CGTSVX(3)      LAPACK routine of NEC Numeric Library Collection      CGTSVX(3)



NAME
       CGTSVX

SYNOPSIS
       SUBROUTINE CGTSVX (FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
           IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)



PURPOSE
            CGTSVX uses the LU factorization to compute the solution to a complex
            system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
            where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
            matrices.

            Error bounds on the solution and a condition estimate are also
            provided.



       Description:


            The following steps are performed:

            1. If FACT = 'N', the LU decomposition is used to factor the matrix A
               as A = L * U, where L is a product of permutation and unit lower
               bidiagonal matrices and U is upper triangular with nonzeros in
               only the main diagonal and first two superdiagonals.

            2. If some U(i,i)=0, so that U is exactly singular, then the routine
               returns with INFO = i. Otherwise, the factored form of A is used
               to estimate the condition number of the matrix A.  If the
               reciprocal of the condition number is less than machine precision,
               INFO = N+1 is returned as a warning, but the routine still goes on
               to solve for X and compute error bounds as described below.

            3. The system of equations is solved for X using the factored form
               of A.

            4. Iterative refinement is applied to improve the computed solution
               matrix and calculate error bounds and backward error estimates
               for it.




ARGUMENTS
           FACT      (input)
                     FACT is CHARACTER*1
                     Specifies whether or not the factored form of A has been
                     supplied on entry.
                     = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored form
                             of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not
                             be modified.
                     = 'N':  The matrix will be copied to DLF, DF, and DUF
                             and factored.

           TRANS     (input)
                     TRANS is CHARACTER*1
                     Specifies the form of the system of equations:
                     = 'N':  A * X = B     (No transpose)
                     = 'T':  A**T * X = B  (Transpose)
                     = 'C':  A**H * X = B  (Conjugate transpose)

           N         (input)
                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           NRHS      (input)
                     NRHS is INTEGER
                     The number of right hand sides, i.e., the number of columns
                     of the matrix B.  NRHS >= 0.

           DL        (input)
                     DL is COMPLEX array, dimension (N-1)
                     The (n-1) subdiagonal elements of A.

           D         (input)
                     D is COMPLEX array, dimension (N)
                     The n diagonal elements of A.

           DU        (input)
                     DU is COMPLEX array, dimension (N-1)
                     The (n-1) superdiagonal elements of A.

           DLF       (input/output)
                     DLF is COMPLEX array, dimension (N-1)
                     If FACT = 'F', then DLF is an input argument and on entry
                     contains the (n-1) multipliers that define the matrix L from
                     the LU factorization of A as computed by CGTTRF.

                     If FACT = 'N', then DLF is an output argument and on exit
                     contains the (n-1) multipliers that define the matrix L from
                     the LU factorization of A.

           DF        (input/output)
                     DF is COMPLEX array, dimension (N)
                     If FACT = 'F', then DF is an input argument and on entry
                     contains the n diagonal elements of the upper triangular
                     matrix U from the LU factorization of A.

                     If FACT = 'N', then DF is an output argument and on exit
                     contains the n diagonal elements of the upper triangular
                     matrix U from the LU factorization of A.

           DUF       (input/output)
                     DUF is COMPLEX array, dimension (N-1)
                     If FACT = 'F', then DUF is an input argument and on entry
                     contains the (n-1) elements of the first superdiagonal of U.

                     If FACT = 'N', then DUF is an output argument and on exit
                     contains the (n-1) elements of the first superdiagonal of U.

           DU2       (input/output)
                     DU2 is COMPLEX array, dimension (N-2)
                     If FACT = 'F', then DU2 is an input argument and on entry
                     contains the (n-2) elements of the second superdiagonal of
                     U.

                     If FACT = 'N', then DU2 is an output argument and on exit
                     contains the (n-2) elements of the second superdiagonal of
                     U.

           IPIV      (input/output)
                     IPIV is INTEGER array, dimension (N)
                     If FACT = 'F', then IPIV is an input argument and on entry
                     contains the pivot indices from the LU factorization of A as
                     computed by CGTTRF.

                     If FACT = 'N', then IPIV is an output argument and on exit
                     contains the pivot indices from the LU factorization of A;
                     row i of the matrix was interchanged with row IPIV(i).
                     IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
                     a row interchange was not required.

           B         (input)
                     B is COMPLEX array, dimension (LDB,NRHS)
                     The N-by-NRHS right hand side matrix B.

           LDB       (input)
                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           X         (output)
                     X is COMPLEX array, dimension (LDX,NRHS)
                     If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

           LDX       (input)
                     LDX is INTEGER
                     The leading dimension of the array X.  LDX >= max(1,N).

           RCOND     (output)
                     RCOND is REAL
                     The estimate of the reciprocal condition number of the matrix
                     A.  If RCOND is less than the machine precision (in
                     particular, if RCOND = 0), the matrix is singular to working
                     precision.  This condition is indicated by a return code of
                     INFO > 0.

           FERR      (output)
                     FERR is REAL array, dimension (NRHS)
                     The estimated forward error bound for each solution vector
                     X(j) (the j-th column of the solution matrix X).
                     If XTRUE is the true solution corresponding to X(j), FERR(j)
                     is an estimated upper bound for the magnitude of the largest
                     element in (X(j) - XTRUE) divided by the magnitude of the
                     largest element in X(j).  The estimate is as reliable as
                     the estimate for RCOND, and is almost always a slight
                     overestimate of the true error.

           BERR      (output)
                     BERR is REAL array, dimension (NRHS)
                     The componentwise relative backward error of each solution
                     vector X(j) (i.e., the smallest relative change in
                     any element of A or B that makes X(j) an exact solution).

           WORK      (output)
                     WORK is COMPLEX array, dimension (2*N)

           RWORK     (output)
                     RWORK is REAL array, dimension (N)

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, and i is
                           <= N:  U(i,i) is exactly zero.  The factorization
                                  has not been completed unless i = N, but the
                                  factor U is exactly singular, so the solution
                                  and error bounds could not be computed.
                                  RCOND = 0 is returned.
                           = N+1: U is nonsingular, but RCOND is less than machine
                                  precision, meaning that the matrix is singular
                                  to working precision.  Nevertheless, the
                                  solution and error bounds are computed because
                                  there are a number of situations where the
                                  computed solution can be more accurate than the
                                  value of RCOND would suggest.



LAPACK routine                  31 October 2017                      CGTSVX(3)