!>
!>    CPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
!>    to compute the solution to a complex system of linear equations
!>    A * X = B, where A is an N-by-N Hermitian positive definite matrix
!>    and X and B are N-by-NRHS matrices.
!>
!>    If requested, both normwise and maximum componentwise error bounds
!>    are returned. CPOSVXX 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.
!>
!>    CPOSVXX 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
!>    CPOSVXX 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 CPOSVXX would itself produce.
!> 

!>
!>    The following steps are performed:
!>
!>    1. If FACT = 'E', real 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 principal minor of order i is not positive,
!>    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.
!> 

!>     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

!>          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.
!> 

UPLO

!>          UPLO is CHARACTER*1
!>       = 'U':  Upper triangle of A is stored;
!>       = 'L':  Lower triangle of A is stored.
!> 

N

!>          N is INTEGER
!>     The number of linear equations, i.e., the order of the
!>     matrix A.  N >= 0.
!> 

NRHS

!>          NRHS is INTEGER
!>     The number of right hand sides, i.e., the number of columns
!>     of the matrices B and X.  NRHS >= 0.
!> 

A

!>          A is COMPLEX array, dimension (LDA,N)
!>     On entry, the Hermitian 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

!>          LDA is INTEGER
!>     The leading dimension of the array A.  LDA >= max(1,N).
!> 

AF

!>          AF is COMPLEX 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

!>          LDAF is INTEGER
!>     The leading dimension of the array AF.  LDAF >= max(1,N).
!> 

EQUED

!>          EQUED is 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

!>          S is REAL 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

!>          B is COMPLEX 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

!>          LDB is INTEGER
!>     The leading dimension of the array B.  LDB >= max(1,N).
!> 

X

!>          X is COMPLEX 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

!>          LDX is INTEGER
!>     The leading dimension of the array X.  LDX >= max(1,N).
!> 

RCOND

!>          RCOND is REAL
!>     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

!>          RPVGRW is REAL
!>     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.
!> 

BERR

!>          BERR is REAL 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).
!> 

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.
!> 

ERR_BNDS_NORM

!>          ERR_BNDS_NORM is REAL 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) * slamch('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) * slamch('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) * slamch('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.
!> 

ERR_BNDS_COMP

!>          ERR_BNDS_COMP is REAL 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) * slamch('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) * slamch('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) * slamch('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.
!> 

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.
!> 

PARAMS

!>          PARAMS is REAL 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: 1.0
!>            = 0.0:  No refinement is performed, and no error bounds are
!>                    computed.
!>            = 1.0:  Use the double-precision refinement algorithm,
!>                    possibly with doubled-single computations if the
!>                    compilation environment does not support DOUBLE
!>                    PRECISION.
!>              (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)
!> 

WORK

!>          WORK is COMPLEX array, dimension (2*N)
!> 

RWORK

!>          RWORK is REAL array, dimension (2*N)
!> 

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.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>    DPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
!>    to compute the solution to a double precision 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. DPOSVXX 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.
!>
!>    DPOSVXX 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
!>    DPOSVXX 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 DPOSVXX would itself produce.
!> 

!>
!>    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 principal minor of order i is not positive,
!>    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.
!> 

!>     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

!>          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.
!> 

UPLO

!>          UPLO is CHARACTER*1
!>       = 'U':  Upper triangle of A is stored;
!>       = 'L':  Lower triangle of A is stored.
!> 

N

!>          N is INTEGER
!>     The number of linear equations, i.e., the order of the
!>     matrix A.  N >= 0.
!> 

NRHS

!>          NRHS is INTEGER
!>     The number of right hand sides, i.e., the number of columns
!>     of the matrices B and X.  NRHS >= 0.
!> 

A

!>          A is DOUBLE PRECISION 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

!>          LDA is INTEGER
!>     The leading dimension of the array A.  LDA >= max(1,N).
!> 

AF

!>          AF is DOUBLE PRECISION 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

!>          LDAF is INTEGER
!>     The leading dimension of the array AF.  LDAF >= max(1,N).
!> 

EQUED

!>          EQUED is 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

!>          S is 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

!>          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 EQUED = 'Y', B is overwritten by diag(S)*B;
!> 

LDB

!>          LDB is INTEGER
!>     The leading dimension of the array B.  LDB >= max(1,N).
!> 

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(S))*X.
!> 

LDX

!>          LDX is INTEGER
!>     The leading dimension of the array X.  LDX >= max(1,N).
!> 

