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cgbsvx(3P)		    Sun Performance Library		    cgbsvx(3P)

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, TRANSA, N, KL, KU, NRHS, A, LDA, AF,
	     LDAF, IPIVOT, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR,
	     BERR, WORK, WORK2, INFO)

       CHARACTER * 1 FACT, TRANSA, EQUED
       COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
       INTEGER N, KL, KU, NRHS, LDA, LDAF, LDB, LDX, INFO
       INTEGER IPIVOT(*)
       REAL RCOND
       REAL R(*), C(*), FERR(*), BERR(*), WORK2(*)

       SUBROUTINE CGBSVX_64(FACT, TRANSA, N, KL, KU, NRHS, A, LDA, AF,
	     LDAF, IPIVOT, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR,
	     BERR, WORK, WORK2, INFO)

       CHARACTER * 1 FACT, TRANSA, EQUED
       COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*)
       INTEGER*8 N, KL, KU, NRHS, LDA, LDAF, LDB, LDX, INFO
       INTEGER*8 IPIVOT(*)
       REAL RCOND
       REAL R(*), C(*), FERR(*), BERR(*), WORK2(*)

   F95 INTERFACE
       SUBROUTINE GBSVX(FACT, [TRANSA], [N], KL, KU, [NRHS], A, [LDA],
	      AF, [LDAF], IPIVOT, EQUED, R, C, B, [LDB], X, [LDX],
	      RCOND, FERR, BERR, [WORK], [WORK2], [INFO])

       CHARACTER(LEN=1) :: FACT, TRANSA, EQUED
       COMPLEX, DIMENSION(:) :: WORK
       COMPLEX, DIMENSION(:,:) :: A, AF, B, X
       INTEGER :: N, KL, KU, NRHS, LDA, LDAF, LDB, LDX, INFO
       INTEGER, DIMENSION(:) :: IPIVOT
       REAL :: RCOND
       REAL, DIMENSION(:) :: R, C, FERR, BERR, WORK2

       SUBROUTINE GBSVX_64(FACT, [TRANSA], [N], KL, KU, [NRHS], A,
	      [LDA], AF, [LDAF], IPIVOT, EQUED, R, C, B, [LDB], X, [LDX],
	      RCOND, FERR, BERR, [WORK], [WORK2], [INFO])

       CHARACTER(LEN=1) :: FACT, TRANSA, EQUED
       COMPLEX, DIMENSION(:) :: WORK
       COMPLEX, DIMENSION(:,:) :: A, AF, B, X
       INTEGER(8) :: N, KL, KU, NRHS, LDA, LDAF, LDB, LDX, INFO
       INTEGER(8), DIMENSION(:) :: IPIVOT
       REAL :: RCOND
       REAL, DIMENSION(:) :: R, C, FERR, BERR, WORK2

   C INTERFACE
       #include <sunperf.h>

       void cgbsvx(char fact, char transa, int n, int kl, int  ku,  int	 nrhs,
		 complex *a, int lda, complex *af, int ldaf, int *ipivot, char
		 *equed, float *r, float *c, complex *b, int ldb, complex  *x,
		 int ldx, float *rcond, float *ferr, float *berr, int *info);

       void  cgbsvx_64(char  fact, char transa, long n, long kl, long ku, long
		 nrhs, complex *a, long lda,  complex  *af,  long  ldaf,  long
		 *ipivot,  char	 *equed,  float *r, float *c, complex *b, long
		 ldb, complex *x, long ldx, float *rcond, float	 *ferr,	 float
		 *berr, long *info);

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 super‐
       diagonals, and X and B are N-by-NRHS matrices.

       Error bounds on the solution and a condition  estimate  are  also  pro‐
       vided.

       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)
		 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
		 IPIVOT 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 IPIVOT are not modified.  = 'N':  The
		 matrix A will be copied to AF and factored.
		 =  'E':  The matrix A will be equilibrated if necessary, then
		 copied to AF and factored.

       TRANSA (input)
		 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)

		 TRANSA is defaulted to 'N' for F95 INTERFACE.

       N (input) The number of linear equations, i.e., the order of the matrix
		 A.  N >= 0.

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

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

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

       A (input/output)
		 COMPLEX  array,  dimension  (LDA,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	A  as follows:
		 A(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.  A 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).

       LDA (input)
		 The leading dimension of the array A.	LDA >= KL+KU+1.

       AF (input or output)
		 COMPLEX array, dimension (LDAF,N) If FACT = 'F', then	AF  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 AF is the	factored  form
		 of the equilibrated matrix A.

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

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

       LDAF (input)
		 The leading dimension of the array AF.	 LDAF >= 2*KL+KU+1.

       IPIVOT (input or output)
		 INTEGER array, dimension (N) If FACT = 'F', then IPIVOT 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 IPIVOT(i).

		 If FACT = 'N', then IPIVOT 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 IPIVOT 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)
		 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 equi‐
		 libration, i.e., A  has  been	replaced  by  diag(R)  *  A  *
		 diag(C).   EQUED  is  an input argument if FACT = 'F'; other‐
		 wise, 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
		 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 (input/output)
		 COMPLEX array, dimension (LDB,NRHS) On entry, the right  hand
		 side  matrix  B.  On exit, if EQUED = 'N', B is not modified;
		 if TRANSA = 'N' and EQUED = 'R' or 'B', B is  overwritten  by
		 diag(R)*B;  if	 TRANSA = 'T' or 'C' and EQUED = 'C' or 'B', B
		 is overwritten by diag(C)*B.

       LDB (input)
		 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 TRANSA = 'N' and EQUED = 'C' or 'B',  or
		 inv(diag(R))*X if TRANSA = 'T' or 'C' and EQUED = 'R' or 'B'.

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

       RCOND (output)
		 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	 indi‐
		 cated by a return code of INFO > 0.

       FERR (output)
		 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 esti‐
		 mated upper bound for the magnitude of the largest element in
		 (X(j)	-  XTRUE) divided by the magnitude of the largest ele‐
		 ment 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 (output)
		 The componentwise relative backward error  of	each  solution
		 vector	 X(j)  (i.e., the smallest relative change in any ele‐
		 ment of A or B that makes X(j) an exact solution).

       WORK (workspace)
		 dimension(2*N)

       WORK2 (workspace)
		 dimension(N) On exit, WORK2(1) contains the reciprocal	 pivot
		 growth	 factor	 norm(A)/norm(U).  The	"max absolute element"
		 norm is used. If WORK2(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 unre‐
		 liable. If factorization fails with 0<INFO<=N, then  WORK2(1)
		 contains  the	reciprocal pivot growth factor for the leading
		 INFO columns of A.

       INFO (output)
		 = 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 solu‐
		 tion 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.

				  6 Mar 2009			    cgbsvx(3P)
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