LAPACK  3.9.0 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)`
Date
December 2016
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 232 of file clar1v.f.

232 *
233 * -- LAPACK auxiliary routine (version 3.7.0) --
234 * -- LAPACK is a software package provided by Univ. of Tennessee, --
235 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
236 * December 2016
237 *
238 * .. Scalar Arguments ..
239  LOGICAL WANTNC
240  INTEGER B1, BN, N, NEGCNT, R
241  REAL GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID,
242  \$ RQCORR, ZTZ
243 * ..
244 * .. Array Arguments ..
245  INTEGER ISUPPZ( * )
246  REAL D( * ), L( * ), LD( * ), LLD( * ),
247  \$ WORK( * )
248  COMPLEX Z( * )
249 * ..
250 *
251 * =====================================================================
252 *
253 * .. Parameters ..
254  REAL ZERO, ONE
255  parameter( zero = 0.0e0, one = 1.0e0 )
256  COMPLEX CONE
257  parameter( cone = ( 1.0e0, 0.0e0 ) )
258
259 * ..
260 * .. Local Scalars ..
261  LOGICAL SAWNAN1, SAWNAN2
262  INTEGER I, INDLPL, INDP, INDS, INDUMN, NEG1, NEG2, R1,
263  \$ R2
264  REAL DMINUS, DPLUS, EPS, S, TMP
265 * ..
266 * .. External Functions ..
267  LOGICAL SISNAN
268  REAL SLAMCH
269  EXTERNAL sisnan, slamch
270 * ..
271 * .. Intrinsic Functions ..
272  INTRINSIC abs, real
273 * ..
274 * .. Executable Statements ..
275 *
276  eps = slamch( 'Precision' )
277
278
279  IF( r.EQ.0 ) THEN
280  r1 = b1
281  r2 = bn
282  ELSE
283  r1 = r
284  r2 = r
285  END IF
286
287 * Storage for LPLUS
288  indlpl = 0
289 * Storage for UMINUS
290  indumn = n
291  inds = 2*n + 1
292  indp = 3*n + 1
293
294  IF( b1.EQ.1 ) THEN
295  work( inds ) = zero
296  ELSE
297  work( inds+b1-1 ) = lld( b1-1 )
298  END IF
299
300 *
301 * Compute the stationary transform (using the differential form)
302 * until the index R2.
303 *
304  sawnan1 = .false.
305  neg1 = 0
306  s = work( inds+b1-1 ) - lambda
307  DO 50 i = b1, r1 - 1
308  dplus = d( i ) + s
309  work( indlpl+i ) = ld( i ) / dplus
310  IF(dplus.LT.zero) neg1 = neg1 + 1
311  work( inds+i ) = s*work( indlpl+i )*l( i )
312  s = work( inds+i ) - lambda
313  50 CONTINUE
314  sawnan1 = sisnan( s )
315  IF( sawnan1 ) GOTO 60
316  DO 51 i = r1, r2 - 1
317  dplus = d( i ) + s
318  work( indlpl+i ) = ld( i ) / dplus
319  work( inds+i ) = s*work( indlpl+i )*l( i )
320  s = work( inds+i ) - lambda
321  51 CONTINUE
322  sawnan1 = sisnan( s )
323 *
324  60 CONTINUE
325  IF( sawnan1 ) THEN
326 * Runs a slower version of the above loop if a NaN is detected
327  neg1 = 0
328  s = work( inds+b1-1 ) - lambda
329  DO 70 i = b1, r1 - 1
330  dplus = d( i ) + s
331  IF(abs(dplus).LT.pivmin) dplus = -pivmin
332  work( indlpl+i ) = ld( i ) / dplus
333  IF(dplus.LT.zero) neg1 = neg1 + 1
334  work( inds+i ) = s*work( indlpl+i )*l( i )
335  IF( work( indlpl+i ).EQ.zero )
336  \$ work( inds+i ) = lld( i )
337  s = work( inds+i ) - lambda
338  70 CONTINUE
339  DO 71 i = r1, r2 - 1
340  dplus = d( i ) + s
341  IF(abs(dplus).LT.pivmin) dplus = -pivmin
342  work( indlpl+i ) = ld( i ) / dplus
343  work( inds+i ) = s*work( indlpl+i )*l( i )
344  IF( work( indlpl+i ).EQ.zero )
345  \$ work( inds+i ) = lld( i )
346  s = work( inds+i ) - lambda
347  71 CONTINUE
348  END IF
349 *
350 * Compute the progressive transform (using the differential form)
351 * until the index R1
352 *
353  sawnan2 = .false.
