  END IF
462  IF( (abs(z(i))+abs(z(i+1)))* abs(ld(i)).LT.gaptol )
463  \$ THEN
464  z( i+1 ) = zero
465  isuppz( 2 ) = i
466  GO TO 280
467  END IF
468  ztz = ztz + real( z( i+1 )*z( i+1 ) )
469  270 CONTINUE
470  280 CONTINUE
471  END IF
472 *
473 * Compute quantities for convergence test
474 *
475  tmp = one / ztz
476  nrminv = sqrt( tmp )
477  resid = abs( mingma )*nrminv
478  rqcorr = mingma*tmp
479 *
480 *
481  RETURN
482 *
483 * End of CLAR1V
484 *
logical function sisnan(SIN)
SISNAN tests input for NaN.
Definition: sisnan.f:59
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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