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    1:       SUBROUTINE ZGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
    2:      $                   DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
    3:      $                   WORK, RWORK, INFO )
    4: *
    5: *  -- LAPACK routine (version 3.2) --
    6: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    7: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    8: *     November 2006
    9: *
   10: *     .. Scalar Arguments ..
   11:       CHARACTER          FACT, TRANS
   12:       INTEGER            INFO, LDB, LDX, N, NRHS
   13:       DOUBLE PRECISION   RCOND
   14: *     ..
   15: *     .. Array Arguments ..
   16:       INTEGER            IPIV( * )
   17:       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
   18:       COMPLEX*16         B( LDB, * ), D( * ), DF( * ), DL( * ),
   19:      $                   DLF( * ), DU( * ), DU2( * ), DUF( * ),
   20:      $                   WORK( * ), X( LDX, * )
   21: *     ..
   22: *
   23: *  Purpose
   24: *  =======
   25: *
   26: *  ZGTSVX uses the LU factorization to compute the solution to a complex
   27: *  system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
   28: *  where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
   29: *  matrices.
   30: *
   31: *  Error bounds on the solution and a condition estimate are also
   32: *  provided.
   33: *
   34: *  Description
   35: *  ===========
   36: *
   37: *  The following steps are performed:
   38: *
   39: *  1. If FACT = 'N', the LU decomposition is used to factor the matrix A
   40: *     as A = L * U, where L is a product of permutation and unit lower
   41: *     bidiagonal matrices and U is upper triangular with nonzeros in
   42: *     only the main diagonal and first two superdiagonals.
   43: *
   44: *  2. If some U(i,i)=0, so that U is exactly singular, then the routine
   45: *     returns with INFO = i. Otherwise, the factored form of A is used
   46: *     to estimate the condition number of the matrix A.  If the
   47: *     reciprocal of the condition number is less than machine precision,
   48: *     INFO = N+1 is returned as a warning, but the routine still goes on
   49: *     to solve for X and compute error bounds as described below.
   50: *
   51: *  3. The system of equations is solved for X using the factored form
   52: *     of A.
   53: *
   54: *  4. Iterative refinement is applied to improve the computed solution
   55: *     matrix and calculate error bounds and backward error estimates
   56: *     for it.
   57: *
   58: *  Arguments
   59: *  =========
   60: *
   61: *  FACT    (input) CHARACTER*1
   62: *          Specifies whether or not the factored form of A has been
   63: *          supplied on entry.
   64: *          = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored form
   65: *                  of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not
   66: *                  be modified.
   67: *          = 'N':  The matrix will be copied to DLF, DF, and DUF
   68: *                  and factored.
   69: *
   70: *  TRANS   (input) CHARACTER*1
   71: *          Specifies the form of the system of equations:
   72: *          = 'N':  A * X = B     (No transpose)
   73: *          = 'T':  A**T * X = B  (Transpose)
   74: *          = 'C':  A**H * X = B  (Conjugate transpose)
   75: *
   76: *  N       (input) INTEGER
   77: *          The order of the matrix A.  N >= 0.
   78: *
   79: *  NRHS    (input) INTEGER
   80: *          The number of right hand sides, i.e., the number of columns
   81: *          of the matrix B.  NRHS >= 0.
   82: *
   83: *  DL      (input) COMPLEX*16 array, dimension (N-1)
   84: *          The (n-1) subdiagonal elements of A.
   85: *
   86: *  D       (input) COMPLEX*16 array, dimension (N)
   87: *          The n diagonal elements of A.
   88: *
   89: *  DU      (input) COMPLEX*16 array, dimension (N-1)
   90: *          The (n-1) superdiagonal elements of A.
   91: *
   92: *  DLF     (input or output) COMPLEX*16 array, dimension (N-1)
   93: *          If FACT = 'F', then DLF is an input argument and on entry
   94: *          contains the (n-1) multipliers that define the matrix L from
   95: *          the LU factorization of A as computed by ZGTTRF.
   96: *
   97: *          If FACT = 'N', then DLF is an output argument and on exit
   98: *          contains the (n-1) multipliers that define the matrix L from
   99: *          the LU factorization of A.
