LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
sgtsvx.f
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00001 *> \brief \b SGTSVX
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download SGTSVX + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgtsvx.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
00022 *                          DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
00023 *                          WORK, IWORK, INFO )
00024 * 
00025 *       .. Scalar Arguments ..
00026 *       CHARACTER          FACT, TRANS
00027 *       INTEGER            INFO, LDB, LDX, N, NRHS
00028 *       REAL               RCOND
00029 *       ..
00030 *       .. Array Arguments ..
00031 *       INTEGER            IPIV( * ), IWORK( * )
00032 *       REAL               B( LDB, * ), BERR( * ), D( * ), DF( * ),
00033 *      $                   DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
00034 *      $                   FERR( * ), WORK( * ), X( LDX, * )
00035 *       ..
00036 *  
00037 *
00038 *> \par Purpose:
00039 *  =============
00040 *>
00041 *> \verbatim
00042 *>
00043 *> SGTSVX uses the LU factorization to compute the solution to a real
00044 *> system of linear equations A * X = B or A**T * X = B,
00045 *> where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
00046 *> matrices.
00047 *>
00048 *> Error bounds on the solution and a condition estimate are also
00049 *> provided.
00050 *> \endverbatim
00051 *
00052 *> \par Description:
00053 *  =================
00054 *>
00055 *> \verbatim
00056 *>
00057 *> The following steps are performed:
00058 *>
00059 *> 1. If FACT = 'N', the LU decomposition is used to factor the matrix A
00060 *>    as A = L * U, where L is a product of permutation and unit lower
00061 *>    bidiagonal matrices and U is upper triangular with nonzeros in
00062 *>    only the main diagonal and first two superdiagonals.
00063 *>
00064 *> 2. If some U(i,i)=0, so that U is exactly singular, then the routine
00065 *>    returns with INFO = i. Otherwise, the factored form of A is used
00066 *>    to estimate the condition number of the matrix A.  If the
00067 *>    reciprocal of the condition number is less than machine precision,
00068 *>    INFO = N+1 is returned as a warning, but the routine still goes on
00069 *>    to solve for X and compute error bounds as described below.
00070 *>
00071 *> 3. The system of equations is solved for X using the factored form
00072 *>    of A.
00073 *>
00074 *> 4. Iterative refinement is applied to improve the computed solution
00075 *>    matrix and calculate error bounds and backward error estimates
00076 *>    for it.
00077 *> \endverbatim
00078 *
00079 *  Arguments:
00080 *  ==========
00081 *
00082 *> \param[in] FACT
00083 *> \verbatim
00084 *>          FACT is CHARACTER*1
00085 *>          Specifies whether or not the factored form of A has been
00086 *>          supplied on entry.
00087 *>          = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored
00088 *>                  form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
00089 *>                  will not be modified.
00090 *>          = 'N':  The matrix will be copied to DLF, DF, and DUF
00091 *>                  and factored.
00092 *> \endverbatim
00093 *>
00094 *> \param[in] TRANS
00095 *> \verbatim
00096 *>          TRANS is CHARACTER*1
00097 *>          Specifies the form of the system of equations:
00098 *>          = 'N':  A * X = B     (No transpose)
00099 *>          = 'T':  A**T * X = B  (Transpose)
00100 *>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
00101 *> \endverbatim
00102 *>
00103 *> \param[in] N
00104 *> \verbatim
00105 *>          N is INTEGER
00106 *>          The order of the matrix A.  N >= 0.
00107 *> \endverbatim
00108 *>
00109 *> \param[in] NRHS
00110 *> \verbatim
00111 *>          NRHS is INTEGER
00112 *>          The number of right hand sides, i.e., the number of columns
00113 *>          of the matrix B.  NRHS >= 0.
00114 *> \endverbatim
00115 *>
00116 *> \param[in] DL
00117 *> \verbatim
00118 *>          DL is REAL array, dimension (N-1)
00119 *>          The (n-1) subdiagonal elements of A.
00120 *> \endverbatim
00121 *>
00122 *> \param[in] D
00123 *> \verbatim
00124 *>          D is REAL array, dimension (N)
00125 *>          The n diagonal elements of A.
00126 *> \endverbatim
00127 *>
00128 *> \param[in] DU
00129 *> \verbatim
00130 *>          DU is REAL array, dimension (N-1)
00131 *>          The (n-1) superdiagonal elements of A.
00132 *> \endverbatim
00133 *>
00134 *> \param[in,out] DLF
00135 *> \verbatim
00136 *>          DLF is REAL array, dimension (N-1)
00137 *>          If FACT = 'F', then DLF is an input argument and on entry
00138 *>          contains the (n-1) multipliers that define the matrix L from
00139 *>          the LU factorization of A as computed by SGTTRF.
00140 *>
00141 *>          If FACT = 'N', then DLF is an output argument and on exit
00142 *>          contains the (n-1) multipliers that define the matrix L from
00143 *>          the LU factorization of A.
