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CGBSVX(l)			       )			     CGBSVX(l)

NAME
       CGBSVX  - use 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,

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

	   CHARACTER	  EQUED, FACT, TRANS

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

	   REAL		  RCOND

	   INTEGER	  IPIV( * )

	   REAL		  BERR( * ), C( * ), FERR( * ), R( * ), RWORK( * )

	   COMPLEX	  AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( *
			  ), X( LDX, * )

PURPOSE
       CGBSVX  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 band matrix of order N with KL subdiagonals and KU super‐
       diagonals, 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  (Conjugate 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) COMPLEX 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) COMPLEX 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 CGBTRF.  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 CGBTRF; 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) REAL 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) 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) COMPLEX 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) COMPLEX 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) 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) 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) REAL 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) COMPLEX array, dimension (2*N)

       RWORK   (workspace/output) REAL array, dimension (N)
	       On  exit,  RWORK(1) contains the reciprocal pivot growth factor
	       norm(A)/norm(U). The "max absolute element" norm	 is  used.  If
	       RWORK(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  RWORK(1) contains the reciprocal
	       pivot growth factor for the leading INFO columns of A.

       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			     CGBSVX(l)
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