--- rpl/lapack/lapack/dgbsvx.f 2010/12/21 13:53:24 1.7
+++ rpl/lapack/lapack/dgbsvx.f 2011/11/21 20:42:50 1.8
@@ -1,11 +1,378 @@
+*> \brief DGBSVX computes the solution to system of linear equations A * X = B for GB matrices
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download DGBSVX + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
+* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
+* RCOND, FERR, BERR, WORK, IWORK, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER EQUED, FACT, TRANS
+* INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
+* DOUBLE PRECISION RCOND
+* ..
+* .. Array Arguments ..
+* INTEGER IPIV( * ), IWORK( * )
+* DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
+* $ BERR( * ), C( * ), FERR( * ), R( * ),
+* $ WORK( * ), X( LDX, * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> DGBSVX uses the LU factorization to compute the solution to a real
+*> system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
+*> where A is a band matrix of order N with KL subdiagonals and KU
+*> superdiagonals, and X and B are N-by-NRHS matrices.
+*>
+*> Error bounds on the solution and a condition estimate are also
+*> provided.
+*> \endverbatim
+*
+*> \par Description:
+* =================
+*>
+*> \verbatim
+*>
+*> The following steps are performed by this subroutine:
+*>
+*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
+*> the system:
+*> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
+*> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
+*> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
+*> Whether or not the system will be equilibrated depends on the
+*> scaling of the matrix A, but if equilibration is used, A is
+*> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
+*> or diag(C)*B (if TRANS = 'T' or 'C').
+*>
+*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
+*> matrix A (after equilibration if FACT = 'E') as
+*> A = L * U,
+*> where L is a product of permutation and unit lower triangular
+*> matrices with KL subdiagonals, and U is upper triangular with
+*> KL+KU superdiagonals.
+*>
+*> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
+*> returns with INFO = i. Otherwise, the factored form of A is used
+*> to estimate the condition number of the matrix A. If the
+*> reciprocal of the condition number is less than machine precision,
+*> INFO = N+1 is returned as a warning, but the routine still goes on
+*> to solve for X and compute error bounds as described below.
+*>
+*> 4. The system of equations is solved for X using the factored form
+*> of A.
+*>
+*> 5. Iterative refinement is applied to improve the computed solution
+*> matrix and calculate error bounds and backward error estimates
+*> for it.
+*>
+*> 6. If equilibration was used, the matrix X is premultiplied by
+*> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
+*> that it solves the original system before equilibration.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] FACT
+*> \verbatim
+*> FACT is CHARACTER*1
+*> Specifies whether or not the factored form of the matrix A is
+*> supplied on entry, and if not, whether the matrix A should be
+*> equilibrated before it is factored.
+*> = 'F': On entry, AFB and IPIV contain the factored form of
+*> A. If EQUED is not 'N', the matrix A has been
+*> equilibrated with scaling factors given by R and C.
+*> AB, AFB, and IPIV are not modified.
+*> = 'N': The matrix A will be copied to AFB and factored.
+*> = 'E': The matrix A will be equilibrated if necessary, then
+*> copied to AFB and factored.
+*> \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 (Transpose)
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The number of linear equations, i.e., 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,out] AB
+*> \verbatim
+*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
+*> On entry, the matrix A in band storage, 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)
+*>
+*> If FACT = 'F' and EQUED is not 'N', then A must have been
+*> equilibrated by the scaling factors in R and/or C. AB is not
+*> modified if FACT = 'F' or 'N', or if FACT = 'E' and
+*> EQUED = 'N' on exit.
+*>
+*> On exit, if EQUED .ne. 'N', A is scaled as follows:
+*> EQUED = 'R': A := diag(R) * A
+*> EQUED = 'C': A := A * diag(C)
+*> EQUED = 'B': A := diag(R) * A * diag(C).
+*> \endverbatim
+*>
+*> \param[in] LDAB
+*> \verbatim
+*> LDAB is INTEGER
+*> The leading dimension of the array AB. LDAB >= KL+KU+1.
+*> \endverbatim
+*>
+*> \param[in,out] AFB
+*> \verbatim
+*> AFB is or output) DOUBLE PRECISION array, dimension (LDAFB,N)
+*> If FACT = 'F', then AFB is an input argument and on entry
+*> contains 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. If EQUED .ne. 'N', then AFB is
+*> the factored form of the equilibrated matrix A.
