```      SUBROUTINE ZTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
\$                   LDX, FERR, BERR, WORK, RWORK, INFO )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
*
*     .. Scalar Arguments ..
CHARACTER          DIAG, TRANS, UPLO
INTEGER            INFO, LDA, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * ),
\$                   X( LDX, * )
*     ..
*
*  Purpose
*  =======
*
*  ZTRRFS provides error bounds and backward error estimates for the
*  solution to a system of linear equations with a triangular
*  coefficient matrix.
*
*  The solution matrix X must be computed by ZTRTRS or some other
*  means before entering this routine.  ZTRRFS 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)
*
*  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.
*
*  A       (input) COMPLEX*16 array, dimension (LDA,N)
*          The triangular matrix A.  If UPLO = 'U', the leading N-by-N
*          upper triangular part of the array A contains the upper
*          triangular matrix, and the strictly lower triangular part of
*          A is not referenced.  If UPLO = 'L', the leading N-by-N lower
*          triangular part of the array A contains the lower triangular
*          matrix, and the strictly upper triangular part of A is not
*          referenced.  If DIAG = 'U', the diagonal elements of A are
*          also not referenced and are assumed to be 1.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  B       (input) COMPLEX*16 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) COMPLEX*16 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
*          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*16 array, dimension (2*N)
*
*  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. Parameters ..
DOUBLE PRECISION   ZERO
PARAMETER          ( ZERO = 0.0D+0 )
COMPLEX*16         ONE
PARAMETER          ( ONE = ( 1.0D+0, 0.0D+0 ) )
*     ..
*     .. Local Scalars ..
LOGICAL            NOTRAN, NOUNIT, UPPER
CHARACTER          TRANSN, TRANST
INTEGER            I, J, K, KASE, NZ
DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
COMPLEX*16         ZDUM
*     ..
*     .. Local Arrays ..
INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
EXTERNAL           XERBLA, ZAXPY, ZCOPY, ZLACN2, ZTRMV, ZTRSV
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          ABS, DBLE, DIMAG, MAX
*     ..
*     .. External Functions ..
LOGICAL            LSAME
DOUBLE PRECISION   DLAMCH
EXTERNAL           LSAME, DLAMCH
*     ..
*     .. Statement Functions ..
DOUBLE PRECISION   CABS1
*     ..
*     .. Statement Function definitions ..
CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
NOTRAN = LSAME( TRANS, 'N' )
NOUNIT = LSAME( DIAG, 'N' )
*
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
\$         LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( NRHS.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZTRRFS', -INFO )
RETURN
END IF
*
*     Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10    CONTINUE
RETURN
END IF
*
IF( NOTRAN ) THEN
TRANSN = 'N'
TRANST = 'C'
ELSE
TRANSN = 'C'
TRANST = 'N'
END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = N + 1
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
*     Do for each right hand side
*
DO 250 J = 1, NRHS
*
*        Compute residual R = B - op(A) * X,
*        where op(A) = A, A**T, or A**H, depending on TRANS.
*
CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
CALL ZTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK, 1 )
CALL ZAXPY( N, -ONE, B( 1, J ), 1, WORK, 1 )
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
DO 20 I = 1, N
RWORK( I ) = CABS1( B( I, J ) )
20    CONTINUE
*
IF( NOTRAN ) THEN
*
*           Compute abs(A)*abs(X) + abs(B).
*
IF( UPPER ) THEN
IF( NOUNIT ) THEN
DO 40 K = 1, N
XK = CABS1( X( K, J ) )
DO 30 I = 1, K
RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
30                CONTINUE
40             CONTINUE
ELSE
DO 60 K = 1, N
XK = CABS1( X( K, J ) )
DO 50 I = 1, K - 1
RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
50                CONTINUE
RWORK( K ) = RWORK( K ) + XK
60             CONTINUE
END IF
ELSE
IF( NOUNIT ) THEN
DO 80 K = 1, N
XK = CABS1( X( K, J ) )
DO 70 I = K, N
RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
70                CONTINUE
80             CONTINUE
ELSE
DO 100 K = 1, N
XK = CABS1( X( K, J ) )
DO 90 I = K + 1, N
RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
90                CONTINUE
RWORK( K ) = RWORK( K ) + XK
100             CONTINUE
END IF
END IF
ELSE
*
*           Compute abs(A**H)*abs(X) + abs(B).
*
IF( UPPER ) THEN
IF( NOUNIT ) THEN
DO 120 K = 1, N
S = ZERO
DO 110 I = 1, K
S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
110                CONTINUE
RWORK( K ) = RWORK( K ) + S
120             CONTINUE
ELSE
DO 140 K = 1, N
S = CABS1( X( K, J ) )
DO 130 I = 1, K - 1
S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
130                CONTINUE
RWORK( K ) = RWORK( K ) + S
140             CONTINUE
END IF
ELSE
IF( NOUNIT ) THEN
DO 160 K = 1, N
S = ZERO
DO 150 I = K, N
S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
150                CONTINUE
RWORK( K ) = RWORK( K ) + S
160             CONTINUE
ELSE
DO 180 K = 1, N
S = CABS1( X( K, J ) )
DO 170 I = K + 1, N
S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
170                CONTINUE
RWORK( K ) = RWORK( K ) + S
180             CONTINUE
END IF
END IF
END IF
S = ZERO
DO 190 I = 1, N
IF( RWORK( I ).GT.SAFE2 ) THEN
S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
ELSE
S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
\$             ( RWORK( I )+SAFE1 ) )
END IF
190    CONTINUE
BERR( J ) = S
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(op(A)))*
*           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(op(A)) is the inverse of op(A)
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
*        Use ZLACN2 to estimate the infinity-norm of the matrix
*           inv(op(A)) * diag(W),
*        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
DO 200 I = 1, N
IF( RWORK( I ).GT.SAFE2 ) THEN
RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
ELSE
RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
\$                      SAFE1
END IF
200    CONTINUE
*
KASE = 0
210    CONTINUE
CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
*              Multiply by diag(W)*inv(op(A)**H).
*
CALL ZTRSV( UPLO, TRANST, DIAG, N, A, LDA, WORK, 1 )
DO 220 I = 1, N
WORK( I ) = RWORK( I )*WORK( I )
220          CONTINUE
ELSE
*
*              Multiply by inv(op(A))*diag(W).
*
DO 230 I = 1, N
WORK( I ) = RWORK( I )*WORK( I )
230          CONTINUE
CALL ZTRSV( UPLO, TRANSN, DIAG, N, A, LDA, WORK, 1 )
END IF
GO TO 210
END IF
*
*        Normalize error.
*
LSTRES = ZERO
DO 240 I = 1, N
LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
240    CONTINUE
IF( LSTRES.NE.ZERO )
\$      FERR( J ) = FERR( J ) / LSTRES
*
250 CONTINUE
*
RETURN
*
*     End of ZTRRFS
*
END

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