dgesvxx()

subroutine dgesvxx	(	character	fact,
		character	trans,
		integer	n,
		integer	nrhs,
		double precision, dimension( lda, * )	a,
		integer	lda,
		double precision, dimension( ldaf, * )	af,
		integer	ldaf,
		integer, dimension( * )	ipiv,
		character	equed,
		double precision, dimension( * )	r,
		double precision, dimension( * )	c,
		double precision, dimension( ldb, * )	b,
		integer	ldb,
		double precision, dimension( ldx , * )	x,
		integer	ldx,
		double precision	rcond,
		double precision	rpvgrw,
		double precision, dimension( * )	berr,
		integer	n_err_bnds,
		double precision, dimension( nrhs, * )	err_bnds_norm,
		double precision, dimension( nrhs, * )	err_bnds_comp,
		integer	nparams,
		double precision, dimension( * )	params,
		double precision, dimension( * )	work,
		integer, dimension( * )	iwork,
		integer	info )

DGESVXX computes the solution to system of linear equations A * X = B for GE matrices

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Purpose:

!>
!>    DGESVXX 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. DGESVXX 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.
!>
!>    DGESVXX 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
!>    DGESVXX 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 DGESVXX 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.
!> 

!>     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.
!> 

Parameters

[in]	FACT	!> 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. !>
[in]	TRANS	!> 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) !>
[in]	N	!> N is INTEGER !> The number of linear equations, i.e., the order of the !> matrix A. N >= 0. !>
[in]	NRHS	!> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !>
[in,out]	A	!> A is DOUBLE PRECISION array, dimension (LDA,N) !> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is !> not 'N', then A must have been equilibrated by the scaling !> factors in R and/or C. A 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). !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !>
[in,out]	AF	!> AF is DOUBLE PRECISION array, dimension (LDAF,N) !> If FACT = 'F', then AF is an input argument and on entry !> contains the factors L and U from the factorization !> A = P*L*U as computed by DGETRF. If EQUED .ne. 'N', then !> AF 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). !>
[in]	LDAF	!> LDAF is INTEGER !> The leading dimension of the array AF. LDAF >= max(1,N). !>
[in,out]	IPIV	!> 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. !>
[in,out]	EQUED	!> 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. !>
[in,out]	R	!> 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. !>
[in,out]	C	!> 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. !>
[in,out]	B	!> 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. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !>
[out]	X	!> 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'. !>
[in]	LDX	!> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !>
[out]	RCOND	!> 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. !>
[out]	RPVGRW	!> RPVGRW is DOUBLE PRECISION !> Reciprocal pivot growth. On exit, this contains the reciprocal !> pivot growth factor norm(A)/norm(U). The !> 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<INFO<=N, then this contains the reciprocal pivot growth factor !> for the leading INFO columns of A. In DGESVX, this quantity is !> returned in WORK(1). !>
[out]	BERR	!> 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). !>
[in]	N_ERR_BNDS	!> 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. !>
[out]	ERR_BNDS_NORM	!> 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 boolean. Trust the answer if the !> reciprocal condition number is less than the threshold !> sqrt(n) * dlamch('Epsilon'). !> !> err = 2 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 . 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. !>
[out]	ERR_BNDS_COMP	!> 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 boolean. Trust the answer if the !> reciprocal condition number is less than the threshold !> sqrt(n) * dlamch('Epsilon'). !> !> err = 2 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 . 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. !>
[in]	NPARAMS	!> NPARAMS is INTEGER !> Specifies the number of parameters set in PARAMS. If <= 0, the !> PARAMS array is never referenced and default values are used. !>
[in,out]	PARAMS	!> 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) !>
[out]	WORK	!> WORK is DOUBLE PRECISION array, dimension (4*N) !>
[out]	IWORK	!> IWORK is INTEGER array, dimension (N) !>
[out]	INFO	!> 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. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.