+*>
+*> If FACT = 'N', then AFB is an output argument and on exit
+*> returns details of the LU factorization of A.
+*>
+*> If FACT = 'E', then AFB is an output argument and on exit
+*> returns details of the LU factorization of the equilibrated
+*> matrix A (see the description of AB for the form of the
+*> equilibrated matrix).
+*> \endverbatim
+*>
+*> \param[in] LDAFB
+*> \verbatim
+*> LDAFB is INTEGER
+*> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
+*> \endverbatim
+*>
+*> \param[in,out] IPIV
+*> \verbatim
+*> IPIV is or output) INTEGER array, dimension (N)
+*> If FACT = 'F', then IPIV is an input argument and on entry
+*> contains the pivot indices from the factorization A = L*U
+*> as computed by DGBTRF; row i of the matrix was interchanged
+*> with row IPIV(i).
+*>
+*> If FACT = 'N', then IPIV is an output argument and on exit
+*> contains the pivot indices from the factorization A = L*U
+*> of the original matrix A.
+*>
+*> If FACT = 'E', then IPIV is an output argument and on exit
+*> contains the pivot indices from the factorization A = L*U
+*> of the equilibrated matrix A.
+*> \endverbatim
+*>
+*> \param[in,out] EQUED
+*> \verbatim
+*> EQUED is or output) CHARACTER*1
+*> Specifies the form of equilibration that was done.
+*> = 'N': No equilibration (always true if FACT = 'N').
+*> = '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).
+*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
+*> output argument.
+*> \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.
+*> \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.
+*> \endverbatim
+*>
+*> \param[in,out] B
+*> \verbatim
+*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
+*> On entry, the right hand side matrix B.
+*> On exit,
+*> if EQUED = 'N', B is not modified;
+*> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
+*> diag(R)*B;
+*> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
+*> overwritten by diag(C)*B.
+*> \endverbatim
+*>
+*> \param[in] LDB
+*> \verbatim
+*> LDB is INTEGER
+*> The leading dimension of the array B. LDB >= max(1,N).
+*> \endverbatim
+*>
+*> \param[out] X
+*> \verbatim
+*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
+*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
+*> to the original system of equations. Note that A and B are
+*> modified on exit if EQUED .ne. 'N', and the solution to the
+*> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
+*> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
+*> and EQUED = 'R' or 'B'.
+*> \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
+*> The estimate of the reciprocal condition number of the matrix
+*> A after equilibration (if done). If RCOND is less than the
+*> machine precision (in particular, if RCOND = 0), the matrix
+*> is singular to working precision. This condition is
+*> indicated by a return code of INFO > 0.
+*> \endverbatim
+*>
+*> \param[out] FERR
+*> \verbatim
+*> FERR is DOUBLE PRECISION array, dimension (NRHS)
+*> The estimated forward error bound for each solution vector
+*> X(j) (the j-th column of the solution matrix X).
+*> If XTRUE is the true solution corresponding to X(j), FERR(j)
+*> is an estimated upper bound for the magnitude of the largest
+*> element in (X(j) - XTRUE) divided by the magnitude of the
+*> largest element in X(j). The estimate is as reliable as
+*> the estimate for RCOND, and is almost always a slight
+*> overestimate of the true error.
+*> \endverbatim
+*>
+*> \param[out] BERR
+*> \verbatim
+*> BERR is DOUBLE PRECISION array, dimension (NRHS)
+*> 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[out] WORK
+*> \verbatim
+*> WORK is DOUBLE PRECISION array, dimension (3*N)
+*> On exit, WORK(1) contains the reciprocal pivot growth
+*> factor norm(A)/norm(U). The "max absolute element" norm is
+*> used. If WORK(1) is much less than 1, then the stability
+*> of the LU factorization of the (equilibrated) matrix A
+*> could be poor. This also means that the solution X, condition
+*> estimator RCOND, and forward error bound FERR could be
+*> unreliable. If factorization fails with 0 WORK(1) contains the reciprocal pivot growth factor for the
+*> leading INFO columns of A.
