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

```
ZGTSVX(3F)							    ZGTSVX(3F)

```

### NAME[Toc][Back]

```     ZGTSVX - 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	ZGTSVX(	FACT, TRANS, N,	NRHS, DL, D, DU, DLF, DF, DUF, DU2,
IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK,
INFO )

CHARACTER	FACT, TRANS

INTEGER	INFO, LDB, LDX,	N, NRHS

DOUBLE		PRECISION RCOND

INTEGER	IPIV( *	)

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

COMPLEX*16	B( LDB,	* ), D(	* ), DF( * ), DL( * ), DLF( * ), DU( *
), DU2(	* ), DUF( * ), WORK( * ), X( LDX, * )
```

### PURPOSE[Toc][Back]

```     ZGTSVX 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 tridiagonal matrix of	order N	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	= 'N', the LU decomposition is used to factor the matrix A
as A = L * U, where L is a product of permutation and unit lower
bidiagonal matrices and	U is upper triangular with nonzeros in
only the main diagonal and first two superdiagonals.

2.	The factored form of A is used to estimate the condition number
of the matrix A.  If the reciprocal of the condition number is
less than machine precision, steps 3 and 4 are skipped.

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

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

Page 1

ZGTSVX(3F)							    ZGTSVX(3F)

```

### ARGUMENTS[Toc][Back]

```     FACT    (input) CHARACTER*1
Specifies whether or not the factored form	of A has been supplied
on	entry.	= 'F':	DLF, DF, DUF, DU2, and IPIV contain the
factored form of A; DL, D,	DU, DLF, DF, DUF, DU2 and IPIV will
not be modified.  = 'N':  The matrix will be copied to DLF, DF,
and DUF 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 order of the matrix A.	 N >= 0.

NRHS    (input) INTEGER
The number	of right hand sides, i.e., the number of columns of
the matrix	B.  NRHS >= 0.

DL	     (input) COMPLEX*16	array, dimension (N-1)
The (n-1) subdiagonal elements of A.

D	     (input) COMPLEX*16	array, dimension (N)
The n diagonal elements of	A.

DU	     (input) COMPLEX*16	array, dimension (N-1)
The (n-1) superdiagonal elements of A.

DLF     (input or output) COMPLEX*16 array, dimension (N-1)
If	FACT = 'F', then DLF is	an input argument and on entry
contains the (n-1)	multipliers that define	the matrix L from the
LU	factorization of A as computed by ZGTTRF.

If	FACT = 'N', then DLF is	an output argument and on exit
contains the (n-1)	multipliers that define	the matrix L from the
LU	factorization of A.

DF	     (input or output) COMPLEX*16 array, dimension (N)
If	FACT = 'F', then DF is an input	argument and on	entry contains
the n diagonal elements of	the upper triangular matrix U from the
LU	factorization of A.

If	FACT = 'N', then DF is an output argument and on exit contains
the n diagonal elements of	the upper triangular matrix U from the
LU	factorization of A.

DUF     (input or output) COMPLEX*16 array, dimension (N-1)
If	FACT = 'F', then DUF is	an input argument and on entry
contains the (n-1)	elements of the	first superdiagonal of U.

Page 2

ZGTSVX(3F)							    ZGTSVX(3F)

If	FACT = 'N', then DUF is	an output argument and on exit
contains the (n-1)	elements of the	first superdiagonal of U.

DU2     (input or output) COMPLEX*16 array, dimension (N-2)
If	FACT = 'F', then DU2 is	an input argument and on entry
contains the (n-2)	elements of the	second superdiagonal of	U.

If	FACT = 'N', then DU2 is	an output argument and on exit
contains the (n-2)	elements of the	second superdiagonal of	U.

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 LU factorization of A as
computed by ZGTTRF.

If	FACT = 'N', then IPIV is an output argument and	on exit
contains the pivot	indices	from the LU factorization of A;	row i
of	the matrix was interchanged with row IPIV(i).  IPIV(i) will
always be either i	or i+1;	IPIV(i)	= i indicates a	row
interchange was not required.

B	     (input) COMPLEX*16	array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix 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.

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.
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

Page 3

ZGTSVX(3F)							    ZGTSVX(3F)

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
> 0:  if INFO = i,	and i is
<=	N:  U(i,i) is exactly zero.  The factorization has not been
completed unless i	= N, 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.
ZGTSVX(3F)							    ZGTSVX(3F)

