```001:       SUBROUTINE SPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
002:      \$                   RCOND, FERR, BERR, WORK, INFO )
003: *
004: *  -- LAPACK routine (version 3.2) --
005: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
006: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
007: *     November 2006
008: *
009: *     .. Scalar Arguments ..
010:       CHARACTER          FACT
011:       INTEGER            INFO, LDB, LDX, N, NRHS
012:       REAL               RCOND
013: *     ..
014: *     .. Array Arguments ..
015:       REAL               B( LDB, * ), BERR( * ), D( * ), DF( * ),
016:      \$                   E( * ), EF( * ), FERR( * ), WORK( * ),
017:      \$                   X( LDX, * )
018: *     ..
019: *
020: *  Purpose
021: *  =======
022: *
023: *  SPTSVX uses the factorization A = L*D*L**T to compute the solution
024: *  to a real system of linear equations A*X = B, where A is an N-by-N
025: *  symmetric positive definite tridiagonal matrix and X and B are
026: *  N-by-NRHS matrices.
027: *
028: *  Error bounds on the solution and a condition estimate are also
029: *  provided.
030: *
031: *  Description
032: *  ===========
033: *
034: *  The following steps are performed:
035: *
036: *  1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L
037: *     is a unit lower bidiagonal matrix and D is diagonal.  The
038: *     factorization can also be regarded as having the form
039: *     A = U**T*D*U.
040: *
041: *  2. If the leading i-by-i principal minor is not positive definite,
042: *     then the routine returns with INFO = i. Otherwise, the factored
043: *     form of A is used to estimate the condition number of the matrix
044: *     A.  If the reciprocal of the condition number is less than machine
045: *     precision, INFO = N+1 is returned as a warning, but the routine
046: *     still goes on to solve for X and compute error bounds as
047: *     described below.
048: *
049: *  3. The system of equations is solved for X using the factored form
050: *     of A.
051: *
052: *  4. Iterative refinement is applied to improve the computed solution
053: *     matrix and calculate error bounds and backward error estimates
054: *     for it.
055: *
056: *  Arguments
057: *  =========
058: *
059: *  FACT    (input) CHARACTER*1
060: *          Specifies whether or not the factored form of A has been
061: *          supplied on entry.
062: *          = 'F':  On entry, DF and EF contain the factored form of A.
063: *                  D, E, DF, and EF will not be modified.
064: *          = 'N':  The matrix A will be copied to DF and EF and
065: *                  factored.
066: *
067: *  N       (input) INTEGER
068: *          The order of the matrix A.  N >= 0.
069: *
070: *  NRHS    (input) INTEGER
071: *          The number of right hand sides, i.e., the number of columns
072: *          of the matrices B and X.  NRHS >= 0.
073: *
074: *  D       (input) REAL array, dimension (N)
075: *          The n diagonal elements of the tridiagonal matrix A.
076: *
077: *  E       (input) REAL array, dimension (N-1)
078: *          The (n-1) subdiagonal elements of the tridiagonal matrix A.
079: *
080: *  DF      (input or output) REAL array, dimension (N)
081: *          If FACT = 'F', then DF is an input argument and on entry
082: *          contains the n diagonal elements of the diagonal matrix D
083: *          from the L*D*L**T factorization of A.
084: *          If FACT = 'N', then DF is an output argument and on exit
085: *          contains the n diagonal elements of the diagonal matrix D
086: *          from the L*D*L**T factorization of A.
087: *
088: *  EF      (input or output) REAL array, dimension (N-1)
089: *          If FACT = 'F', then EF is an input argument and on entry
090: *          contains the (n-1) subdiagonal elements of the unit
091: *          bidiagonal factor L from the L*D*L**T factorization of A.
092: *          If FACT = 'N', then EF is an output argument and on exit
093: *          contains the (n-1) subdiagonal elements of the unit
094: *          bidiagonal factor L from the L*D*L**T factorization of A.
095: *
096: *  B       (input) REAL array, dimension (LDB,NRHS)
097: *          The N-by-NRHS right hand side matrix B.
098: *
099: *  LDB     (input) INTEGER
100: *          The leading dimension of the array B.  LDB >= max(1,N).
101: *
102: *  X       (output) REAL array, dimension (LDX,NRHS)
103: *          If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.
104: *
105: *  LDX     (input) INTEGER
106: *          The leading dimension of the array X.  LDX >= max(1,N).
107: *
108: *  RCOND   (output) REAL
109: *          The reciprocal condition number of the matrix A.  If RCOND
110: *          is less than the machine precision (in particular, if
111: *          RCOND = 0), the matrix is singular to working precision.
