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DSYSVX(l)			       )			     DSYSVX(l)

NAME
       DSYSVX  -  use the diagonal pivoting factorization to compute the solu‐
       tion to a real system of linear equations A * X = B,

SYNOPSIS
       SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
			  X,  LDX, RCOND, FERR, BERR, WORK, LWORK, IWORK, INFO
			  )

	   CHARACTER	  FACT, UPLO

	   INTEGER	  INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS

	   DOUBLE	  PRECISION RCOND

	   INTEGER	  IPIV( * ), IWORK( * )

	   DOUBLE	  PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB,	 *  ),
			  BERR( * ), FERR( * ), WORK( * ), X( LDX, * )

PURPOSE
       DSYSVX uses the diagonal pivoting factorization to compute the solution
       to a real system of linear equations A * X = B, where A	is  an	N-by-N
       symmetric matrix 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:

       1. If FACT = 'N', the diagonal pivoting method is used to factor A.
	  The form of the factorization is
	     A = U * D * U**T,	if UPLO = 'U', or
	     A = L * D * L**T,	if UPLO = 'L',
	  where U (or L) is a product of permutation and unit upper (lower)
	  triangular matrices, and D is symmetric and block diagonal with
	  1-by-1 and 2-by-2 diagonal blocks.

       2. If some D(i,i)=0, so that D 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.

       3. The system of equations is solved for X using the factored form
	  of A.

       4. Iterative refinement is applied to improve the computed solution
	  matrix and calculate error bounds and backward error estimates
	  for it.

ARGUMENTS
       FACT    (input) CHARACTER*1
	       Specifies whether or not the factored form of A has  been  sup‐
	       plied on entry.	= 'F':	On entry, AF and IPIV contain the fac‐
	       tored form of A.	 AF and IPIV will not  be  modified.   =  'N':
	       The matrix A will be copied to AF and factored.

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

       N       (input) INTEGER
	       The  number  of linear equations, i.e., 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) DOUBLE PRECISION 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 or output) DOUBLE PRECISION array, dimension (LDAF,N)
	       If  FACT	 = 'F', then AF is an input argument and on entry con‐
	       tains the block diagonal 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 DSYTRF.

	       If FACT = 'N', then AF  is  an  output  argument	 and  on  exit
	       returns the block diagonal 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.

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

       IPIV    (input or output) INTEGER array, dimension (N)
	       If FACT = 'F', then IPIV is an input argument and on entry con‐
	       tains details of the interchanges and the block structure of D,
	       as determined by DSYTRF.	 If IPIV(k) > 0, then rows and columns
	       k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal
	       block.	If  UPLO  = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows
	       and  columns   k-1   and	  -IPIV(k)   were   interchanged   and
	       D(k-1:k,k-1:k)  is  a 2-by-2 diagonal block.  If UPLO = 'L' and
	       IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k)
	       were  interchanged  and	D(k:k+1,k:k+1)	is  a  2-by-2 diagonal
	       block.

	       If FACT = 'N', then IPIV is an output argument and on exit con‐
	       tains details of the interchanges and the block structure of D,
	       as determined by DSYTRF.

       B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
	       The N-by-NRHS right hand side matrix 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 or INFO = N+1, the N-by-NRHS solution matrix X.

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

       RCOND   (output) DOUBLE PRECISION
	       The estimate of the reciprocal condition number of  the	matrix
	       A.  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) DOUBLE PRECISION 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) DOUBLE PRECISION 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/output) DOUBLE PRECISION array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The  length  of	WORK.	LWORK >= 3*N, and for best performance
	       LWORK >= N*NB, where NB is the optimal blocksize for DSYTRF.

	       If LWORK = -1, then a workspace query is assumed;  the  routine
	       only  calculates	 the  optimal  size of the WORK array, returns
	       this value as the first entry of the WORK array, and  no	 error
	       message related to LWORK is issued by XERBLA.

       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:  D(i,i) is exactly zero.  The factorization has been com‐
	       pleted but the factor D is exactly singular,  so	 the  solution
	       and  error bounds could not be computed. RCOND = 0 is returned.
	       = N+1: D 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			     DSYSVX(l)
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