zcgesv zgbsv zgbsvx zgesv zgesvx zgtsv zgtsvx zspsv zspsvx zsysv zsysvx

zgtsvx() void zgtsvx ( char  fact, char  trans, int  n, int  nrhs, doublecomplex  dl[], doublecomplex  d[], doublecomplex  du[], doublecomplex  dlf[], doublecomplex  df[], doublecomplex  duf[], doublecomplex  du2[], int  ipiv[], int  ldb, doublecomplex  b[], int  ldx, doublecomplex  x[], double *  rcond, double  ferr[], double  berr[], doublecomplex  work[], double  rwork[], int *  info  ) (Expert driver) Solution to system of linear equations AX = B for a complex tridiagonal matrix) Purpose This routine uses the LU factorization to computes the solution to a complex system of linear equations A * X = B, A^T * X = B or A^H * X = B where A is a tridiagonal matrix of order n and X and B are n x nrhs matrices. Error bounds on the solution and a condition estimate are also provided. Description The following steps are performed: If fact = 'N', the LU decomposition is used to factor the matrix A as A = L * U where L is a product of permutation and unit lower bidiagonal matrices and U is upper triangular with nonzeros in only the main diagonal and first two super-diagonals. If i-th diagonal element of U = 0, so that U is exactly singular, then the routine returns with info = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, info = n+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below. The system of equations is solved for X using the factored form of A. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. Parameters [in] fact Specifies whether or not the factored form of A has been supplied on entry. = 'F': dlf[], df[], duf[], du2[] and ipiv[] contain the factored form of A; dlf[], df[], duf[], du2[] and ipiv[] will not be modified. = 'N': The matrix will be copied to dlf[], df[] and duf[] and factored. [in] trans 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) [in] n Order of the matrix A. (n >= 0) (If n = 0, returns without computation) [in] nrhs Number of right hand sides, i.e., number of columns of the matrix B. (nrhs >= 0) (if nrhs = 0, returns without computation) [in] dl[] Array dl[ldl] (ldl >= n - 1) n-1 sub-diagonal elements of A. [in] d[] Array d[ld] (ld >= n) n diagonal elements of A. [in] du[] Array du[ldu] (ldu >= n - 1) n-1 super-diagonal elements of A. [in,out] dlf[] Array dlf[ldlf] (ldlf >= n - 1) [in] If fact = 'F', (n-1) multipliers that define the matrix L from the LU factorization of A as computed by zgttrf, are to be stored. [out] If fact = 'N', (n-1) multipliers that define the matrix L from the LU factorization of A, are returned. [in,out] df[] Array df[ldf] (ldf >= n) [in] If fact = 'F', diagonal elements of the upper triangular matrix U from the LU factorization of A, are to be stored. [out] If fact = 'N', diagonal elements of the upper triangular matrix U from the LU factorization of A, are returned. [in,out] duf[] Array duf[lduf] (lduf >= n - 1) [in] If fact = 'F', (n-1) elements of the first super-diagonal of U, are to be stored. [out] If fact = 'N', (n-1) elements of the first super-diagonal of U, are returned. [in,out] du2[] Array du2[ldu2] (ldu2 >= n - 2) [in] If fact = 'F', (n-2) elements of the second super-diagonal of U, are to be stored. [out] If fact = 'N', (n-2) elements of the second super-diagonal of U, are returned. [in,out] ipiv[] Array ipiv[lipiv] (lipiv >= n) [in] If fact = 'F', the pivot indices from the LU factorization of A as computed by zgttrf, are to be stored. [out] If fact = 'N', the pivot indices from the LU factorization of A are returned; row i of the matrix was interchanged with row ipiv[i-1]. ipiv[i-1] will always be either i or i+1; ipiv[i-1] = i indicates a row interchange was not required. [in] ldb Leading dimension of the two dimensional array b[][]. (ldb >= max(1, n)) [in] b[][] Array b[lb][ldb] (lb >= nrhs) n x nrhs right hand side matrix B. [in] ldx Leading dimension of the two dimensional array x[][]. (ldx >= max(1, n)) [out] x[][] Array x[lx][ldx] (lx >= nrhs) If info = 0 or info = n+1, n x nrhs solution matrix X. [out] rcond The estimate of the reciprocal condition number of the matrix A. If rcond is less than the machine precision (in particular, if rcond = 0), the matrix is singular to working precision. This condition is indicated by a return code of info > 0. [out] ferr[] Array ferr[lferr] (lferr >= 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-1] 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 estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error. [out] berr[] Array berr[lberr] (lberr >= 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 solution). [out] work[] Array work[lwork] (lwork >= 2*n) Work array. [out] rwork[] Array rwork[lrwork] (lrwork >= n) Work array. [out] info = 0: Successful exit = -1: The argument fact had an illegal value (fact != 'F' nor 'N') = -2: The argument trans had an illegal value (trans != 'N', 'T' nor 'C') = -3: The argument n had an illegal value (n < 0) = -4: The argument nrhs had an illegal value (nrhs < 0) = -13: The argument ldb had an illegal value (ldb < max(1, n)) = -15: The argument ldx had an illegal value (ldx < max(1, n)) = i (0 < i <= n): The i-th diagonal element of the factor U is exactly zero. The factorization has not been completed unless i = n, but the factor U is exactly singular, so the solution and error bounds could not be computed. rcond = 0 is returned. = n+1: U is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest. Reference LAPACK