--- rpl/lapack/lapack/zgbrfsx.f 2010/12/21 13:53:42 1.4
+++ rpl/lapack/lapack/zgbrfsx.f 2011/11/21 20:43:08 1.5
@@ -1,19 +1,450 @@
+*> \brief \b ZGBRFSX
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download ZGBRFSX + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE ZGBRFSX( 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 )
+*
+* .. Scalar Arguments ..
+* CHARACTER TRANS, EQUED
+* INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS,
+* $ NPARAMS, N_ERR_BNDS
+* DOUBLE PRECISION RCOND
+* ..
+* .. Array Arguments ..
+* INTEGER IPIV( * )
+* COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
+* $ X( LDX , * ),WORK( * )
+* DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
+* $ ERR_BNDS_NORM( NRHS, * ),
+* $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> ZGBRFSX 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.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[in] TRANS
+*> \verbatim
+*> 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)
+*> \endverbatim
+*>
+*> \param[in] EQUED
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The order of the matrix A. N >= 0.
+*> \endverbatim
+*>
+*> \param[in] KL
+*> \verbatim
+*> KL is INTEGER
+*> The number of subdiagonals within the band of A. KL >= 0.
+*> \endverbatim
+*>
+*> \param[in] KU
+*> \verbatim
+*> KU is INTEGER
+*> The number of superdiagonals within the band of A. KU >= 0.
+*> \endverbatim
+*>
+*> \param[in] NRHS
+*> \verbatim
+*> NRHS is INTEGER
+*> The number of right hand sides, i.e., the number of columns
+*> of the matrices B and X. NRHS >= 0.
+*> \endverbatim
+*>
+*> \param[in] AB
+*> \verbatim
+*> AB is 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).
+*> \endverbatim
+*>
+*> \param[in] LDAB
+*> \verbatim
+*> LDAB is INTEGER
+*> The leading dimension of the array AB. LDAB >= KL+KU+1.
+*> \endverbatim
+*>
+*> \param[in] AFB
+*> \verbatim
+*> AFB is DOUBLE PRECISION 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.
+*> \endverbatim
+*>
+*> \param[in] LDAFB
+*> \verbatim
+*> LDAFB is INTEGER
+*> The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
+*> \endverbatim
+*>
+*> \param[in] IPIV
+*> \verbatim
+*> IPIV is INTEGER array, dimension (N)
+*> The pivot indices from DGETRF; for 1<=i<=N, row i of the
+*> matrix was interchanged with row IPIV(i).
+*> \endverbatim
+*>
+*> \param[in,out] R
+*> \verbatim
+*> R is or output) DOUBLE PRECISION 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.
+*> \endverbatim
+*>
+*> \param[in,out] C
+*> \verbatim
+*> C is 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 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.
+*> \endverbatim
+*>
+*> \param[in] B
+*> \verbatim
+*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
+*> The right hand side matrix B.
+*> \endverbatim
+*>
+*> \param[in] LDB
+*> \verbatim
+*> LDB is INTEGER
+*> The leading dimension of the array B. LDB >= max(1,N).
+*> \endverbatim
+*>
+*> \param[in,out] X
+*> \verbatim
+*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
+*> On entry, the solution matrix X, as computed by DGETRS.
+*> On exit, the improved solution matrix X.
+*> \endverbatim
+*>
+*> \param[in] LDX
+*> \verbatim
+*> LDX is INTEGER
+*> The leading dimension of the array X. LDX >= max(1,N).
+*> \endverbatim
+*>
+*> \param[out] RCOND
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[out] BERR
+*> \verbatim
+*> BERR is DOUBLE PRECISION 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).
+*> \endverbatim
+*>
+*> \param[in] N_ERR_BNDS
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[out] ERR_BNDS_NORM
+*> \verbatim
+*> 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) * 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
+*> reciprocal 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 absolute row sums of Z are approximately 1.
+*>
+*> See Lapack Working Note 165 for further details and extra
+*> cautions.
+*> \endverbatim
+*>
+*> \param[out] ERR_BNDS_COMP
+*> \verbatim
+*> 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) * 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 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*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.
+*> \endverbatim
+*>
+*> \param[in] NPARAMS
+*> \verbatim
+*> 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.
+*> \endverbatim
+*>
+*> \param[in,out] PARAMS
+*> \verbatim
+*> PARAMS is / 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 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)
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*> WORK is COMPLEX*16 array, dimension (2*N)
+*> \endverbatim
+*>
+*> \param[out] RWORK
+*> \verbatim
+*> RWORK is DOUBLE PRECISION array, dimension (2*N)
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> 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.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date November 2011
+*
+*> \ingroup complex16GBcomputational
+*
+* =====================================================================
SUBROUTINE ZGBRFSX( 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 )
*
-* -- LAPACK routine (version 3.2.2) --
-* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
-* -- Jason Riedy of Univ. of California Berkeley. --
-* -- June 2010 --
-*
-* -- LAPACK is a software package provided by Univ. of Tennessee, --
-* -- Univ. of California Berkeley and NAG Ltd. --
+* -- LAPACK computational routine (version 3.4.0) --
+* -- LAPACK is a software package provided by Univ. of Tennessee, --
+* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
+* November 2011
*
- IMPLICIT NONE
-* ..
* .. Scalar Arguments ..
CHARACTER TRANS, EQUED
INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS,
@@ -29,302 +460,7 @@
$ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
* ..
*
-* Purpose
-* =======
-*
-* ZGBRFSX 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.
-*
-* Arguments
-* =========
-*
-* 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.
-*
-* 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 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 (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
-* 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 (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
-* 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 (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 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 (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
-* 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 (output) DOUBLE PRECISION 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 (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 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) * 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
-* reciprocal 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 absolute 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 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) * 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 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*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 (input) INTEGER
-* 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 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 (workspace) COMPLEX*16 array, dimension (2*N)
-*
-* RWORK (workspace) DOUBLE PRECISION array, dimension (2*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 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.
-*
-* ==================================================================
+* ==================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE