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CSYRFSX(1) LAPACK routine (version 3.2)				    CSYRFSX(1)

NAME
       CSYRFSX	-  CSYRFSX improve the computed solution to a system of linear
       equations when the coefficient  matrix  is  symmetric  indefinite,  and
       provides error bounds and backward error estimates for the  solution

SYNOPSIS
       Subroutine CSYRFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B,
			   LDB,	  X,	LDX,	RCOND,	  BERR,	   N_ERR_BNDS,
			   ERR_BNDS_NORM,   ERR_BNDS_COMP,   NPARAMS,  PARAMS,
			   WORK, RWORK, INFO )

	   IMPLICIT	   NONE

	   CHARACTER	   UPLO, EQUED

	   INTEGER	   INFO,  LDA,	LDAF,  LDB,  LDX,  N,  NRHS,  NPARAMS,
			   N_ERR_BNDS

	   REAL		   RCOND

	   INTEGER	   IPIV( * )

	   COMPLEX	   A(  LDA, * ), AF( LDAF, * ), B( LDB, * ), X( LDX, *
			   ), WORK( * )

	   REAL		   S( * ), PARAMS(  *  ),  BERR(  *  ),	 RWORK(	 *  ),
			   ERR_BNDS_NORM( NRHS, * ), ERR_BNDS_COMP( NRHS, * )

PURPOSE
	  CSYRFSX improves the computed solution to a system of linear
	  equations when the coefficient matrix is symmetric indefinite, 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 and S
	  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.

       UPLO    (input) CHARACTER*1
	       = 'U':  Upper triangle of A is stored;
	       = 'L':  Lower triangle of A is stored.

       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
	       = 'Y':  Both row and column equilibration,  i.e.,  A  has  been
	       replaced	 by  diag(S) * A * diag(S).  The right hand side B has
	       been changed accordingly.

       N       (input) INTEGER
	       The order of the matrix A.  N >= 0.

       NRHS    (input) INTEGER
	       The number of right hand sides, i.e., the number of columns  of
	       the matrices B and X.  NRHS >= 0.

       A       (input) COMPLEX array, dimension (LDA,N)
	       The  symmetric  matrix  A.   If	UPLO = 'U', the leading N-by-N
	       upper triangular part of A contains the upper  triangular  part
	       of the matrix A, and the strictly lower triangular part of A is
	       not referenced.	If UPLO = 'L', the leading N-by-N lower trian‐
	       gular  part  of	A  contains  the  lower triangular part of the
	       matrix A, and the strictly upper triangular part of  A  is  not
	       referenced.

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

       AF      (input) COMPLEX array, dimension (LDAF,N)
	       The factored form of the matrix A.  AF contains the block diag‐
	       onal matrix D and the multipliers used to obtain the  factor  U
	       or  L  from  the	 factorization A = U*D*U**T or A = L*D*L**T as
	       computed by SSYTRF.

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

       IPIV    (input) INTEGER array, dimension (N)
	       Details of the interchanges and the block  structure  of	 D  as
	       determined by SSYTRF.

       S       (input or output) REAL array, dimension (N)
	       The  scale  factors  for A.  If EQUED = 'Y', A is multiplied on
	       the left and right by diag(S).  S is an input argument if  FACT
	       =  'F';	otherwise, S is an output argument.  If FACT = 'F' and
	       EQUED = 'Y', each element of S must be positive.	 If S is  out‐
	       put,  each element of S is a power of the radix. If S is input,
	       each element of S should be a power of the radix	 to  ensure  a
	       reliable solution and error estimates. Scaling by powers of the
	       radix does not cause rounding errors unless the	result	under‐
	       flows  or  overflows.   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) COMPLEX 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) COMPLEX 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			    CSYRFSX(1)
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