zpbsv zpbsvx zptsv zptsvx

zptsvx() void zptsvx ( char  fact, int  n, int  nrhs, double  d[], doublecomplex  e[], double  df[], doublecomplex  ef[], 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 Hermitian positive definite tridiagonal matrix Purpose This routine uses the factorization A = L*D*L^H to compute the solution to a complex system of linear equations A * X = B, where A is an n x n Hermitian positive definite tridiagonal matrix, 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 matrix A is factored as A = L*D*L^H, where L is a unit lower bidiagonal matrix and D is diagonal. The factorization can also be regarded as having the form A = U^H*D*U. If the leading i x i principal minor is not positive definite, 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 the matrix A is supplied on entry. = 'F': df[] and ef[] contain the factored form of A. df[] and ef[] will not be modified. = 'N': The matrix A will be copied to df[] and ef[] and factored. [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] d[] Array d[ld] (ld >= n) n diagonal elements of the Hermitian positive definite tridiagonal matrix A. [in] e[] Array e[le] (le >= n - 1) n-1 sub-diagonal elements of the Hermitian positive definite tridiagonal matrix A. [in,out] df[] Array df[ldf] (ldf >= n) [in] If fact = 'F', n diagonal elements of the diagonal matrix D from the L*D*L^H factorization of A, are to be stored. [out] If fact = 'N', n diagonal elements of the diagonal matrix D from the L*D*L^H factorization of A, are returned. [in,out] ef[] Array ef[lef] (lef >= n - 1) [in] If fact = 'F', n-1 sub-diagonal elements of the unit bidiagonal factor L from the L*D*L^H factorization of A, are to be stored. [out] If fact = 'N', n-1 sub-diagonal elements of the unit bidiagonal factor L from the L*D*L^H factorization of A, are returned. [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 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 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). [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 >= 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 n had an illegal value (n < 0) = -3: The argument nrhs had an illegal value (nrhs < 0) = -8: The argument ldb had an illegal value (ldb < max(1, n)) = -10: The argument ldx had an illegal value (ldx < max(1, n)) = i (0 < i <= n): The leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been 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