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

DGESVX(l)					)					DGESVX(l)

NAME
       DGESVX - use the LU factorization to compute the solution to a real system of linear equa-
       tions A * X = B,

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

	   CHARACTER	  EQUED, FACT, TRANS

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

	   DOUBLE	  PRECISION RCOND

	   INTEGER	  IPIV( * ), IWORK( * )

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

PURPOSE
       DGESVX uses the LU factorization to compute the solution to a real system of linear  equa-
       tions 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  (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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DGETRF.  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  DGETRF;  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) DOUBLE PRECISION 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) DOUBLE PRECISION 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/output) DOUBLE PRECISION array, dimension (4*N)
	       On  exit, WORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The
	       "max absolute element" norm is used. If WORK(1) is much less than 1, then the sta-
	       bility  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 WORK(1) con-
	       tains the reciprocal pivot growth factor for the leading INFO columns of A.

       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:  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 				DGESVX(l)


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