LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ slasq3()

 subroutine slasq3 ( integer I0, integer N0, real, dimension( * ) Z, integer PP, real DMIN, real SIGMA, real DESIG, real QMAX, integer NFAIL, integer ITER, integer NDIV, logical IEEE, integer TTYPE, real DMIN1, real DMIN2, real DN, real DN1, real DN2, real G, real TAU )

SLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr.

Purpose:
``` SLASQ3 checks for deflation, computes a shift (TAU) and calls dqds.
In case of failure it changes shifts, and tries again until output
is positive.```
Parameters
 [in] I0 ``` I0 is INTEGER First index.``` [in,out] N0 ``` N0 is INTEGER Last index.``` [in,out] Z ``` Z is REAL array, dimension ( 4*N0 ) Z holds the qd array.``` [in,out] PP ``` PP is INTEGER PP=0 for ping, PP=1 for pong. PP=2 indicates that flipping was applied to the Z array and that the initial tests for deflation should not be performed.``` [out] DMIN ``` DMIN is REAL Minimum value of d.``` [out] SIGMA ``` SIGMA is REAL Sum of shifts used in current segment.``` [in,out] DESIG ``` DESIG is REAL Lower order part of SIGMA``` [in] QMAX ``` QMAX is REAL Maximum value of q.``` [in,out] NFAIL ``` NFAIL is INTEGER Increment NFAIL by 1 each time the shift was too big.``` [in,out] ITER ``` ITER is INTEGER Increment ITER by 1 for each iteration.``` [in,out] NDIV ``` NDIV is INTEGER Increment NDIV by 1 for each division.``` [in] IEEE ``` IEEE is LOGICAL Flag for IEEE or non IEEE arithmetic (passed to SLASQ5).``` [in,out] TTYPE ``` TTYPE is INTEGER Shift type.``` [in,out] DMIN1 ` DMIN1 is REAL` [in,out] DMIN2 ` DMIN2 is REAL` [in,out] DN ` DN is REAL` [in,out] DN1 ` DN1 is REAL` [in,out] DN2 ` DN2 is REAL` [in,out] G ` G is REAL` [in,out] TAU ``` TAU is REAL These are passed as arguments in order to save their values between calls to SLASQ3.```

Definition at line 179 of file slasq3.f.

182*
183* -- LAPACK computational routine --
184* -- LAPACK is a software package provided by Univ. of Tennessee, --
185* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
186*
187* .. Scalar Arguments ..
188 LOGICAL IEEE
189 INTEGER I0, ITER, N0, NDIV, NFAIL, PP
190 REAL DESIG, DMIN, DMIN1, DMIN2, DN, DN1, DN2, G,
191 \$ QMAX, SIGMA, TAU
192* ..
193* .. Array Arguments ..
194 REAL Z( * )
195* ..
196*
197* =====================================================================
198*
199* .. Parameters ..
200 REAL CBIAS
201 parameter( cbias = 1.50e0 )
202 REAL ZERO, QURTR, HALF, ONE, TWO, HUNDRD
203 parameter( zero = 0.0e0, qurtr = 0.250e0, half = 0.5e0,
204 \$ one = 1.0e0, two = 2.0e0, hundrd = 100.0e0 )
205* ..
206* .. Local Scalars ..
207 INTEGER IPN4, J4, N0IN, NN, TTYPE
208 REAL EPS, S, T, TEMP, TOL, TOL2
209* ..
210* .. External Subroutines ..
211 EXTERNAL slasq4, slasq5, slasq6
212* ..
213* .. External Function ..
214 REAL SLAMCH
215 LOGICAL SISNAN
216 EXTERNAL sisnan, slamch
217* ..
218* .. Intrinsic Functions ..
219 INTRINSIC abs, max, min, sqrt
220* ..
221* .. Executable Statements ..
222*
223 n0in = n0
224 eps = slamch( 'Precision' )
225 tol = eps*hundrd
226 tol2 = tol**2
227*
228* Check for deflation.
