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

```
CGBSVX(3F)							    CGBSVX(3F)

```

### NAME[Toc][Back]

```     CGBSVX - use the LU factorization to compute the solution to a complex
system of linear equations	A * X =	B, A**T	* X = B, or A**H * X = B,
```

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	CGBSVX(	FACT, TRANS, N,	KL, KU,	NRHS, AB, LDAB,	AFB, LDAFB,
IPIV, EQUED, R,	C, B, LDB, X, LDX, RCOND, FERR,	BERR,
WORK, RWORK, INFO )

CHARACTER	EQUED, FACT, TRANS

INTEGER	INFO, KL, KU, LDAB, LDAFB, LDB,	LDX, N,	NRHS

REAL		RCOND

INTEGER	IPIV( *	)

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

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

### PURPOSE[Toc][Back]

```     CGBSVX uses the LU	factorization to compute the solution to a complex
system of linear equations	A * X =	B, A**T	* X = B, or A**H * X = B,
where A is	a band matrix of order N with KL subdiagonals and KU
superdiagonals, and X and B are N-by-NRHS matrices.

Error bounds on the solution and a	condition estimate are also provided.

```

### DESCRIPTION[Toc][Back]

```     The following steps are performed by this subroutine:

1.	If FACT	= 'E', real scaling factors are	computed to equilibrate
the system:
TRANS = 'N':	 diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
TRANS = 'T':	(diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C':	(diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
Whether	or not the system will be equilibrated depends on the
scaling	of the matrix A, but if	equilibration is used, A is
overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if	TRANS='N')
or diag(C)*B (if TRANS = 'T' or	'C').

2.	If FACT	= 'N' or 'E', the LU decomposition is used to factor the
matrix A (after	equilibration if FACT =	'E') as
A = L * U,
where L	is a product of	permutation and	unit lower triangular
matrices with KL subdiagonals, and U is	upper triangular with
KL+KU superdiagonals.

3.	The factored form of A is used to estimate the condition number

Page 1

CGBSVX(3F)							    CGBSVX(3F)

of the matrix A.  If the reciprocal of the condition number is
less than machine precision, steps 4-6 are skipped.

4.	The system of equations	is solved for X	using the factored form
of A.

5.	Iterative refinement is	applied	to improve the computed	solution
matrix and calculate error bounds and backward error estimates
for it.

6.	If equilibration was used, the matrix X	is premultiplied by
diag(C)	(if TRANS = 'N') or diag(R) (if	TRANS =	'T' or 'C') so
that it	solves the original system before equilibration.

