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

```
ZGESVX(3F)							    ZGESVX(3F)

```

### NAME[Toc][Back]

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

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	ZGESVX(	FACT, TRANS, N,	NRHS, A, LDA, AF, LDAF,	IPIV, EQUED,
R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK,
INFO )

CHARACTER	EQUED, FACT, TRANS

INTEGER	INFO, LDA, LDAF, LDB, LDX, N, NRHS

DOUBLE		PRECISION RCOND

INTEGER	IPIV( *	)

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

COMPLEX*16	A( LDA,	* ), AF( LDAF, * ), B( LDB, * ), WORK( * ), X(
LDX, * )
```

### PURPOSE[Toc][Back]

```     ZGESVX uses the LU	factorization to compute the solution to a complex
system of linear equations
A * X =	B, where A is an N-by-N	matrix 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:

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 = P * L * U,
where P	is a permutation matrix, L is a	unit lower triangular
matrix,	and U is upper triangular.

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

Page 1

ZGESVX(3F)							    ZGESVX(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, AF 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.  A,	AF, and	IPIV are not modified.	= 'N':	The matrix A
will be copied to AF and factored.
= 'E':  The matrix	A will be equilibrated if necessary, then
copied to AF 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.

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/output) COMPLEX*16 array, dimension	(LDA,N)
On	entry, the N-by-N matrix A.  If	FACT = 'F' and EQUED is	not
'N', then A must have been	equilibrated by	the scaling factors in
R and/or C.  A is not modified if FACT = 'F' or

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).

Page 2

ZGESVX(3F)							    ZGESVX(3F)

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

AF	     (input or output) COMPLEX*16 array, dimension (LDAF,N)
If	FACT = 'F', then AF is an input	argument and on	entry contains
the factors L and U from the factorization	A = P*L*U as computed
by	ZGETRF.	 If EQUED .ne. 'N', then AF is the factored form of
the equilibrated matrix A.

If	FACT = 'N', then AF is an output argument and on exit returns
the factors L and U from the factorization	A = P*L*U of the
original matrix A.

If	FACT = 'E', then AF is an output argument and on exit returns
the factors L and U from the factorization	A = P*L*U of the
equilibrated matrix A (see	the description	of A for the form of
the equilibrated matrix).

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

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 = P*L*U as
computed by ZGETRF; 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 = P*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 = P*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
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) DOUBLE PRECISION	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.

Page 3

ZGESVX(3F)							    ZGESVX(3F)

C	     (input or output) DOUBLE PRECISION	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*16 array, dimension	(LDB,NRHS)
On	entry, the N-by-NRHS 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*16 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) DOUBLE PRECISION
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) 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/output)	DOUBLE PRECISION array,	dimension (2*N)
On	exit, RWORK(1) contains	the reciprocal pivot growth factor
norm(A)/norm(U). The "max absolute	element" norm is used. If

Page 4

ZGESVX(3F)							    ZGESVX(3F)

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 is singular	to working precision, and the solution
and error bounds have not been computed.
ZGESVX(3F)							    ZGESVX(3F)

```

### NAME[Toc][Back]

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

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	ZGESVX(	FACT, TRANS, N,	NRHS, A, LDA, AF, LDAF,	IPIV, EQUED,
R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK,
INFO )

CHARACTER	EQUED, FACT, TRANS

INTEGER	INFO, LDA, LDAF, LDB, LDX, N, NRHS

DOUBLE		PRECISION RCOND

INTEGER	IPIV( *	)

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

COMPLEX*16	A( LDA,	* ), AF( LDAF, * ), B( LDB, * ), WORK( * ), X(
LDX, * )
```

### PURPOSE[Toc][Back]

```     ZGESVX uses the LU	factorization to compute the solution to a complex
system of linear equations
A * X =	B, where A is an N-by-N	matrix 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:

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 = P * L * U,
where P	is a permutation matrix, L is a	unit lower triangular
matrix,	and U is upper triangular.

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

Page 1

ZGESVX(3F)							    ZGESVX(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, AF 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.  A,	AF, and	IPIV are not modified.	= 'N':	The matrix A
will be copied to AF and factored.
= 'E':  The matrix	A will be equilibrated if necessary, then
copied to AF 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.

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/output) COMPLEX*16 array, dimension	(LDA,N)
On	entry, the N-by-N matrix A.  If	FACT = 'F' and EQUED is	not
'N', then A must have been	equilibrated by	the scaling factors in
R and/or C.  A is not modified if FACT = 'F' or

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).

Page 2

ZGESVX(3F)							    ZGESVX(3F)

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

AF	     (input or output) COMPLEX*16 array, dimension (LDAF,N)
If	FACT = 'F', then AF is an input	argument and on	entry contains
the factors L and U from the factorization	A = P*L*U as computed
by	ZGETRF.	 If EQUED .ne. 'N', then AF is the factored form of
the equilibrated matrix A.

If	FACT = 'N', then AF is an output argument and on exit returns
the factors L and U from the factorization	A = P*L*U of the
original matrix A.

If	FACT = 'E', then AF is an output argument and on exit returns
the factors L and U from the factorization	A = P*L*U of the
equilibrated matrix A (see	the description	of A for the form of
the equilibrated matrix).

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

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 = P*L*U as
computed by ZGETRF; 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 = P*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 = P*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
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) DOUBLE PRECISION	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.

Page 3

ZGESVX(3F)							    ZGESVX(3F)

C	     (input or output) DOUBLE PRECISION	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*16 array, dimension	(LDB,NRHS)
On	entry, the N-by-NRHS 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*16 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) DOUBLE PRECISION
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) 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/output)	DOUBLE PRECISION array,	dimension (2*N)
On	exit, RWORK(1) contains	the reciprocal pivot growth factor
norm(A)/norm(U). The "max absolute	element" norm is used. If

Page 4

ZGESVX(3F)							    ZGESVX(3F)

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 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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