1*> \brief <b> ZGTSVX computes the solution to system of linear equations A * X = B for GT matrices </b>
2*
3*  =========== DOCUMENTATION ===========
4*
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7*
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17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE ZGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
22*                          DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
23*                          WORK, RWORK, INFO )
24*
25*       .. Scalar Arguments ..
26*       CHARACTER          FACT, TRANS
27*       INTEGER            INFO, LDB, LDX, N, NRHS
28*       DOUBLE PRECISION   RCOND
29*       ..
30*       .. Array Arguments ..
31*       INTEGER            IPIV( * )
32*       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
33*       COMPLEX*16         B( LDB, * ), D( * ), DF( * ), DL( * ),
34*      $                   DLF( * ), DU( * ), DU2( * ), DUF( * ),
35*      $                   WORK( * ), X( LDX, * )
36*       ..
37*
38*
39*> \par Purpose:
40*  =============
41*>
42*> \verbatim
43*>
44*> ZGTSVX uses the LU factorization to compute the solution to a complex
45*> system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
46*> where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
47*> matrices.
48*>
49*> Error bounds on the solution and a condition estimate are also
50*> provided.
51*> \endverbatim
52*
53*> \par Description:
54*  =================
55*>
56*> \verbatim
57*>
58*> The following steps are performed:
59*>
60*> 1. If FACT = 'N', the LU decomposition is used to factor the matrix A
61*>    as A = L * U, where L is a product of permutation and unit lower
62*>    bidiagonal matrices and U is upper triangular with nonzeros in
63*>    only the main diagonal and first two superdiagonals.
64*>
65*> 2. If some U(i,i)=0, so that U is exactly singular, then the routine
66*>    returns with INFO = i. Otherwise, the factored form of A is used
67*>    to estimate the condition number of the matrix A.  If the
68*>    reciprocal of the condition number is less than machine precision,
69*>    INFO = N+1 is returned as a warning, but the routine still goes on
70*>    to solve for X and compute error bounds as described below.
71*>
72*> 3. The system of equations is solved for X using the factored form
73*>    of A.
74*>
75*> 4. Iterative refinement is applied to improve the computed solution
76*>    matrix and calculate error bounds and backward error estimates
77*>    for it.
78*> \endverbatim
79*
80*  Arguments:
81*  ==========
82*
83*> \param[in] FACT
84*> \verbatim
85*>          FACT is CHARACTER*1
86*>          Specifies whether or not the factored form of A has been
87*>          supplied on entry.
88*>          = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored form
89*>                  of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not
90*>                  be modified.
91*>          = 'N':  The matrix will be copied to DLF, DF, and DUF
92*>                  and factored.
93*> \endverbatim
94*>
95*> \param[in] TRANS
96*> \verbatim
97*>          TRANS is CHARACTER*1
98*>          Specifies the form of the system of equations:
99*>          = 'N':  A * X = B     (No transpose)
100*>          = 'T':  A**T * X = B  (Transpose)
101*>          = 'C':  A**H * X = B  (Conjugate transpose)
102*> \endverbatim
103*>
104*> \param[in] N
105*> \verbatim
106*>          N is INTEGER
107*>          The order of the matrix A.  N >= 0.
108*> \endverbatim
109*>
110*> \param[in] NRHS
111*> \verbatim
112*>          NRHS is INTEGER
113*>          The number of right hand sides, i.e., the number of columns
114*>          of the matrix B.  NRHS >= 0.
115*> \endverbatim
116*>
117*> \param[in] DL
118*> \verbatim
119*>          DL is COMPLEX*16 array, dimension (N-1)
120*>          The (n-1) subdiagonal elements of A.
121*> \endverbatim
122*>
123*> \param[in] D
124*> \verbatim
125*>          D is COMPLEX*16 array, dimension (N)
126*>          The n diagonal elements of A.
127*> \endverbatim
128*>
129*> \param[in] DU
130*> \verbatim
131*>          DU is COMPLEX*16 array, dimension (N-1)
132*>          The (n-1) superdiagonal elements of A.
133*> \endverbatim
134*>
135*> \param[in,out] DLF
136*> \verbatim
137*>          DLF is COMPLEX*16 array, dimension (N-1)
138*>          If FACT = 'F', then DLF is an input argument and on entry
139*>          contains the (n-1) multipliers that define the matrix L from
140*>          the LU factorization of A as computed by ZGTTRF.
141*>
142*>          If FACT = 'N', then DLF is an output argument and on exit
143*>          contains the (n-1) multipliers that define the matrix L from
144*>          the LU factorization of A.