354  neg2 = 0
355  work( indp+bn-1 ) = d( bn ) - lambda
356  DO 80 i = bn - 1, r1, -1
357  dminus = lld( i ) + work( indp+i )
358  tmp = d( i ) / dminus
359  IF(dminus.LT.zero) neg2 = neg2 + 1
360  work( indumn+i ) = l( i )*tmp
361  work( indp+i-1 ) = work( indp+i )*tmp - lambda
362  80 CONTINUE
363  tmp = work( indp+r1-1 )
364  sawnan2 = sisnan( tmp )
365
366  IF( sawnan2 ) THEN
367 * Runs a slower version of the above loop if a NaN is detected
368  neg2 = 0
369  DO 100 i = bn-1, r1, -1
370  dminus = lld( i ) + work( indp+i )
371  IF(abs(dminus).LT.pivmin) dminus = -pivmin
372  tmp = d( i ) / dminus
373  IF(dminus.LT.zero) neg2 = neg2 + 1
374  work( indumn+i ) = l( i )*tmp
375  work( indp+i-1 ) = work( indp+i )*tmp - lambda
376  IF( tmp.EQ.zero )
377  \$ work( indp+i-1 ) = d( i ) - lambda
378  100 CONTINUE
379  END IF
380 *
381 * Find the index (from R1 to R2) of the largest (in magnitude)
382 * diagonal element of the inverse
383 *
384  mingma = work( inds+r1-1 ) + work( indp+r1-1 )
385  IF( mingma.LT.zero ) neg1 = neg1 + 1
386  IF( wantnc ) THEN
387  negcnt = neg1 + neg2
388  ELSE
389  negcnt = -1
390  ENDIF
391  IF( abs(mingma).EQ.zero )
392  \$ mingma = eps*work( inds+r1-1 )
393  r = r1
394  DO 110 i = r1, r2 - 1
395  tmp = work( inds+i ) + work( indp+i )
396  IF( tmp.EQ.zero )
397  \$ tmp = eps*work( inds+i )
398  IF( abs( tmp ).LE.abs( mingma ) ) THEN
399  mingma = tmp
400  r = i + 1
401  END IF
402  110 CONTINUE
403 *
404 * Compute the FP vector: solve N^T v = e_r
405 *
406  isuppz( 1 ) = b1
407  isuppz( 2 ) = bn
408  z( r ) = cone
409  ztz = one
410 *
411 * Compute the FP vector upwards from R
412 *
413  IF( .NOT.sawnan1 .AND. .NOT.sawnan2 ) THEN
414  DO 210 i = r-1, b1, -1
415  z( i ) = -( work( indlpl+i )*z( i+1 ) )
416  IF( (abs(z(i))+abs(z(i+1)))* abs(ld(i)).LT.gaptol )
417  \$ THEN
418  z( i ) = zero
419  isuppz( 1 ) = i + 1
420  GOTO 220
421  ENDIF
422  ztz = ztz + real( z( i )*z( i ) )
423  210 CONTINUE
424  220 CONTINUE
425  ELSE
426 * Run slower loop if NaN occurred.
427  DO 230 i = r - 1, b1, -1
428  IF( z( i+1 ).EQ.zero ) THEN
429  z( i ) = -( ld( i+1 ) / ld( i ) )*z( i+2 )
430  ELSE
431  z( i ) = -( work( indlpl+i )*z( i+1 ) )
432  END IF
433  IF( (abs(z(i))+abs(z(i+1)))* abs(ld(i)).LT.gaptol )
434  \$ THEN
435  z( i ) = zero
436  isuppz( 1 ) = i + 1
437  GO TO 240
438  END IF
439  ztz = ztz + real( z( i )*z( i ) )
440  230 CONTINUE
441  240 CONTINUE
442  ENDIF
443
444 * Compute the FP vector downwards from R in blocks of size BLKSIZ
445  IF( .NOT.sawnan1 .AND. .NOT.sawnan2 ) THEN
446  DO 250 i = r, bn-1
447  z( i+1 ) = -( work( indumn+i )*z( i ) )
448  IF( (abs(z(i))+abs(z(i+1)))* abs(ld(i)).LT.gaptol )
449  \$ THEN
450  z( i+1 ) = zero
451  isuppz( 2 ) = i
452  GO TO 260
453  END IF
454  ztz = ztz + real( z( i+1 )*z( i+1 ) )
455  250 CONTINUE
456  260 CONTINUE
457  ELSE
458 * Run slower loop if NaN occurred.
459  DO 270 i = r, bn - 1
460  IF( z( i ).EQ.zero ) THEN
461  z( i+1 ) = -( ld( i-1 ) / ld( i ) )*z( i-1 )
462  ELSE
463  z( i+1 ) = -( work( indumn+i )*z( i ) )
464  END IF
465  IF( (abs(z(i))+abs(z(i+1)))* abs(ld(i)).LT.gaptol )
466  \$ THEN
467  z( i+1 ) = zero
468  isuppz( 2 ) = i
469  GO TO 280
470  END IF
471  ztz = ztz + real( z( i+1 )*z( i+1 ) )
472  270 CONTINUE
473  280 CONTINUE
474  END IF
475 *
476 * Compute quantities for convergence test
477 *
478  tmp = one / ztz
479  nrminv = sqrt( tmp )
480  resid = abs( mingma )*nrminv
481  rqcorr = mingma*tmp
482 *
483 *
484  RETURN
485 *
486 * End of CLAR1V
487 *
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sisnan
logical function sisnan(SIN)
SISNAN tests input for NaN.
Definition: sisnan.f:61
slamch
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:70