CGBRFSX(1) LAPACK routine (version 3.2) CGBRFSX(1)[top]NAMECGBRFSX - CGBRFSX improve the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solutionSYNOPSISSUBROUTINE CGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO ) IMPLICIT NONE CHARACTER TRANS, EQUED INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, NPARAMS, N_ERR_BNDS REAL RCOND INTEGER IPIV( * ) COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), X( LDX , * ),WORK( * ) REAL R( * ), C( * ), PARAMS( * ), BERR( * ), ERR_BNDS_NORM( NRHS, * ), ERR_BNDS_COMP( NRHS, * ), RWORK( * )PURPOSECGBRFSX 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_N and ERR_BNDS_C 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.ARGUMENTSSome optional parameters are bundled in the PARAMS array. These set‐ tings 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 (input) 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 (input) 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 post‐ multiplied by diag(C). = 'B': Both row and column equilibra‐ tion, i.e., A has been replaced by diag(R) * A * diag(C). The right hand side B has been changed accordingly. N (input) INTEGER The order of the matrix A. N >= 0. KL (input) INTEGER The number of subdiagonals within the band of A. KL >= 0. KU (input) INTEGER The number of superdiagonals within the band of A. KU >= 0. NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. AB (input) 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 (input) INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1. AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N) Details of the LU factorization of the band matrix A, as com‐ puted 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 (input) INTEGER The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. IPIV (input) 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 (input or output) REAL array, dimension (N) The row scale factors for A. If EQUED = 'R' or 'B', A is mul‐ tiplied 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 ele‐ ment 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 solu‐ tion and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or over‐ flows. 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 (input or output) 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 ele‐ ment 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 solu‐ tion and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or over‐ flows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. B (input) REAL array, dimension (LDB,NRHS) The right hand side matrix B. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). X (input/output) REAL array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by SGETRS. On exit, the improved solution matrix X. LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,N). RCOND (output) REAL Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equili‐ bration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to work‐ ing precision. Note that the error may still be small even if this number is very small and the matrix appears ill- condi‐ tioned. BERR (output) REAL array, dimension (NRHS) Componentwise relative backward error. This is the component‐ wise 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 (input) 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 (output) REAL array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains informa‐ tion 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 recipro‐ cal 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 abso‐ lute row sums of Z are approximately 1. See Lapack Working Note 165 for further details and extra cautions. ERR_BNDS_COMP (output) REAL array, dimension (NRHS, N_ERR_BNDS) For each right-hand side, this array contains informa‐ tion 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 informa‐ tion 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 thresh‐ old sqrt(n) * slamch('Epsilon') to determine if the error estimate is "guaranteed". These reciprocal condi‐ tion 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 fur‐ ther details and extra cautions. NPARAMS (input) INTE‐ GER Specifies the number of parameters set in PARAMS. If .LE. 0, the PARAMS array is never referenced and default values are used. PARAMS (input / output) REAL 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: 1.0 = 0.0 : No refinement is performed, and no error bounds are computed. = 1.0 : Use the double-precision refinement algo‐ rithm, possibly with doubled-single computations if the compi‐ lation 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 factor‐ ization 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 compo‐ nentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt compo‐ nentwise convergence) WORK (workspace) REAL array, dimension (4*N) IWORK (workspace) INTEGER array, dimension (N) INFO (output) INTEGER = 0: Successful exit. The solution to every right-hand side is guaranteed. < 0: If INFO =----------------------, the i-th argument had an ille‐ gal 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 compo‐ nentwise 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. LAPACK routine (version 3.2) November 2008 CGBRFSX(1)-i

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