229*
230 10 CONTINUE
231*
232 IF( n0.LT.i0 )
233 \$ RETURN
234 IF( n0.EQ.i0 )
235 \$ GO TO 20
236 nn = 4*n0 + pp
237 IF( n0.EQ.( i0+1 ) )
238 \$ GO TO 40
239*
240* Check whether E(N0-1) is negligible, 1 eigenvalue.
241*
242 IF( z( nn-5 ).GT.tol2*( sigma+z( nn-3 ) ) .AND.
243 \$ z( nn-2*pp-4 ).GT.tol2*z( nn-7 ) )
244 \$ GO TO 30
245*
246 20 CONTINUE
247*
248 z( 4*n0-3 ) = z( 4*n0+pp-3 ) + sigma
249 n0 = n0 - 1
250 GO TO 10
251*
252* Check whether E(N0-2) is negligible, 2 eigenvalues.
253*
254 30 CONTINUE
255*
256 IF( z( nn-9 ).GT.tol2*sigma .AND.
257 \$ z( nn-2*pp-8 ).GT.tol2*z( nn-11 ) )
258 \$ GO TO 50
259*
260 40 CONTINUE
261*
262 IF( z( nn-3 ).GT.z( nn-7 ) ) THEN
263 s = z( nn-3 )
264 z( nn-3 ) = z( nn-7 )
265 z( nn-7 ) = s
266 END IF
267 t = half*( ( z( nn-7 )-z( nn-3 ) )+z( nn-5 ) )
268 IF( z( nn-5 ).GT.z( nn-3 )*tol2.AND.t.NE.zero ) THEN
269 s = z( nn-3 )*( z( nn-5 ) / t )
270 IF( s.LE.t ) THEN
271 s = z( nn-3 )*( z( nn-5 ) /
272 \$ ( t*( one+sqrt( one+s / t ) ) ) )
273 ELSE
274 s = z( nn-3 )*( z( nn-5 ) / ( t+sqrt( t )*sqrt( t+s ) ) )
275 END IF
276 t = z( nn-7 ) + ( s+z( nn-5 ) )
277 z( nn-3 ) = z( nn-3 )*( z( nn-7 ) / t )
278 z( nn-7 ) = t
279 END IF
280 z( 4*n0-7 ) = z( nn-7 ) + sigma
281 z( 4*n0-3 ) = z( nn-3 ) + sigma
282 n0 = n0 - 2
283 GO TO 10
284*
285 50 CONTINUE
286 IF( pp.EQ.2 )
287 \$ pp = 0
288*
289* Reverse the qd-array, if warranted.
290*
291 IF( dmin.LE.zero .OR. n0.LT.n0in ) THEN
292 IF( cbias*z( 4*i0+pp-3 ).LT.z( 4*n0+pp-3 ) ) THEN
293 ipn4 = 4*( i0+n0 )
294 DO 60 j4 = 4*i0, 2*( i0+n0-1 ), 4
295 temp = z( j4-3 )
296 z( j4-3 ) = z( ipn4-j4-3 )
297 z( ipn4-j4-3 ) = temp
298 temp = z( j4-2 )
299 z( j4-2 ) = z( ipn4-j4-2 )
300 z( ipn4-j4-2 ) = temp
301 temp = z( j4-1 )
302 z( j4-1 ) = z( ipn4-j4-5 )
303 z( ipn4-j4-5 ) = temp
304 temp = z( j4 )
305 z( j4 ) = z( ipn4-j4-4 )
306 z( ipn4-j4-4 ) = temp
307 60 CONTINUE
308 IF( n0-i0.LE.4 ) THEN
309 z( 4*n0+pp-1 ) = z( 4*i0+pp-1 )
310 z( 4*n0-pp ) = z( 4*i0-pp )
311 END IF
312 dmin2 = min( dmin2, z( 4*n0+pp-1 ) )
313 z( 4*n0+pp-1 ) = min( z( 4*n0+pp-1 ), z( 4*i0+pp-1 ),
314 \$ z( 4*i0+pp+3 ) )
315 z( 4*n0-pp ) = min( z( 4*n0-pp ), z( 4*i0-pp ),
316 \$ z( 4*i0-pp+4 ) )
317 qmax = max( qmax, z( 4*i0+pp-3 ), z( 4*i0+pp+1 ) )
318 dmin = -zero
319 END IF
320 END IF
321*
322* Choose a shift.
