Home Man
Search
Today's Posts
Register

Linux & Unix Commands - Search Man Pages

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

CGBSVX(l)					)					CGBSVX(l)

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

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

	   CHARACTER	  EQUED, FACT, TRANS

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

	   REAL 	  RCOND

	   INTEGER	  IPIV( * )

	   REAL 	  BERR( * ), C( * ), FERR( * ), R( * ), RWORK( * )

	   COMPLEX	  AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( * ), X( LDX, * )

PURPOSE
       CGBSVX  uses  the  LU  factorization to compute the solution to a complex 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.

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  (Conjugate 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) COMPLEX 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) COMPLEX 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 CGBTRF.  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 CGBTRF; 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) REAL 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) REAL 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 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) COMPLEX 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) REAL
	       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) REAL 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) REAL 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 array, dimension (2*N)

       RWORK   (workspace/output) REAL array, dimension (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 				CGBSVX(l)


All times are GMT -4. The time now is 08:58 AM.

Unix & Linux Forums Content Copyrightę1993-2018. All Rights Reserved.
UNIX.COM Login
Username:
Password:  
Show Password