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