323*
324 CALL slasq4( i0, n0, z, pp, n0in, dmin, dmin1, dmin2, dn, dn1,
325 \$ dn2, tau, ttype, g )
326*
327* Call dqds until DMIN > 0.
328*
329 70 CONTINUE
330*
331 CALL slasq5( i0, n0, z, pp, tau, sigma, dmin, dmin1, dmin2, dn,
332 \$ dn1, dn2, ieee, eps )
333*
334 ndiv = ndiv + ( n0-i0+2 )
335 iter = iter + 1
336*
337* Check status.
338*
339 IF( dmin.GE.zero .AND. dmin1.GE.zero ) THEN
340*
341* Success.
342*
343 GO TO 90
344*
345 ELSE IF( dmin.LT.zero .AND. dmin1.GT.zero .AND.
346 \$ z( 4*( n0-1 )-pp ).LT.tol*( sigma+dn1 ) .AND.
347 \$ abs( dn ).LT.tol*sigma ) THEN
348*
349* Convergence hidden by negative DN.
350*
351 z( 4*( n0-1 )-pp+2 ) = zero
352 dmin = zero
353 GO TO 90
354 ELSE IF( dmin.LT.zero ) THEN
355*
356* TAU too big. Select new TAU and try again.
357*
358 nfail = nfail + 1
359 IF( ttype.LT.-22 ) THEN
360*
361* Failed twice. Play it safe.
362*
363 tau = zero
364 ELSE IF( dmin1.GT.zero ) THEN
365*
366* Late failure. Gives excellent shift.
367*
368 tau = ( tau+dmin )*( one-two*eps )
369 ttype = ttype - 11
370 ELSE
371*
372* Early failure. Divide by 4.
373*
374 tau = qurtr*tau
375 ttype = ttype - 12
376 END IF
377 GO TO 70
378 ELSE IF( sisnan( dmin ) ) THEN
379*
380* NaN.
381*
382 IF( tau.EQ.zero ) THEN
383 GO TO 80
384 ELSE
385 tau = zero
386 GO TO 70
387 END IF
388 ELSE
389*
390* Possible underflow. Play it safe.
391*
392 GO TO 80
393 END IF
394*
395* Risk of underflow.
396*
397 80 CONTINUE
398 CALL slasq6( i0, n0, z, pp, dmin, dmin1, dmin2, dn, dn1, dn2 )
399 ndiv = ndiv + ( n0-i0+2 )
400 iter = iter + 1
401 tau = zero
402*
403 90 CONTINUE
404 IF( tau.LT.sigma ) THEN
405 desig = desig + tau
406 t = sigma + desig
407 desig = desig - ( t-sigma )
408 ELSE
409 t = sigma + tau
410 desig = sigma - ( t-tau ) + desig
411 END IF
412 sigma = t
413*
414 RETURN
415*
416* End of SLASQ3
417*
logical function sisnan(SIN)
SISNAN tests input for NaN.
Definition: sisnan.f:59
subroutine slasq4(I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN, DN1, DN2, TAU, TTYPE, G)
SLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous trans...
Definition: slasq4.f:151
subroutine slasq5(I0, N0, Z, PP, TAU, SIGMA, DMIN, DMIN1, DMIN2, DN, DNM1, DNM2, IEEE, EPS)
SLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and sstegr.
Definition: slasq5.f:144
subroutine slasq6(I0, N0, Z, PP, DMIN, DMIN1, DMIN2, DN, DNM1, DNM2)
SLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and sstegr.
Definition: slasq6.f:119
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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