CGBRFSX improves the computed solution to a system of linear


    equations 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, R


    and C 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.

TRANS

          TRANS is CHARACTER*1


     Specifies the form of the system of equations:


       = 'N':  A * X = B     (No transpose)


       = 'T':  A**T * X = B  (Transpose)


       = 'C':  A**H * X = B  (Conjugate transpose)

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


       = 'R':  Row equilibration, i.e., A has been premultiplied by


               diag(R).


       = 'C':  Column equilibration, i.e., A has been postmultiplied


               by diag(C).


       = 'B':  Both row and column equilibration, i.e., A has been


               replaced by diag(R) * A * diag(C).


               The right hand side B has been changed accordingly.

N

          N is INTEGER


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

KL

          KL is INTEGER


     The number of subdiagonals within the band of A.  KL >= 0.

KU

          KU is INTEGER


     The number of superdiagonals within the band of A.  KU >= 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.

AB

          AB is COMPLEX array, dimension (LDAB,N)


     The original band matrix A, stored in rows 1 to KL+KU+1.


     The j-th column of A is stored in the j-th column of the


     array AB as follows:


     AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

LDAB

          LDAB is INTEGER


     The leading dimension of the array AB.  LDAB >= KL+KU+1.

AFB

          AFB is COMPLEX array, dimension (LDAFB,N)


     Details of the LU factorization of the band matrix A, as


     computed by CGBTRF.  U is stored as an upper triangular band


     matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and


     the multipliers used during the factorization are stored in


     rows KL+KU+2 to 2*KL+KU+1.

LDAFB

          LDAFB is INTEGER


     The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.

IPIV

          IPIV is INTEGER array, dimension (N)


     The pivot indices from CGETRF; for 1<=i<=N, row i of the


     matrix was interchanged with row IPIV(i).

R

          R is REAL array, dimension (N)


     The row scale factors for A.  If EQUED = 'R' or 'B', A is


     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R


     is not accessed.  R is an input argument if FACT = 'F';


     otherwise, R is an output argument.  If FACT = 'F' and


     EQUED = 'R' or 'B', each element of R must be positive.


     If R is output, each element of R is a power of the radix.


     If R is input, each element of R should be a power of the radix


     to ensure a reliable solution and error estimates. Scaling by


     powers of the radix does not cause rounding errors unless the


     result underflows or overflows. Rounding errors during scaling


     lead to refining with a matrix that is not equivalent to the


     input matrix, producing error estimates that may not be


     reliable.

C

          C is REAL array, dimension (N)


     The column scale factors for A.  If EQUED = 'C' or 'B', A is


     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C


     is not accessed.  C is an input argument if FACT = 'F';


     otherwise, C is an output argument.  If FACT = 'F' and


     EQUED = 'C' or 'B', each element of C must be positive.


     If C is output, each element of C is a power of the radix.


     If C is input, each element of C should be a power of the radix


     to ensure a reliable solution and error estimates. Scaling by


     powers of the radix does not cause rounding errors unless the


     result underflows or overflows. Rounding errors during scaling


     lead to refining with a matrix that is not equivalent to the


     input matrix, producing error estimates that may not be


     reliable.

B

          B is COMPLEX 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 COMPLEX array, dimension (LDX,NRHS)


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


     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 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.

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 '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.


     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 '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.


     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.

    DGBRFSX improves the computed solution to a system of linear


    equations 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, R


    and C 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.

TRANS

          TRANS is CHARACTER*1


     Specifies the form of the system of equations:


       = 'N':  A * X = B     (No transpose)


       = 'T':  A**T * X = B  (Transpose)


       = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

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


       = 'R':  Row equilibration, i.e., A has been premultiplied by


               diag(R).


       = 'C':  Column equilibration, i.e., A has been postmultiplied


               by diag(C).


       = 'B':  Both row and column equilibration, i.e., A has been


               replaced by diag(R) * A * diag(C).


               The right hand side B has been changed accordingly.

N

          N is INTEGER


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

KL

          KL is INTEGER


     The number of subdiagonals within the band of A.  KL >= 0.

KU

          KU is INTEGER


     The number of superdiagonals within the band of A.  KU >= 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.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)


     The original band matrix A, stored in rows 1 to KL+KU+1.


