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DGBSVXX(1) LAPACK driver routine (version 3.2)			    DGBSVXX(1)

NAME
       DGBSVXX - DGBSVXX use the LU factorization to compute the solution to a
       double precision system of linear equations A * X = B, where  A	is  an
       N-by-N matrix and X and B are N-by-NRHS matrices

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

	   IMPLICIT	   NONE

	   CHARACTER	   EQUED, FACT, TRANS

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

	   DOUBLE	   PRECISION RCOND, RPVGRW

	   INTEGER	   IPIV( * ), IWORK( * )

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

	   DOUBLE	   PRECISION R( * ), C( * ), PARAMS( * ), BERR(	 *  ),
			   ERR_BNDS_NORM( NRHS, * ), ERR_BNDS_COMP( NRHS, * )

PURPOSE
	  DGBSVXX uses the LU factorization to compute the solution to a
	  double  precision system of linear equations	A * X = B,  where A is
       an
	  N-by-N matrix and X and B are N-by-NRHS matrices.
	  If requested, both normwise and maximum componentwise error bounds
	  are returned. DGBSVXX will return a solution with a tiny
	  guaranteed error (O(eps) where eps is the working machine
	  precision) unless the matrix is very ill-conditioned, in which
	  case a warning is returned. Relevant condition numbers also are
	  calculated and returned.
	  DGBSVXX accepts user-provided factorizations and equilibration
	  factors; see the definitions of the FACT and EQUED options.
	  Solving with refinement and using a factorization from a previous
	  DGBSVXX call will also produce a solution with either O(eps)
	  errors or warnings, but we cannot make that claim for general
	  user-provided factorizations and equilibration factors if they
	  differ from what DGBSVXX would itself produce.

DESCRIPTION
	  The following steps are performed:
	  1. If FACT = 'E', double precision 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 = P * L * U,
	  where P is a permutation matrix, L is a unit lower triangular
	  matrix, and U is upper triangular.
	  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 (see
	  argument RCOND). If the reciprocal of the condition number is less
	  than machine precision, 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. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
	  the routine will use iterative refinement to try to get a small
	  error and error bounds.  Refinement calculates the residual to at
	  least twice the working precision.
	  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
       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.

       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,  AF  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.  A, AF, and IPIV are not modified.  = 'N':  The matrix A
	       will be copied to AF and factored.
	       = 'E':  The matrix A will be equilibrated  if  necessary,  then
	       copied to AF 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 = 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 AB
	       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  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
	       AF  is an output argument and on exit returns the factors L and
	       U from the factorization A = P*L*U of the  original  matrix  A.
	       If  FACT	 =  'E',  then	AF  is	an output argument and on exit
	       returns the factors L and U from the factorization A = P*L*U of
	       the  equilibrated  matrix  A  (see the description of A 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 con‐
	       tains  the  pivot  indices  from the factorization A = P*L*U as
	       computed by DGETRF; 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
	       =  P*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 = P*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.  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) 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.  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/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
	       On  entry, the N-by-NRHS 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, 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
	       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.

       RPVGRW  (output) DOUBLE PRECISION
	       Reciprocal pivot growth.	 On exit, this contains the reciprocal
	       pivot growth factor norm(A)/norm(U). The "max absolute element"
	       norm is used.  If this 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, estimated condition
	       numbers, and error bounds could be unreliable. If factorization
	       fails with 0<INFO<=N, then this contains the  reciprocal	 pivot
	       growth  factor  for  the leading INFO columns of A.  In DGESVX,
	       this quantity is returned in WORK(1).

       BERR    (output) DOUBLE PRECISION 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)	 DOUBLE	 PRECISION  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)   *
		      dlamch('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)    *
		      dlamch('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)  *  dlamch('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)  DOUBLE  PRECISION  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) * dlamch('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) * dlamch('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) *  dlamch('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) DOUBLE PRECISION 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: 1.0D+0
	       = 0.0 : No refinement is performed, and	no  error  bounds  are
	       computed.   = 1.0 : Use the extra-precise refinement algorithm.
	       (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) DOUBLE PRECISION 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 driver routine (versioNovember 2008			    DGBSVXX(1)
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