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.
!> 

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.
!> 

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).
!> 

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.
!> 

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.
!> 

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.
!> 

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.
!> 

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)
!> 

WORK

!>          WORK is DOUBLE PRECISION array, dimension (4*N)
!> 

IWORK

!>          IWORK is INTEGER array, dimension (N)
!> 

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.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>    SPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
!>    to compute the solution to a real 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. SPOSVXX 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.
!>
!>    SPOSVXX 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
!>    SPOSVXX 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 SPOSVXX would itself produce.
!> 

!>
!>    The following steps are performed:
!>
!>    1. If FACT = 'E', real 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 principal minor of order i is not positive,
!>    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.
!> 

!>     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

!>          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.
!> 

UPLO

!>          UPLO is CHARACTER*1
!>       = 'U':  Upper triangle of A is stored;
!>       = 'L':  Lower triangle of A is stored.
!> 

N

!>          N is INTEGER
!>     The number of linear equations, i.e., the order of the
!>     matrix A.  N >= 0.
!> 

NRHS

!>          NRHS is INTEGER
!>     The number of right hand sides, i.e., the number of columns
!>     of the matrices B and X.  NRHS >= 0.
!> 

A

!>          A is REAL 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

!>          LDA is INTEGER
!>     The leading dimension of the array A.  LDA >= max(1,N).
!> 

AF

!>          AF is REAL 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

!>          LDAF is INTEGER
!>     The leading dimension of the array AF.  LDAF >= max(1,N).
!> 

EQUED

!>          EQUED is 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

!>          S is REAL 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

!>          B is REAL 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

!>          LDB is INTEGER
!>     The leading dimension of the array B.  LDB >= max(1,N).
!> 

X

!>          X is REAL 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

!>          LDX is INTEGER
!>     The leading dimension of the array X.  LDX >= max(1,N).
!> 

RCOND

!>          RCOND is REAL
!>     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

!>          RPVGRW is REAL
!>     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.
!> 

BERR

!>          BERR is REAL 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).
!> 

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.
!> 

ERR_BNDS_NORM

!>          ERR_BNDS_NORM is REAL 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) * slamch('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) * slamch('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) * slamch('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.
!> 

ERR_BNDS_COMP

!>          ERR_BNDS_COMP is REAL 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) * slamch('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) * slamch('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) * slamch('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.
!> 

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.
!> 

PARAMS

!>          PARAMS is REAL 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: 1.0
!>            = 0.0:  No refinement is performed, and no error bounds are
!>                    computed.
!>            = 1.0:  Use the double-precision refinement algorithm,
!>                    possibly with doubled-single computations if the
!>                    compilation environment does not support DOUBLE
!>                    PRECISION.
!>              (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)
!> 

WORK

!>          WORK is REAL array, dimension (4*N)
!> 

IWORK

!>          IWORK is INTEGER array, dimension (N)
!> 

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.
!> 

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!>
!>    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 Hermitian 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.
!> 

!>
!>    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 principal minor of order i is not positive,
!>    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.
!> 

!>     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

!>          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.
!> 

UPLO

!>          UPLO is CHARACTER*1
!>       = 'U':  Upper triangle of A is stored;
!>       = 'L':  Lower triangle of A is stored.
!> 

N

!>          N is INTEGER
!>     The number of linear equations, i.e., the order of the
!>     matrix A.  N >= 0.
!> 

NRHS

!>          NRHS is INTEGER
!>     The number of right hand sides, i.e., the number of columns
!>     of the matrices B and X.  NRHS >= 0.
!> 

A

!>          A is COMPLEX*16 array, dimension (LDA,N)
!>     On entry, the Hermitian 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

!>          LDA is INTEGER
!>     The leading dimension of the array A.  LDA >= max(1,N).
!> 

AF

!>          AF is 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

!>          LDAF is INTEGER
!>     The leading dimension of the array AF.  LDAF >= max(1,N).
!> 

EQUED

!>          EQUED is 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

!>          S is 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

!>          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;
!> 

LDB

!>          LDB is INTEGER
!>     The leading dimension of the array B.  LDB >= max(1,N).
!> 

X

!>          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.
!> 

LDX

!>          LDX is INTEGER
!>     The leading dimension of the array X.  LDX >= max(1,N).
!> 

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.
!> 

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.
!> 

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).
!> 

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.
!> 

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.
!> 

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.
!> 

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.
!> 

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)
!> 

WORK

!>          WORK is COMPLEX*16 array, dimension (2*N)
!> 

RWORK

!>          RWORK is DOUBLE PRECISION array, dimension (2*N)
!> 

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.
!> 

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NAG Ltd.