Provided by: liblapack-doc_3.8.0-2_all bug

NAME

       doublePOcomputational

SYNOPSIS

   Functions
       double precision function dla_porcond (UPLO, N, A, LDA, AF, LDAF, CMODE, C, INFO, WORK,
           IWORK)
           DLA_PORCOND estimates the Skeel condition number for a symmetric positive-definite
           matrix.
       subroutine dla_porfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, COLEQU, C, B,
           LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL,
           RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
           DLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for
           symmetric or Hermitian positive-definite matrices by performing extra-precise
           iterative refinement and provides error bounds and backward error estimates for the
           solution.
       double precision function dla_porpvgrw (UPLO, NCOLS, A, LDA, AF, LDAF, WORK)
           DLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a
           symmetric or Hermitian positive-definite matrix.
       subroutine dpocon (UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
           DPOCON
       subroutine dpoequ (N, A, LDA, S, SCOND, AMAX, INFO)
           DPOEQU
       subroutine dpoequb (N, A, LDA, S, SCOND, AMAX, INFO)
           DPOEQUB
       subroutine dporfs (UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK,
           IWORK, INFO)
           DPORFS
       subroutine dporfsx (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, LDB, X, LDX, RCOND,
           BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
           DPORFSX
       subroutine dpotf2 (UPLO, N, A, LDA, INFO)
           DPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite
           matrix (unblocked algorithm).
       subroutine dpotrf (UPLO, N, A, LDA, INFO)
           DPOTRF
       recursive subroutine dpotrf2 (UPLO, N, A, LDA, INFO)
           DPOTRF2
       subroutine dpotri (UPLO, N, A, LDA, INFO)
           DPOTRI
       subroutine dpotrs (UPLO, N, NRHS, A, LDA, B, LDB, INFO)
           DPOTRS

Detailed Description

       This is the group of double computational functions for PO matrices

Function Documentation

   double precision function dla_porcond (character UPLO, integer N, double precision, dimension(
       lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer
       CMODE, double precision, dimension( * ) C, integer INFO, double precision, dimension( * )
       WORK, integer, dimension( * ) IWORK)
       DLA_PORCOND estimates the Skeel condition number for a symmetric positive-definite matrix.

       Purpose:

               DLA_PORCOND Estimates the Skeel condition number of  op(A) * op2(C)
               where op2 is determined by CMODE as follows
               CMODE =  1    op2(C) = C
               CMODE =  0    op2(C) = I
               CMODE = -1    op2(C) = inv(C)
               The Skeel condition number  cond(A) = norminf( |inv(A)||A| )
               is computed by computing scaling factors R such that
               diag(R)*A*op2(C) is row equilibrated and computing the standard
               infinity-norm condition number.

       Parameters:
           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.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                On entry, the N-by-N matrix A.

           LDA

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

           AF

                     AF is DOUBLE PRECISION array, dimension (LDAF,N)
                The triangular factor U or L from the Cholesky factorization
                A = U**T*U or A = L*L**T, as computed by DPOTRF.

           LDAF

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

           CMODE

                     CMODE is INTEGER
                Determines op2(C) in the formula op(A) * op2(C) as follows:
                CMODE =  1    op2(C) = C
                CMODE =  0    op2(C) = I
                CMODE = -1    op2(C) = inv(C)

           C

                     C is DOUBLE PRECISION array, dimension (N)
                The vector C in the formula op(A) * op2(C).

           INFO

                     INFO is INTEGER
                  = 0:  Successful exit.
                i > 0:  The ith argument is invalid.

           WORK

                     WORK is DOUBLE PRECISION array, dimension (3*N).
                Workspace.

           IWORK

                     IWORK is INTEGER array, dimension (N).
                Workspace.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dla_porfsx_extended (integer PREC_TYPE, character UPLO, integer N, integer NRHS,
       double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, *
       ) AF, integer LDAF, logical COLEQU, double precision, dimension( * ) C, double precision,
       dimension( ldb, * ) B, integer LDB, double precision, dimension( ldy, * ) Y, integer LDY,
       double precision, dimension( * ) BERR_OUT, integer N_NORMS, double precision, dimension(
       nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, double
       precision, dimension( * ) RES, double precision, dimension(*) AYB, double precision,
       dimension( * ) DY, double precision, dimension( * ) Y_TAIL, double precision RCOND,
       integer ITHRESH, double precision RTHRESH, double precision DZ_UB, logical IGNORE_CWISE,
       integer INFO)
       DLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for
       symmetric or Hermitian positive-definite matrices by performing extra-precise iterative
       refinement and provides error bounds and backward error estimates for the solution.