112: *          This condition is indicated by a return code of INFO > 0.
113: *
114: *  FERR    (output) REAL array, dimension (NRHS)
115: *          The forward error bound for each solution vector
116: *          X(j) (the j-th column of the solution matrix X).
117: *          If XTRUE is the true solution corresponding to X(j), FERR(j)
118: *          is an estimated upper bound for the magnitude of the largest
119: *          element in (X(j) - XTRUE) divided by the magnitude of the
120: *          largest element in X(j).
121: *
122: *  BERR    (output) REAL array, dimension (NRHS)
123: *          The componentwise relative backward error of each solution
124: *          vector X(j) (i.e., the smallest relative change in any
125: *          element of A or B that makes X(j) an exact solution).
126: *
127: *  WORK    (workspace) REAL array, dimension (2*N)
128: *
129: *  INFO    (output) INTEGER
130: *          = 0:  successful exit
131: *          < 0:  if INFO = -i, the i-th argument had an illegal value
132: *          > 0:  if INFO = i, and i is
133: *                <= N:  the leading minor of order i of A is
134: *                       not positive definite, so the factorization
135: *                       could not be completed, and the solution has not
136: *                       been computed. RCOND = 0 is returned.
137: *                = N+1: U is nonsingular, but RCOND is less than machine
138: *                       precision, meaning that the matrix is singular
139: *                       to working precision.  Nevertheless, the
140: *                       solution and error bounds are computed because
141: *                       there are a number of situations where the
142: *                       computed solution can be more accurate than the
143: *                       value of RCOND would suggest.
144: *
145: *  =====================================================================
146: *
147: *     .. Parameters ..
148:       REAL               ZERO
149:       PARAMETER          ( ZERO = 0.0E+0 )
150: *     ..
151: *     .. Local Scalars ..
152:       LOGICAL            NOFACT
153:       REAL               ANORM
154: *     ..
155: *     .. External Functions ..
156:       LOGICAL            LSAME
157:       REAL               SLAMCH, SLANST
158:       EXTERNAL           LSAME, SLAMCH, SLANST
159: *     ..
160: *     .. External Subroutines ..
161:       EXTERNAL           SCOPY, SLACPY, SPTCON, SPTRFS, SPTTRF, SPTTRS,
162:      \$                   XERBLA
163: *     ..
164: *     .. Intrinsic Functions ..
165:       INTRINSIC          MAX
166: *     ..
167: *     .. Executable Statements ..
168: *
169: *     Test the input parameters.
170: *
171:       INFO = 0
172:       NOFACT = LSAME( FACT, 'N' )
173:       IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
174:          INFO = -1
175:       ELSE IF( N.LT.0 ) THEN
176:          INFO = -2
177:       ELSE IF( NRHS.LT.0 ) THEN
178:          INFO = -3
179:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
180:          INFO = -9
181:       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
182:          INFO = -11
183:       END IF
184:       IF( INFO.NE.0 ) THEN
185:          CALL XERBLA( 'SPTSVX', -INFO )
186:          RETURN
187:       END IF
188: *
189:       IF( NOFACT ) THEN
190: *
191: *        Compute the L*D*L' (or U'*D*U) factorization of A.
192: *
193:          CALL SCOPY( N, D, 1, DF, 1 )
194:          IF( N.GT.1 )
195:      \$      CALL SCOPY( N-1, E, 1, EF, 1 )
196:          CALL SPTTRF( N, DF, EF, INFO )
197: *
198: *        Return if INFO is non-zero.
199: *
200:          IF( INFO.GT.0 )THEN
201:             RCOND = ZERO
202:             RETURN
203:          END IF
204:       END IF
205: *
206: *     Compute the norm of the matrix A.
207: *
208:       ANORM = SLANST( '1', N, D, E )
209: *
210: *     Compute the reciprocal of the condition number of A.
211: *
212:       CALL SPTCON( N, DF, EF, ANORM, RCOND, WORK, INFO )
213: *
214: *     Compute the solution vectors X.
215: *
216:       CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
217:       CALL SPTTRS( N, NRHS, DF, EF, X, LDX, INFO )
218: *
219: *     Use iterative refinement to improve the computed solutions and
220: *     compute error bounds and backward error estimates for them.
221: *
222:       CALL SPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR,
223:      \$             WORK, INFO )
224: *
225: *     Set INFO = N+1 if the matrix is singular to working precision.
226: *
227:       IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
228:      \$   INFO = N + 1
229: *
230:       RETURN
231: *
232: *     End of SPTSVX
233: *
234:       END
235: ```