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DTPRFS(1)		 LAPACK routine (version 3.2)		     DTPRFS(1)

NAME
       DTPRFS  -  provides  error  bounds and backward error estimates for the
       solution to a system of linear equations with a triangular packed coef‐
       ficient matrix

SYNOPSIS
       SUBROUTINE DTPRFS( UPLO,	 TRANS,	 DIAG,	N,  NRHS,  AP, B, LDB, X, LDX,
			  FERR, BERR, WORK, IWORK, INFO )

	   CHARACTER	  DIAG, TRANS, UPLO

	   INTEGER	  INFO, LDB, LDX, N, NRHS

	   INTEGER	  IWORK( * )

	   DOUBLE	  PRECISION AP( * ), B( LDB, * ), BERR( * ),  FERR(  *
			  ), WORK( * ), X( LDX, * )

PURPOSE
       DTPRFS provides error bounds and backward error estimates for the solu‐
       tion to a system of linear equations with a triangular  packed  coeffi‐
       cient matrix.  The solution matrix X must be computed by DTPTRS or some
       other means before entering this routine.  DTPRFS does not do iterative
       refinement because doing so cannot improve the backward error.

ARGUMENTS
       UPLO    (input) CHARACTER*1
	       = 'U':  A is upper triangular;
	       = 'L':  A is lower triangular.

       TRANS   (input) 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  (Conjugate transpose = Transpose)

       DIAG    (input) CHARACTER*1
	       = 'N':  A is non-unit triangular;
	       = 'U':  A is unit triangular.

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

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

       AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
	       The upper or lower triangular matrix A, packed columnwise in  a
	       linear  array.	The j-th column of A is stored in the array AP
	       as follows: if UPLO = 'U',  AP(i	 +  (j-1)*j/2)	=  A(i,j)  for
	       1<=i<=j;	 if  UPLO  = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for
	       j<=i<=n.	 If DIAG = 'U', the diagonal elements  of  A  are  not
	       referenced and are assumed to be 1.

       B       (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS)
	       The solution matrix X.

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

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

       BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
	       The componentwise relative backward error of each solution vec‐
	       tor X(j) (i.e., the smallest relative change in any element  of
	       A or B that makes X(j) an exact solution).

       WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

       IWORK   (workspace) INTEGER array, dimension (N)

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

 LAPACK routine (version 3.2)	 November 2008			     DTPRFS(1)
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