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NAME ZGBRFS - improve the computed solution to a system of linear equations when the coefficient matrix is banded, and pro- vides error bounds and backward error estimates for the solution SYNOPSIS SUBROUTINE ZGBRFS( 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( * ) DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), WORK( * ), X( LDX, * ) PURPOSE ZGBRFS improves the computed solution to a system of linear equations when the coefficient matrix is banded, and pro- vides error bounds and backward error estimates for the solution. ARGUMENTS 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) 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*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 (input) INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1. AFB (input) 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 (input) INTEGER The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. IPIV (input) INTEGER array, dimension (N) The pivot indices from ZGBTRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i). B (input) COMPLEX*16 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*16 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by ZGBTRS. On exit, the improved solution matrix X. LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,N). FERR (output) DOUBLE PRECISION array, dimension (NRHS) The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution, FERR(j) bounds the magnitude of the largest entry in (X(j) - XTRUE) divided by the magnitude of the largest entry in X(j). The quality of the error bound depends on the quality of the estimate of norm(inv(A)) computed in the code; if the estimate of norm(inv(A)) is accu- rate, the error bound is guaranteed. BERR (output) DOUBLE PRECISION array, dimension (NRHS) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any entry of A or B that makes X(j) an exact solution). WORK (workspace) COMPLEX*16 array, dimension (2*N) RWORK (workspace) DOUBLE PRECISION array, dimension (N) INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value PARAMETERS ITMAX is the maximum number of steps of iterative refine- ment.