1*> \brief <b> ZPTSVX computes the solution to system of linear equations A * X = B for PT 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 ZPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
22*                          RCOND, FERR, BERR, WORK, RWORK, INFO )
23*
24*       .. Scalar Arguments ..
25*       CHARACTER          FACT
26*       INTEGER            INFO, LDB, LDX, N, NRHS
27*       DOUBLE PRECISION   RCOND
28*       ..
29*       .. Array Arguments ..
30*       DOUBLE PRECISION   BERR( * ), D( * ), DF( * ), FERR( * ),
31*      $                   RWORK( * )
32*       COMPLEX*16         B( LDB, * ), E( * ), EF( * ), WORK( * ),
33*      $                   X( LDX, * )
34*       ..
35*
36*
37*> \par Purpose:
38*  =============
39*>
40*> \verbatim
41*>
42*> ZPTSVX uses the factorization A = L*D*L**H to compute the solution
43*> to a complex system of linear equations A*X = B, where A is an
44*> N-by-N Hermitian positive definite tridiagonal matrix and X and B
45*> are N-by-NRHS matrices.
46*>
47*> Error bounds on the solution and a condition estimate are also
48*> provided.
49*> \endverbatim
50*
51*> \par Description:
52*  =================
53*>
54*> \verbatim
55*>
56*> The following steps are performed:
57*>
58*> 1. If FACT = 'N', the matrix A is factored as A = L*D*L**H, where L
59*>    is a unit lower bidiagonal matrix and D is diagonal.  The
60*>    factorization can also be regarded as having the form
61*>    A = U**H*D*U.
62*>
63*> 2. If the leading i-by-i principal minor is not positive definite,
64*>    then the routine returns with INFO = i. Otherwise, the factored
65*>    form of A is used to estimate the condition number of the matrix
66*>    A.  If the reciprocal of the condition number is less than machine
67*>    precision, INFO = N+1 is returned as a warning, but the routine
68*>    still goes on to solve for X and compute error bounds as
69*>    described below.
70*>
71*> 3. The system of equations is solved for X using the factored form
72*>    of A.
73*>
74*> 4. Iterative refinement is applied to improve the computed solution
75*>    matrix and calculate error bounds and backward error estimates
76*>    for it.
77*> \endverbatim
78*
79*  Arguments:
80*  ==========
81*
82*> \param[in] FACT
83*> \verbatim
84*>          FACT is CHARACTER*1
85*>          Specifies whether or not the factored form of the matrix
86*>          A is supplied on entry.
87*>          = 'F':  On entry, DF and EF contain the factored form of A.
88*>                  D, E, DF, and EF will not be modified.
89*>          = 'N':  The matrix A will be copied to DF and EF and
90*>                  factored.
91*> \endverbatim
92*>
93*> \param[in] N
94*> \verbatim
95*>          N is INTEGER
96*>          The order of the matrix A.  N >= 0.
97*> \endverbatim
98*>
99*> \param[in] NRHS
100*> \verbatim
101*>          NRHS is INTEGER
102*>          The number of right hand sides, i.e., the number of columns
103*>          of the matrices B and X.  NRHS >= 0.
104*> \endverbatim
105*>
106*> \param[in] D
107*> \verbatim
108*>          D is DOUBLE PRECISION array, dimension (N)
109*>          The n diagonal elements of the tridiagonal matrix A.
110*> \endverbatim
111*>
112*> \param[in] E
113*> \verbatim
114*>          E is COMPLEX*16 array, dimension (N-1)
115*>          The (n-1) subdiagonal elements of the tridiagonal matrix A.
116*> \endverbatim
117*>
118*> \param[in,out] DF
119*> \verbatim
120*>          DF is DOUBLE PRECISION array, dimension (N)
121*>          If FACT = 'F', then DF is an input argument and on entry
122*>          contains the n diagonal elements of the diagonal matrix D
123*>          from the L*D*L**H factorization of A.
