--- rpl/lapack/lapack/zposvxx.f 2010/12/21 13:53:53 1.4
+++ rpl/lapack/lapack/zposvxx.f 2011/11/21 20:43:19 1.5
@@ -1,18 +1,503 @@
+*> \brief ZPOSVXX computes the solution to system of linear equations A * X = B for PO matrices
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download ZPOSVXX + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE ZPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
+* S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
+* N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
+* NPARAMS, PARAMS, WORK, RWORK, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER EQUED, FACT, UPLO
+* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
+* $ N_ERR_BNDS
+* DOUBLE PRECISION RCOND, RPVGRW
+* ..
+* .. Array Arguments ..
+* COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
+* $ WORK( * ), X( LDX, * )
+* DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
+* $ ERR_BNDS_NORM( NRHS, * ),
+* $ ERR_BNDS_COMP( NRHS, * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
+*> to compute the solution to a complex*16 system of linear equations
+*> A * X = B, where A is an N-by-N symmetric positive definite matrix
+*> and X and B are N-by-NRHS matrices.
+*>
+*> If requested, both normwise and maximum componentwise error bounds
+*> are returned. ZPOSVXX 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.
+*>
+*> ZPOSVXX 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
+*> ZPOSVXX 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 ZPOSVXX 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:
+*>
+*> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*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(S)*A*diag(S) and B by diag(S)*B.
+*>
+*> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
+*> factor the matrix A (after equilibration if FACT = 'E') as
+*> A = U**T* U, if UPLO = 'U', or
+*> A = L * L**T, if UPLO = 'L',
+*> where U is an upper triangular matrix and L is a lower triangular
+*> matrix.
+*>
+*> 3. If the leading i-by-i principal minor is not positive definite,
+*> 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(S) 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 contains the factored form of A.
+*> If EQUED is not 'N', the matrix A has been
+*> equilibrated with scaling factors given by S.
+*> A and AF 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] UPLO
+*> \verbatim
+*> UPLO is CHARACTER*1
+*> = 'U': Upper triangle of A is stored;
+*> = 'L': Lower triangle of A is stored.
+*> \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] 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] A
+*> \verbatim
+*> A is COMPLEX*16 array, dimension (LDA,N)
+*> On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
+*> 'Y', then A must contain the equilibrated matrix
+*> diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
+*> triangular part of A contains the upper triangular part of the
+*> matrix A, and the strictly lower triangular part of A is not
+*> referenced. If UPLO = 'L', the leading N-by-N lower triangular
+*> part of A contains the lower triangular part of the matrix A, and
+*> the strictly upper triangular part of A is not referenced. A is
+*> not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
+*> 'N' on exit.
+*>
+*> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
+*> diag(S)*A*diag(S).
+*> \endverbatim
+*>
+*> \param[in] LDA
+*> \verbatim
+*> LDA is INTEGER
+*> The leading dimension of the array A. LDA >= max(1,N).
+*> \endverbatim
+*>
+*> \param[in,out] AF
+*> \verbatim
+*> AF is or output) COMPLEX*16 array, dimension (LDAF,N)
+*> If FACT = 'F', then AF is an input argument and on entry
+*> contains the triangular factor U or L from the Cholesky
+*> factorization A = U**T*U or A = L*L**T, in the same storage
+*> format as A. If EQUED .ne. 'N', then AF is the factored
+*> form of the equilibrated matrix diag(S)*A*diag(S).
+*>
+*> If FACT = 'N', then AF is an output argument and on exit
+*> returns the triangular factor U or L from the Cholesky
+*> factorization A = U**T*U or A = L*L**T of the original
+*> matrix A.
+*>
+*> If FACT = 'E', then AF is an output argument and on exit
+*> returns the triangular factor U or L from the Cholesky
+*> factorization A = U**T*U or A = L*L**T of the equilibrated
+*> matrix A (see the description of A for the form of the
+*> equilibrated matrix).