```

### ARGUMENTS[Toc][Back]

```     FACT    (input) CHARACTER*1
Specifies whether or not the factored form	of the matrix A	is
supplied on entry,	and if not, whether the	matrix A should	be
equilibrated before it is factored.  = 'F':  On entry, AFB	and
IPIV contain the factored form of A.  If EQUED is not 'N',	the
matrix A has been equilibrated with scaling factors given by R
and C.  AB, AFB, and IPIV are not modified.  = 'N':  The matrix A
will be copied to AFB and factored.
= 'E':  The matrix	A will be equilibrated if necessary, then
copied to AFB and factored.

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)

N	     (input) INTEGER
The number	of linear equations, i.e., the order of	the matrix A.
N >= 0.

KL	     (input) INTEGER
The number	of subdiagonals	within the band	of A.  KL >= 0.

KU	     (input) INTEGER
The number	of superdiagonals within the band of A.	 KU >= 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/output) COMPLEX array, dimension (LDAB,N)
On	entry, the matrix A in band storage, in	rows 1 to KL+KU+1.
The j-th column of	A is stored in the j-th	column of the array AB
as	follows:  AB(KU+1+i-j,j) = A(i,j) for max(1,jKU)<=i<=min(N,j+kl)

Page 2

CGBSVX(3F)							    CGBSVX(3F)

If	FACT = 'F' and EQUED is	not 'N', then A	must have been
equilibrated by the scaling factors in R and/or C.	 AB is not
modified if FACT =	'F' or 'N', or if FACT = 'E' and EQUED = 'N'
on	exit.

On	exit, if EQUED .ne. 'N', A is scaled as	follows:  EQUED	= 'R':
A := diag(R) * A
EQUED = 'C':  A :=	A * diag(C)
EQUED = 'B':  A :=	diag(R)	* A * diag(C).

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

AFB     (input or output) COMPLEX array, dimension	(LDAFB,N)
If	FACT = 'F', then AFB is	an input argument and on entry
contains details of the LU	factorization of the band matrix A, as
computed by CGBTRF.  U is stored as an upper triangular band
matrix with KL+KU superdiagonals in rows 1	to KL+KU+1, and	the
multipliers used during the factorization are stored in rows
KL+KU+2 to	2*KL+KU+1.  If EQUED .ne. 'N', then AFB	is the
factored form of the equilibrated matrix A.

If	FACT = 'N', then AFB is	an output argument and on exit returns
details of	the LU factorization of	A.

If	FACT = 'E', then AFB is	an output argument and on exit returns
details of	the LU factorization of	the equilibrated matrix	A (see
the description of	AB for the form	of the equilibrated matrix).

LDAFB   (input) INTEGER
The leading dimension of the array	AFB.  LDAFB >= 2*KL+KU+1.

IPIV    (input or output) INTEGER array, dimension	(N)
If	FACT = 'F', then IPIV is an input argument and on entry
contains the pivot	indices	from the factorization A = L*U as
computed by CGBTRF; row i of the matrix was interchanged with row
IPIV(i).

If	FACT = 'N', then IPIV is an output argument and	on exit
contains the pivot	indices	from the factorization A = L*U of the
original matrix A.

If	FACT = 'E', then IPIV is an output argument and	on exit
contains the pivot	indices	from the factorization A = L*U of the
equilibrated matrix A.

EQUED   (input or output) CHARACTER*1
Specifies the form	of equilibration that was done.	 = 'N':	 No
equilibration (always true	if FACT	= 'N').
= 'R':  Row equilibration,	i.e., A	has been premultiplied by
diag(R).  = 'C':  Column equilibration, i.e., A has been
postmultiplied by diag(C).	 = 'B':	 Both row and column

Page 3

CGBSVX(3F)							    CGBSVX(3F)

equilibration, i.e., A has	been replaced by diag(R) * A *
diag(C).  EQUED is	an input argument if FACT = 'F'; otherwise, it
is	an output argument.

R	     (input or output) REAL array, dimension (N)
The row scale factors for A.  If EQUED = 'R' or 'B', A is
multiplied	on the left by diag(R);	if EQUED = 'N' or 'C', R is
not accessed.  R is an input argument if FACT = 'F'; otherwise, R
is	an output argument.  If	FACT = 'F' and EQUED = 'R' or 'B',
each element of R must be positive.

C	     (input or output) REAL array, dimension (N)
The column	scale factors for A.  If EQUED = 'C' or	'B', A is
multiplied	on the right by	diag(C); if EQUED = 'N'	or 'R',	C is
not accessed.  C is an input argument if FACT = 'F'; otherwise, C
is	an output argument.  If	FACT = 'F' and EQUED = 'C' or 'B',
each element of C must be positive.

B	     (input/output) COMPLEX array, dimension (LDB,NRHS)
On	entry, the right hand side matrix B.  On exit, if EQUED	= 'N',
B is not modified;	if TRANS = 'N' and EQUED = 'R' or 'B', B is
overwritten by diag(R)*B; if TRANS	= 'T' or 'C' and EQUED = 'C'
or	'B', B is overwritten by diag(C)*B.

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

X	     (output) COMPLEX array, dimension (LDX,NRHS)
If	INFO = 0, the n-by-nrhs	solution matrix	X to the original
system of equations.  Note	that A and B are modified on exit if
EQUED .ne.	'N', and the solution to the equilibrated system is
inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or or 'B'.

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

RCOND   (output) REAL
The estimate of the reciprocal condition number of	the matrix A
after equilibration (if done).  If	RCOND is less than the machine
precision (in particular, if RCOND	= 0), the matrix is singular
to	working	precision.  This condition is indicated	by a return
code of INFO > 0, and the solution	and error bounds are not
computed.

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.

Page 4

CGBSVX(3F)							    CGBSVX(3F)

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/output)	REAL array, dimension (N)
On	exit, RWORK(1) contains	the reciprocal pivot growth factor
norm(A)/norm(U). The "max absolute	element" norm is used. If
RWORK(1) is much less than	1, then	the stability of the LU
factorization of the (equilibrated) matrix	A could	be poor. This
also means	that the solution X, condition estimator RCOND,	and
forward error bound FERR could be unreliable. If factorization
fails with	0<INFO<=N, then	RWORK(1) contains the reciprocal pivot
growth factor for the leading INFO	columns	of A.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i,	and i is
<=	N:  U(i,i) is exactly zero.  The factorization has been
completed,	but the	factor U is exactly singular, so the solution
and error bounds could not	be computed.  =	N+1: RCOND is less
than machine precision.  The factorization	has been completed,
but the matrix A is singular to working precision,	and the
solution and error	bounds have not	been computed.
CGBSVX(3F)							    CGBSVX(3F)

