zhesv zhesvx zhpsv zhpsvx

zhesvx() void zhesvx ( char  fact, char  uplo, int  n, int  nrhs, int  lda, doublecomplex  a[], int  ldaf, doublecomplex  af[], int  ipiv[], int  ldb, doublecomplex  b[], int  ldx, doublecomplex  x[], double *  rcond, double  ferr[], double  berr[], doublecomplex  work[], int  lwork, double  rwork[], int *  info  ) (Expert driver) Solution to system of linear equations AX = B for a Hermitian matrix Purpose This routine uses the diagonal pivoting factorization to computes the solution to a complex system of linear equations A * X = B where A is an n x n Hermitian 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 diagonal pivoting method is used to factor A. The form of the factorization is A = U * D * U^H, if uplo = 'U', or A = L * D * L^H, if uplo = 'L', where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1 x 1 and 2 x 2 diagonal blocks. If some i-th diagonal element of D = 0, so that D 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': af[][] and ipiv[] contain the factored form of A. af[][] and ipiv[] will not be modified. = 'N': The matrix A will be copied to af[][] and factored. [in] uplo = 'U': Upper triangle of A is stored. = 'L': Lower triangle of A is stored. [in] n Number of linear equations, i.e., 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 and X. (nrhs >= 0) (if nrhs = 0, returns without computation) [in] lda Leading dimension of the two dimensional array a[][]. (lda >= max(1, n)) [in] a[][] Array a[la][lda] (la >= n) n x n Hermitian matrix A. The upper or lower triangular part is to be referenced in accordance with uplo. [in] ldaf Leading dimension of the two dimensional array af[][]. (ldaf >= max(1, n)) [in,out] af[][] Array af[laf][ldaf] (laf >= n) [in] If fact = 'F', the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U^H or A = L*D*L^H as computed by zhetrf. [out] If fact = 'N', the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U^H or A = L*D*L^H. [in,out] ipiv[] Array ipiv[lipiv] (lipiv >= n) [in] If fact = 'F', details of the interchanges and the block structure of D, as determined by zhetrf, are to be stored.   If ipiv[k-1] > 0, then rows and columns k and ipiv[k-1] were interchanged, and k-th diagonal of D is a 1 x 1 diagonal block.   If uplo = 'U' and ipiv[k-1] = ipiv[k-2] < 0, then rows and columns k-1 and -ipiv[k-1] were interchanged and (k-1)-th diagonal of D is a 2 x 2 diagonal block.   If uplo = 'L' and ipiv[k-1] = ipiv[k] < 0, then rows and columns k+1 and -ipiv[k-1] were interchanged and k-th diagonal of D is a 2 x 2 diagonal block. [out] If fact = 'N', details of the interchanges and the block structure of D, as determined by zhetrf, 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 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] Work array On exit, if info = 0, work[0] returns the optimal lwork. [in] lwork The length of work[] (lwork >= max(1, 2*n)) For best performance, when fact = 'N', lwork >= max(1, 2*n, n*nb), where nb is the optimal blocksize for zhetrf. If lwork = -1, then a workspace query is assumed. The routine only calculates the optimal size of the work array and returns the value in work[0]. [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 uplo had an illegal value (uplo != 'U' nor 'L') = -3: The argument n had an illegal value (n < 0) = -4: The argument nrhs had an illegal value (nrhs < 0) = -5: The argument lda had an illegal value (lda < max(1, n)) = -7: The argument ldaf had an illegal value (ldaf < max(1, n)) = -10: The argument ldb had an illegal value (ldb < max(1, n)) = -12: The argument ldx had an illegal value (ldx < max(1, n)) = -18: The argument lwork had an illegal value (lwork too small) = i (0 < i <= n): The i-th element of the factor D is exactly zero. The factorization has been completed, but the factor D is exactly singular, so the solution and error bounds could not be computed. rcond = 0 is returned. = n+1: D 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