1*> \brief <b> ZCPOSV computes the solution to system of linear equations A * X = B for PO matrices</b>
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
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7*
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17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE ZCPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,
22*                          SWORK, RWORK, ITER, INFO )
23*
24*       .. Scalar Arguments ..
25*       CHARACTER          UPLO
26*       INTEGER            INFO, ITER, LDA, LDB, LDX, N, NRHS
27*       ..
28*       .. Array Arguments ..
29*       DOUBLE PRECISION   RWORK( * )
30*       COMPLEX            SWORK( * )
31*       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( N, * ),
32*      $                   X( LDX, * )
33*       ..
34*
35*
36*> \par Purpose:
37*  =============
38*>
39*> \verbatim
40*>
41*> ZCPOSV computes the solution to a complex system of linear equations
42*>    A * X = B,
43*> where A is an N-by-N Hermitian positive definite matrix and X and B
44*> are N-by-NRHS matrices.
45*>
46*> ZCPOSV first attempts to factorize the matrix in COMPLEX and use this
47*> factorization within an iterative refinement procedure to produce a
48*> solution with COMPLEX*16 normwise backward error quality (see below).
49*> If the approach fails the method switches to a COMPLEX*16
50*> factorization and solve.
51*>
52*> The iterative refinement is not going to be a winning strategy if
53*> the ratio COMPLEX performance over COMPLEX*16 performance is too
54*> small. A reasonable strategy should take the number of right-hand
55*> sides and the size of the matrix into account. This might be done
56*> with a call to ILAENV in the future. Up to now, we always try
57*> iterative refinement.
58*>
59*> The iterative refinement process is stopped if
60*>     ITER > ITERMAX
61*> or for all the RHS we have:
62*>     RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
63*> where
64*>     o ITER is the number of the current iteration in the iterative
65*>       refinement process
66*>     o RNRM is the infinity-norm of the residual
67*>     o XNRM is the infinity-norm of the solution
68*>     o ANRM is the infinity-operator-norm of the matrix A
69*>     o EPS is the machine epsilon returned by DLAMCH('Epsilon')
70*> The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00
71*> respectively.
72*> \endverbatim
73*
74*  Arguments:
75*  ==========
76*
77*> \param[in] UPLO
78*> \verbatim
79*>          UPLO is CHARACTER*1
80*>          = 'U':  Upper triangle of A is stored;
81*>          = 'L':  Lower triangle of A is stored.
82*> \endverbatim
83*>
84*> \param[in] N
85*> \verbatim
86*>          N is INTEGER
87*>          The number of linear equations, i.e., the order of the
88*>          matrix A.  N >= 0.
89*> \endverbatim
90*>
91*> \param[in] NRHS
92*> \verbatim
93*>          NRHS is INTEGER
94*>          The number of right hand sides, i.e., the number of columns
95*>          of the matrix B.  NRHS >= 0.
96*> \endverbatim
97*>
98*> \param[in,out] A
99*> \verbatim
100*>          A is COMPLEX*16 array,
101*>          dimension (LDA,N)
102*>          On entry, the Hermitian matrix A. If UPLO = 'U', the leading
103*>          N-by-N upper triangular part of A contains the upper
104*>          triangular part of the matrix A, and the strictly lower
105*>          triangular part of A is not referenced.  If UPLO = 'L', the
106*>          leading N-by-N lower triangular part of A contains the lower
107*>          triangular part of the matrix A, and the strictly upper
108*>          triangular part of A is not referenced.
109*>
110*>          Note that the imaginary parts of the diagonal
111*>          elements need not be set and are assumed to be zero.
112*>
113*>          On exit, if iterative refinement has been successfully used
114*>          (INFO = 0 and ITER >= 0, see description below), then A is
115*>          unchanged, if double precision factorization has been used
116*>          (INFO = 0 and ITER < 0, see description below), then the
117*>          array A contains the factor U or L from the Cholesky
118*>          factorization A = U**H*U or A = L*L**H.
