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man pages->IRIX man pages -> complib/zpbrfs (3)
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### Contents

```
ZPBRFS(3F)							    ZPBRFS(3F)

```

### NAME[Toc][Back]

```     ZPBRFS - 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	ZPBRFS(	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

DOUBLE		PRECISION BERR(	* ), FERR( * ),	RWORK( * )

COMPLEX*16	AB( LDAB, * ), AFB( LDAFB, * ),	B( LDB,	* ), WORK( *
), X( LDX, * )
```

### PURPOSE[Toc][Back]

```     ZPBRFS 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) DOUBLE PRECISION 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

ZPBRFS(3F)							    ZPBRFS(3F)

AFB     (input) COMPLEX*16	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 ZPBTRF,
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*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/output) COMPLEX*16 array, dimension	(LDX,NRHS)
On	entry, the solution matrix X, as computed by ZPBTRS.  On exit,
the improved 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[Toc][Back]

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

```

### NAME[Toc][Back]

```     ZPBRFS - 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	ZPBRFS(	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

DOUBLE		PRECISION BERR(	* ), FERR( * ),	RWORK( * )

COMPLEX*16	AB( LDAB, * ), AFB( LDAFB, * ),	B( LDB,	* ), WORK( *
), X( LDX, * )
```

### PURPOSE[Toc][Back]

```     ZPBRFS 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) DOUBLE PRECISION 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

ZPBRFS(3F)							    ZPBRFS(3F)

AFB     (input) COMPLEX*16	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 ZPBTRF,
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*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/output) COMPLEX*16 array, dimension	(LDX,NRHS)
On	entry, the solution matrix X, as computed by ZPBTRS.  On exit,
the improved 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[Toc][Back]

```     ITMAX is the maximum number of steps of iterative refinement.

PPPPaaaaggggeeee 2222```
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