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

DGBSVX(l)					)					DGBSVX(l)

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

SYNOPSIS
       SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R,  C,
			  B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )

	   CHARACTER	  EQUED, FACT, TRANS

	   INTEGER	  INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS

	   DOUBLE	  PRECISION RCOND

	   INTEGER	  IPIV( * ), IWORK( * )

	   DOUBLE	  PRECISION  AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( * ), C( *
			  ), FERR( * ), R( * ), WORK( * ), X( LDX, * )

PURPOSE
       DGBSVX uses the LU factorization to compute the solution to a real system of linear  equa-
       tions  A  * X = B, A**T * X = B, or A**H * X = B, where A is a band matrix of order N with
       KL subdiagonals and KU superdiagonals, 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 by this subroutine:

       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 = L * U,
	  where L is a product of permutation and unit lower triangular
	  matrices with KL subdiagonals, and U is upper triangular with
	  KL+KU superdiagonals.

       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, AFB 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.  AB,
	       AFB, and IPIV are not modified.	= 'N':	The matrix A will be copied  to  AFB  and
	       factored.
	       =  'E':	 The  matrix  A will be equilibrated if necessary, then copied to AFB and
	       factored.

       TRANS   (input) CHARACTER*1
	       Specifies the form of the system of equations.  = 'N':  A * X = B      (No  trans-
	       pose)
	       = '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.

       KL      (input) INTEGER
	       The number of subdiagonals within the band of A.  KL >= 0.

       KU      (input) INTEGER
	       The number of superdiagonals within the band of A.  KU >= 0.

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

       AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
	       On entry, the matrix A in band storage, in rows 1 to KL+KU+1.  The j-th column  of
	       A is stored in the j-th column of the array AB as follows: AB(KU+1+i-j,j) = A(i,j)
	       for max(1,j-KU)<=i<=min(N,j+kl)

	       If FACT = 'F' and EQUED is not 'N', then A must	have  been  equilibrated  by  the
	       scaling	factors  in  R	and/or C.  AB is not modified if FACT = 'F' or 'N', or if
	       FACT = 'E' and EQUED = 'N' on exit.

	       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).

       LDAB    (input) INTEGER
	       The leading dimension of the array AB.  LDAB >= KL+KU+1.

       AFB     (input or output) DOUBLE PRECISION array, dimension (LDAFB,N)
	       If FACT = 'F', then AFB is an input argument and on entry contains details of  the
	       LU  factorization  of the band matrix A, as computed by DGBTRF.	U is stored as an
	       upper triangular band matrix with KL+KU superdiagonals in rows 1 to  KL+KU+1,  and
	       the  multipliers  used  during  the  factorization  are	stored in rows KL+KU+2 to
	       2*KL+KU+1.  If EQUED .ne. 'N', then AFB is the factored form of	the  equilibrated
	       matrix A.

	       If  FACT  = 'N', then AFB is an output argument and on exit returns details of the
	       LU factorization of A.

	       If FACT = 'E', then AFB is an output argument and on exit returns details  of  the
	       LU  factorization  of the equilibrated matrix A (see the description of AB for the
	       form of the equilibrated matrix).

       LDAFB   (input) INTEGER
	       The leading dimension of the array AFB.	LDAFB >= 2*KL+KU+1.

       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 = L*U as computed by DGBTRF; 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 = 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 = 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 right hand side matrix B.	On exit, if EQUED = 'N', B is  not  modi-
	       fied;  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 (3*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 				DGBSVX(l)


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