*> \brief ** DGBSVXX computes the solution to system of linear equations A * X = B for GB matrices**
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE DGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
* LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
* RCOND, RPVGRW, BERR, N_ERR_BNDS,
* ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
* WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER EQUED, FACT, TRANS
* INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
* $ N_ERR_BNDS, KL, KU
* DOUBLE PRECISION RCOND, RPVGRW
* ..
* .. Array Arguments ..
* INTEGER IPIV( * ), IWORK( * )
* DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
* $ X( LDX , * ),WORK( * )
* DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
* $ ERR_BNDS_NORM( NRHS, * ),
* $ ERR_BNDS_COMP( NRHS, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGBSVXX uses the LU factorization to compute the solution to a
*> double precision system of linear equations A * X = B, where A is an
*> N-by-N matrix and X and B are N-by-NRHS matrices.
*>
*> If requested, both normwise and maximum componentwise error bounds
*> are returned. DGBSVXX will return a solution with a tiny
*> guaranteed error (O(eps) where eps is the working machine
*> precision) unless the matrix is very ill-conditioned, in which
*> case a warning is returned. Relevant condition numbers also are
*> calculated and returned.
*>
*> DGBSVXX accepts user-provided factorizations and equilibration
*> factors; see the definitions of the FACT and EQUED options.
*> Solving with refinement and using a factorization from a previous
*> DGBSVXX call will also produce a solution with either O(eps)
*> errors or warnings, but we cannot make that claim for general
*> user-provided factorizations and equilibration factors if they
*> differ from what DGBSVXX would itself produce.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'E', double precision 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 = P * L * U,
*>
*> where P is a permutation matrix, L is a unit lower triangular
*> matrix, and U is upper triangular.
*>
*> 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 (see
*> argument RCOND). If the reciprocal of the condition number is less
*> than machine precision, 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. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
*> the routine will use iterative refinement to try to get a small
*> error and error bounds. Refinement calculates the residual to at
*> least twice the working precision.
*>
*> 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:
* ==========
*
*> \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] 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, AF 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.
*> A, AF, and IPIV are not modified.
*> = 'N': The matrix A will be copied to AF and factored.
*> = 'E': The matrix A will be equilibrated if necessary, then
*> copied to AF 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 (Conjugate Transpose = 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 AB 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 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 AF is an output argument and on exit
*> returns the factors L and U from the factorization A = P*L*U
*> of the original matrix A.
*>
*> If FACT = 'E', then AF is an output argument and on exit
*> returns the factors L and U from the factorization A = P*L*U
*> of the equilibrated matrix A (see the description of A 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 INTEGER array, dimension (N)
*> If FACT = 'F', then IPIV is an input argument and on entry
*> contains the pivot indices from the factorization A = P*L*U
*> as computed by DGETRF; 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 = P*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 = P*L*U
*> of the equilibrated matrix A.
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*> EQUED is 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 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 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,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS 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, 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
*> 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] RPVGRW
*> \verbatim
*> RPVGRW is DOUBLE PRECISION
*> Reciprocal pivot growth. On exit, this contains the reciprocal
*> pivot growth factor norm(A)/norm(U). The "max absolute element"
*> norm is used. If this 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, estimated condition numbers,
*> and error bounds could be unreliable. If factorization fails with
*> 0 for the leading INFO columns of A. In DGESVX, this quantity is
*> returned in WORK(1).
*> \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 < 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 <= 0, the
*> PARAMS array is never referenced and default values are used.
*> \endverbatim
*>
*> \param[in,out] PARAMS
*> \verbatim
*> PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)
*> Specifies algorithm parameters. If an entry is < 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 extra-precise refinement algorithm.
*> (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 DOUBLE PRECISION array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (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.
*
*> \ingroup doubleGBsolve
*
* =====================================================================
SUBROUTINE DGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
$ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
$ RCOND, RPVGRW, BERR, N_ERR_BNDS,
$ ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
$ WORK, IWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
$ N_ERR_BNDS, KL, KU
DOUBLE PRECISION RCOND, RPVGRW
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
$ X( LDX , * ),WORK( * )
DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
$ ERR_BNDS_NORM( NRHS, * ),
$ ERR_BNDS_COMP( NRHS, * )
* ..
*
* ==================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
INTEGER CMP_ERR_I, PIV_GROWTH_I
PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
$ BERR_I = 3 )
PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
$ PIV_GROWTH_I = 9 )
* ..
* .. Local Scalars ..
LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
INTEGER INFEQU, I, J
DOUBLE PRECISION AMAX, BIGNUM, COLCND, RCMAX, RCMIN,
$ ROWCND, SMLNUM
* ..