```

### NAME[Toc][Back]

```     ZGTSVX - 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	ZGTSVX(	FACT, TRANS, N,	NRHS, DL, D, DU, DLF, DF, DUF, DU2,
IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK,
INFO )

CHARACTER	FACT, TRANS

INTEGER	INFO, LDB, LDX,	N, NRHS

DOUBLE		PRECISION RCOND

INTEGER	IPIV( *	)

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

COMPLEX*16	B( LDB,	* ), D(	* ), DF( * ), DL( * ), DLF( * ), DU( *
), DU2(	* ), DUF( * ), WORK( * ), X( LDX, * )
```

### PURPOSE[Toc][Back]

```     ZGTSVX 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 tridiagonal matrix of	order N	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	= 'N', the LU decomposition is used to factor the matrix A
as A = L * U, where L is a product of permutation and unit lower
bidiagonal matrices and	U is upper triangular with nonzeros in
only the main diagonal and first two superdiagonals.

2.	The factored form of A is used to estimate the condition number
of the matrix A.  If the reciprocal of the condition number is
less than machine precision, steps 3 and 4 are skipped.

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

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

Page 1

ZGTSVX(3F)							    ZGTSVX(3F)

```

### ARGUMENTS[Toc][Back]

```     FACT    (input) CHARACTER*1
Specifies whether or not the factored form	of A has been supplied
on	entry.	= 'F':	DLF, DF, DUF, DU2, and IPIV contain the
factored form of A; DL, D,	DU, DLF, DF, DUF, DU2 and IPIV will
not be modified.  = 'N':  The matrix will be copied to DLF, DF,
and DUF 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 order of the matrix A.	 N >= 0.

NRHS    (input) INTEGER
The number	of right hand sides, i.e., the number of columns of
the matrix	B.  NRHS >= 0.

DL	     (input) COMPLEX*16	array, dimension (N-1)
The (n-1) subdiagonal elements of A.

D	     (input) COMPLEX*16	array, dimension (N)
The n diagonal elements of	A.

DU	     (input) COMPLEX*16	array, dimension (N-1)
The (n-1) superdiagonal elements of A.

DLF     (input or output) COMPLEX*16 array, dimension (N-1)
If	FACT = 'F', then DLF is	an input argument and on entry
contains the (n-1)	multipliers that define	the matrix L from the
LU	factorization of A as computed by ZGTTRF.

If	FACT = 'N', then DLF is	an output argument and on exit
contains the (n-1)	multipliers that define	the matrix L from the
LU	factorization of A.

DF	     (input or output) COMPLEX*16 array, dimension (N)
If	FACT = 'F', then DF is an input	argument and on	entry contains
the n diagonal elements of	the upper triangular matrix U from the
LU	factorization of A.

If	FACT = 'N', then DF is an output argument and on exit contains
the n diagonal elements of	the upper triangular matrix U from the
LU	factorization of A.

DUF     (input or output) COMPLEX*16 array, dimension (N-1)
If	FACT = 'F', then DUF is	an input argument and on entry
contains the (n-1)	elements of the	first superdiagonal of U.

Page 2

ZGTSVX(3F)							    ZGTSVX(3F)

If	FACT = 'N', then DUF is	an output argument and on exit
contains the (n-1)	elements of the	first superdiagonal of U.

DU2     (input or output) COMPLEX*16 array, dimension (N-2)
If	FACT = 'F', then DU2 is	an input argument and on entry
contains the (n-2)	elements of the	second superdiagonal of	U.

If	FACT = 'N', then DU2 is	an output argument and on exit
contains the (n-2)	elements of the	second superdiagonal of	U.

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 LU factorization of A as
computed by ZGTTRF.

If	FACT = 'N', then IPIV is an output argument and	on exit
contains the pivot	indices	from the LU factorization of A;	row i
of	the matrix was interchanged with row IPIV(i).  IPIV(i) will
always be either i	or i+1;	IPIV(i)	= i indicates a	row
interchange was not required.

B	     (input) COMPLEX*16	array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix 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.

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.
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

Page 3

ZGTSVX(3F)							    ZGTSVX(3F)

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
> 0:  if INFO = i,	and i is
<=	N:  U(i,i) is exactly zero.  The factorization has not been
completed unless i	= N, 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 4444```
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