       Purpose:

            DLA_PORFSX_EXTENDED improves the computed solution to a system of
            linear equations by performing extra-precise iterative refinement
            and provides error bounds and backward error estimates for the solution.
            This subroutine is called by DPORFSX to perform iterative refinement.
            In addition to normwise error bound, the code provides maximum
            componentwise error bound if possible. See comments for ERR_BNDS_NORM
            and ERR_BNDS_COMP for details of the error bounds. Note that this
            subroutine is only resonsible for setting the second fields of
            ERR_BNDS_NORM and ERR_BNDS_COMP.

       Parameters:
           PREC_TYPE

                     PREC_TYPE is INTEGER
                Specifies the intermediate precision to be used in refinement.
                The value is defined by ILAPREC(P) where P is a CHARACTER and
                P    = 'S':  Single
                     = 'D':  Double
                     = 'I':  Indigenous
                     = 'X', 'E':  Extra

           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
                matrix B.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                On entry, the N-by-N matrix A.

           LDA

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

           AF

                     AF is DOUBLE PRECISION array, dimension (LDAF,N)
                The triangular factor U or L from the Cholesky factorization
                A = U**T*U or A = L*L**T, as computed by DPOTRF.

           LDAF

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

           COLEQU

                     COLEQU is LOGICAL
                If .TRUE. then column equilibration was done to A before calling
                this routine. This is needed to compute the solution and error
                bounds correctly.

           C

                     C is DOUBLE PRECISION array, dimension (N)
                The column scale factors for A. If COLEQU = .FALSE., C
                is not accessed. If C is input, each element of C should be a power
                of the radix to ensure a reliable solution and error estimates.
                Scaling by powers of the radix does not cause rounding errors unless
                the result underflows or overflows. Rounding errors during scaling
                lead to refining with a matrix that is not equivalent to the
                input matrix, producing error estimates that may not be
                reliable.

           B

                     B is DOUBLE PRECISION array, dimension (LDB,NRHS)
                The right-hand-side matrix B.

           LDB

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

           Y

                     Y is DOUBLE PRECISION array, dimension (LDY,NRHS)
                On entry, the solution matrix X, as computed by DPOTRS.
                On exit, the improved solution matrix Y.

           LDY

                     LDY is INTEGER
                The leading dimension of the array Y.  LDY >= max(1,N).

           BERR_OUT

                     BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
                On exit, BERR_OUT(j) contains the componentwise relative backward
                error for right-hand-side j from the formula
                    max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
                where abs(Z) is the componentwise absolute value of the matrix
                or vector Z. This is computed by DLA_LIN_BERR.

           N_NORMS

                     N_NORMS is INTEGER
                Determines which error bounds to return (see ERR_BNDS_NORM
                and ERR_BNDS_COMP).
                If N_NORMS >= 1 return normwise error bounds.
                If N_NORMS >= 2 return componentwise error bounds.

           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 "Trust/don't trust" boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * slamch('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) * 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 "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.

                This subroutine is only responsible for setting the second field
                above.
                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 .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) * slamch('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) * 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 "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.

                This subroutine is only responsible for setting the second field
                above.
                See Lapack Working Note 165 for further details and extra
                cautions.

           RES

                     RES is DOUBLE PRECISION array, dimension (N)
                Workspace to hold the intermediate residual.

           AYB

                     AYB is DOUBLE PRECISION array, dimension (N)
                Workspace. This can be the same workspace passed for Y_TAIL.

           DY

                     DY is DOUBLE PRECISION array, dimension (N)
                Workspace to hold the intermediate solution.

           Y_TAIL

                     Y_TAIL is DOUBLE PRECISION array, dimension (N)
                Workspace to hold the trailing bits of the intermediate solution.

           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.

           ITHRESH

                     ITHRESH is INTEGER
                The maximum number of residual computations allowed for
                refinement. The default is 10. For '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.

           RTHRESH

                     RTHRESH is DOUBLE PRECISION
                Determines when to stop refinement if the error estimate stops
                decreasing. Refinement will stop when the next solution no longer
                satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
                the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
                default value is 0.5. For 'aggressive' set to 0.9 to permit
                convergence on extremely ill-conditioned matrices. See LAWN 165
                for more details.

           DZ_UB

                     DZ_UB is DOUBLE PRECISION
                Determines when to start considering componentwise convergence.
                Componentwise convergence is only considered after each component
                of the solution Y is stable, which we definte as the relative
                change in each component being less than DZ_UB. The default value
                is 0.25, requiring the first bit to be stable. See LAWN 165 for
                more details.