124*>          If FACT = 'N', then DF is an output argument and on exit
125*>          contains the n diagonal elements of the diagonal matrix D
126*>          from the L*D*L**H factorization of A.
127*> \endverbatim
128*>
129*> \param[in,out] EF
130*> \verbatim
131*>          EF is COMPLEX*16 array, dimension (N-1)
132*>          If FACT = 'F', then EF is an input argument and on entry
133*>          contains the (n-1) subdiagonal elements of the unit
134*>          bidiagonal factor L from the L*D*L**H factorization of A.
135*>          If FACT = 'N', then EF is an output argument and on exit
136*>          contains the (n-1) subdiagonal elements of the unit
137*>          bidiagonal factor L from the L*D*L**H factorization of A.
138*> \endverbatim
139*>
140*> \param[in] B
141*> \verbatim
142*>          B is COMPLEX*16 array, dimension (LDB,NRHS)
143*>          The N-by-NRHS right hand side matrix B.
144*> \endverbatim
145*>
146*> \param[in] LDB
147*> \verbatim
148*>          LDB is INTEGER
149*>          The leading dimension of the array B.  LDB >= max(1,N).
150*> \endverbatim
151*>
152*> \param[out] X
153*> \verbatim
154*>          X is COMPLEX*16 array, dimension (LDX,NRHS)
155*>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
156*> \endverbatim
157*>
158*> \param[in] LDX
159*> \verbatim
160*>          LDX is INTEGER
161*>          The leading dimension of the array X.  LDX >= max(1,N).
162*> \endverbatim
163*>
164*> \param[out] RCOND
165*> \verbatim
166*>          RCOND is DOUBLE PRECISION
167*>          The reciprocal condition number of the matrix A.  If RCOND
168*>          is less than the machine precision (in particular, if
169*>          RCOND = 0), the matrix is singular to working precision.
170*>          This condition is indicated by a return code of INFO > 0.
171*> \endverbatim
172*>
173*> \param[out] FERR
174*> \verbatim
175*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
176*>          The forward error bound for each solution vector
177*>          X(j) (the j-th column of the solution matrix X).
178*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
179*>          is an estimated upper bound for the magnitude of the largest
180*>          element in (X(j) - XTRUE) divided by the magnitude of the
181*>          largest element in X(j).
182*> \endverbatim
183*>
184*> \param[out] BERR
185*> \verbatim
186*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
187*>          The componentwise relative backward error of each solution
188*>          vector X(j) (i.e., the smallest relative change in any
189*>          element of A or B that makes X(j) an exact solution).
190*> \endverbatim
191*>
192*> \param[out] WORK
193*> \verbatim
194*>          WORK is COMPLEX*16 array, dimension (N)
195*> \endverbatim
196*>
197*> \param[out] RWORK
198*> \verbatim
199*>          RWORK is DOUBLE PRECISION array, dimension (N)
200*> \endverbatim
201*>
202*> \param[out] INFO
203*> \verbatim
204*>          INFO is INTEGER
205*>          = 0:  successful exit
206*>          < 0:  if INFO = -i, the i-th argument had an illegal value
207*>          > 0:  if INFO = i, and i is
208*>                <= N:  the leading minor of order i of A is
209*>                       not positive definite, so the factorization
210*>                       could not be completed, and the solution has not
211*>                       been computed. RCOND = 0 is returned.
212*>                = N+1: U is nonsingular, but RCOND is less than machine
213*>                       precision, meaning that the matrix is singular
214*>                       to working precision.  Nevertheless, the
215*>                       solution and error bounds are computed because
216*>                       there are a number of situations where the
217*>                       computed solution can be more accurate than the
218*>                       value of RCOND would suggest.
219*> \endverbatim
220*
221*  Authors:
222*  ========
223*
224*> \author Univ. of Tennessee
225*> \author Univ. of California Berkeley
226*> \author Univ. of Colorado Denver
227*> \author NAG Ltd.
