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dgbsvx.f(3)				      LAPACK				      dgbsvx.f(3)

NAME
       dgbsvx.f -

SYNOPSIS
   Functions/Subroutines
       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)
	    DGBSVX computes the solution to system of linear equations A * X = B for GB matrices

Function/Subroutine Documentation
   subroutine dgbsvx (characterFACT, characterTRANS, integerN, integerKL, integerKU, integerNRHS,
       double precision, dimension( ldab, * )AB, integerLDAB, double precision, dimension( ldafb,
       * )AFB, integerLDAFB, integer, dimension( * )IPIV, characterEQUED, double precision,
       dimension( * )R, double precision, dimension( * )C, double precision, dimension( ldb, *
       )B, integerLDB, double precision, dimension( ldx, * )X, integerLDX, double precisionRCOND,
       double precision, dimension( * )FERR, double precision, dimension( * )BERR, double
       precision, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)
	DGBSVX computes the solution to system of linear equations A * X = B for GB matrices

       Purpose:

	    DGBSVX uses the LU factorization to compute the solution to a real
	    system of linear equations 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.

       Parameters:
	   FACT

		     FACT is 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

		     TRANS is 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

		     N is INTEGER
		     The number of linear equations, i.e., the order of the
		     matrix A.	N >= 0.

	   KL

		     KL is INTEGER
		     The number of subdiagonals within the band of A.  KL >= 0.

	   KU

		     KU is INTEGER
		     The number of superdiagonals within the band of A.  KU >= 0.

	   NRHS

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

	   AB

		     AB is 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

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

	   AFB

		     AFB is 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

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

	   IPIV

		     IPIV is 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

		     EQUED is 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

		     R is 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

		     C is 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 argument 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

		     B is DOUBLE PRECISION array, dimension (LDB,NRHS)
		     On entry, the 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

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

	   X

		     X is 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

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

	   RCOND

		     RCOND is DOUBLE PRECISION
		     The estimate of the reciprocal condition number of the matrix
		     A after equilibration (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
		     indicated by a return code of INFO > 0.

	   FERR

		     FERR is 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 estimate is as reliable as
		     the estimate for RCOND, and is almost always a slight
		     overestimate of the true error.

	   BERR

		     BERR is 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 solution).

	   WORK

		     WORK is 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 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
		     WORK(1) contains the reciprocal pivot growth factor for the
		     leading INFO columns of A.

	   IWORK

		     IWORK is INTEGER array, dimension (N)

	   INFO

		     INFO is 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 factor 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.  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.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   April 2012

       Definition at line 368 of file dgbsvx.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      dgbsvx.f(3)
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