  100: *
  101: *  DF      (input or output) COMPLEX*16 array, dimension (N)
  102: *          If FACT = 'F', then DF is an input argument and on entry
  103: *          contains the n diagonal elements of the upper triangular
  104: *          matrix U from the LU factorization of A.
  105: *
  106: *          If FACT = 'N', then DF is an output argument and on exit
  107: *          contains the n diagonal elements of the upper triangular
  108: *          matrix U from the LU factorization of A.
  109: *
  110: *  DUF     (input or output) COMPLEX*16 array, dimension (N-1)
  111: *          If FACT = 'F', then DUF is an input argument and on entry
  112: *          contains the (n-1) elements of the first superdiagonal of U.
  113: *
  114: *          If FACT = 'N', then DUF is an output argument and on exit
  115: *          contains the (n-1) elements of the first superdiagonal of U.
  116: *
  117: *  DU2     (input or output) COMPLEX*16 array, dimension (N-2)
  118: *          If FACT = 'F', then DU2 is an input argument and on entry
  119: *          contains the (n-2) elements of the second superdiagonal of
  120: *          U.
  121: *
  122: *          If FACT = 'N', then DU2 is an output argument and on exit
  123: *          contains the (n-2) elements of the second superdiagonal of
  124: *          U.
  125: *
  126: *  IPIV    (input or output) INTEGER array, dimension (N)
  127: *          If FACT = 'F', then IPIV is an input argument and on entry
  128: *          contains the pivot indices from the LU factorization of A as
  129: *          computed by ZGTTRF.
  130: *
  131: *          If FACT = 'N', then IPIV is an output argument and on exit
  132: *          contains the pivot indices from the LU factorization of A;
  133: *          row i of the matrix was interchanged with row IPIV(i).
  134: *          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
  135: *          a row interchange was not required.
  136: *
  137: *  B       (input) COMPLEX*16 array, dimension (LDB,NRHS)
  138: *          The N-by-NRHS right hand side matrix B.
  139: *
  140: *  LDB     (input) INTEGER
  141: *          The leading dimension of the array B.  LDB >= max(1,N).
  142: *
  143: *  X       (output) COMPLEX*16 array, dimension (LDX,NRHS)
  144: *          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
  145: *
  146: *  LDX     (input) INTEGER
  147: *          The leading dimension of the array X.  LDX >= max(1,N).
  148: *
  149: *  RCOND   (output) DOUBLE PRECISION
  150: *          The estimate of the reciprocal condition number of the matrix
  151: *          A.  If RCOND is less than the machine precision (in
  152: *          particular, if RCOND = 0), the matrix is singular to working
  153: *          precision.  This condition is indicated by a return code of
  154: *          INFO > 0.
  155: *
  156: *  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
  157: *          The estimated forward error bound for each solution vector
  158: *          X(j) (the j-th column of the solution matrix X).
  159: *          If XTRUE is the true solution corresponding to X(j), FERR(j)
  160: *          is an estimated upper bound for the magnitude of the largest
  161: *          element in (X(j) - XTRUE) divided by the magnitude of the
  162: *          largest element in X(j).  The estimate is as reliable as
  163: *          the estimate for RCOND, and is almost always a slight
  164: *          overestimate of the true error.
  165: *
  166: *  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
  167: *          The componentwise relative backward error of each solution
  168: *          vector X(j) (i.e., the smallest relative change in
  169: *          any element of A or B that makes X(j) an exact solution).
  170: *
  171: *  WORK    (workspace) COMPLEX*16 array, dimension (2*N)
  172: *
  173: *  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
  174: *
  175: *  INFO    (output) INTEGER
  176: *          = 0:  successful exit
  177: *          < 0:  if INFO = -i, the i-th argument had an illegal value
  178: *          > 0:  if INFO = i, and i is
  179: *                <= N:  U(i,i) is exactly zero.  The factorization
  180: *                       has not been completed unless i = N, but the
  181: *                       factor U is exactly singular, so the solution
  182: *                       and error bounds could not be computed.