119*> \endverbatim
120*>
121*> \param[in] LDA
122*> \verbatim
123*>          LDA is INTEGER
124*>          The leading dimension of the array A.  LDA >= max(1,N).
125*> \endverbatim
126*>
127*> \param[in] B
128*> \verbatim
129*>          B is COMPLEX*16 array, dimension (LDB,NRHS)
130*>          The N-by-NRHS right hand side matrix B.
131*> \endverbatim
132*>
133*> \param[in] LDB
134*> \verbatim
135*>          LDB is INTEGER
136*>          The leading dimension of the array B.  LDB >= max(1,N).
137*> \endverbatim
138*>
139*> \param[out] X
140*> \verbatim
141*>          X is COMPLEX*16 array, dimension (LDX,NRHS)
142*>          If INFO = 0, the N-by-NRHS solution matrix X.
143*> \endverbatim
144*>
145*> \param[in] LDX
146*> \verbatim
147*>          LDX is INTEGER
148*>          The leading dimension of the array X.  LDX >= max(1,N).
149*> \endverbatim
150*>
151*> \param[out] WORK
152*> \verbatim
153*>          WORK is COMPLEX*16 array, dimension (N,NRHS)
154*>          This array is used to hold the residual vectors.
155*> \endverbatim
156*>
157*> \param[out] SWORK
158*> \verbatim
159*>          SWORK is COMPLEX array, dimension (N*(N+NRHS))
160*>          This array is used to use the single precision matrix and the
161*>          right-hand sides or solutions in single precision.
162*> \endverbatim
163*>
164*> \param[out] RWORK
165*> \verbatim
166*>          RWORK is DOUBLE PRECISION array, dimension (N)
167*> \endverbatim
168*>
169*> \param[out] ITER
170*> \verbatim
171*>          ITER is INTEGER
172*>          < 0: iterative refinement has failed, COMPLEX*16
173*>               factorization has been performed
174*>               -1 : the routine fell back to full precision for
175*>                    implementation- or machine-specific reasons
176*>               -2 : narrowing the precision induced an overflow,
177*>                    the routine fell back to full precision
178*>               -3 : failure of CPOTRF
179*>               -31: stop the iterative refinement after the 30th
180*>                    iterations
181*>          > 0: iterative refinement has been successfully used.
182*>               Returns the number of iterations
183*> \endverbatim
184*>
185*> \param[out] INFO
186*> \verbatim
187*>          INFO is INTEGER
188*>          = 0:  successful exit
189*>          < 0:  if INFO = -i, the i-th argument had an illegal value
190*>          > 0:  if INFO = i, the leading minor of order i of
191*>                (COMPLEX*16) A is not positive definite, so the
192*>                factorization could not be completed, and the solution
193*>                has not been computed.
194*> \endverbatim
195*
196*  Authors:
197*  ========
198*
199*> \author Univ. of Tennessee
200*> \author Univ. of California Berkeley
201*> \author Univ. of Colorado Denver
202*> \author NAG Ltd.
203*
204*> \date June 2016
205*
206*> \ingroup complex16POsolve
207*
208*  =====================================================================
209      SUBROUTINE ZCPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,
210     $                   SWORK, RWORK, ITER, INFO )
211*
212*  -- LAPACK driver routine (version 3.8.0) --
213*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
214*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
215*     June 2016
216*
217*     .. Scalar Arguments ..
218      CHARACTER          UPLO
219      INTEGER            INFO, ITER, LDA, LDB, LDX, N, NRHS
220*     ..
221*     .. Array Arguments ..
222      DOUBLE PRECISION   RWORK( * )
223      COMPLEX            SWORK( * )
224      COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( N, * ),
225     $                   X( LDX, * )
226*     ..
227*
228*  =====================================================================
229*
230*     .. Parameters ..
