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.

 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.

FACT

          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

          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

          N is INTEGER


          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrix B.  NRHS >= 0.

DL

          DL is COMPLEX array, dimension (N-1)


          The (n-1) subdiagonal elements of A.

D

          D is COMPLEX array, dimension (N)


          The n diagonal elements of A.

DU

          DU is COMPLEX array, dimension (N-1)


          The (n-1) superdiagonal elements of A.

DLF

          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

          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

          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

          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

          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

          B is COMPLEX array, dimension (LDB,NRHS)


          The N-by-NRHS right hand side matrix B.

LDB

          LDB is INTEGER


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

X

          X is COMPLEX array, dimension (LDX,NRHS)


          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

LDX

          LDX is INTEGER


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

RCOND

          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

          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

          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

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (N)

INFO

          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016