+*> \endverbatim
+*>
+*> \param[out] IWORK
+*> \verbatim
+*> IWORK is INTEGER array, dimension (N)
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> INFO is INTEGER
+*> = 0: successful exit
+*> < 0: if INFO = -i, the i-th argument had an illegal value
+*> > 0: if INFO = i, and i is
+*> <= N: U(i,i) 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+1: U is nonsingular, but RCOND is less than machine
+*> precision, meaning that the matrix is singular
+*> to working precision. Nevertheless, the
+*> solution and error bounds are computed because
+*> there are a number of situations where the
+*> computed solution can be more accurate than the
+*> value of RCOND would suggest.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date November 2011
+*
+*> \ingroup doubleGBsolve
+*
+* =====================================================================
SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
$ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
$ RCOND, FERR, BERR, WORK, IWORK, INFO )
*
-* -- LAPACK driver routine (version 3.2) --
+* -- LAPACK driver 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 2006
+* November 2011
*
* .. Scalar Arguments ..
CHARACTER EQUED, FACT, TRANS
@@ -19,245 +386,6 @@
$ WORK( * ), X( LDX, * )
* ..
*
-* Purpose
-* =======
-*
-* DGBSVX uses the LU factorization to compute the solution to a real
-* system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
-* where A is a band matrix of order N with KL subdiagonals and KU
-* superdiagonals, and X and B are N-by-NRHS matrices.
-*
-* Error bounds on the solution and a condition estimate are also
-* provided.
-*
-* Description
-* ===========
-*
-* The following steps are performed by this subroutine:
-*
-* 1. If FACT = 'E', real scaling factors are computed to equilibrate
-* the system:
-* TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
-* TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
-* TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
-* Whether or not the system will be equilibrated depends on the
-* scaling of the matrix A, but if equilibration is used, A is
-* overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
-* or diag(C)*B (if TRANS = 'T' or 'C').
-*
-* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
-* matrix A (after equilibration if FACT = 'E') as
-* A = L * U,
-* where L is a product of permutation and unit lower triangular
-* matrices with KL subdiagonals, and U is upper triangular with
-* KL+KU superdiagonals.
-*
-* 3. If some U(i,i)=0, so that U is exactly singular, then the routine
-* returns with INFO = i. Otherwise, the factored form of A is used
-* to estimate the condition number of the matrix A. If the
-* reciprocal of the condition number is less than machine precision,
-* INFO = N+1 is returned as a warning, but the routine still goes on
-* to solve for X and compute error bounds as described below.
-*
-* 4. The system of equations is solved for X using the factored form
-* of A.
-*
-* 5. Iterative refinement is applied to improve the computed solution
-* matrix and calculate error bounds and backward error estimates
-* for it.
-*
-* 6. If equilibration was used, the matrix X is premultiplied by
-* diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
-* that it solves the original system before equilibration.
-*
-* Arguments
-* =========
-*
-* FACT (input) CHARACTER*1
-* Specifies whether or not the factored form of the matrix A is
-* supplied on entry, and if not, whether the matrix A should be
-* equilibrated before it is factored.
-* = 'F': On entry, AFB and IPIV contain the factored form of
-* A. If EQUED is not 'N', the matrix A has been
-* equilibrated with scaling factors given by R and C.
-* AB, AFB, and IPIV are not modified.
-* = 'N': The matrix A will be copied to AFB and factored.
-* = 'E': The matrix A will be equilibrated if necessary, then
-* copied to AFB and factored.
-*
-* 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 (Transpose)
-*
-* N (input) INTEGER
-* The number of linear equations, i.e., 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/output) DOUBLE PRECISION array, dimension (LDAB,N)
-* On entry, the matrix A in band storage, 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)
-*
-* If FACT = 'F' and EQUED is not 'N', then A must have been
-* equilibrated by the scaling factors in R and/or C. AB is not
-* modified if FACT = 'F' or 'N', or if FACT = 'E' and
-* EQUED = 'N' on exit.
-*
-* On exit, if EQUED .ne. 'N', A is scaled as follows:
-* EQUED = 'R': A := diag(R) * A
-* EQUED = 'C': A := A * diag(C)
-* EQUED = 'B': A := diag(R) * A * diag(C).
