# Ubuntu Feisty 7.04 manual page repository

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### NAME

```        ZGBSVX  - 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

```        SUBROUTINE ZGBSVX( 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

DOUBLE         PRECISION RCOND

INTEGER        IPIV( * )

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

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

### PURPOSE

```        ZGBSVX 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  super‐
diagonals, and X and B are N-by-NRHS matrices.

Error  bounds  on  the  solution and a condition estimate are also pro‐
vided.
```

### DESCRIPTION

```        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. If some U(i,i)=0, so that U is exactly singular, then the routine
returns with INFO = i. Otherwise, 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,
INFO = N+1 is returned as a warning, but the routine still goes on
to solve for X and compute error bounds as described below.

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

```        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*16 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,j-
KU)<=i<=min(N,j+kl)

If FACT = ’F’ and EQUED is not ’N’, then A must have been equi‐
librated by the scaling factors in R and/or C.  AB is not modi‐
fied  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*16 array, dimension (LDAFB,N)
If  FACT = ’F’, then AFB is an input argument and on entry con‐
tains details of the LU factorization of the band matrix A,  as
computed  by  ZGBTRF.   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  fac‐
tored 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 equili‐
brated 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 con‐
tains the pivot indices from the factorization A = L*U as  com‐
puted  by ZGBTRF; row i of the matrix was interchanged with row
IPIV(i).

If FACT = ’N’, then IPIV is an output argument and on exit con‐
tains  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 con‐
tains  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  post‐
multiplied  by diag(C).  = ’B’:  Both row and column equilibra‐
tion, 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  mul‐
tiplied 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) 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 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 or INFO = N+1, the N-by-NRHS solution matrix X to
the original system of equations.  Note that A and B are  modi‐
fied  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 ’B’, or inv(diag(R))*X if TRANS = ’T’ or ’C’ and EQUED
= ’R’ 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.

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 vec‐
tor  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 (N)
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 fac‐
torization  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
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 com‐
pleted, but the factor U is exactly singular, so  the  solution
and  error bounds could not be computed. RCOND = 0 is returned.
= N+1: U is nonsingular, but RCOND is less than machine  preci‐
sion, meaning that the matrix is singular to working precision.
Nevertheless,  the  solution  and  error  bounds  are  computed
because  there  are  a  number of situations where the computed
solution can be more accurate than the  value  of  RCOND  would
suggest.

```
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