CXML
        	    
LAPACK version 3.0
dgbsvx(3)

PURPOSE
  DGBSVX - use 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,

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

      CHARACTER	     EQUED, FACT, TRANS

      INTEGER	     INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS

      DOUBLE	     PRECISION RCOND

      INTEGER	     IPIV( * ), IWORK( * )

      DOUBLE	     PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
		     BERR( * ), C( * ), FERR( * ), R( * ), WORK( * ), X( LDX,
		     * )

DESCRIPTION
  DGBSVX 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) 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) 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) INTEGER
	  The number of linear equations, i.e., the order of the matrix A.  N
	  >= 0.

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

  KU	  (input) INTEGER
	  The number of superdiagonals within the band of A.  KU >= 0.

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

  AB	  (input/output) DOUBLE PRECISION 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) INTEGER
	  The leading dimension of the array AB.  LDAB >= KL+KU+1.

  AFB	  (input or output) DOUBLE PRECISION 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 DGBTRF.  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) INTEGER
	  The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.

  IPIV	  (input or output) 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
	  DGBTRF; 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 or output) 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 or output) DOUBLE PRECISION 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 or output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) INTEGER
	  The leading dimension of the array B.	 LDB >= max(1,N).

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

  RCOND	  (output) DOUBLE PRECISION
	  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) DOUBLE PRECISION 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) DOUBLE PRECISION 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	  (workspace/output) DOUBLE PRECISION 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	  (workspace) INTEGER array, dimension (N)

  INFO	  (output) 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 value of RCOND would suggest.