ZHERFSX(1) LAPACK routine (version 3.2) ZHERFSX(1)NAME
ZHERFSX - ZHERFSX improve the computed solution to a system of linear
equations when the coefficient matrix is Hermitian indefinite, and
provides error bounds and backward error estimates for the solution
SYNOPSIS
Subroutine ZHERFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B,
LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
WORK, RWORK, INFO )
IMPLICIT NONE
CHARACTER UPLO, EQUED
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
N_ERR_BNDS
DOUBLE PRECISION RCOND
INTEGER IPIV( * )
COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), X( LDX, *
), WORK( * )
DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( *
), ERR_BNDS_NORM( NRHS, * ), ERR_BNDS_COMP( NRHS, *
)
PURPOSE
ZHERFSX improves the computed solution to a system of linear
equations when the coefficient matrix is Hermitian indefinite, 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 and S
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.
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
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
= 'Y': Both row and column equilibration, i.e., A has been
replaced by diag(S) * A * diag(S). The right hand side B has
been changed accordingly.
N (input) INTEGER
The order of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of
the matrices B and X. NRHS >= 0.
A (input) COMPLEX*16 array, dimension (LDA,N)
The symmetric matrix A. If UPLO = 'U', the leading N-by-N
upper triangular part of A contains the upper triangular part
of the matrix A, and the strictly lower triangular part of A is
not referenced. If UPLO = 'L', the leading N-by-N lower trian‐
gular part of A contains the lower triangular part of the
matrix A, and the strictly upper triangular part of A is not
referenced.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF (input) COMPLEX*16 array, dimension (LDAF,N)
The factored form of the matrix A. AF contains the block diag‐
onal matrix D and the multipliers used to obtain the factor U
or L from the factorization A = U*D*U**T or A = L*D*L**T as
computed by DSYTRF.
LDAF (input) INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV (input) INTEGER array, dimension (N)
Details of the interchanges and the block structure of D as
determined by DSYTRF.
S (input or output) DOUBLE PRECISION array, dimension (N)
The scale factors for A. If EQUED = 'Y', A is multiplied on
the left and right by diag(S). S is an input argument if FACT
= 'F'; otherwise, S is an output argument. If FACT = 'F' and
EQUED = 'Y', each element of S must be positive. If S is out‐
put, each element of S is a power of the radix. If S is input,
each element of S 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 under‐
flows 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) COMPLEX*16 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) COMPLEX*16 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 ZHERFSX(1)
