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CPBRFS(3F)							    CPBRFS(3F)


NAME    [Toc]    [Back]

     CPBRFS - improve the computed solution to a system	of linear equations
     when the coefficient matrix is Hermitian positive definite	and banded,
     and provides error	bounds and backward error estimates for	the solution

SYNOPSIS    [Toc]    [Back]

     SUBROUTINE	CPBRFS(	UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X,
			LDX, FERR, BERR, WORK, RWORK, INFO )

	 CHARACTER	UPLO

	 INTEGER	INFO, KD, LDAB,	LDAFB, LDB, LDX, N, NRHS

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

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

PURPOSE    [Toc]    [Back]

     CPBRFS improves the computed solution to a	system of linear equations
     when the coefficient matrix is Hermitian positive definite	and banded,
     and provides error	bounds and backward error estimates for	the solution.

ARGUMENTS    [Toc]    [Back]

     UPLO    (input) CHARACTER*1
	     = 'U':  Upper triangle of A is stored;
	     = 'L':  Lower triangle of A is stored.

     N	     (input) INTEGER
	     The order of the matrix A.	 N >= 0.

     KD	     (input) INTEGER
	     The number	of superdiagonals of the matrix	A if UPLO = 'U', or
	     the number	of subdiagonals	if UPLO	= 'L'.	KD >= 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) REAL array, dimension (LDAB,N)
	     The upper or lower	triangle of the	Hermitian band matrix A,
	     stored in the first KD+1 rows of the array.  The j-th column of A
	     is	stored in the j-th column of the array AB as follows:  if UPLO
	     = 'U', AB(kd+1+i-j,j) = A(i,j) for	max(1,j-kd)<=i<=j; if UPLO =
	     'L', AB(1+i-j,j)	 = A(i,j) for j<=i<=min(n,j+kd).

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






									Page 1






CPBRFS(3F)							    CPBRFS(3F)



     AFB     (input) COMPLEX array, dimension (LDAFB,N)
	     The triangular factor U or	L from the Cholesky factorization A =
	     U**H*U or A = L*L**H of the band matrix A as computed by CPBTRF,
	     in	the same storage format	as A (see AB).

     LDAFB   (input) INTEGER
	     The leading dimension of the array	AFB.  LDAFB >= KD+1.

     B	     (input) COMPLEX array, dimension (LDB,NRHS)
	     The right hand side matrix	B.

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

     X	     (input/output) COMPLEX array, dimension (LDX,NRHS)
	     On	entry, the solution matrix X, as computed by CPBTRS.  On exit,
	     the improved solution matrix X.

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

     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
	     estimate is as reliable as	the estimate for RCOND,	and is almost
	     always a slight overestimate 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 solution).

     WORK    (workspace) COMPLEX array,	dimension (2*N)

     RWORK   (workspace) REAL array, dimension (N)

     INFO    (output) INTEGER
	     = 0:  successful exit
	     < 0:  if INFO = -i, the i-th argument had an illegal value

PARAMETERS    [Toc]    [Back]

     ITMAX is the maximum number of steps of iterative refinement.
CPBRFS(3F)							    CPBRFS(3F)


NAME    [Toc]    [Back]

     CPBRFS - improve the computed solution to a system	of linear equations
     when the coefficient matrix is Hermitian positive definite	and banded,
     and provides error	bounds and backward error estimates for	the solution

SYNOPSIS    [Toc]    [Back]

     SUBROUTINE	CPBRFS(	UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X,
			LDX, FERR, BERR, WORK, RWORK, INFO )

	 CHARACTER	UPLO

	 INTEGER	INFO, KD, LDAB,	LDAFB, LDB, LDX, N, NRHS

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

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

PURPOSE    [Toc]    [Back]

     CPBRFS improves the computed solution to a	system of linear equations
     when the coefficient matrix is Hermitian positive definite	and banded,
     and provides error	bounds and backward error estimates for	the solution.

ARGUMENTS    [Toc]    [Back]

     UPLO    (input) CHARACTER*1
	     = 'U':  Upper triangle of A is stored;
	     = 'L':  Lower triangle of A is stored.

     N	     (input) INTEGER
	     The order of the matrix A.	 N >= 0.

     KD	     (input) INTEGER
	     The number	of superdiagonals of the matrix	A if UPLO = 'U', or
	     the number	of subdiagonals	if UPLO	= 'L'.	KD >= 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) REAL array, dimension (LDAB,N)
	     The upper or lower	triangle of the	Hermitian band matrix A,
	     stored in the first KD+1 rows of the array.  The j-th column of A
	     is	stored in the j-th column of the array AB as follows:  if UPLO
	     = 'U', AB(kd+1+i-j,j) = A(i,j) for	max(1,j-kd)<=i<=j; if UPLO =
	     'L', AB(1+i-j,j)	 = A(i,j) for j<=i<=min(n,j+kd).

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






									Page 1






CPBRFS(3F)							    CPBRFS(3F)



     AFB     (input) COMPLEX array, dimension (LDAFB,N)
	     The triangular factor U or	L from the Cholesky factorization A =
	     U**H*U or A = L*L**H of the band matrix A as computed by CPBTRF,
	     in	the same storage format	as A (see AB).

     LDAFB   (input) INTEGER
	     The leading dimension of the array	AFB.  LDAFB >= KD+1.

     B	     (input) COMPLEX array, dimension (LDB,NRHS)
	     The right hand side matrix	B.

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

     X	     (input/output) COMPLEX array, dimension (LDX,NRHS)
	     On	entry, the solution matrix X, as computed by CPBTRS.  On exit,
	     the improved solution matrix X.

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

     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
	     estimate is as reliable as	the estimate for RCOND,	and is almost
	     always a slight overestimate 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 solution).

     WORK    (workspace) COMPLEX array,	dimension (2*N)

     RWORK   (workspace) REAL array, dimension (N)

     INFO    (output) INTEGER
	     = 0:  successful exit
	     < 0:  if INFO = -i, the i-th argument had an illegal value

PARAMETERS    [Toc]    [Back]

     ITMAX is the maximum number of steps of iterative refinement.


									PPPPaaaaggggeeee 2222
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