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



NAME
       SGBSVX

SYNOPSIS
       SUBROUTINE SGBSVX (FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
           IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK,
           INFO)



PURPOSE
            SGBSVX uses the LU factorization to compute the solution to a real
            system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
            where A is a band matrix of order N with KL subdiagonals and KU
            superdiagonals, 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 by this subroutine:

            1. If FACT = 'E', real scaling factors are computed to equilibrate
               the system:
                  TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
                  TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
                  TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
               Whether or not the system will be equilibrated depends on the
               scaling of the matrix A, but if equilibration is used, A is
               overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
               or diag(C)*B (if TRANS = 'T' or 'C').

            2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
               matrix A (after equilibration if FACT = 'E') as
                  A = L * U,
               where L is a product of permutation and unit lower triangular
               matrices with KL subdiagonals, and U is upper triangular with
               KL+KU superdiagonals.

            3. 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.

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

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

            6. If equilibration was used, the matrix X is premultiplied by
               diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
               that it solves the original system before equilibration.




ARGUMENTS
           FACT      (input)
                     FACT is CHARACTER*1
                     Specifies whether or not the factored form of the matrix A is
                     supplied on entry, and if not, whether the matrix A should be
                     equilibrated before it is factored.
                     = 'F':  On entry, AFB and IPIV contain the factored form of
                             A.  If EQUED is not 'N', the matrix A has been
                             equilibrated with scaling factors given by R and C.
                             AB, AFB, and IPIV are not modified.
                     = 'N':  The matrix A will be copied to AFB and factored.
                     = 'E':  The matrix A will be equilibrated if necessary, then
                             copied to AFB 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  (Transpose)

           N         (input)
                     N is INTEGER
                     The number of linear equations, i.e., the order of the
                     matrix A.  N >= 0.

           KL        (input)
                     KL is INTEGER
                     The number of subdiagonals within the band of A.  KL >= 0.

           KU        (input)
                     KU is INTEGER
                     The number of superdiagonals within the band of A.  KU >= 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.

           AB        (input/output)
                     AB is REAL array, dimension (LDAB,N)
                     On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
                     The j-th column of A is stored in the j-th column of the
                     array AB as follows:
                     AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

                     If FACT = 'F' and EQUED is not 'N', then A must have been
                     equilibrated by the scaling factors in R and/or C.  AB is not
                     modified if FACT = 'F' or 'N', or if FACT = 'E' and
                     EQUED = 'N' on exit.

                     On exit, if EQUED .ne. 'N', A is scaled as follows:
                     EQUED = 'R':  A := diag(R) * A
                     EQUED = 'C':  A := A * diag(C)
                     EQUED = 'B':  A := diag(R) * A * diag(C).

           LDAB      (input)
                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= KL+KU+1.

           AFB       (input/output)
                     AFB is REAL array, dimension (LDAFB,N)
                     If FACT = 'F', then AFB is an input argument and on entry
                     contains details of the LU factorization of the band matrix
                     A, as computed by SGBTRF.  U is stored as an upper triangular
                     band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
                     and the multipliers used during the factorization are stored
                     in rows KL+KU+2 to 2*KL+KU+1.  If EQUED .ne. 'N', then AFB is
                     the factored form of the equilibrated matrix A.

                     If FACT = 'N', then AFB is an output argument and on exit
                     returns details of the LU factorization of A.

                     If FACT = 'E', then AFB is an output argument and on exit
                     returns details of the LU factorization of the equilibrated
                     matrix A (see the description of AB for the form of the
                     equilibrated matrix).

           LDAFB     (input)
                     LDAFB is INTEGER
                     The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.

           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 factorization A = L*U
                     as computed by SGBTRF; row i of the matrix was interchanged
                     with row IPIV(i).

                     If FACT = 'N', then IPIV is an output argument and on exit
                     contains the pivot indices from the factorization A = L*U
                     of the original matrix A.

                     If FACT = 'E', then IPIV is an output argument and on exit
                     contains the pivot indices from the factorization A = L*U
                     of the equilibrated matrix A.

           EQUED     (input/output)
                     EQUED is CHARACTER*1
                     Specifies the form of equilibration that was done.
                     = 'N':  No equilibration (always true if FACT = 'N').
                     = 'R':  Row equilibration, i.e., A has been premultiplied by
                             diag(R).
                     = 'C':  Column equilibration, i.e., A has been postmultiplied
                             by diag(C).
                     = 'B':  Both row and column equilibration, i.e., A has been
                             replaced by diag(R) * A * diag(C).
                     EQUED is an input argument if FACT = 'F'; otherwise, it is an
                     output argument.

           R         (input/output)
                     R is REAL array, dimension (N)
                     The row scale factors for A.  If EQUED = 'R' or 'B', A is
                     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
                     is not accessed.  R is an input argument if FACT = 'F';
                     otherwise, R is an output argument.  If FACT = 'F' and
                     EQUED = 'R' or 'B', each element of R must be positive.

           C         (input/output)
                     C is REAL array, dimension (N)
                     The column scale factors for A.  If EQUED = 'C' or 'B', A is
                     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
                     is not accessed.  C is an input argument if FACT = 'F';
                     otherwise, C is an output argument.  If FACT = 'F' and
                     EQUED = 'C' or 'B', each element of C must be positive.

           B         (input/output)
                     B is REAL array, dimension (LDB,NRHS)
                     On entry, the right hand side matrix B.
                     On exit,
                     if EQUED = 'N', B is not modified;
                     if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
                     diag(R)*B;
                     if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
                     overwritten by diag(C)*B.

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

           X         (output)
                     X is REAL array, dimension (LDX,NRHS)
                     If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
                     to the original system of equations.  Note that A and B are
                     modified on exit if EQUED .ne. 'N', and the solution to the
                     equilibrated system is inv(diag(C))*X if TRANS = 'N' and
                     EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
                     and EQUED = 'R' or 'B'.

           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 after equilibration (if done).  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 REAL array, dimension (3*N)
                     On exit, WORK(1) contains the reciprocal pivot growth
                     factor norm(A)/norm(U). The "max absolute element" norm is
                     used. If WORK(1) is much less than 1, then the stability
                     of the LU factorization of the (equilibrated) matrix A
                     could be poor. This also means that the solution X, condition
                     estimator RCOND, and forward error bound FERR could be
                     unreliable. If factorization fails with 0<INFO<=N, then
                     WORK(1) contains the reciprocal pivot growth factor for the
                     leading INFO columns of A.

           IWORK     (output)
                     IWORK is INTEGER 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 been completed, 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



LAPACK routine                  31 October 2017                      SGBSVX(3)