-*
-* LDAB (input) INTEGER
-* The leading dimension of the array AB. LDAB >= KL+KU+1.
-*
-* AFB (input or output) DOUBLE PRECISION array, dimension (LDAFB,N)
-* If FACT = 'F', then AFB is an input argument and on entry
-* contains 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. If EQUED .ne. 'N', then AFB is
-* the factored form of the equilibrated matrix A.
-*
-* If FACT = 'N', then AFB is an output argument and on exit
-* returns details of the LU factorization of A.
-*
-* If FACT = 'E', then AFB is an output argument and on exit
-* returns details of the LU factorization of the equilibrated
-* matrix A (see the description of AB for the form of the
-* equilibrated matrix).
-*
-* LDAFB (input) INTEGER
-* The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
-*
-* IPIV (input or output) INTEGER array, dimension (N)
-* If FACT = 'F', then IPIV is an input argument and on entry
-* contains the pivot indices from the factorization A = L*U
-* as computed by DGBTRF; row i of the matrix was interchanged
-* with row IPIV(i).
-*
-* If FACT = 'N', then IPIV is an output argument and on exit
-* contains the pivot indices from the factorization A = L*U
-* of the original matrix A.
-*
-* If FACT = 'E', then IPIV is an output argument and on exit
-* contains the pivot indices from the factorization A = L*U
-* of the equilibrated matrix A.
-*
-* EQUED (input or output) CHARACTER*1
-* Specifies the form of equilibration that was done.
-* = 'N': No equilibration (always true if FACT = 'N').
-* = '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).
-* EQUED is an input argument if FACT = 'F'; otherwise, it is an
-* output argument.
-*
-* 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.
-*
-* 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.
-*
-* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
-* On entry, the right hand side matrix B.
-* On exit,
-* if EQUED = 'N', B is not modified;
-* if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
-* diag(R)*B;
-* if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
-* overwritten by diag(C)*B.
-*
-* LDB (input) INTEGER
-* The leading dimension of the array B. LDB >= max(1,N).
-*
-* X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
-* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
-* to the original system of equations. Note that A and B are
-* modified on exit if EQUED .ne. 'N', and the solution to the
-* equilibrated system is inv(diag(C))*X if TRANS = 'N' and
-* EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
-* and EQUED = 'R' or 'B'.
-*
-* LDX (input) INTEGER
-* The leading dimension of the array X. LDX >= max(1,N).
-*
-* RCOND (output) DOUBLE PRECISION
-* The estimate of the reciprocal condition number of the matrix
-* A after equilibration (if done). If RCOND is less than the
-* machine precision (in particular, if RCOND = 0), the matrix
-* is singular to working precision. This condition is
-* indicated by a return code of INFO > 0.
-*
-* FERR (output) DOUBLE PRECISION array, dimension (NRHS)
-* The estimated forward error bound for each solution vector
-* X(j) (the j-th column of the solution matrix X).
-* If XTRUE is the true solution corresponding to X(j), FERR(j)
-* is an estimated upper bound for the magnitude of the largest
-* element in (X(j) - XTRUE) divided by the magnitude of the
-* largest element in X(j). The estimate is as reliable as
-* the estimate for RCOND, and is almost always a slight
-* overestimate of the true error.
-*
-* BERR (output) DOUBLE PRECISION array, dimension (NRHS)
-* 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).
-*
-* WORK (workspace/output) DOUBLE PRECISION array, dimension (3*N)
-* On exit, WORK(1) contains the reciprocal pivot growth
-* factor norm(A)/norm(U). The "max absolute element" norm is
-* used. If WORK(1) is much less than 1, then the stability
-* of the LU factorization of the (equilibrated) matrix A
-* could be poor. This also means that the solution X, condition
-* estimator RCOND, and forward error bound FERR could be
-* unreliable. If factorization fails with 0 0: if INFO = i, and i is
-* <= N: U(i,i) 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+1: U is nonsingular, but RCOND is less than machine
-* precision, meaning that the matrix is singular
-* to working precision. Nevertheless, the
-* solution and error bounds are computed because
-* there are a number of situations where the
-* computed solution can be more accurate than the
-* value of RCOND would suggest.
-*
* =====================================================================
*
* .. Parameters ..