* .. External Functions ..
EXTERNAL LSAME, DLAMCH, DLA_GBRPVGRW
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLA_GBRPVGRW
* ..
* .. External Subroutines ..
EXTERNAL DGBEQUB, DGBTRF, DGBTRS, DLACPY, DLAQGB,
$ XERBLA, DLASCL2, DGBRFSX
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
EQUIL = LSAME( FACT, 'E' )
NOTRAN = LSAME( TRANS, 'N' )
SMLNUM = DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
IF( NOFACT .OR. EQUIL ) THEN
EQUED = 'N'
ROWEQU = .FALSE.
COLEQU = .FALSE.
ELSE
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
END IF
*
* Default is failure. If an input parameter is wrong or
* factorization fails, make everything look horrible. Only the
* pivot growth is set here, the rest is initialized in DGBRFSX.
*
RPVGRW = ZERO
*
* Test the input parameters. PARAMS is not tested until DGBRFSX.
*
IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
$ LSAME( FACT, 'F' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( KL.LT.0 ) THEN
INFO = -4
ELSE IF( KU.LT.0 ) THEN
INFO = -5
ELSE IF( NRHS.LT.0 ) THEN
INFO = -6
ELSE IF( LDAB.LT.KL+KU+1 ) THEN
INFO = -8
ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
INFO = -10
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
INFO = -12
ELSE
IF( ROWEQU ) THEN
RCMIN = BIGNUM
RCMAX = ZERO
DO 10 J = 1, N
RCMIN = MIN( RCMIN, R( J ) )
RCMAX = MAX( RCMAX, R( J ) )
10 CONTINUE
IF( RCMIN.LE.ZERO ) THEN
INFO = -13
ELSE IF( N.GT.0 ) THEN
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
ELSE
ROWCND = ONE
END IF
END IF
IF( COLEQU .AND. INFO.EQ.0 ) THEN
RCMIN = BIGNUM
RCMAX = ZERO
DO 20 J = 1, N
RCMIN = MIN( RCMIN, C( J ) )
RCMAX = MAX( RCMAX, C( J ) )
20 CONTINUE
IF( RCMIN.LE.ZERO ) THEN
INFO = -14
ELSE IF( N.GT.0 ) THEN
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
ELSE
COLCND = ONE
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -15
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -16
END IF
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBSVXX', -INFO )
RETURN
END IF
*
IF( EQUIL ) THEN
*
* Compute row and column scalings to equilibrate the matrix A.
*
CALL DGBEQUB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
$ AMAX, INFEQU )
IF( INFEQU.EQ.0 ) THEN
*
* Equilibrate the matrix.
*
CALL DLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
$ AMAX, EQUED )
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
END IF
*
* If the scaling factors are not applied, set them to 1.0.
*
IF ( .NOT.ROWEQU ) THEN
DO J = 1, N
R( J ) = 1.0D+0
END DO
END IF
IF ( .NOT.COLEQU ) THEN
DO J = 1, N
C( J ) = 1.0D+0
END DO
END IF
END IF
*
* Scale the right hand side.
*
IF( NOTRAN ) THEN
IF( ROWEQU ) CALL DLASCL2(N, NRHS, R, B, LDB)
ELSE
IF( COLEQU ) CALL DLASCL2(N, NRHS, C, B, LDB)
END IF
*
IF( NOFACT .OR. EQUIL ) THEN
*
* Compute the LU factorization of A.
*
DO 40, J = 1, N
DO 30, I = KL+1, 2*KL+KU+1
AFB( I, J ) = AB( I-KL, J )
30 CONTINUE
40 CONTINUE
CALL DGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 ) THEN
*
* Pivot in column INFO is exactly 0
* Compute the reciprocal pivot growth factor of the
* leading rank-deficient INFO columns of A.
*
RPVGRW = DLA_GBRPVGRW( N, KL, KU, INFO, AB, LDAB, AFB,
$ LDAFB )
RETURN
END IF
END IF
*
* Compute the reciprocal pivot growth factor RPVGRW.
*
RPVGRW = DLA_GBRPVGRW( N, KL, KU, N, AB, LDAB, AFB, LDAFB )
*
* Compute the solution matrix X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
$ INFO )
*
* Use iterative refinement to improve the computed solution and
* compute error bounds and backward error estimates for it.
*
CALL DGBRFSX( 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, IWORK, INFO )
*
* Scale solutions.
*
IF ( COLEQU .AND. NOTRAN ) THEN
CALL DLASCL2 ( N, NRHS, C, X, LDX )
ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN
CALL DLASCL2 ( N, NRHS, R, X, LDX )
END IF
*
RETURN
*
* End of DGBSVXX
*
END