145*> \endverbatim
146*>
147*> \param[in,out] DF
148*> \verbatim
149*>          DF is COMPLEX*16 array, dimension (N)
150*>          If FACT = 'F', then DF is an input argument and on entry
151*>          contains the n diagonal elements of the upper triangular
152*>          matrix U from the LU factorization of A.
153*>
154*>          If FACT = 'N', then DF is an output argument and on exit
155*>          contains the n diagonal elements of the upper triangular
156*>          matrix U from the LU factorization of A.
157*> \endverbatim
158*>
159*> \param[in,out] DUF
160*> \verbatim
161*>          DUF is COMPLEX*16 array, dimension (N-1)
162*>          If FACT = 'F', then DUF is an input argument and on entry
163*>          contains the (n-1) elements of the first superdiagonal of U.
164*>
165*>          If FACT = 'N', then DUF is an output argument and on exit
166*>          contains the (n-1) elements of the first superdiagonal of U.
167*> \endverbatim
168*>
169*> \param[in,out] DU2
170*> \verbatim
171*>          DU2 is COMPLEX*16 array, dimension (N-2)
172*>          If FACT = 'F', then DU2 is an input argument and on entry
173*>          contains the (n-2) elements of the second superdiagonal of
174*>          U.
175*>
176*>          If FACT = 'N', then DU2 is an output argument and on exit
177*>          contains the (n-2) elements of the second superdiagonal of
178*>          U.
179*> \endverbatim
180*>
181*> \param[in,out] IPIV
182*> \verbatim
183*>          IPIV is INTEGER array, dimension (N)
184*>          If FACT = 'F', then IPIV is an input argument and on entry
185*>          contains the pivot indices from the LU factorization of A as
186*>          computed by ZGTTRF.
187*>
188*>          If FACT = 'N', then IPIV is an output argument and on exit
189*>          contains the pivot indices from the LU factorization of A;
190*>          row i of the matrix was interchanged with row IPIV(i).
191*>          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
192*>          a row interchange was not required.
193*> \endverbatim
194*>
195*> \param[in] B
196*> \verbatim
197*>          B is COMPLEX*16 array, dimension (LDB,NRHS)
198*>          The N-by-NRHS right hand side matrix B.
199*> \endverbatim
200*>
201*> \param[in] LDB
202*> \verbatim
203*>          LDB is INTEGER
204*>          The leading dimension of the array B.  LDB >= max(1,N).
205*> \endverbatim
206*>
207*> \param[out] X
208*> \verbatim
209*>          X is COMPLEX*16 array, dimension (LDX,NRHS)
210*>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
211*> \endverbatim
212*>
213*> \param[in] LDX
214*> \verbatim
215*>          LDX is INTEGER
216*>          The leading dimension of the array X.  LDX >= max(1,N).
217*> \endverbatim
218*>
219*> \param[out] RCOND
220*> \verbatim
221*>          RCOND is DOUBLE PRECISION
222*>          The estimate of the reciprocal condition number of the matrix
223*>          A.  If RCOND is less than the machine precision (in
224*>          particular, if RCOND = 0), the matrix is singular to working
225*>          precision.  This condition is indicated by a return code of
226*>          INFO > 0.
227*> \endverbatim
228*>
229*> \param[out] FERR
230*> \verbatim
231*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
232*>          The estimated forward error bound for each solution vector
233*>          X(j) (the j-th column of the solution matrix X).
234*>          If XTRUE is the true solution corresponding to X(j), FERR(j)
235*>          is an estimated upper bound for the magnitude of the largest
236*>          element in (X(j) - XTRUE) divided by the magnitude of the
237*>          largest element in X(j).  The estimate is as reliable as
238*>          the estimate for RCOND, and is almost always a slight
239*>          overestimate of the true error.
240*> \endverbatim
241*>
242*> \param[out] BERR
243*> \verbatim
244*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
245*>          The componentwise relative backward error of each solution
246*>          vector X(j) (i.e., the smallest relative change in
247*>          any element of A or B that makes X(j) an exact solution).
248*> \endverbatim
249*>
250*> \param[out] WORK
251*> \verbatim
252*>          WORK is COMPLEX*16 array, dimension (2*N)
253*> \endverbatim
254*>
255*> \param[out] RWORK
256*> \verbatim
257*>          RWORK is DOUBLE PRECISION array, dimension (N)
258*> \endverbatim
259*>
260*> \param[out] INFO
261*> \verbatim
262*>          INFO is INTEGER
263*>          = 0:  successful exit
264*>          < 0:  if INFO = -i, the i-th argument had an illegal value
265*>          > 0:  if INFO = i, and i is
266*>                <= N:  U(i,i) is exactly zero.  The factorization
267*>                       has not been completed unless i = N, but the
268*>                       factor U is exactly singular, so the solution
269*>                       and error bounds could not be computed.
