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csysvx(3P)		    Sun Performance Library		    csysvx(3P)

NAME
       csysvx  -  use 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, IPIVOT, B,
	     LDB, X, LDX, RCOND, FERR, BERR, WORK, LDWORK, WORK2, INFO)

       CHARACTER * 1 FACT, UPLO
       COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
       INTEGER N, NRHS, LDA, LDAF, LDB, LDX, LDWORK, INFO
       INTEGER IPIVOT(*)
       REAL RCOND
       REAL FERR(*), BERR(*), WORK2(*)

       SUBROUTINE CSYSVX_64(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIVOT,
	     B, LDB, X, LDX, RCOND, FERR, BERR, WORK, LDWORK, WORK2, INFO)

       CHARACTER * 1 FACT, UPLO
       COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
       INTEGER*8 N, NRHS, LDA, LDAF, LDB, LDX, LDWORK, INFO
       INTEGER*8 IPIVOT(*)
       REAL RCOND
       REAL FERR(*), BERR(*), WORK2(*)

   F95 INTERFACE
       SUBROUTINE SYSVX(FACT, UPLO, [N], [NRHS], A, [LDA], AF, [LDAF],
	      IPIVOT, B, [LDB], X, [LDX], RCOND, FERR, BERR, [WORK], [LDWORK],
	      [WORK2], [INFO])

       CHARACTER(LEN=1) :: FACT, UPLO
       COMPLEX, DIMENSION(:) :: WORK
       COMPLEX, DIMENSION(:,:) :: A, AF, B, X
       INTEGER :: N, NRHS, LDA, LDAF, LDB, LDX, LDWORK, INFO
       INTEGER, DIMENSION(:) :: IPIVOT
       REAL :: RCOND
       REAL, DIMENSION(:) :: FERR, BERR, WORK2

       SUBROUTINE SYSVX_64(FACT, UPLO, [N], [NRHS], A, [LDA], AF, [LDAF],
	      IPIVOT, B, [LDB], X, [LDX], RCOND, FERR, BERR, [WORK], [LDWORK],
	      [WORK2], [INFO])

       CHARACTER(LEN=1) :: FACT, UPLO
       COMPLEX, DIMENSION(:) :: WORK
       COMPLEX, DIMENSION(:,:) :: A, AF, B, X
       INTEGER(8) :: N, NRHS, LDA, LDAF, LDB, LDX, LDWORK, INFO
       INTEGER(8), DIMENSION(:) :: IPIVOT
       REAL :: RCOND
       REAL, DIMENSION(:) :: FERR, BERR, WORK2

   C INTERFACE
       #include <sunperf.h>

       void csysvx(char fact, char uplo, int n, int nrhs, complex *a, int lda,
		 complex *af, int ldaf, int *ipivot, complex *b, int ldb, com‐
		 plex *x, int ldx, float *rcond, float *ferr, float *berr, int
		 *info);

       void  csysvx_64(char  fact,  char  uplo, long n, long nrhs, complex *a,
		 long lda, complex *af, long ldaf, long *ipivot,  complex  *b,
		 long  ldb,  complex  *x, long ldx, float *rcond, float *ferr,
		 float *berr, long *info);

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.

       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)
		 Specifies whether or not the factored form of A has been sup‐
		 plied on entry.  = 'F':  On entry, AF and IPIVOT contain  the
		 factored  form	 of A.	A, AF and IPIVOT will not be modified.
		 = 'N':	 The matrix A will be copied to AF and factored.

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

       N (input) The number of linear equations, i.e., the order of the matrix
		 A.  N >= 0.

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

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

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

       IPIVOT (input or output)
		 If FACT = 'F', then IPIVOT is an input argument and on	 entry
		 contains  details of the interchanges and the block structure
		 of D, as determined by CSYTRF.	 If IPIVOT(k) > 0,  then  rows
		 and columns k and IPIVOT(k) were interchanged and D(k,k) is a
		 1-by-1 diagonal  block.   If  UPLO  =	'U'  and  IPIVOT(k)  =
		 IPIVOT(k-1)  <	 0,  then  rows and columns k-1 and -IPIVOT(k)
		 were interchanged and D(k-1:k,k-1:k)  is  a  2-by-2  diagonal
		 block.	  If  UPLO = 'L' and IPIVOT(k) = IPIVOT(k+1) < 0, then
		 rows and columns k+1 and  -IPIVOT(k)  were  interchanged  and
		 D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

		 If  FACT = 'N', then IPIVOT is an output argument and on exit
		 contains details of the interchanges and the block  structure
		 of D, as determined by CSYTRF.

       B (input) The N-by-NRHS right hand side matrix B.

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

       X (output)
		 If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

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

       RCOND (output)
		 The estimate of the reciprocal condition number of the matrix
		 A.  If RCOND is less than the machine precision (in  particu‐
		 lar,  if RCOND = 0), the matrix is singular to working preci‐
		 sion.	This condition is indicated by a return code of INFO >
		 0.

       FERR (output)
		 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 esti‐
		 mated upper bound for the magnitude of the largest element in
		 (X(j)	-  XTRUE) divided by the magnitude of the largest ele‐
		 ment 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)
		 The componentwise relative backward error  of	each  solution
		 vector	 X(j)  (i.e., the smallest relative change in any ele‐
		 ment of A or B that makes X(j) an exact solution).

       WORK (workspace)
		 On exit, if INFO = 0, WORK(1) returns the optimal LDWORK.

       LDWORK (input)
		 The length of WORK.  LDWORK >= 2*N, and for best  performance
		 LDWORK >= N*NB, where NB is the optimal blocksize for CSYTRF.

		 If  LDWORK  = -1, then a workspace query is assumed; the rou‐
		 tine 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 LDWORK is issued by XERBLA.

       WORK2 (workspace)
		 dimension(N)

       INFO (output)
		 = 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
		 completed  but the factor D is exactly singular, so the solu‐
		 tion and error bounds could not be computed.  RCOND  =	 0  is
		 returned.   =	N+1:  D is nonsingular, but RCOND is less than
		 machine precision, 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.

				  6 Mar 2009			    csysvx(3P)
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