228*
229*> \date September 2012
230*
231*> \ingroup complex16PTsolve
232*
233*  =====================================================================
234      SUBROUTINE ZPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
235     $                   RCOND, FERR, BERR, WORK, RWORK, INFO )
236*
237*  -- LAPACK driver routine (version 3.4.2) --
238*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
239*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
240*     September 2012
241*
242*     .. Scalar Arguments ..
243      CHARACTER          FACT
244      INTEGER            INFO, LDB, LDX, N, NRHS
245      DOUBLE PRECISION   RCOND
246*     ..
247*     .. Array Arguments ..
248      DOUBLE PRECISION   BERR( * ), D( * ), DF( * ), FERR( * ),
249     $                   RWORK( * )
250      COMPLEX*16         B( LDB, * ), E( * ), EF( * ), WORK( * ),
251     $                   X( LDX, * )
252*     ..
253*
254*  =====================================================================
255*
256*     .. Parameters ..
257      DOUBLE PRECISION   ZERO
258      PARAMETER          ( ZERO = 0.0D+0 )
259*     ..
260*     .. Local Scalars ..
261      LOGICAL            NOFACT
262      DOUBLE PRECISION   ANORM
263*     ..
264*     .. External Functions ..
265      LOGICAL            LSAME
266      DOUBLE PRECISION   DLAMCH, ZLANHT
267      EXTERNAL           LSAME, DLAMCH, ZLANHT
268*     ..
269*     .. External Subroutines ..
270      EXTERNAL           DCOPY, XERBLA, ZCOPY, ZLACPY, ZPTCON, ZPTRFS,
271     $                   ZPTTRF, ZPTTRS
272*     ..
273*     .. Intrinsic Functions ..
274      INTRINSIC          MAX
275*     ..
276*     .. Executable Statements ..
277*
278*     Test the input parameters.
279*
280      INFO = 0
281      NOFACT = LSAME( FACT, 'N' )
282      IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
283         INFO = -1
284      ELSE IF( N.LT.0 ) THEN
285         INFO = -2
286      ELSE IF( NRHS.LT.0 ) THEN
287         INFO = -3
288      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
289         INFO = -9
290      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
291         INFO = -11
292      END IF
293      IF( INFO.NE.0 ) THEN
294         CALL XERBLA( 'ZPTSVX', -INFO )
295         RETURN
296      END IF
297*
298      IF( NOFACT ) THEN
299*
300*        Compute the L*D*L**H (or U**H*D*U) factorization of A.
301*
302         CALL DCOPY( N, D, 1, DF, 1 )
303         IF( N.GT.1 )
304     $      CALL ZCOPY( N-1, E, 1, EF, 1 )
305         CALL ZPTTRF( N, DF, EF, INFO )
306*
307*        Return if INFO is non-zero.
308*
309         IF( INFO.GT.0 )THEN
310            RCOND = ZERO
311            RETURN
312         END IF
313      END IF
314*
315*     Compute the norm of the matrix A.
316*
317      ANORM = ZLANHT( '1', N, D, E )
318*
319*     Compute the reciprocal of the condition number of A.
320*
321      CALL ZPTCON( N, DF, EF, ANORM, RCOND, RWORK, INFO )
322*
323*     Compute the solution vectors X.
324*
325      CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
326      CALL ZPTTRS( 'Lower', N, NRHS, DF, EF, X, LDX, INFO )
327*
328*     Use iterative refinement to improve the computed solutions and
329*     compute error bounds and backward error estimates for them.
330*
331      CALL ZPTRFS( 'Lower', N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
332     $             BERR, WORK, RWORK, INFO )
333*
334*     Set INFO = N+1 if the matrix is singular to working precision.
335*
336      IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
337     $   INFO = N + 1
338*
339      RETURN
340*
341*     End of ZPTSVX
342*
343      END
344