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

NAME
       CSYSVX  - uses the diagonal pivoting factorization to compute the solu‐
       tion to a complex system of linear equations A * X = B,

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

	   CHARACTER	  FACT, UPLO

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

	   REAL		  RCOND

	   INTEGER	  IPIV( * )

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

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

PURPOSE
       CSYSVX uses the diagonal pivoting factorization to compute the solution
       to a complex 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.  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) 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 or output) COMPLEX 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 CSYTRF.  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 CSYTRF.	 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 contains details of the interchanges and the block	struc‐
	       ture of D, as determined by CSYTRF.

       B       (input) COMPLEX 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) COMPLEX 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) REAL
	       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) 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/output) COMPLEX array, dimension (MAX(1,LWORK))
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The length of WORK.  LWORK >= max(1,2*N), and for best  perfor‐
	       mance,  when  FACT = 'N', LWORK >= max(1,2*N,N*NB), where NB is
	       the optimal blocksize for  CSYTRF.   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.

       RWORK   (workspace) REAL 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 driver routine (version 3November 2008			     CSYSVX(1)
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