           IGNORE_CWISE

                     IGNORE_CWISE is LOGICAL
                If .TRUE. then ignore componentwise convergence. Default value
                is .FALSE..

           INFO

                     INFO is INTEGER
                  = 0:  Successful exit.
                  < 0:  if INFO = -i, the ith argument to DPOTRS had an illegal
                        value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           June 2017

   double precision function dla_porpvgrw (character*1 UPLO, integer NCOLS, double precision,
       dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer
       LDAF, double precision, dimension( * ) WORK)
       DLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric
       or Hermitian positive-definite matrix.

       Purpose:

            DLA_PORPVGRW computes the reciprocal pivot growth factor
            norm(A)/norm(U). The "max absolute element" norm is used. If this is
            much less than 1, 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.

       Parameters:
           UPLO

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

           NCOLS

                     NCOLS is INTEGER
                The number of columns of the matrix A. NCOLS >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                On entry, the N-by-N matrix A.

           LDA

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

           AF

                     AF is DOUBLE PRECISION array, dimension (LDAF,N)
                The triangular factor U or L from the Cholesky factorization
                A = U**T*U or A = L*L**T, as computed by DPOTRF.

           LDAF

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

           WORK

                     WORK is DOUBLE PRECISION array, dimension (2*N)

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dpocon (character UPLO, integer N, double precision, dimension( lda, * ) A, integer
       LDA, double precision ANORM, double precision RCOND, double precision, dimension( * )
       WORK, integer, dimension( * ) IWORK, integer INFO)
       DPOCON

       Purpose:

            DPOCON estimates the reciprocal of the condition number (in the
            1-norm) of a real symmetric positive definite matrix using the
            Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF.

            An estimate is obtained for norm(inv(A)), and the reciprocal of the
            condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

       Parameters:
           UPLO

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

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     The triangular factor U or L from the Cholesky factorization
                     A = U**T*U or A = L*L**T, as computed by DPOTRF.

           LDA

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

           ANORM

                     ANORM is DOUBLE PRECISION
                     The 1-norm (or infinity-norm) of the symmetric matrix A.

           RCOND

                     RCOND is DOUBLE PRECISION
                     The reciprocal of the condition number of the matrix A,
                     computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
                     estimate of the 1-norm of inv(A) computed in this routine.

           WORK

                     WORK is DOUBLE PRECISION array, dimension (3*N)

           IWORK

                     IWORK is INTEGER array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dpoequ (integer N, double precision, dimension( lda, * ) A, integer LDA, double
       precision, dimension( * ) S, double precision SCOND, double precision AMAX, integer INFO)
       DPOEQU

       Purpose:

            DPOEQU computes row and column scalings intended to equilibrate a
            symmetric positive definite matrix A and reduce its condition number
            (with respect to the two-norm).  S contains the scale factors,
            S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
            elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
            choice of S puts the condition number of B within a factor N of the
            smallest possible condition number over all possible diagonal
            scalings.

       Parameters:
           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     The N-by-N symmetric positive definite matrix whose scaling
                     factors are to be computed.  Only the diagonal elements of A
                     are referenced.

           LDA

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

           S

                     S is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, S contains the scale factors for A.

           SCOND

                     SCOND is DOUBLE PRECISION
                     If INFO = 0, S contains the ratio of the smallest S(i) to
                     the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
                     large nor too small, it is not worth scaling by S.

           AMAX

                     AMAX is DOUBLE PRECISION
                     Absolute value of largest matrix element.  If AMAX is very
                     close to overflow or very close to underflow, the matrix
                     should be scaled.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, the i-th diagonal element is nonpositive.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dpoequb (integer N, double precision, dimension( lda, * ) A, integer LDA, double
       precision, dimension( * ) S, double precision SCOND, double precision AMAX, integer INFO)
       DPOEQUB

       Purpose:

            DPOEQUB computes row and column scalings intended to equilibrate a
            symmetric positive definite matrix A and reduce its condition number
            (with respect to the two-norm).  S contains the scale factors,
            S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
            elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
            choice of S puts the condition number of B within a factor N of the
            smallest possible condition number over all possible diagonal
            scalings.

            This routine differs from DPOEQU by restricting the scaling factors
            to a power of the radix.  Barring over- and underflow, scaling by
            these factors introduces no additional rounding errors.  However, the
            scaled diagonal entries are no longer approximately 1 but lie
            between sqrt(radix) and 1/sqrt(radix).

       Parameters:
           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     The N-by-N symmetric positive definite matrix whose scaling
                     factors are to be computed.  Only the diagonal elements of A
                     are referenced.