00144 *> \endverbatim
00145 *>
00146 *> \param[in,out] DF
00147 *> \verbatim
00148 *>          DF is REAL array, dimension (N)
00149 *>          If FACT = 'F', then DF is an input argument and on entry
00150 *>          contains the n diagonal elements of the upper triangular
00151 *>          matrix U from the LU factorization of A.
00152 *>
00153 *>          If FACT = 'N', then DF is an output argument and on exit
00154 *>          contains the n diagonal elements of the upper triangular
00155 *>          matrix U from the LU factorization of A.
00156 *> \endverbatim
00157 *>
00158 *> \param[in,out] DUF
00159 *> \verbatim
00160 *>          DUF is REAL array, dimension (N-1)
00161 *>          If FACT = 'F', then DUF is an input argument and on entry
00162 *>          contains the (n-1) elements of the first superdiagonal of U.
00163 *>
00164 *>          If FACT = 'N', then DUF is an output argument and on exit
00165 *>          contains the (n-1) elements of the first superdiagonal of U.
00166 *> \endverbatim
00167 *>
00168 *> \param[in,out] DU2
00169 *> \verbatim
00170 *>          DU2 is REAL array, dimension (N-2)
00171 *>          If FACT = 'F', then DU2 is an input argument and on entry
00172 *>          contains the (n-2) elements of the second superdiagonal of
00173 *>          U.
00174 *>
00175 *>          If FACT = 'N', then DU2 is an output argument and on exit
00176 *>          contains the (n-2) elements of the second superdiagonal of
00177 *>          U.
00178 *> \endverbatim
00179 *>
00180 *> \param[in,out] IPIV
00181 *> \verbatim
00182 *>          IPIV is INTEGER array, dimension (N)
00183 *>          If FACT = 'F', then IPIV is an input argument and on entry
00184 *>          contains the pivot indices from the LU factorization of A as
00185 *>          computed by SGTTRF.
00186 *>
00187 *>          If FACT = 'N', then IPIV is an output argument and on exit
00188 *>          contains the pivot indices from the LU factorization of A;
00189 *>          row i of the matrix was interchanged with row IPIV(i).
00190 *>          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
00191 *>          a row interchange was not required.
00192 *> \endverbatim
00193 *>
00194 *> \param[in] B
00195 *> \verbatim
00196 *>          B is REAL array, dimension (LDB,NRHS)
00197 *>          The N-by-NRHS right hand side matrix B.
00198 *> \endverbatim
00199 *>
00200 *> \param[in] LDB
00201 *> \verbatim
00202 *>          LDB is INTEGER
00203 *>          The leading dimension of the array B.  LDB >= max(1,N).
00204 *> \endverbatim
00205 *>
00206 *> \param[out] X
00207 *> \verbatim
00208 *>          X is REAL array, dimension (LDX,NRHS)
00209 *>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
00210 *> \endverbatim
00211 *>
00212 *> \param[in] LDX
00213 *> \verbatim
00214 *>          LDX is INTEGER
00215 *>          The leading dimension of the array X.  LDX >= max(1,N).
00216 *> \endverbatim
00217 *>
00218 *> \param[out] RCOND
00219 *> \verbatim
00220 *>          RCOND is REAL
00221 *>          The estimate of the reciprocal condition number of the matrix
00222 *>          A.  If RCOND is less than the machine precision (in
00223 *>          particular, if RCOND = 0), the matrix is singular to working
00224 *>          precision.  This condition is indicated by a return code of
00225 *>          INFO > 0.
00226 *> \endverbatim
00227 *>
00228 *> \param[out] FERR
00229 *> \verbatim
00230 *>          FERR is REAL array, dimension (NRHS)
00231 *>          The estimated forward error bound for each solution vector
00232 *>          X(j) (the j-th column of the solution matrix X).
00233 *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
00234 *>          is an estimated upper bound for the magnitude of the largest
00235 *>          element in (X(j) - XTRUE) divided by the magnitude of the
00236 *>          largest element in X(j).  The estimate is as reliable as
00237 *>          the estimate for RCOND, and is almost always a slight
00238 *>          overestimate of the true error.
00239 *> \endverbatim
00240 *>
00241 *> \param[out] BERR
00242 *> \verbatim
00243 *>          BERR is REAL array, dimension (NRHS)
00244 *>          The componentwise relative backward error of each solution
00245 *>          vector X(j) (i.e., the smallest relative change in
00246 *>          any element of A or B that makes X(j) an exact solution).
00247 *> \endverbatim
00248 *>
00249 *> \param[out] WORK
00250 *> \verbatim
00251 *>          WORK is REAL array, dimension (3*N)
00252 *> \endverbatim
00253 *>
00254 *> \param[out] IWORK
00255 *> \verbatim
00256 *>          IWORK is INTEGER array, dimension (N)
00257 *> \endverbatim
00258 *>
00259 *> \param[out] INFO
00260 *> \verbatim
00261 *>          INFO is INTEGER
00262 *>          = 0:  successful exit
00263 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00264 *>          > 0:  if INFO = i, and i is
00265 *>                <= N:  U(i,i) is exactly zero.  The factorization
00266 *>                       has not been completed unless i = N, but the
00267 *>                       factor U is exactly singular, so the solution
00268 *>                       and error bounds could not be computed.
