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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: ZGBSVXX use the LU factorization to compute the solution to a complex*16 system of linear equations A * X = B, where A

```NAME
LAPACK-3  - ZGBSVXX use the LU factorization to compute the solution to
a  complex*16 system of linear equations A * X = B, where A is  an   N-
by-N matrix and X and B are N-by-NRHS matrices

SYNOPSIS
SUBROUTINE ZGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
IPIV, EQUED, R, C, B, LDB, X, LDX,  RCOND,  RPVGRW,
BERR,   N_ERR_BNDS,  ERR_BNDS_NORM,  ERR_BNDS_COMP,
NPARAMS, PARAMS, WORK, RWORK, INFO )

IMPLICIT        NONE

CHARACTER       EQUED, FACT, TRANS

INTEGER         INFO, LDAB, LDAFB,  LDB,  LDX,  N,  NRHS,  NPARAMS,
N_ERR_BNDS

DOUBLE          PRECISION RCOND, RPVGRW

INTEGER         IPIV( * )

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

DOUBLE          PRECISION R( * ), C( * ), PARAMS( * ), BERR(  *  ),
ERR_BNDS_NORM( NRHS, * ), ERR_BNDS_COMP( NRHS, * ),
RWORK( * )

PURPOSE
ZGBSVXX uses the LU factorization to compute the solution to a
complex*16 system of linear equations  A * X = B,  where A is an
N-by-N matrix and X and B are N-by-NRHS matrices.
If requested, both normwise and maximum componentwise error bounds
are returned. ZGBSVXX will return a solution with a tiny
guaranteed error (O(eps) where eps is the working machine
precision) unless the matrix is very ill-conditioned, in which
case a warning is returned. Relevant condition numbers also are
calculated and returned.
ZGBSVXX accepts user-provided factorizations and equilibration
factors; see the definitions of the FACT and EQUED options.
Solving with refinement and using a factorization from a previous
ZGBSVXX call will also produce a solution with either O(eps)
errors or warnings, but we cannot make that claim for general
user-provided factorizations and equilibration factors if they
differ from what ZGBSVXX would itself produce.

DESCRIPTION
The following steps are performed:
1. If FACT = 'E', double precision 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. 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 (see
argument RCOND). If the reciprocal of the condition number is less
than machine precision, 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. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
the routine will use iterative refinement to try to get a small
error and error bounds.  Refinement calculates the residual to at
least twice the working precision.
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
Some optional parameters are bundled in the PARAMS array.  These
settings determine how refinement is performed, but often the
defaults are acceptable.  If the defaults are acceptable, users
can pass NPARAMS = 0 which prevents the source code from accessing
the PARAMS argument.

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 = 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 AB 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*16 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 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 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).

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 = P*L*U
as computed by DGETRF; 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.
If R is output, each element of R is a power of the radix.
If  R  is  input,  each  element of R should be a power of the
to ensure a reliable solution and error estimates. Scaling by
powers of the radix does not cause rounding errors unless the
result underflows or overflows. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable.

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.
If C is output, each element of C is a power of the radix.
If C is input, each element of C should  be  a  power  of  the
to ensure a reliable solution and error estimates. Scaling by
powers of the radix does not cause rounding errors unless the
result underflows or overflows. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable.

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 '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
Reciprocal scaled condition number.  This is  an  estimate  of
the
reciprocal Skeel condition number of the matrix A after
equilibration (if done).  If this is less than the machine
precision  (in  particular,  if  it  is  zero),  the matrix is
singular
to working precision.  Note that the error may still be  small
even
if this number is very small and the matrix appears ill-
conditioned.

RPVGRW  (output) DOUBLE PRECISION
Reciprocal   pivot   growth.    On  exit,  this  contains  the
reciprocal
pivot  growth  factor  norm(A)/norm(U).  The   "max   absolute
element"
norm is used.  If this 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, estimated condition
numbers,
and error bounds could be unreliable. If  factorization  fails
with
0<INFO<=N,  then  this  contains  the  reciprocal pivot growth
factor
for the leading INFO columns of A.  In DGESVX,  this  quantity
is
returned in WORK(1).

BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
Componentwise relative backward error.  This is 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).
N_ERR_BNDS (input) INTEGER
Number of error bounds to return for each right hand side
and  each type (normwise or componentwise).  See ERR_BNDS_NORM
and
ERR_BNDS_COMP below.

ERR_BNDS_NORM   (output)  DOUBLE  PRECISION  array,  dimension  (NRHS,
N_ERR_BNDS)
For   each   right-hand   side,   this  array  contains
various   error   bounds    and    condition    numbers
corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))
The  array  is indexed by the type of error information
as described
below. There  currently  are  up  to  three  pieces  of
information
returned.
The  first  index  in ERR_BNDS_NORM(i,:) corresponds to
the ith
right-hand side.