           LDA

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

           S

                     S is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, S contains the scale factors for A.

           SCOND

                     SCOND is DOUBLE PRECISION
                     If INFO = 0, S contains the ratio of the smallest S(i) to
                     the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
                     large nor too small, it is not worth scaling by S.

           AMAX

                     AMAX is DOUBLE PRECISION
                     Absolute value of largest matrix element.  If AMAX is very
                     close to overflow or very close to underflow, the matrix
                     should be scaled.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, the i-th diagonal element is nonpositive.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dporfs (character UPLO, integer N, integer NRHS, double precision, dimension( lda,
       * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, double
       precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X,
       integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR,
       double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
       DPORFS

       Purpose:

            DPORFS improves the computed solution to a system of linear
            equations when the coefficient matrix is symmetric positive definite,
            and provides error bounds and backward error estimates for the
            solution.

       Parameters:
           UPLO

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

           N

                     N is INTEGER
                     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)
                     The symmetric matrix A.  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.

           LDA

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

           AF

                     AF is DOUBLE PRECISION array, dimension (LDAF,N)
                     The triangular factor U or L from the Cholesky factorization
                     A = U**T*U or A = L*L**T, as computed by DPOTRF.

           LDAF

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

           B

                     B is DOUBLE PRECISION array, dimension (LDB,NRHS)
                     The right hand side matrix 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)
                     On entry, the solution matrix X, as computed by DPOTRS.
                     On exit, the improved solution matrix X.

           LDX

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

           FERR

                     FERR is DOUBLE PRECISION array, dimension (NRHS)
                     The estimated forward error bound for each solution vector
                     X(j) (the j-th column of the solution matrix X).
                     If XTRUE is the true solution corresponding to X(j), FERR(j)
                     is an estimated upper bound for the magnitude of the largest
                     element in (X(j) - XTRUE) divided by the magnitude of the
                     largest element in X(j).  The estimate is as reliable as
                     the estimate for RCOND, and is almost always a slight
                     overestimate of the true error.

           BERR

                     BERR is DOUBLE PRECISION array, dimension (NRHS)
                     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).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (3*N)

           IWORK

                     IWORK is INTEGER array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Internal Parameters:

             ITMAX is the maximum number of steps of iterative refinement.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dporfsx (character UPLO, character EQUED, integer N, integer NRHS, double
       precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF,
       integer LDAF, double precision, dimension( * ) S, double precision, dimension( ldb, * ) B,
       integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision RCOND,
       double precision, dimension( * ) BERR, integer N_ERR_BNDS, double precision, dimension(
       nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, integer
       NPARAMS, double precision, dimension( * ) PARAMS, double precision, dimension( * ) WORK,
       integer, dimension( * ) IWORK, integer INFO)
       DPORFSX

       Purpose:

               DPORFSX improves the computed solution to a system of linear
               equations when the coefficient matrix is symmetric positive
               definite, and provides error bounds and backward error estimates
               for the solution.  In addition to normwise error bound, the code
               provides maximum componentwise error bound if possible.  See
               comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
               error bounds.

               The original system of linear equations may have been equilibrated
               before calling this routine, as described by arguments EQUED and S
               below. In this case, the solution and error bounds returned are
               for the original unequilibrated system.

                Some optional parameters are bundled in the PARAMS array.  These
                settings determine how refinement is performed, but often the
                defaults are acceptable.  If the defaults are acceptable, users
                can pass NPARAMS = 0 which prevents the source code from accessing
                the PARAMS argument.

       Parameters:
           UPLO

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

           EQUED

                     EQUED is CHARACTER*1
                Specifies the form of equilibration that was done to A
                before calling this routine. This is needed to compute
                the solution and error bounds correctly.
                  = 'N':  No equilibration
                  = 'Y':  Both row and column equilibration, i.e., A has been
                          replaced by diag(S) * A * diag(S).
                          The right hand side B has been changed accordingly.

           N

                     N is INTEGER
                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)
                The symmetric matrix A.  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.

           LDA

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

           AF

                     AF is DOUBLE PRECISION array, dimension (LDAF,N)
                The triangular factor U or L from the Cholesky factorization
                A = U**T*U or A = L*L**T, as computed by DPOTRF.

           LDAF

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

           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)
                The right hand side matrix 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)
                On entry, the solution matrix X, as computed by DGETRS.
                On exit, the improved solution matrix 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.

           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 "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.

           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 .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.

           NPARAMS

                     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.

           PARAMS

                     PARAMS is 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 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 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.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           April 2012

   subroutine dpotf2 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer
       LDA, integer INFO)
       DPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite
       matrix (unblocked algorithm).