     The j-th column of A is stored in the j-th column of the


     array AB as follows:


     AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

LDAB

          LDAB is INTEGER


     The leading dimension of the array AB.  LDAB >= KL+KU+1.

AFB

          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)


     Details of the LU factorization of the band matrix A, as


     computed by DGBTRF.  U is stored as an upper triangular band


     matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and


     the multipliers used during the factorization are stored in


     rows KL+KU+2 to 2*KL+KU+1.

LDAFB

          LDAFB is INTEGER


     The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.

IPIV

          IPIV is INTEGER array, dimension (N)


     The pivot indices from DGETRF; for 1<=i<=N, row i of the


     matrix was interchanged with row IPIV(i).

R

          R is DOUBLE PRECISION array, dimension (N)


     The row scale factors for A.  If EQUED = 'R' or 'B', A is


     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R


     is not accessed.  R is an input argument if FACT = 'F';


     otherwise, R is an output argument.  If FACT = 'F' and


     EQUED = 'R' or 'B', each element of R must be positive.


     If R is output, each element of R is a power of the radix.


     If R is input, each element of R should be a power of the radix


     to ensure a reliable solution and error estimates. Scaling by


     powers of the radix does not cause rounding errors unless the


     result underflows or overflows. Rounding errors during scaling


     lead to refining with a matrix that is not equivalent to the


     input matrix, producing error estimates that may not be


     reliable.

C

          C is DOUBLE PRECISION array, dimension (N)


     The column scale factors for A.  If EQUED = 'C' or 'B', A is


     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C


     is not accessed.  C is an input argument if FACT = 'F';


     otherwise, C is an output argument.  If FACT = 'F' and


     EQUED = 'C' or 'B', each element of C must be positive.


     If C is output, each element of C is a power of the radix.


     If C is input, each element of C should be a power of the radix


     to ensure a reliable solution and error estimates. Scaling by


     powers of the radix does not cause rounding errors unless the


     result underflows or overflows. Rounding errors during scaling


     lead to refining with a matrix that is not equivalent to the


     input matrix, producing error estimates that may not be


     reliable.

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 < 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 <= 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 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

    SGBRFSX improves the computed solution to a system of linear


    equations 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, R


    and C 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.

TRANS

          TRANS is CHARACTER*1


     Specifies the form of the system of equations:


       = 'N':  A * X = B     (No transpose)


       = 'T':  A**T * X = B  (Transpose)


       = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

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


       = 'R':  Row equilibration, i.e., A has been premultiplied by


               diag(R).


       = 'C':  Column equilibration, i.e., A has been postmultiplied


               by diag(C).


       = 'B':  Both row and column equilibration, i.e., A has been


               replaced by diag(R) * A * diag(C).


               The right hand side B has been changed accordingly.

N

          N is INTEGER


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

KL

          KL is INTEGER


     The number of subdiagonals within the band of A.  KL >= 0.

KU

          KU is INTEGER


     The number of superdiagonals within the band of A.  KU >= 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.

AB

          AB is REAL array, dimension (LDAB,N)


     The original band matrix A, stored in rows 1 to KL+KU+1.


     The j-th column of A is stored in the j-th column of the


     array AB as follows:


     AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

LDAB

          LDAB is INTEGER


     The leading dimension of the array AB.  LDAB >= KL+KU+1.

AFB

          AFB is REAL array, dimension (LDAFB,N)


     Details of the LU factorization of the band matrix A, as


     computed by SGBTRF.  U is stored as an upper triangular band


     matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and


     the multipliers used during the factorization are stored in


     rows KL+KU+2 to 2*KL+KU+1.

LDAFB

          LDAFB is INTEGER


     The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.

IPIV

          IPIV is INTEGER array, dimension (N)


     The pivot indices from SGETRF; for 1<=i<=N, row i of the


     matrix was interchanged with row IPIV(i).

R

          R is REAL array, dimension (N)


     The row scale factors for A.  If EQUED = 'R' or 'B', A is


     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R


     is not accessed.  R is an input argument if FACT = 'F';


     otherwise, R is an output argument.  If FACT = 'F' and


     EQUED = 'R' or 'B', each element of R must be positive.


     If R is output, each element of R is a power of the radix.


     If R is input, each element of R should be a power of the radix


     to ensure a reliable solution and error estimates. Scaling by


     powers of the radix does not cause rounding errors unless the


     result underflows or overflows. Rounding errors during scaling


     lead to refining with a matrix that is not equivalent to the


     input matrix, producing error estimates that may not be


     reliable.