```

### NAME[Toc][Back]

```     CGBSVX - use the LU factorization to compute the solution to a complex
system of linear equations	A * X =	B, A**T	* X = B, or A**H * X = B,
```

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	CGBSVX(	FACT, TRANS, N,	KL, KU,	NRHS, AB, LDAB,	AFB, LDAFB,
IPIV, EQUED, R,	C, B, LDB, X, LDX, RCOND, FERR,	BERR,
WORK, RWORK, INFO )

CHARACTER	EQUED, FACT, TRANS

INTEGER	INFO, KL, KU, LDAB, LDAFB, LDB,	LDX, N,	NRHS

REAL		RCOND

INTEGER	IPIV( *	)

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

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

### PURPOSE[Toc][Back]

```     CGBSVX uses the LU	factorization to compute the solution to a complex
system of linear equations	A * X =	B, A**T	* X = B, or A**H * X = B,
where A is	a band matrix of order N with KL subdiagonals and KU
superdiagonals, and X and B are N-by-NRHS matrices.

Error bounds on the solution and a	condition estimate are also provided.

```

### DESCRIPTION[Toc][Back]

```     The following steps are performed by this subroutine:

1.	If FACT	= 'E', real scaling factors are	computed to equilibrate
the system:
TRANS = 'N':	 diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
TRANS = 'T':	(diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C':	(diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
Whether	or not the system will be equilibrated depends on the
scaling	of the matrix A, but if	equilibration is used, A is
overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if	TRANS='N')
or diag(C)*B (if TRANS = 'T' or	'C').

2.	If FACT	= 'N' or 'E', the LU decomposition is used to factor the
matrix A (after	equilibration if FACT =	'E') as
A = L * U,
where L	is a product of	permutation and	unit lower triangular
matrices with KL subdiagonals, and U is	upper triangular with
KL+KU superdiagonals.

3.	The factored form of A is used to estimate the condition number

Page 1

CGBSVX(3F)							    CGBSVX(3F)

of the matrix A.  If the reciprocal of the condition number is
less than machine precision, steps 4-6 are skipped.

4.	The system of equations	is solved for X	using the factored form
of A.

5.	Iterative refinement is	applied	to improve the computed	solution
matrix and calculate error bounds and backward error estimates
for it.

6.	If equilibration was used, the matrix X	is premultiplied by
diag(C)	(if TRANS = 'N') or diag(R) (if	TRANS =	'T' or 'C') so
that it	solves the original system before equilibration.

