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



NAME
       CPTSVX

SYNOPSIS
       SUBROUTINE CPTSVX (FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND,
           FERR, BERR, WORK, RWORK, INFO)



PURPOSE
            CPTSVX uses the factorization A = L*D*L**H to compute the solution
            to a complex system of linear equations A*X = B, where A is an
            N-by-N Hermitian positive definite tridiagonal matrix 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 matrix A is factored as A = L*D*L**H, where L
               is a unit lower bidiagonal matrix and D is diagonal.  The
               factorization can also be regarded as having the form
               A = U**H*D*U.

            2. If the leading i-by-i principal minor is not positive definite,
               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 the matrix
                     A is supplied on entry.
                     = 'F':  On entry, DF and EF contain the factored form of A.
                             D, E, DF, and EF will not be modified.
                     = 'N':  The matrix A will be copied to DF and EF and
                             factored.

           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 matrices B and X.  NRHS >= 0.

           D         (input)
                     D is REAL array, dimension (N)
                     The n diagonal elements of the tridiagonal matrix A.

           E         (input)
                     E is COMPLEX array, dimension (N-1)
                     The (n-1) subdiagonal elements of the tridiagonal matrix A.

           DF        (input/output)
                     DF is REAL array, dimension (N)
                     If FACT = 'F', then DF is an input argument and on entry
                     contains the n diagonal elements of the diagonal matrix D
                     from the L*D*L**H factorization of A.
                     If FACT = 'N', then DF is an output argument and on exit
                     contains the n diagonal elements of the diagonal matrix D
                     from the L*D*L**H factorization of A.

           EF        (input/output)
                     EF is COMPLEX array, dimension (N-1)
                     If FACT = 'F', then EF is an input argument and on entry
                     contains the (n-1) subdiagonal elements of the unit
                     bidiagonal factor L from the L*D*L**H factorization of A.
                     If FACT = 'N', then EF is an output argument and on exit
                     contains the (n-1) subdiagonal elements of the unit
                     bidiagonal factor L from the L*D*L**H factorization of A.

           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 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 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).

           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 (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:  the leading minor of order i of A is
                                  not positive definite, so the factorization
                                  could not be completed, and the solution has not
                                  been 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                      CPTSVX(3)