dgbrfsx man page on Scientific

Printed from http://www.polarhome.com/service/man/?qf=dgbrfsx&af=0&tf=2&of=Scientific

```DGBRFSX(1) LAPACK routine (version 3.2)				    DGBRFSX(1)

NAME
DGBRFSX	-  DGBRFSX improve the computed solution to a system of linear
equations and provides error bounds and backward error  estimates   for
the solution

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

IMPLICIT	   NONE

CHARACTER	   TRANS, EQUED

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

DOUBLE	   PRECISION RCOND

INTEGER	   IPIV( * ), IWORK( * )

DOUBLE	   PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, *
), X( LDX , * ),WORK( * )

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

PURPOSE
DGBRFSX improves the computed solution to a system of linear
equations and provides error bounds and backward error estimates
for the solution.  In addition to normwise error bound, the code
provides maximum componentwise error bound if possible.  See
comments for ERR_BNDS_N and ERR_BNDS_C for details of the error
bounds.
The original system of linear equations may have been equilibrated
before calling this routine, as described by arguments EQUED, R
and C below. In this case, the solution and error bounds returned
are for the original unequilibrated system.

ARGUMENTS
Some  optional  parameters are bundled in the PARAMS array.  These set‐
tings 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.

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)

EQUED   (input) CHARACTER*1
Specifies the form of equilibration that was done to  A	before
calling	this  routine.	This is needed to compute the solution
and error bounds correctly.  = 'N':  No equilibration
= '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tion,  i.e., A has been replaced by diag(R) * A * diag(C).  The
right hand side B has been changed accordingly.

N       (input) INTEGER
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) DOUBLE PRECISION array, dimension (LDAB,N)
The  original  band matrix A, stored 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).

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

AFB     (input) DOUBLE PRECISION array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix A,  as  com‐
puted  by  DGBTRF.   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.

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

IPIV    (input) INTEGER array, dimension (N)
The pivot indices from DGETRF; for 1<=i<=N, row i of the matrix
was interchanged with row IPIV(i).

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.  If R is output, each ele‐
ment of R is a power of the radix.  If R is input, each element
of  R should be a power of the radix to ensure a reliable solu‐
tion and error estimates. Scaling by powers of the  radix  does
not cause rounding errors unless the result underflows or over‐
flows. 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  ele‐
ment of C is a power of the radix.  If C is input, each element
of C should be a power of the radix to ensure a reliable	 solu‐
tion  and  error estimates. Scaling by powers of the radix does
not cause rounding errors unless the result underflows or over‐
flows.  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) DOUBLE PRECISION array, dimension (LDB,NRHS)
The right hand side matrix B.

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

X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
On  entry,  the	solution  matrix X, as computed by DGETRS.  On
exit, the improved solution matrix X.

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 equili‐
bration (if done).  If this is less than the machine  precision
(in particular, if it is zero), the matrix is singular to work‐
ing precision.  Note that the error may still be small even  if
this  number  is	 very small and the matrix appears ill- condi‐
tioned.

BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
Componentwise relative backward error.  This is the  component‐
wise  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 informa‐
tion about 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 recipro‐
cal  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 abso‐
lute 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  informa‐
tion  about  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 informa‐
tion 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 thresh‐
old sqrt(n) *  dlamch('Epsilon')	to  determine  if  the
error  estimate is "guaranteed". These reciprocal condi‐
tion 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 fur‐
ther details and extra cautions.	NPARAMS (input)	 INTE‐
GER  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 double-precision refinement algo‐
rithm, possibly with doubled-single computations if the	compi‐
lation  environment  does not support DOUBLE PRECISION.	(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 factor‐
ization 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 compo‐
nentwise relative  error	 in  the  double-precision  algorithm.
Positive	 is  true, 0.0 is false.  Default: 1.0 (attempt compo‐
nentwise convergence)

WORK    (workspace) DOUBLE PRECISION array, dimension (4*N)

IWORK   (workspace) INTEGER array, dimension (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 ille‐
gal 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 compo‐
nentwise 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 routine (version 3.2) November 2008			    DGBRFSX(1)
```
[top]

List of man pages available for Scientific

Copyright (c) for man pages and the logo by the respective OS vendor.

For those who want to learn more, the polarhome community provides shell access and support.

[legal] [privacy] [GNU] [policy] [cookies] [netiquette] [sponsors] [FAQ]

Polarhome, production since 1999.
Member of Polarhome portal.
Based on Fawad Halim's script.
....................................................................
 Vote for polarhome