```

### ARGUMENTS[Toc][Back]

```     FACT    (input) CHARACTER*1
Specifies whether or not the factored form	of the matrix A	is
supplied on entry,	and if not, whether the	matrix A should	be
equilibrated before it is factored.  = 'F':  On entry, AFB	and
IPIV contain the factored form of A.  If EQUED is not 'N',	the
matrix A has been equilibrated with scaling factors given by R
and C.  AB, AFB, and IPIV are not modified.  = 'N':  The matrix A
will be copied to AFB and factored.
= 'E':  The matrix	A will be equilibrated if necessary, then
copied to AFB and factored.

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)

N	     (input) INTEGER
The number	of linear equations, i.e., the order of	the matrix A.
N >= 0.

KL	     (input) INTEGER
The number	of subdiagonals	within the band	of A.  KL >= 0.

KU	     (input) INTEGER
The number	of superdiagonals within the band of A.	 KU >= 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/output) COMPLEX array, dimension (LDAB,N)
On	entry, the matrix A in band storage, in	rows 1 to KL+KU+1.
The j-th column of	A is stored in the j-th	column of the array AB
as	follows:  AB(KU+1+i-j,j) = A(i,j) for max(1,jKU)<=i<=min(N,j+kl)

Page 2

CGBSVX(3F)							    CGBSVX(3F)

If	FACT = 'F' and EQUED is	not 'N', then A	must have been
equilibrated by the scaling factors in R and/or C.	 AB is not
modified if FACT =	'F' or 'N', or if FACT = 'E' and EQUED = 'N'
on	exit.

On	exit, if EQUED .ne. 'N', A is scaled as	follows:  EQUED	= 'R':
A := diag(R) * A
EQUED = 'C':  A :=	A * diag(C)
EQUED = 'B':  A :=	diag(R)	* A * diag(C).

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

AFB     (input or output) COMPLEX array, dimension	(LDAFB,N)
If	FACT = 'F', then AFB is	an input argument and on entry
contains details of the LU	factorization of the band matrix A, as
computed by CGBTRF.  U is stored as an upper triangular band
matrix with KL+KU superdiagonals in rows 1	to KL+KU+1, and	the
multipliers used during the factorization are stored in rows
KL+KU+2 to	2*KL+KU+1.  If EQUED .ne. 'N', then AFB	is the
factored form of the equilibrated matrix A.

If	FACT = 'N', then AFB is	an output argument and on exit returns
details of	the LU factorization of	A.

If	FACT = 'E', then AFB is	an output argument and on exit returns
details of	the LU factorization of	the equilibrated matrix	A (see
the description of	AB for the form	of the equilibrated matrix).

LDAFB   (input) INTEGER
The leading dimension of the array	AFB.  LDAFB >= 2*KL+KU+1.

IPIV    (input or output) INTEGER array, dimension	(N)
If	FACT = 'F', then IPIV is an input argument and on entry
contains the pivot	indices	from the factorization A = L*U as
computed by CGBTRF; row i of the matrix was interchanged with row
IPIV(i).

If	FACT = 'N', then IPIV is an output argument and	on exit
contains the pivot	indices	from the factorization A = L*U of the
original matrix A.

If	FACT = 'E', then IPIV is an output argument and	on exit
contains the pivot	indices	from the factorization A = L*U of the
equilibrated matrix A.

EQUED   (input or output) CHARACTER*1
Specifies the form	of equilibration that was done.	 = 'N':	 No
equilibration (always true	if FACT	= 'N').
= 'R':  Row equilibration,	i.e., A	has been premultiplied by
diag(R).  = 'C':  Column equilibration, i.e., A has been
postmultiplied by diag(C).	 = 'B':	 Both row and column

Page 3

CGBSVX(3F)							    CGBSVX(3F)

equilibration, i.e., A has	been replaced by diag(R) * A *
diag(C).  EQUED is	an input argument if FACT = 'F'; otherwise, it
is	an output argument.

R	     (input or output) REAL array, dimension (N)
The row scale factors for A.  If EQUED = 'R' or 'B', A is
multiplied	on the left by diag(R);	if EQUED = 'N' or 'C', R is
not accessed.  R is an input argument if FACT = 'F'; otherwise, R
is	an output argument.  If	FACT = 'F' and EQUED = 'R' or 'B',
each element of R must be positive.

C	     (input or output) REAL array, dimension (N)
The column	scale factors for A.  If EQUED = 'C' or	'B', A is
multiplied	on the right by	diag(C); if EQUED = 'N'	or 'R',	C is
not accessed.  C is an input argument if FACT = 'F'; otherwise, C
is	an output argument.  If	FACT = 'F' and EQUED = 'C' or 'B',
each element of C must be positive.

B	     (input/output) COMPLEX array, dimension (LDB,NRHS)
On	entry, the right hand side matrix B.  On exit, if EQUED	= 'N',
B is not modified;	if TRANS = 'N' and EQUED = 'R' or 'B', B is
overwritten by diag(R)*B; if TRANS	= 'T' or 'C' and EQUED = 'C'
or	'B', B is overwritten by diag(C)*B.

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

X	     (output) COMPLEX array, dimension (LDX,NRHS)
If	INFO = 0, the n-by-nrhs	solution matrix	X to the original
system of equations.  Note	that A and B are modified on exit if
EQUED .ne.	'N', and the solution to the equilibrated system is
inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or or 'B'.

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

RCOND   (output) REAL
The estimate of the reciprocal condition number of	the matrix A
after equilibration (if done).  If	RCOND is less than the machine
precision (in particular, if RCOND	= 0), the matrix is singular
to	working	precision.  This condition is indicated	by a return
code of INFO > 0, and the solution	and error bounds are not
computed.

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.

Page 4

CGBSVX(3F)							    CGBSVX(3F)

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/output)	REAL array, dimension (N)
On	exit, RWORK(1) contains	the reciprocal pivot growth factor
norm(A)/norm(U). The "max absolute	element" norm is used. If
RWORK(1) is much less than	1, then	the stability of the LU
factorization of the (equilibrated) matrix	A could	be poor. This
also means	that the solution X, condition estimator RCOND,	and
forward error bound FERR could be unreliable. If factorization
fails with	0<INFO<=N, then	RWORK(1) contains the reciprocal pivot
growth factor for the leading INFO	columns	of A.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i,	and i is
<=	N:  U(i,i) is exactly zero.  The factorization has been
completed,	but the	factor U is exactly singular, so the solution
and error bounds could not	be computed.  =	N+1: RCOND is less
than machine precision.  The factorization	has been completed,
but the matrix A is singular to working precision,	and the
solution and error	bounds have not	been computed.

PPPPaaaaggggeeee 5555```
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