00269 *>                       RCOND = 0 is returned.
00270 *>                = N+1: U is nonsingular, but RCOND is less than machine
00271 *>                       precision, meaning that the matrix is singular
00272 *>                       to working precision.  Nevertheless, the
00273 *>                       solution and error bounds are computed because
00274 *>                       there are a number of situations where the
00275 *>                       computed solution can be more accurate than the
00276 *>                       value of RCOND would suggest.
00277 *> \endverbatim
00278 *
00279 *  Authors:
00280 *  ========
00281 *
00282 *> \author Univ. of Tennessee 
00283 *> \author Univ. of California Berkeley 
00284 *> \author Univ. of Colorado Denver 
00285 *> \author NAG Ltd. 
00286 *
00287 *> \date April 2012
00288 *
00289 *> \ingroup realOTHERcomputational
00290 *
00291 *  =====================================================================
00292       SUBROUTINE SGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
00293      $                   DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
00294      $                   WORK, IWORK, INFO )
00295 *
00296 *  -- LAPACK computational routine (version 3.4.1) --
00297 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00298 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00299 *     April 2012
00300 *
00301 *     .. Scalar Arguments ..
00302       CHARACTER          FACT, TRANS
00303       INTEGER            INFO, LDB, LDX, N, NRHS
00304       REAL               RCOND
00305 *     ..
00306 *     .. Array Arguments ..
00307       INTEGER            IPIV( * ), IWORK( * )
00308       REAL               B( LDB, * ), BERR( * ), D( * ), DF( * ),
00309      $                   DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
00310      $                   FERR( * ), WORK( * ), X( LDX, * )
00311 *     ..
00312 *
00313 *  =====================================================================
00314 *
00315 *     .. Parameters ..
00316       REAL               ZERO
00317       PARAMETER          ( ZERO = 0.0E+0 )
00318 *     ..
00319 *     .. Local Scalars ..
00320       LOGICAL            NOFACT, NOTRAN
00321       CHARACTER          NORM
00322       REAL               ANORM
00323 *     ..
00324 *     .. External Functions ..
00325       LOGICAL            LSAME
00326       REAL               SLAMCH, SLANGT
00327       EXTERNAL           LSAME, SLAMCH, SLANGT
00328 *     ..
00329 *     .. External Subroutines ..
00330       EXTERNAL           SCOPY, SGTCON, SGTRFS, SGTTRF, SGTTRS, SLACPY,
00331      $                   XERBLA
00332 *     ..
00333 *     .. Intrinsic Functions ..
00334       INTRINSIC          MAX
00335 *     ..
00336 *     .. Executable Statements ..
00337 *
00338       INFO = 0
00339       NOFACT = LSAME( FACT, 'N' )
00340       NOTRAN = LSAME( TRANS, 'N' )
00341       IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
00342          INFO = -1
00343       ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
00344      $         LSAME( TRANS, 'C' ) ) THEN
00345          INFO = -2
00346       ELSE IF( N.LT.0 ) THEN
00347          INFO = -3
00348       ELSE IF( NRHS.LT.0 ) THEN
00349          INFO = -4
00350       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00351          INFO = -14
00352       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00353          INFO = -16
00354       END IF
00355       IF( INFO.NE.0 ) THEN
00356          CALL XERBLA( 'SGTSVX', -INFO )
00357          RETURN
00358       END IF
00359 *
00360       IF( NOFACT ) THEN
00361 *
00362 *        Compute the LU factorization of A.
00363 *
00364          CALL SCOPY( N, D, 1, DF, 1 )
00365          IF( N.GT.1 ) THEN
00366             CALL SCOPY( N-1, DL, 1, DLF, 1 )
00367             CALL SCOPY( N-1, DU, 1, DUF, 1 )
00368          END IF
00369          CALL SGTTRF( N, DLF, DF, DUF, DU2, IPIV, INFO )
00370 *
00371 *        Return if INFO is non-zero.
00372 *
00373          IF( INFO.GT.0 )THEN
00374             RCOND = ZERO
00375             RETURN
00376          END IF
00377       END IF
00378 *
00379 *     Compute the norm of the matrix A.
00380 *
00381       IF( NOTRAN ) THEN
00382          NORM = '1'
00383       ELSE
00384          NORM = 'I'
00385       END IF
00386       ANORM = SLANGT( NORM, N, DL, D, DU )
00387 *
00388 *     Compute the reciprocal of the condition number of A.
00389 *
00390       CALL SGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK,
00391      $             IWORK, INFO )
00392 *
00393 *     Compute the solution vectors X.
00394 *
00395       CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
00396       CALL SGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX,
00397      $             INFO )
00398 *
00399 *     Use iterative refinement to improve the computed solutions and
00400 *     compute error bounds and backward error estimates for them.
00401 *
00402       CALL SGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV,
00403      $             B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
00404 *
00405 *     Set INFO = N+1 if the matrix is singular to working precision.
00406 *
00407       IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
00408      $   INFO = N + 1
00409 *
00410       RETURN
00411 *
00412 *     End of SGTSVX
00413 *
00414       END
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