The second index in ERR_BNDS_NORM(:,err)  contains  the
following
three fields:
err  =  1 "Trust/don't trust" boolean. Trust the answer
if the
reciprocal condition number is less than the threshold
sqrt(n) * dlamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward
error,
almost  certainly  within  a  factor  of 10 of the true
error
so long as the next entry is greater than the threshold
sqrt(n) * dlamch('Epsilon'). This  error  bound  should
only
be trusted if the previous boolean is true.
err   =   3   Reciprocal  condition  number:  Estimated
normwise
reciprocal  condition  number.    Compared   with   the
threshold
sqrt(n) * dlamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers  are  1  / (norm(Z^{-1},inf) * norm(Z,inf)) for
some
appropriately scaled matrix Z.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z  are  approximately
1.
See  Lapack  Working  Note  165 for further details and
extra
cautions.

ERR_BNDS_COMP   (output)  DOUBLE  PRECISION  array,  dimension  (NRHS,
N_ERR_BNDS)
For   each   right-hand   side,   this  array  contains
various   error   bounds    and    condition    numbers
corresponding to the
componentwise  relative  error,  which  is  defined  as
follows:
Componentwise  relative  error  in  the  ith   solution
vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))
The array is indexed by the right-hand side i (on which
the
componentwise relative error depends), and the type  of
error
information  as described below. There currently are up
to three
pieces of  information  returned  for  each  right-hand
side. If
componentwise  accuracy  is  not requested (PARAMS(3) =
0.0), then
ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS  .LT.  3,
then at most
the first (:,N_ERR_BNDS) entries are returned.
The  first  index  in ERR_BNDS_COMP(i,:) corresponds to
the ith
right-hand side.
The second index in ERR_BNDS_COMP(:,err)  contains  the
following
three fields:
err  =  1 "Trust/don't trust" boolean. Trust the answer
if the
reciprocal condition number is less than the threshold
sqrt(n) * dlamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward
error,
almost  certainly  within  a  factor  of 10 of the true
error
so long as the next entry is greater than the threshold
sqrt(n) * dlamch('Epsilon'). This  error  bound  should
only
be trusted if the previous boolean is true.
err   =   3   Reciprocal  condition  number:  Estimated
componentwise
reciprocal  condition  number.    Compared   with   the
threshold
sqrt(n) * dlamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers  are  1  / (norm(Z^{-1},inf) * norm(Z,inf)) for
some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.
See Lapack Working Note 165  for  further  details  and
extra
cautions.
NPARAMS (input) INTEGER
Specifies  the  number of parameters set in PARAMS.  If
.LE. 0, the
PARAMS array is never referenced and default values are
used.

PARAMS  (input / output) DOUBLE PRECISION array, dimension NPARAMS
Specifies algorithm parameters.  If an entry is .LT. 0.0, then
that entry will be filled with default value used for that
parameter.    Only  positions  up  to  NPARAMS  are  accessed;
defaults
are used for higher-numbered parameters.
PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
refinement or not.
Default: 1.0D+0
= 0.0 : No refinement is performed, and no error bounds are
computed.
= 1.0 : Use the extra-precise refinement algorithm.
(other values are reserved for future use)
PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
computations allowed for refinement.
Default: 10
Aggressive: Set to 100 to permit convergence using approximate
factorizations or factorizations other than LU. If
the factorization uses a technique other than
Gaussian elimination, the guarantees in
err_bnds_norm and err_bnds_comp may no longer be
trustworthy.
PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
will attempt to find a solution with small componentwise
relative error in the double-precision algorithm.  Positive
is true, 0.0 is false.
Default: 1.0 (attempt componentwise convergence)

WORK    (workspace) COMPLEX*16 array, dimension (2*N)

RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N)

INFO    (output) INTEGER
= 0:  Successful exit. The solution to every  right-hand  side
is
guaranteed.
< 0:  If INFO = -i, the i-th argument had an illegal value
>   0   and   <=   N:   U(INFO,INFO)  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. RCOND = 0
is returned.
= N+J: The solution corresponding to the Jth  right-hand  side
is
not guaranteed. The solutions corresponding to other right-
hand sides K with K > J may not be guaranteed as well, but
only the first such right-hand side is reported. If a small
componentwise error is not requested (PARAMS(3) = 0.0) then
the Jth right-hand side is the first with a normwise error
bound that is not guaranteed (the smallest J such
that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
the Jth right-hand side is the first with either a normwise or
componentwise error bound that is not guaranteed (the smallest
J such that either ERR_BNDS_NORM(J,1) = 0.0 or
ERR_BNDS_COMP(J,1) = 0.0). See the definition of
ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
about all of the right-hand sides check ERR_BNDS_NORM or
ERR_BNDS_COMP.

LAPACK driver routine (versionMarch22011                       LAPACK-3(3)
```