C

          C is REAL array, dimension (N)


     The column scale factors for A.  If EQUED = 'C' or 'B', A is


     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C


     is not accessed.  C is an input argument if FACT = 'F';


     otherwise, C is an output argument.  If FACT = 'F' and


     EQUED = 'C' or 'B', each element of C must be positive.


     If C is output, each element of C is a power of the radix.


     If C is input, each element of C should be a power of the radix


     to ensure a reliable solution and error estimates. Scaling by


     powers of the radix does not cause rounding errors unless the


     result underflows or overflows. Rounding errors during scaling


     lead to refining with a matrix that is not equivalent to the


     input matrix, producing error estimates that may not be


     reliable.

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).

X

          X is REAL array, dimension (LDX,NRHS)


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


     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 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.

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 '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.


     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 '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.


     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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

    ZGBRFSX improves the computed solution to a system of linear


    equations 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, R


    and C 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.

TRANS

          TRANS is CHARACTER*1


     Specifies the form of the system of equations:


       = 'N':  A * X = B     (No transpose)


       = 'T':  A**T * X = B  (Transpose)


       = 'C':  A**H * X = B  (Conjugate transpose)

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


       = 'R':  Row equilibration, i.e., A has been premultiplied by


               diag(R).


       = 'C':  Column equilibration, i.e., A has been postmultiplied


               by diag(C).


       = 'B':  Both row and column equilibration, i.e., A has been


               replaced by diag(R) * A * diag(C).


               The right hand side B has been changed accordingly.

N

          N is INTEGER


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

KL

          KL is INTEGER


     The number of subdiagonals within the band of A.  KL >= 0.

KU

          KU is INTEGER


     The number of superdiagonals within the band of A.  KU >= 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.

AB

          AB is COMPLEX*16 array, dimension (LDAB,N)


     The original band matrix A, stored in rows 1 to KL+KU+1.


     The j-th column of A is stored in the j-th column of the


     array AB as follows:


     AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

LDAB

          LDAB is INTEGER


     The leading dimension of the array AB.  LDAB >= KL+KU+1.

AFB

          AFB is COMPLEX*16 array, dimension (LDAFB,N)


     Details of the LU factorization of the band matrix A, as


     computed by ZGBTRF.  U is stored as an upper triangular band


     matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and


     the multipliers used during the factorization are stored in


     rows KL+KU+2 to 2*KL+KU+1.

LDAFB

          LDAFB is INTEGER


     The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.

IPIV

          IPIV is INTEGER array, dimension (N)


     The pivot indices from ZGETRF; for 1<=i<=N, row i of the


     matrix was interchanged with row IPIV(i).

R

          R is DOUBLE PRECISION array, dimension (N)


     The row scale factors for A.  If EQUED = 'R' or 'B', A is


     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R


     is not accessed.  R is an input argument if FACT = 'F';


     otherwise, R is an output argument.  If FACT = 'F' and


     EQUED = 'R' or 'B', each element of R must be positive.


     If R is output, each element of R is a power of the radix.


     If R is input, each element of R should be a power of the radix


     to ensure a reliable solution and error estimates. Scaling by


     powers of the radix does not cause rounding errors unless the


     result underflows or overflows. Rounding errors during scaling


     lead to refining with a matrix that is not equivalent to the


     input matrix, producing error estimates that may not be


     reliable.

C

          C is DOUBLE PRECISION array, dimension (N)


     The column scale factors for A.  If EQUED = 'C' or 'B', A is


     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C


     is not accessed.  C is an input argument if FACT = 'F';


     otherwise, C is an output argument.  If FACT = 'F' and


     EQUED = 'C' or 'B', each element of C must be positive.


     If C is output, each element of C is a power of the radix.


     If C is input, each element of C should be a power of the radix


     to ensure a reliable solution and error estimates. Scaling by


     powers of the radix does not cause rounding errors unless the


     result underflows or overflows. Rounding errors during scaling


     lead to refining with a matrix that is not equivalent to the


     input matrix, producing error estimates that may not be


     reliable.

B

          B is COMPLEX*16 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 COMPLEX*16 array, dimension (LDX,NRHS)


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


     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 < 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 <= 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 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*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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.