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RedHat 9 (Linux i386) - man page for zgesvx (redhat section l)

ZGESVX(l)					)					ZGESVX(l)

NAME
       ZGESVX  -  use  the LU factorization to compute the solution to a complex system of linear
       equations A * X = B,

SYNOPSIS
       SUBROUTINE ZGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B,  LDB,  X,
			  LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )

	   CHARACTER	  EQUED, FACT, TRANS

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

	   DOUBLE	  PRECISION RCOND

	   INTEGER	  IPIV( * )

	   DOUBLE	  PRECISION BERR( * ), C( * ), FERR( * ), R( * ), RWORK( * )

	   COMPLEX*16	  A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( * ), X( LDX, * )

PURPOSE
       ZGESVX  uses  the  LU  factorization to compute the solution to a complex system of linear
       equations A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

       Error bounds on the solution and a condition estimate are also provided.

DESCRIPTION
       The following steps are performed:

       1. If FACT = 'E', real 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.  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.

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

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

       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
       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  fac-
	       tored.
	       = 'E':  The matrix A will be equilibrated if necessary, then copied to AF and fac-
	       tored.

       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)

       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/output) COMPLEX*16 array, dimension (LDA,N)
	       On  entry,  the	N-by-N matrix A.  If FACT = 'F' and EQUED is not 'N', then A must
	       have been equilibrated by the scaling factors in R and/or C.  A is not modified if
	       FACT = 'F' or

	       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).

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

       AF      (input or output) COMPLEX*16 array, dimension (LDAF,N)
	       If  FACT  =  'F', then AF is an input argument and on entry contains the factors L
	       and U from the factorization A = P*L*U as computed by ZGETRF.  If EQUED .ne.  'N',
	       then AF is the factored 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 descrip-
	       tion of A for the form of the equilibrated matrix).

       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  contains  the  pivot
	       indices	from  the  factorization  A  =	P*L*U as computed by ZGETRF; 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 postmultiplied by diag(C).   =  'B':   Both
	       row  and column equilibration, 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 multiplied 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.

       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  argu-
	       ment if FACT = 'F'; otherwise, C is an output argument.	If FACT = 'F' and EQUED =
	       'C' or 'B', each element of C must be positive.

       B       (input/output) COMPLEX*16 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) COMPLEX*16 array, dimension (LDX,NRHS)
	       If  INFO = 0 or INFO = N+1, 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
	       The estimate of the reciprocal condition number of the matrix A	after  equilibra-
	       tion  (if  done).   If RCOND is less than the machine precision (in particular, if
	       RCOND = 0), the matrix is singular to working precision.  This condition is  indi-
	       cated 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 esti-
	       mate is as reliable as the estimate for RCOND, and is almost always a slight over-
	       estimate of the true error.

       BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
	       The componentwise 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 solu-
	       tion).

       WORK    (workspace) COMPLEX*16 array, dimension (2*N)

       RWORK   (workspace/output) DOUBLE PRECISION array, dimension (2*N)
	       On exit, RWORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The
	       "max  absolute  element"  norm  is used. If RWORK(1) 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, condition estimator RCOND, and forward error
	       bound FERR could be  unreliable.  If  factorization  fails  with  0<INFO<=N,  then
	       RWORK(1)  contains the reciprocal pivot growth factor for the leading INFO columns
	       of A.

       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:  U(i,i) is exactly zero.  The factorization has been completed, but the fac-
	       tor U is exactly singular, so the solution and error bounds could not be computed.
	       RCOND = 0 is returned.  = N+1: U is nonsingular, but RCOND is  less  than  machine
	       precision,  meaning  that  the matrix is singular to working precision.	Neverthe-
	       less, 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 				ZGESVX(l)


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