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CGBRFSX(1) LAPACK routine (version 3.2)				    CGBRFSX(1)

NAME
       CGBRFSX	-  CGBRFSX improve the computed solution to a system of linear
       equations and provides error bounds and backward error  estimates   for
       the solution

SYNOPSIS
       SUBROUTINE CGBRFSX( TRANS,  EQUED,  N,  KL,  KU,	 NRHS,	AB, LDAB, AFB,
			   LDAFB, IPIV, R, C, B, LDB,  X,  LDX,	 RCOND,	 BERR,
			   N_ERR_BNDS,	ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS,
			   PARAMS, WORK, RWORK, INFO )

	   IMPLICIT	   NONE

	   CHARACTER	   TRANS, EQUED

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

	   REAL		   RCOND

	   INTEGER	   IPIV( * )

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

	   REAL		   R(  *  ),  C(  *  ),	 PARAMS(  *  ),	 BERR(	*   ),
			   ERR_BNDS_NORM( NRHS, * ), ERR_BNDS_COMP( NRHS, * ),
			   RWORK( * )

PURPOSE
	  CGBRFSX improves the computed solution to a system of linear
	  equations and provides error bounds and backward error estimates
	  for the solution.  In addition to normwise error bound, the code
	  provides maximum componentwise error bound if possible.  See
	  comments for ERR_BNDS_N and ERR_BNDS_C for details of the error
	  bounds.
	  The original system of linear equations may have been equilibrated
	  before calling this routine, as described by arguments EQUED, R
	  and C below. In this case, the solution and error bounds returned
	  are for the original unequilibrated system.

ARGUMENTS
       Some optional parameters are bundled in the PARAMS array.   These  set‐
       tings determine how refinement is performed, but often the defaults are
       acceptable.  If the defaults are acceptable, users can pass NPARAMS = 0
       which prevents the source code from accessing the PARAMS argument.

       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 = Transpose)

       EQUED   (input) CHARACTER*1
	       Specifies  the  form of equilibration that was done to A before
	       calling this routine. This is needed to	compute	 the  solution
	       and error bounds correctly.  = 'N':  No equilibration
	       =  '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).   The
	       right hand side B has been changed accordingly.

       N       (input) INTEGER
	       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) DOUBLE PRECISION array, dimension (LDAB,N)
	       The original band matrix A, stored 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).

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

       AFB     (input) DOUBLE PRECISION array, dimension (LDAFB,N)
	       Details	of  the LU factorization of the band matrix A, as com‐
	       puted 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.

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

       IPIV    (input) INTEGER array, dimension (N)
	       The pivot indices from SGETRF; for 1<=i<=N, row i of the matrix
	       was interchanged with row IPIV(i).

       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.  If R is output, each  ele‐
	       ment of R is a power of the radix.  If R is input, each element
	       of R should be a power of the radix to ensure a reliable	 solu‐
	       tion  and  error estimates. Scaling by powers of the radix does
	       not cause rounding errors unless the result underflows or over‐
	       flows.  Rounding	 errors during scaling lead to refining with a
	       matrix that is not equivalent to the  input  matrix,  producing
	       error estimates that may not be reliable.

       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.  If C is output, each ele‐
	       ment of C is a power of the radix.  If C is input, each element
	       of  C should be a power of the radix to ensure a reliable solu‐
	       tion and error estimates. Scaling by powers of the  radix  does
	       not cause rounding errors unless the result underflows or over‐
	       flows. Rounding errors during scaling lead to refining  with  a
	       matrix  that  is	 not equivalent to the input matrix, producing
	       error estimates that may not be reliable.

       B       (input) REAL array, dimension (LDB,NRHS)
	       The right hand side matrix B.

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

       X       (input/output) REAL array, dimension (LDX,NRHS)
	       On entry, the solution matrix X, as  computed  by  SGETRS.   On
	       exit, the improved solution matrix X.