231      LOGICAL            DOITREF
232      PARAMETER          ( DOITREF = .TRUE. )
233*
234      INTEGER            ITERMAX
235      PARAMETER          ( ITERMAX = 30 )
236*
237      DOUBLE PRECISION   BWDMAX
238      PARAMETER          ( BWDMAX = 1.0E+00 )
239*
240      COMPLEX*16         NEGONE, ONE
241      PARAMETER          ( NEGONE = ( -1.0D+00, 0.0D+00 ),
242     $                   ONE = ( 1.0D+00, 0.0D+00 ) )
243*
244*     .. Local Scalars ..
245      INTEGER            I, IITER, PTSA, PTSX
246      DOUBLE PRECISION   ANRM, CTE, EPS, RNRM, XNRM
247      COMPLEX*16         ZDUM
248*
249*     .. External Subroutines ..
250      EXTERNAL           ZAXPY, ZHEMM, ZLACPY, ZLAT2C, ZLAG2C, CLAG2Z,
251     $                   CPOTRF, CPOTRS, XERBLA, ZPOTRF, ZPOTRS
252*     ..
253*     .. External Functions ..
254      INTEGER            IZAMAX
255      DOUBLE PRECISION   DLAMCH, ZLANHE
256      LOGICAL            LSAME
257      EXTERNAL           IZAMAX, DLAMCH, ZLANHE, LSAME
258*     ..
259*     .. Intrinsic Functions ..
260      INTRINSIC          ABS, DBLE, MAX, SQRT
261*     .. Statement Functions ..
262      DOUBLE PRECISION   CABS1
263*     ..
264*     .. Statement Function definitions ..
265      CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
266*     ..
267*     .. Executable Statements ..
268*
269      INFO = 0
270      ITER = 0
271*
272*     Test the input parameters.
273*
274      IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
275         INFO = -1
276      ELSE IF( N.LT.0 ) THEN
277         INFO = -2
278      ELSE IF( NRHS.LT.0 ) THEN
279         INFO = -3
280      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
281         INFO = -5
282      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
283         INFO = -7
284      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
285         INFO = -9
286      END IF
287      IF( INFO.NE.0 ) THEN
288         CALL XERBLA( 'ZCPOSV', -INFO )
289         RETURN
290      END IF
291*
292*     Quick return if (N.EQ.0).
293*
294      IF( N.EQ.0 )
295     $   RETURN
296*
297*     Skip single precision iterative refinement if a priori slower
298*     than double precision factorization.
299*
300      IF( .NOT.DOITREF ) THEN
301         ITER = -1
302         GO TO 40
303      END IF
304*
305*     Compute some constants.
306*
307      ANRM = ZLANHE( 'I', UPLO, N, A, LDA, RWORK )
308      EPS = DLAMCH( 'Epsilon' )
309      CTE = ANRM*EPS*SQRT( DBLE( N ) )*BWDMAX
310*
311*     Set the indices PTSA, PTSX for referencing SA and SX in SWORK.
312*
313      PTSA = 1
314      PTSX = PTSA + N*N
315*
316*     Convert B from double precision to single precision and store the
317*     result in SX.
318*
319      CALL ZLAG2C( N, NRHS, B, LDB, SWORK( PTSX ), N, INFO )
320*
321      IF( INFO.NE.0 ) THEN
322         ITER = -2
323         GO TO 40
324      END IF
325*
326*     Convert A from double precision to single precision and store the
327*     result in SA.
328*
329      CALL ZLAT2C( UPLO, N, A, LDA, SWORK( PTSA ), N, INFO )
330*
331      IF( INFO.NE.0 ) THEN
332         ITER = -2
333         GO TO 40
334      END IF
335*
336*     Compute the Cholesky factorization of SA.
337*
338      CALL CPOTRF( UPLO, N, SWORK( PTSA ), N, INFO )
339*
340      IF( INFO.NE.0 ) THEN
341         ITER = -3
342         GO TO 40
343      END IF
344*
345*     Solve the system SA*SX = SB.
