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CGBRFS(l)) CGBRFS(l)NAMECGBRFS - improve the computed solution to a system of linear equations when the coeffi- cient matrix is banded, and provides error bounds and backward error estimates for the solutionSYNOPSISSUBROUTINE CGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO ) CHARACTER TRANS INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS INTEGER IPIV( * ) REAL BERR( * ), FERR( * ), RWORK( * ) COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( * ), X( LDX, * )PURPOSECGBRFS improves the computed solution to a system of linear equations when the coefficient matrix is banded, and provides error bounds and backward error estimates for the solution.ARGUMENTSTRANS (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) 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) 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 (input) INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1. AFB (input) 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 (input) INTEGER The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. IPIV (input) INTEGER array, dimension (N) The pivot indices from CGBTRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i). B (input) COMPLEX 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) COMPLEX array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by CGBTRS. On exit, the improved solution matrix X. LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,N). FERR (output) REAL 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 esti- mate is as reliable as the estimate for RCOND, and is almost always a slight over- estimate of the true error. BERR (output) REAL 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 solu- tion). WORK (workspace) COMPLEX array, dimension (2*N) RWORK (workspace) REAL array, dimension (N) INFO (output) INTEGER = 0: successful exit < 0: if INFO =, the i-th argument had an illegal value-iPARAMETERSITMAX is the maximum number of steps of iterative refinement.LAPACK version 3.015 June 2000 CGBRFS(l)

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