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

       RCOND   (output) REAL
	       Reciprocal scaled condition number.  This is an estimate of the
	       reciprocal Skeel condition number of the matrix A after equili‐
	       bration	(if done).  If this is less than the machine precision
	       (in particular, if it is zero), the matrix is singular to work‐
	       ing  precision.	Note that the error may still be small even if
	       this number is very small and the matrix	 appears  ill-	condi‐
	       tioned.

       BERR    (output) REAL array, dimension (NRHS)
	       Componentwise  relative backward error.	This is the component‐
	       wise 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).  N_ERR_BNDS (input) INTEGER
	       Number  of  error bounds to return for each right hand side and
	       each type (normwise or componentwise).  See  ERR_BNDS_NORM  and
	       ERR_BNDS_COMP below.

       ERR_BNDS_NORM  (output) REAL array, dimension (NRHS, N_ERR_BNDS)
		      For  each	 right-hand side, this array contains informa‐
		      tion about various error bounds  and  condition  numbers
		      corresponding  to	 the normwise relative error, which is
		      defined as follows: Normwise relative error in  the  ith
		      solution	 vector:   max_j  (abs(XTRUE(j,i)  -  X(j,i)))
		      ------------------------------  max_j  abs(X(j,i))   The
		      array  is	 indexed  by  the type of error information as
		      described below. There currently are up to three	pieces
		      of   information	 returned.    The   first   index   in
		      ERR_BNDS_NORM(i,:) corresponds  to  the  ith  right-hand
		      side.  The second index in ERR_BNDS_NORM(:,err) contains
		      the following three fields: err = 1 "Trust/don't	trust"
		      boolean.	Trust  the  answer if the reciprocal condition
		      number  is   less	  than	 the   threshold   sqrt(n)   *
		      slamch('Epsilon').   err	=  2 "Guaranteed" error bound:
		      The estimated forward error, almost certainly  within  a
		      factor of 10 of the true error so long as the next entry
		      is   greater    than    the    threshold	  sqrt(n)    *
		      slamch('Epsilon').  This	error  bound  should  only  be
		      trusted if the  previous	boolean	 is  true.   err  =  3
		      Reciprocal condition number: Estimated normwise recipro‐
		      cal  condition  number.	Compared  with	the  threshold
		      sqrt(n)  *  slamch('Epsilon')  to determine if the error
		      estimate is  "guaranteed".  These	 reciprocal  condition
		      numbers  are  1  /  (norm(Z^{-1},inf) * norm(Z,inf)) for
		      some appropriately scaled matrix Z.  Let Z = S*A,	 where
		      S	 scales	 each row by a power of the radix so all abso‐
		      lute row sums of Z  are  approximately  1.   See	Lapack
		      Working Note 165 for further details and extra cautions.

       ERR_BNDS_COMP  (output) REAL array, dimension (NRHS, N_ERR_BNDS)
		      For  each	 right-hand side, this array contains informa‐
		      tion about various error bounds  and  condition  numbers
		      corresponding to the componentwise relative error, which
		      is defined as follows: Componentwise relative  error  in
		      the  ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j
		      ---------------------- abs(X(j,i)) The array is  indexed
		      by  the  right-hand  side	 i (on which the componentwise
		      relative error depends), and the type of error  informa‐
		      tion as described below. There currently are up to three
		      pieces of information returned for each right-hand side.
		      If  componentwise accuracy is not requested (PARAMS(3) =
		      0.0), then ERR_BNDS_COMP is not accessed.	 If N_ERR_BNDS
		      .LT.  3,	then  at most the first (:,N_ERR_BNDS) entries
		      are returned.  The  first	 index	in  ERR_BNDS_COMP(i,:)
		      corresponds  to  the  ith	 right-hand  side.  The second
		      index in	ERR_BNDS_COMP(:,err)  contains	the  following
		      three fields: err = 1 "Trust/don't trust" boolean. Trust
		      the answer if the reciprocal condition  number  is  less
		      than the threshold sqrt(n) * slamch('Epsilon').  err = 2
		      "Guaranteed" error bound: The estimated  forward	error,
		      almost certainly within a factor of 10 of the true error
		      so long as the next entry is greater than the  threshold
		      sqrt(n)  *  slamch('Epsilon').  This  error bound should
		      only be trusted if the previous boolean is true.	err  =
		      3	  Reciprocal condition number: Estimated componentwise
		      reciprocal condition number.  Compared with the  thresh‐
		      old  sqrt(n)  *  slamch('Epsilon')  to  determine if the
		      error estimate is "guaranteed". These reciprocal	condi‐
		      tion  numbers  are  1 / (norm(Z^{-1},inf) * norm(Z,inf))
		      for  some	 appropriately	scaled	matrix	Z.   Let  Z  =
		      S*(A*diag(x)),  where  x is the solution for the current
		      right-hand side and S scales each row of A*diag(x) by  a
		      power  of	 the  radix  so all absolute row sums of Z are
		      approximately 1.	See Lapack Working Note 165  for  fur‐
		      ther  details and extra cautions.	 NPARAMS (input) INTE‐
		      GER Specifies the number of parameters  set  in  PARAMS.
		      If  .LE.	0,  the	 PARAMS	 array is never referenced and
		      default values are used.