270*>                       RCOND = 0 is returned.
271*>                = N+1: U is nonsingular, but RCOND is less than machine
272*>                       precision, meaning that the matrix is singular
273*>                       to working precision.  Nevertheless, the
274*>                       solution and error bounds are computed because
275*>                       there are a number of situations where the
276*>                       computed solution can be more accurate than the
277*>                       value of RCOND would suggest.
278*> \endverbatim
279*
280*  Authors:
281*  ========
282*
283*> \author Univ. of Tennessee
284*> \author Univ. of California Berkeley
285*> \author Univ. of Colorado Denver
286*> \author NAG Ltd.
287*
288*> \ingroup complex16GTsolve
289*
290*  =====================================================================
291      SUBROUTINE ZGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
292     $                   DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
293     $                   WORK, RWORK, INFO )
294*
295*  -- LAPACK driver routine --
296*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
297*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
298*
299*     .. Scalar Arguments ..
300      CHARACTER          FACT, TRANS
301      INTEGER            INFO, LDB, LDX, N, NRHS
302      DOUBLE PRECISION   RCOND
303*     ..
304*     .. Array Arguments ..
305      INTEGER            IPIV( * )
306      DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
307      COMPLEX*16         B( LDB, * ), D( * ), DF( * ), DL( * ),
308     $                   DLF( * ), DU( * ), DU2( * ), DUF( * ),
309     $                   WORK( * ), X( LDX, * )
310*     ..
311*
312*  =====================================================================
313*
314*     .. Parameters ..
315      DOUBLE PRECISION   ZERO
316      PARAMETER          ( ZERO = 0.0D+0 )
317*     ..
318*     .. Local Scalars ..
319      LOGICAL            NOFACT, NOTRAN
320      CHARACTER          NORM
321      DOUBLE PRECISION   ANORM
322*     ..
323*     .. External Functions ..
324      LOGICAL            LSAME
325      DOUBLE PRECISION   DLAMCH, ZLANGT
326      EXTERNAL           LSAME, DLAMCH, ZLANGT
327*     ..
328*     .. External Subroutines ..
329      EXTERNAL           XERBLA, ZCOPY, ZGTCON, ZGTRFS, ZGTTRF, ZGTTRS,
330     $                   ZLACPY
331*     ..
332*     .. Intrinsic Functions ..
333      INTRINSIC          MAX
334*     ..
335*     .. Executable Statements ..
336*
337      INFO = 0
338      NOFACT = LSAME( FACT, 'N' )
339      NOTRAN = LSAME( TRANS, 'N' )
340      IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
341         INFO = -1
342      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
343     $         LSAME( TRANS, 'C' ) ) THEN
344         INFO = -2
345      ELSE IF( N.LT.0 ) THEN
346         INFO = -3
347      ELSE IF( NRHS.LT.0 ) THEN
348         INFO = -4
349      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
350         INFO = -14
351      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
352         INFO = -16
353      END IF
354      IF( INFO.NE.0 ) THEN
355         CALL XERBLA( 'ZGTSVX', -INFO )
356         RETURN
357      END IF
358*
359      IF( NOFACT ) THEN
360*
361*        Compute the LU factorization of A.
362*
363         CALL ZCOPY( N, D, 1, DF, 1 )
364         IF( N.GT.1 ) THEN
365            CALL ZCOPY( N-1, DL, 1, DLF, 1 )
366            CALL ZCOPY( N-1, DU, 1, DUF, 1 )
367         END IF
368         CALL ZGTTRF( N, DLF, DF, DUF, DU2, IPIV, INFO )
369*
370*        Return if INFO is non-zero.
371*
372         IF( INFO.GT.0 )THEN
373            RCOND = ZERO
374            RETURN
375         END IF
376      END IF
377*
378*     Compute the norm of the matrix A.
379*
380      IF( NOTRAN ) THEN
381         NORM = '1'
382      ELSE
383         NORM = 'I'
384      END IF
385      ANORM = ZLANGT( NORM, N, DL, D, DU )
386*
387*     Compute the reciprocal of the condition number of A.
388*
389      CALL ZGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK,
390     $             INFO )
391*
392*     Compute the solution vectors X.
393*
394      CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
395      CALL ZGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX,
396     $             INFO )
397*
398*     Use iterative refinement to improve the computed solutions and
399*     compute error bounds and backward error estimates for them.
400*
401      CALL ZGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV,
402     $             B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
403*
404*     Set INFO = N+1 if the matrix is singular to working precision.
405*
406      IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
407     $   INFO = N + 1
408*
409      RETURN
410*
411*     End of ZGTSVX
412*
413      END
414