+*> \endverbatim
+*>
+*> \param[in] LDAF
+*> \verbatim
+*> LDAF is INTEGER
+*> The leading dimension of the array AF. LDAF >= max(1,N).
+*> \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').
+*> = 'Y': Both row and column equilibration, i.e., A has been
+*> replaced by diag(S) * A * diag(S).
+*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
+*> output argument.
+*> \endverbatim
+*>
+*> \param[in,out] S
+*> \verbatim
+*> S is or output) DOUBLE PRECISION array, dimension (N)
+*> The row scale factors for A. If EQUED = 'Y', A is multiplied on
+*> the left and right by diag(S). S is an input argument if FACT =
+*> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
+*> = 'Y', each element of S must be positive. If S is output, each
+*> element of S is a power of the radix. If S is input, each element
+*> of S 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 COMPLEX*16 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 EQUED = 'Y', B is overwritten by diag(S)*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 COMPLEX*16 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(S))*X.
+*> \endverbatim
+*>
+*> \param[in] LDX
+*> \verbatim
+*> LDX is INTEGER
+*> The leading dimension of the array X. LDX >= max(1,N).
+*> \endverbatim
+*>
+*> \param[out] RCOND
+*> \verbatim
+*> RCOND is DOUBLE PRECISION
+*> Reciprocal scaled condition number. This is an estimate of the
+*> reciprocal Skeel condition number of the matrix A after
+*> equilibration (if done). If this is less than the machine
+*> precision (in particular, if it is zero), the matrix is singular
+*> to working precision. Note that the error may still be small even
+*> if this number is very small and the matrix appears ill-
+*> conditioned.
+*> \endverbatim
+*>
+*> \param[out] 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.
+*> \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 COMPLEX*16 array, dimension (2*N)
+*> \endverbatim
+*>
+*> \param[out] RWORK
+*> \verbatim
+*> RWORK is DOUBLE PRECISION array, dimension (2*N)
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> INFO is INTEGER
+*> = 0: Successful exit. The solution to every right-hand side is
+*> guaranteed.
+*> < 0: If INFO = -i, the i-th argument had an illegal value
+*> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
+*> has been completed, but the factor U is exactly singular, so
+*> the solution and error bounds could not be computed. RCOND = 0
+*> is returned.
+*> = N+J: The solution corresponding to the Jth right-hand side is
+*> not guaranteed. The solutions corresponding to other right-
+*> hand sides K with K > J may not be guaranteed as well, but
+*> only the first such right-hand side is reported. If a small
+*> componentwise error is not requested (PARAMS(3) = 0.0) then
+*> the Jth right-hand side is the first with a normwise error
+*> bound that is not guaranteed (the smallest J such
+*> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
+*> the Jth right-hand side is the first with either a normwise or
+*> componentwise error bound that is not guaranteed (the smallest
+*> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
+*> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
+*> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
+*> about all of the right-hand sides check ERR_BNDS_NORM or
+*> ERR_BNDS_COMP.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date November 2011
+*
+*> \ingroup complex16POsolve
+*
+* =====================================================================
SUBROUTINE ZPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
$ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
$ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
$ NPARAMS, PARAMS, WORK, RWORK, 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, UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
@@ -27,361 +512,7 @@
$ ERR_BNDS_COMP( NRHS, * )
* ..
*
-* Purpose
-* =======
-*
-* ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
-* to compute the solution to a complex*16 system of linear equations
-* A * X = B, where A is an N-by-N symmetric positive definite matrix
-* and X and B are N-by-NRHS matrices.
-*
-* If requested, both normwise and maximum componentwise error bounds
-* are returned. ZPOSVXX 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.
-*
-* ZPOSVXX 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
-* ZPOSVXX 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 ZPOSVXX would itself produce.
-*
-* Description
-* ===========
-*
-* The following steps are performed:
-*
-* 1. If FACT = 'E', double precision scaling factors are computed to equilibrate
-* the system:
-*
-* diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*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(S)*A*diag(S) and B by diag(S)*B.