346*
347      CALL CPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N,
348     $             INFO )
349*
350*     Convert SX back to COMPLEX*16
351*
352      CALL CLAG2Z( N, NRHS, SWORK( PTSX ), N, X, LDX, INFO )
353*
354*     Compute R = B - AX (R is WORK).
355*
356      CALL ZLACPY( 'All', N, NRHS, B, LDB, WORK, N )
357*
358      CALL ZHEMM( 'Left', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE,
359     $            WORK, N )
360*
361*     Check whether the NRHS normwise backward errors satisfy the
362*     stopping criterion. If yes, set ITER=0 and return.
363*
364      DO I = 1, NRHS
365         XNRM = CABS1( X( IZAMAX( N, X( 1, I ), 1 ), I ) )
366         RNRM = CABS1( WORK( IZAMAX( N, WORK( 1, I ), 1 ), I ) )
367         IF( RNRM.GT.XNRM*CTE )
368     $      GO TO 10
369      END DO
370*
371*     If we are here, the NRHS normwise backward errors satisfy the
372*     stopping criterion. We are good to exit.
373*
374      ITER = 0
375      RETURN
376*
377   10 CONTINUE
378*
379      DO 30 IITER = 1, ITERMAX
380*
381*        Convert R (in WORK) from double precision to single precision
382*        and store the result in SX.
383*
384         CALL ZLAG2C( N, NRHS, WORK, N, SWORK( PTSX ), N, INFO )
385*
386         IF( INFO.NE.0 ) THEN
387            ITER = -2
388            GO TO 40
389         END IF
390*
391*        Solve the system SA*SX = SR.
392*
393         CALL CPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N,
394     $                INFO )
395*
396*        Convert SX back to double precision and update the current
397*        iterate.
398*
399         CALL CLAG2Z( N, NRHS, SWORK( PTSX ), N, WORK, N, INFO )
400*
401         DO I = 1, NRHS
402            CALL ZAXPY( N, ONE, WORK( 1, I ), 1, X( 1, I ), 1 )
403         END DO
404*
405*        Compute R = B - AX (R is WORK).
406*
407         CALL ZLACPY( 'All', N, NRHS, B, LDB, WORK, N )
408*
409         CALL ZHEMM( 'L', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE,
410     $               WORK, N )
411*
412*        Check whether the NRHS normwise backward errors satisfy the
413*        stopping criterion. If yes, set ITER=IITER>0 and return.
414*
415         DO I = 1, NRHS
416            XNRM = CABS1( X( IZAMAX( N, X( 1, I ), 1 ), I ) )
417            RNRM = CABS1( WORK( IZAMAX( N, WORK( 1, I ), 1 ), I ) )
418            IF( RNRM.GT.XNRM*CTE )
419     $         GO TO 20
420         END DO
421*
422*        If we are here, the NRHS normwise backward errors satisfy the
423*        stopping criterion, we are good to exit.
424*
425         ITER = IITER
426*
427         RETURN
428*
429   20    CONTINUE
430*
431   30 CONTINUE
432*
433*     If we are at this place of the code, this is because we have
434*     performed ITER=ITERMAX iterations and never satisfied the
435*     stopping criterion, set up the ITER flag accordingly and follow
436*     up on double precision routine.
437*
438      ITER = -ITERMAX - 1
439*
440   40 CONTINUE
441*
442*     Single-precision iterative refinement failed to converge to a
443*     satisfactory solution, so we resort to double precision.
444*
445      CALL ZPOTRF( UPLO, N, A, LDA, INFO )
446*
447      IF( INFO.NE.0 )
448     $   RETURN
449*
450      CALL ZLACPY( 'All', N, NRHS, B, LDB, X, LDX )
451      CALL ZPOTRS( UPLO, N, NRHS, A, LDA, X, LDX, INFO )
452*
453      RETURN
454*
455*     End of ZCPOSV.
456*
457      END
458