SLA_GERFSX_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 SGERFSX to perform iterative refinement.


 In addition to normwise error bound, the code provides maximum


 componentwise error bound if possible. See comments for ERRS_N


 and ERRS_C for details of the error bounds. Note that this


 subroutine is only resonsible for setting the second fields of


 ERRS_N and ERRS_C.

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

TRANS_TYPE

          TRANS_TYPE is INTEGER


     Specifies the transposition operation on A.


     The value is defined by ILATRANS(T) where T is a CHARACTER and


     T    = 'N':  No transpose


          = 'T':  Transpose


          = 'C':  Conjugate transpose

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 REAL 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 REAL array, dimension (LDAF,N)


     The factors L and U from the factorization


     A = P*L*U as computed by SGETRF.

LDAF

          LDAF is INTEGER


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

IPIV

          IPIV is INTEGER array, dimension (N)


     The pivot indices from the factorization A = P*L*U


     as computed by SGETRF; row i of the matrix was interchanged


     with row IPIV(i).

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 REAL 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 REAL 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 REAL array, dimension (LDY,NRHS)


     On entry, the solution matrix X, as computed by SGETRS.


     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 REAL 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 SLA_LIN_BERR.

N_NORMS

          N_NORMS is INTEGER


     Determines which error bounds to return (see ERRS_N


     and ERRS_C).


     If N_NORMS >= 1 return normwise error bounds.


     If N_NORMS >= 2 return componentwise error bounds.

ERRS_N

          ERRS_N 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 ERRS_N(i,:) corresponds to the ith


     right-hand side.


     The second index in ERRS_N(:,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.

ERRS_C

          ERRS_C 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


     ERRS_C is not accessed.  If N_ERR_BNDS .LT. 3, then at most


     the first (:,N_ERR_BNDS) entries are returned.


     The first index in ERRS_C(i,:) corresponds to the ith


     right-hand side.


     The second index in ERRS_C(:,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 REAL array, dimension (N)


     Workspace to hold the intermediate residual.

AYB

          AYB is REAL array, dimension (N)


     Workspace. This can be the same workspace passed for Y_TAIL.

DY

          DY is REAL array, dimension (N)


     Workspace to hold the intermediate solution.

Y_TAIL

          Y_TAIL is REAL array, dimension (N)


     Workspace to hold the trailing bits of the intermediate solution.

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.

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


     ERRS_N and ERRS_C may no longer be trustworthy.

RTHRESH

          RTHRESH is REAL


     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 REAL


     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 SGETRS had an illegal


             value

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December 2016