
dla_porfsx_extended.f(3) LAPACK dla_porfsx_extended.f(3)
NAME
dla_porfsx_extended.f 
SYNOPSIS
Functions/Subroutines
subroutine dla_porfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, 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)
DLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for
symmetric or Hermitian positivedefinite matrices by performing extraprecise
iterative refinement and provides error bounds and backward error estimates for the
solution.
Function/Subroutine Documentation
subroutine dla_porfsx_extended (integerPREC_TYPE, characterUPLO, integerN, integerNRHS, double
precision, dimension( lda, * )A, integerLDA, double precision, dimension( ldaf, * )AF,
integerLDAF, logicalCOLEQU, double precision, dimension( * )C, double precision,
dimension( ldb, * )B, integerLDB, double precision, 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, double
precision, dimension( * )RES, double precision, dimension(*)AYB, double precision,
dimension( * )DY, double precision, dimension( * )Y_TAIL, double precisionRCOND,
integerITHRESH, double precisionRTHRESH, double precisionDZ_UB, logicalIGNORE_CWISE,
integerINFO)
DLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for
symmetric or Hermitian positivedefinite matrices by performing extraprecise iterative
refinement and provides error bounds and backward error estimates for the solution.
Purpose:
DLA_PORFSX_EXTENDED improves the computed solution to a system of
linear equations by performing extraprecise iterative refinement
and provides error bounds and backward error estimates for the solution.
This subroutine is called by DPORFSX 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
UPLO
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of righthandsides, i.e., the number of columns of the
matrix B.
A
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the NbyN matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF
AF is DOUBLE PRECISION array, dimension (LDAF,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, as computed by DPOTRF.
LDAF
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
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 DOUBLE PRECISION array, dimension (LDB,NRHS)
The righthandside matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Y
Y is DOUBLE PRECISION array, dimension
(LDY,NRHS)
On entry, the solution matrix X, as computed by DPOTRS.
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 righthandside 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 DLA_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 righthand 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
righthand 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 righthand 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 righthand 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 righthand 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
righthand 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 righthand 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 DOUBLE PRECISION array, dimension (N)
Workspace to hold the intermediate residual.
AYB
AYB is DOUBLE PRECISION array, dimension (N)
Workspace. This can be the same workspace passed for Y_TAIL.
DY
DY is DOUBLE PRECISION array, dimension (N)
Workspace to hold the intermediate solution.
Y_TAIL
Y_TAIL is DOUBLE PRECISION 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 illconditioned 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 DPOTRS had an illegal
value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012
Definition at line 385 of file dla_porfsx_extended.f.
Author
Generated automatically by Doxygen for LAPACK from the source code.
Version 3.4.2 Tue Sep 25 2012 dla_porfsx_extended.f(3) 
