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# ctbrfs

```
NAME
CTBRFS - provide error bounds and backward error estimates
for the solution to a system of linear equations with a tri-
angular band coefficient matrix

SYNOPSIS
SUBROUTINE CTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB,
B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
INFO )

CHARACTER      DIAG, TRANS, UPLO

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

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

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

PURPOSE
CTBRFS provides error bounds and backward error estimates
for the solution to a system of linear equations with a tri-
angular band coefficient matrix.

The solution matrix X must be computed by CTBTRS or some
other means before entering this routine.  CTBRFS 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.

KD      (input) INTEGER
The number of superdiagonals or subdiagonals of the
triangular band matrix A.  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) COMPLEX array, dimension (LDAB,N)
The upper or lower triangular 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).  If DIAG = 'U', the
diagonal elements of A are not referenced and are
assumed to be 1.

LDAB    (input) INTEGER
The leading dimension of the array AB.  LDAB >=
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) COMPLEX array, dimension (LDX,NRHS)
The 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 bounds for each solution
vector X(j) (the j-th column of the solution matrix
X).  If XTRUE is the true solution, FERR(j) bounds
the magnitude of the largest entry in (X(j) - XTRUE)
divided by the magnitude of the largest entry in
X(j).  The quality of the error bound depends on the
quality of the estimate of norm(inv(A)) computed in
the code; if the estimate of norm(inv(A)) is accu-
rate, the error bound is guaranteed.

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 entry 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
```