-*
-* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
-* factor the matrix A (after equilibration if FACT = 'E') as
-* A = U**T* U, if UPLO = 'U', or
-* A = L * L**T, if UPLO = 'L',
-* where U is an upper triangular matrix and L is a lower triangular
-* matrix.
-*
-* 3. If the leading i-by-i principal minor is not positive definite,
-* 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(S) 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 contains the factored form of A.
-* If EQUED is not 'N', the matrix A has been
-* equilibrated with scaling factors given by S.
-* A and AF 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.
-*
-* UPLO (input) CHARACTER*1
-* = 'U': Upper triangle of A is stored;
-* = 'L': Lower triangle of A is stored.
-*
-* N (input) INTEGER
-* The number of linear equations, i.e., the order of the
-* matrix A. N >= 0.
-*
-* NRHS (input) INTEGER
-* The number of right hand sides, i.e., the number of columns
-* of the matrices B and X. NRHS >= 0.
-*
-* A (input/output) COMPLEX*16 array, dimension (LDA,N)
-* On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
-* 'Y', then A must contain the equilibrated matrix
-* diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
-* triangular part of A contains the upper triangular part of the
-* matrix A, and the strictly lower triangular part of A is not
-* referenced. If UPLO = 'L', the leading N-by-N lower triangular
-* part of A contains the lower triangular part of the matrix A, and
-* the strictly upper triangular part of A is not referenced. A is
-* not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
-* 'N' on exit.
-*
-* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
-* diag(S)*A*diag(S).
-*
-* LDA (input) INTEGER
-* The leading dimension of the array A. LDA >= max(1,N).
-*
-* AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
-* If FACT = 'F', then AF is an input argument and on entry
-* contains the triangular factor U or L from the Cholesky
-* factorization A = U**T*U or A = L*L**T, in the same storage
-* format as A. If EQUED .ne. 'N', then AF is the factored
-* form of the equilibrated matrix diag(S)*A*diag(S).
-*
-* If FACT = 'N', then AF is an output argument and on exit
-* returns the triangular factor U or L from the Cholesky
-* factorization A = U**T*U or A = L*L**T of the original
-* matrix A.
-*
-* If FACT = 'E', then AF is an output argument and on exit
-* returns the triangular factor U or L from the Cholesky
-* factorization A = U**T*U or A = L*L**T of the equilibrated
-* matrix A (see the description of A for the form of the
-* equilibrated matrix).
-*
-* LDAF (input) INTEGER
-* The leading dimension of the array AF. LDAF >= max(1,N).
-*
-* EQUED (input or output) CHARACTER*1
-* Specifies the form of equilibration that was done.
-* = 'N': No equilibration (always true if FACT = 'N').
-* = 'Y': Both row and column equilibration, i.e., A has been
-* replaced by diag(S) * A * diag(S).
-* EQUED is an input argument if FACT = 'F'; otherwise, it is an
-* output argument.
-*
-* S (input or output) DOUBLE PRECISION array, dimension (N)
-* The row scale factors for A. If EQUED = 'Y', A is multiplied on
-* the left and right by diag(S). S is an input argument if FACT =
-* 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
-* = 'Y', each element of S must be positive. If S is output, each
-* element of S is a power of the radix. If S is input, each element
-* of S 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) COMPLEX*16 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 EQUED = 'Y', B is overwritten by diag(S)*B;
-*
-* LDB (input) INTEGER
-* The leading dimension of the array B. LDB >= max(1,N).
-*
-* X (output) COMPLEX*16 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(S))*X.
-*
-* LDX (input) INTEGER
-* The leading dimension of the array X. LDX >= max(1,N).
-*
-* RCOND (output) DOUBLE PRECISION
-* Reciprocal scaled condition number. This is an estimate of the
-* reciprocal Skeel condition number of the matrix A after
-* equilibration (if done). If this is less than the machine
-* precision (in particular, if it is zero), the matrix is singular
-* to working precision. Note that the error may still be small even
-* if this number is very small and the matrix appears ill-
-* conditioned.
-*
-* 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