       Purpose:

            DPOTF2 computes the Cholesky factorization of a real symmetric
            positive definite matrix A.

            The factorization has the form
               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 lower triangular.

            This is the unblocked version of the algorithm, calling Level 2 BLAS.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     symmetric matrix A is stored.
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the symmetric matrix A.  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.

                     On exit, if INFO = 0, the factor U or L from the Cholesky
                     factorization A = U**T *U  or A = L*L**T.

           LDA

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

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -k, the k-th argument had an illegal value
                     > 0: if INFO = k, the leading minor of order k is not
                          positive definite, and the factorization could not be
                          completed.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dpotrf (character UPLO, integer N, double precision, dimension( lda, * ) A, integer
       LDA, integer INFO)
       DPOTRF DPOTRF VARIANT: top-looking block version of the algorithm, calling Level 3 BLAS.

       Purpose:

            DPOTRF computes the Cholesky factorization of a real symmetric
            positive definite matrix A.

            The factorization has the form
               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 lower triangular.

            This is the block version of the algorithm, calling Level 3 BLAS.

       Parameters:
           UPLO

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

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the symmetric matrix A.  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.

                     On exit, if INFO = 0, the factor U or L from the Cholesky
                     factorization A = U**T*U or A = L*L**T.

           LDA

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

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, the leading minor of order i is not
                           positive definite, and the factorization could not be
                           completed.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

       Purpose:

        DPOTRF computes the Cholesky factorization of a real symmetric
        positive definite matrix A.

        The factorization has the form
           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 lower triangular.

        This is the top-looking block version of the algorithm, calling Level 3 BLAS.

       Parameters:
           UPLO

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

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the symmetric matrix A.  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.

                     On exit, if INFO = 0, the factor U or L from the Cholesky
                     factorization A = U**T*U or A = L*L**T.

           LDA

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

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, the leading minor of order i is not
                           positive definite, and the factorization could not be
                           completed.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   recursive subroutine dpotrf2 (character UPLO, integer N, double precision, dimension( lda, * )
       A, integer LDA, integer INFO)
       DPOTRF2

       Purpose:

            DPOTRF2 computes the Cholesky factorization of a real symmetric
            positive definite matrix A using the recursive algorithm.

            The factorization has the form
               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 lower triangular.

            This is the recursive version of the algorithm. It divides
            the matrix into four submatrices:

                   [  A11 | A12  ]  where A11 is n1 by n1 and A22 is n2 by n2
               A = [ -----|----- ]  with n1 = n/2
                   [  A21 | A22  ]       n2 = n-n1

            The subroutine calls itself to factor A11. Update and scale A21
            or A12, update A22 then calls itself to factor A22.

       Parameters:
           UPLO

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

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the symmetric matrix A.  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.

                     On exit, if INFO = 0, the factor U or L from the Cholesky
                     factorization A = U**T*U or A = L*L**T.

           LDA

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

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, the leading minor of order i is not
                           positive definite, and the factorization could not be
                           completed.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dpotri (character UPLO, integer N, double precision, dimension( lda, * ) A, integer
       LDA, integer INFO)
       DPOTRI

       Purpose:

            DPOTRI computes the inverse of a real symmetric positive definite
            matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
            computed by DPOTRF.

       Parameters:
           UPLO

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

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     On entry, the triangular factor U or L from the Cholesky
                     factorization A = U**T*U or A = L*L**T, as computed by
                     DPOTRF.
                     On exit, the upper or lower triangle of the (symmetric)
                     inverse of A, overwriting the input factor U or L.

           LDA

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

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, the (i,i) element of the factor U or L is
                           zero, and the inverse could not be computed.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

   subroutine dpotrs (character UPLO, integer N, integer NRHS, double precision, dimension( lda,
       * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)
       DPOTRS

       Purpose:

            DPOTRS solves a system of linear equations A*X = B with a symmetric
            positive definite matrix A using the Cholesky factorization
            A = U**T*U or A = L*L**T computed by DPOTRF.

       Parameters:
           UPLO

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

           N

                     N is INTEGER
                     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 matrix B.  NRHS >= 0.

           A

                     A is DOUBLE PRECISION array, dimension (LDA,N)
                     The triangular factor U or L from the Cholesky factorization
                     A = U**T*U or A = L*L**T, as computed by DPOTRF.

           LDA

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

           B

                     B is DOUBLE PRECISION array, dimension (LDB,NRHS)
                     On entry, the right hand side matrix B.
                     On exit, the solution matrix X.

           LDB

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

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           December 2016

Author

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