       PARAMS  (input / output) REAL array, dimension NPARAMS
	       Specifies algorithm parameters.	If an entry is .LT. 0.0,  then
	       that  entry  will  be  filled  with default value used for that
	       parameter.  Only positions up to NPARAMS are accessed; defaults
	       are	 used	    for	      higher-numbered	   parameters.
	       PARAMS(LA_LINRX_ITREF_I = 1) :  Whether	to  perform  iterative
	       refinement or not.  Default: 1.0
	       =  0.0  :  No  refinement is performed, and no error bounds are
	       computed.  = 1.0 : Use the  double-precision  refinement	 algo‐
	       rithm,  possibly with doubled-single computations if the compi‐
	       lation environment does not support DOUBLE  PRECISION.	(other
	       values are reserved for future use) PARAMS(LA_LINRX_ITHRESH_I =
	       2) :  Maximum  number  of  residual  computations  allowed  for
	       refinement.  Default: 10
	       Aggressive:  Set to 100 to permit convergence using approximate
	       factorizations or factorizations other than LU. If the  factor‐
	       ization	uses  a technique other than Gaussian elimination, the
	       guarantees in err_bnds_norm and err_bnds_comp may no longer  be
	       trustworthy.   PARAMS(LA_LINRX_CWISE_I  = 3) : Flag determining
	       if the code will attempt to find a solution with	 small	compo‐
	       nentwise	 relative  error  in  the  double-precision algorithm.
	       Positive is true, 0.0 is false.	Default: 1.0  (attempt	compo‐
	       nentwise convergence)

       WORK    (workspace) REAL array, dimension (4*N)

       IWORK   (workspace) INTEGER array, dimension (N)

       INFO    (output) INTEGER
	       = 0:  Successful exit. The solution to every right-hand side is
	       guaranteed.  < 0:  If INFO = -i, the i-th argument had an ille‐
	       gal value
	       > 0 and <= N:  U(INFO,INFO) 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+J: The	 solution  corresponding  to  the  Jth
	       right-hand  side is not guaranteed. The solutions corresponding
	       to other right- hand sides K with K > J may not	be  guaranteed
	       as  well,  but only the first such right-hand side is reported.
	       If a small componentwise error is not  requested	 (PARAMS(3)  =
	       0.0)  then the Jth right-hand side is the first with a normwise
	       error bound that is not guaranteed (the smallest	 J  such  that
	       ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth
	       right-hand side is the first with either a normwise  or	compo‐
	       nentwise	 error	bound  that  is not guaranteed (the smallest J
	       such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1)
	       =   0.0).   See	 the   definition  of  ERR_BNDS_NORM(:,1)  and
	       ERR_BNDS_COMP(:,1). To get information about all of the	right-
	       hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.

    LAPACK routine (version 3.2) November 2008			    CGBRFSX(1)
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