zla_gbrfsx_extended.f man page

zla_gbrfsx_extended.f —

Synopsis

Functions/Subroutines

subroutine zla_gbrfsx_extended (PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
ZLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Function/Subroutine Documentation

subroutine zla_gbrfsx_extended (integerPREC_TYPE, integerTRANS_TYPE, integerN, integerKL, integerKU, integerNRHS, complex*16, dimension( ldab, * )AB, integerLDAB, complex*16, dimension( ldafb, * )AFB, integerLDAFB, integer, dimension( * )IPIV, logicalCOLEQU, double precision, dimension( * )C, complex*16, dimension( ldb, * )B, integerLDB, complex*16, dimension( ldy, * )Y, integerLDY, double precision, dimension( * )BERR_OUT, integerN_NORMS, double precision, dimension( nrhs, * )ERR_BNDS_NORM, double precision, dimension( nrhs, * )ERR_BNDS_COMP, complex*16, dimension( * )RES, double precision, dimension(*)AYB, complex*16, dimension( * )DY, complex*16, dimension( * )Y_TAIL, double precisionRCOND, integerITHRESH, double precisionRTHRESH, double precisionDZ_UB, logicalIGNORE_CWISE, integerINFO)

ZLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose:

ZLA_GBRFSX_EXTENDED improves the computed solution to a system of
linear equations by performing extra-precise iterative refinement
and provides error bounds and backward error estimates for the solution.
This subroutine is called by ZGBRFSX to perform iterative refinement.
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. Note that this
subroutine is only resonsible for setting the second fields of
ERR_BNDS_NORM and ERR_BNDS_COMP.

Parameters:

PREC_TYPE

     PREC_TYPE is INTEGER
Specifies the intermediate precision to be used in refinement.
The value is defined by ILAPREC(P) where P is a CHARACTER and
P    = 'S':  Single
     = 'D':  Double
     = 'I':  Indigenous
     = 'X', 'E':  Extra

TRANS_TYPE

     TRANS_TYPE is INTEGER
Specifies the transposition operation on A.
The value is defined by ILATRANS(T) where T is a CHARACTER and
T    = 'N':  No transpose
     = 'T':  Transpose
     = 'C':  Conjugate transpose

N

     N is INTEGER
The number of linear equations, i.e., 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
matrix B.

AB

     AB is COMPLEX*16 array, dimension (LDAB,N)
On entry, the N-by-N matrix A.

LDAB

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

AFB

     AFB is COMPLEX*16 array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by ZGBTRF.

LDAFB

     LDAFB is INTEGER
The leading dimension of the array AF.  LDAF >= max(1,N).

IPIV

     IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by ZGBTRF; row i of the matrix was interchanged
with row IPIV(i).

COLEQU

     COLEQU is LOGICAL
If .TRUE. then column equilibration was done to A before calling
this routine. This is needed to compute the solution and error
bounds correctly.

C

     C is DOUBLE PRECISION array, dimension (N)
The column scale factors for A. If COLEQU = .FALSE., C
is not accessed. 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*16 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).

Y

     Y is COMPLEX*16 array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by ZGBTRS.
On exit, the improved solution matrix Y.

LDY

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

BERR_OUT

     BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
On exit, BERR_OUT(j) contains the componentwise relative backward
error for right-hand-side j from the formula
    max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. This is computed by ZLA_LIN_BERR.

N_NORMS

     N_NORMS is INTEGER
Determines which error bounds to return (see ERR_BNDS_NORM
and ERR_BNDS_COMP).
If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds.

ERR_BNDS_NORM

     ERR_BNDS_NORM is DOUBLE PRECISION 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.

This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.

ERR_BNDS_COMP

     ERR_BNDS_COMP is DOUBLE PRECISION 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.

This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.

RES

     RES is COMPLEX*16 array, dimension (N)
Workspace to hold the intermediate residual.

AYB

     AYB is DOUBLE PRECISION array, dimension (N)
Workspace.

DY

     DY is COMPLEX*16 array, dimension (N)
Workspace to hold the intermediate solution.

Y_TAIL

     Y_TAIL is COMPLEX*16 array, dimension (N)
Workspace to hold the trailing bits of the intermediate solution.

RCOND

     RCOND is 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.

ITHRESH

     ITHRESH is INTEGER
The maximum number of residual computations allowed for
refinement. The default is 10. For '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.

RTHRESH

     RTHRESH is DOUBLE PRECISION
Determines when to stop refinement if the error estimate stops
decreasing. Refinement will stop when the next solution no longer
satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
default value is 0.5. For 'aggressive' set to 0.9 to permit
convergence on extremely ill-conditioned matrices. See LAWN 165
for more details.

DZ_UB

     DZ_UB is DOUBLE PRECISION
Determines when to start considering componentwise convergence.
Componentwise convergence is only considered after each component
of the solution Y is stable, which we definte as the relative
change in each component being less than DZ_UB. The default value
is 0.25, requiring the first bit to be stable. See LAWN 165 for
more details.

IGNORE_CWISE

     IGNORE_CWISE is LOGICAL
If .TRUE. then ignore componentwise convergence. Default value
is .FALSE..

INFO

   INFO is INTEGER
= 0:  Successful exit.
< 0:  if INFO = -i, the ith argument to ZGBTRS had an illegal
      value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

Definition at line 407 of file zla_gbrfsx_extended.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

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

Sat Nov 16 2013 Version 3.4.2 LAPACK