cgbrfsx.f man page

cgbrfsx.f —

Synopsis

Functions/Subroutines

subroutine cgbrfsx (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, RWORK, INFO)
CGBRFSX

Function/Subroutine Documentation

subroutine cgbrfsx (characterTRANS, characterEQUED, integerN, integerKL, integerKU, integerNRHS, complex, dimension( ldab, * )AB, integerLDAB, complex, dimension( ldafb, * )AFB, integerLDAFB, integer, dimension( * )IPIV, real, dimension( * )R, real, dimension( * )C, complex, dimension( ldb, * )B, integerLDB, complex, dimension( ldx , * )X, integerLDX, realRCOND, real, dimension( * )BERR, integerN_ERR_BNDS, real, dimension( nrhs, * )ERR_BNDS_NORM, real, dimension( nrhs, * )ERR_BNDS_COMP, integerNPARAMS, real, dimension( * )PARAMS, complex, dimension( * )WORK, real, dimension( * )RWORK, integerINFO)

CGBRFSX

Purpose:

CGBRFSX 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_NORM and ERR_BNDS_COMP 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.
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.

Parameters:

TRANS

     TRANS is 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

     EQUED is 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 postmultiplied
          by diag(C).
  = 'B':  Both row and column equilibration, i.e., A has been
          replaced by diag(R) * A * diag(C).
          The right hand side B has been changed accordingly.

N

     N is INTEGER
The order of the matrix A.  N >= 0.

KL

     KL is INTEGER
The number of subdiagonals within the band of A.  KL >= 0.

KU

     KU is INTEGER
The number of superdiagonals within the band of A.  KU >= 0.

NRHS

     NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X.  NRHS >= 0.

AB

     AB is COMPLEX 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

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

AFB

     AFB is COMPLEX array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix A, as
computed 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

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

IPIV

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

R

     R is REAL 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 radix
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

     C is REAL 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 radix
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

     B is COMPLEX array, dimension (LDB,NRHS)
The right hand side matrix B.

LDB

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

X

     X is COMPLEX array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SGETRS.
On exit, the improved solution matrix X.

LDX

     LDX is INTEGER
The leading dimension of the array X.  LDX >= max(1,N).

RCOND

     RCOND is REAL
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.

BERR

     BERR is REAL 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

     N_ERR_BNDS is 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

     ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information 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) * slamch('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) * slamch('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) * slamch('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

     ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information 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
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) * slamch('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) * slamch('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) * slamch('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

     NPARAMS is INTEGER
Specifies the number of parameters set in PARAMS.  If .LE. 0, the
PARAMS array is never referenced and default values are used.

PARAMS

     PARAMS is REAL 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: 1.0
       = 0.0 : No refinement is performed, and no error bounds are
               computed.
       = 1.0 : Use the double-precision refinement algorithm,
               possibly with doubled-single computations if the
               compilation 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 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

WORK is COMPLEX array, dimension (2*N)

RWORK

RWORK is REAL array, dimension (2*N)

INFO

   INFO is 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.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

April 2012

Definition at line 437 of file cgbrfsx.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

cgbrfsx(3) is an alias of cgbrfsx.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK