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DGBSVX(l)			       )			     DGBSVX(l)

NAME
       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,

SYNOPSIS
       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, * )

PURPOSE
       DGBSVX uses the LU factorization to compute the solution to a real sys‐
       tem 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  superdiago‐
       nals, and X and B are N-by-NRHS matrices.

       Error  bounds  on  the  solution and a condition estimate are also pro‐
       vided.

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 equi‐
	       librated by the scaling factors in R and/or C.  AB is not modi‐
	       fied  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 con‐
	       tains 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  fac‐
	       tored 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 equili‐
	       brated 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 con‐
	       tains the pivot indices from the factorization A = L*U as  com‐
	       puted  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 con‐
	       tains  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 con‐
	       tains  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	 post‐
	       multiplied  by diag(C).	= 'B':	Both row and column equilibra‐
	       tion, 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  mul‐
	       tiplied 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	 modi‐
	       fied on exit if EQUED .ne. 'N', and the solution to the equili‐
	       brated 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 vec‐
	       tor  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  fac‐
	       torization  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 com‐
	       pleted, 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	preci‐
	       sion, 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 version 3.0		 15 June 2000			     DGBSVX(l)
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