--- rpl/lapack/lapack/dgbsvxx.f 2010/12/21 13:53:24 1.4
+++ rpl/lapack/lapack/dgbsvxx.f 2011/11/21 20:42:50 1.5
@@ -1,19 +1,570 @@
+*> \brief DGBSVXX 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 DGBSVXX + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* 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 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 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 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 = 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 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.
+*> 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,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 .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 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.
+*
+*> \date November 2011
+*
+*> \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 (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 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 2011
*
- IMPLICIT NONE
-* ..
* .. Scalar Arguments ..
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
@@ -29,413 +580,7 @@
$ ERR_BNDS_COMP( NRHS, * )
* ..
*
-* Purpose
-* =======
-*
-* 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.
-*
-* Description
-* ===========
-*
-* 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.
-*
-* 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.
-*
-* 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, 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.
-*
-* 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)
-*
-* 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 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).
-*
-* 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 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).
-*
-* 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 = 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.
-*
-* 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.
-* 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/output) 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.
-*
-* 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, 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
-* 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.
-*
-* RPVGRW (output) 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 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