  183: *                       RCOND = 0 is returned.
  184: *                = N+1: U is nonsingular, but RCOND is less than machine
  185: *                       precision, meaning that the matrix is singular
  186: *                       to working precision.  Nevertheless, the
  187: *                       solution and error bounds are computed because
  188: *                       there are a number of situations where the
  189: *                       computed solution can be more accurate than the
  190: *                       value of RCOND would suggest.
  191: *
  192: *  =====================================================================
  193: *
  194: *     .. Parameters ..
  195:       DOUBLE PRECISION   ZERO
  196:       PARAMETER          ( ZERO = 0.0D+0 )
  197: *     ..
  198: *     .. Local Scalars ..
  199:       LOGICAL            NOFACT, NOTRAN
  200:       CHARACTER          NORM
  201:       DOUBLE PRECISION   ANORM
  202: *     ..
  203: *     .. External Functions ..
  204:       LOGICAL            LSAME
  205:       DOUBLE PRECISION   DLAMCH, ZLANGT
  206:       EXTERNAL           LSAME, DLAMCH, ZLANGT
  207: *     ..
  208: *     .. External Subroutines ..
  209:       EXTERNAL           XERBLA, ZCOPY, ZGTCON, ZGTRFS, ZGTTRF, ZGTTRS,
  210:      $                   ZLACPY
  211: *     ..
  212: *     .. Intrinsic Functions ..
  213:       INTRINSIC          MAX
  214: *     ..
  215: *     .. Executable Statements ..
  216: *
  217:       INFO = 0
  218:       NOFACT = LSAME( FACT, 'N' )
  219:       NOTRAN = LSAME( TRANS, 'N' )
  220:       IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
  221:          INFO = -1
  222:       ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
  223:      $         LSAME( TRANS, 'C' ) ) THEN
  224:          INFO = -2
  225:       ELSE IF( N.LT.0 ) THEN
  226:          INFO = -3
  227:       ELSE IF( NRHS.LT.0 ) THEN
  228:          INFO = -4
  229:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
  230:          INFO = -14
  231:       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
  232:          INFO = -16
  233:       END IF
  234:       IF( INFO.NE.0 ) THEN
  235:          CALL XERBLA( 'ZGTSVX', -INFO )
  236:          RETURN
  237:       END IF
  238: *
  239:       IF( NOFACT ) THEN
  240: *
  241: *        Compute the LU factorization of A.
  242: *
  243:          CALL ZCOPY( N, D, 1, DF, 1 )
  244:          IF( N.GT.1 ) THEN
  245:             CALL ZCOPY( N-1, DL, 1, DLF, 1 )
  246:             CALL ZCOPY( N-1, DU, 1, DUF, 1 )
  247:          END IF
  248:          CALL ZGTTRF( N, DLF, DF, DUF, DU2, IPIV, INFO )
  249: *
  250: *        Return if INFO is non-zero.
  251: *
  252:          IF( INFO.GT.0 )THEN
  253:             RCOND = ZERO
  254:             RETURN
  255:          END IF
  256:       END IF
  257: *
  258: *     Compute the norm of the matrix A.
  259: *
  260:       IF( NOTRAN ) THEN
  261:          NORM = '1'
  262:       ELSE
  263:          NORM = 'I'
  264:       END IF
  265:       ANORM = ZLANGT( NORM, N, DL, D, DU )
  266: *
  267: *     Compute the reciprocal of the condition number of A.
  268: *
  269:       CALL ZGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK,
  270:      $             INFO )
  271: *
  272: *     Compute the solution vectors X.
  273: *
  274:       CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
  275:       CALL ZGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX,
  276:      $             INFO )
  277: *
  278: *     Use iterative refinement to improve the computed solutions and
  279: *     compute error bounds and backward error estimates for them.
  280: *
  281:       CALL ZGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV,
  282:      $             B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
  283: *
  284: *     Set INFO = N+1 if the matrix is singular to working precision.
  285: *
  286:       IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
  287:      $   INFO = N + 1
  288: *
  289:       RETURN
  290: *
  291: *     End of ZGTSVX
  292: *
  293:       END