SGBSVX(l)	     LAPACK driver routine (version 1.1)	    SGBSVX(l)

NAME
  SGBSVX - 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 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 )

      CHARACTER	     EQUED, FACT, TRANS

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

      REAL	     RCOND

      INTEGER	     IPIV( * ),	IWORK( * )

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

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. 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, steps	4-6 are	skipped.

  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	sup-
	  plied	on entry, and if not, whether the matrix A should be equili-
	  brated before	it is factored.	 = 'F':	 On entry, AFB and IPIV	con-
	  tain 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) 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 equili-
	  brated 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) 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 equili-
	  brated 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
	  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 equili-
	  brated matrix	A.

  EQUED	  (input/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 postmulti-
	  plied	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
	  variable if FACT = 'F'; otherwise, it	is an output variable.

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

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

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

  X	  (output) REAL	array, dimension (LDX,NRHS)
	  If INFO = 0, 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 or 'B'.

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

  RCOND	  (output) 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, and	the solution and error bounds are not computed.

  FERR	  (output) REAL	array, dimension (NRHS)
	  The estimated	forward	error bounds for each solution vector X(j)
	  (the j-th column of the solution matrix X).  If XTRUE	is the true
	  solution, FERR(j) bounds the magnitude of the	largest	entry in
	  (X(j)	- XTRUE) divided by the	magnitude of the largest entry in
	  X(j).	 The quality of	the error bound	depends	on the quality of the
	  estimate of norm(inv(A)) computed in the code; if the	estimate of
	  norm(inv(A)) is accurate, the	error bound is guaranteed.

  BERR	  (output) REAL	array, dimension (NRHS)
	  The componentwise relative backward error of each solution vector
	  X(j) (i.e., the smallest relative change in any entry	of A or	B
	  that makes X(j) an exact solution).

  WORK	  (workspace) REAL array, dimension (3*N)

  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. =	N+1: RCOND is less than
	  machine precision.  The factorization	has been completed, but	the
	  matrix A is singular to working precision, and the solution and
	  error	bounds have not	been computed.