dgecon dgecov dgels dgesv dlange dlansy dpocon dposv dsyev

dgels() def dgels ( trans  , m  , n  , a  , b  , nrhs  = 1  ) Solution to overdetermined or underdetermined linear equations Ax = b (full rank) Purpose dgels solves overdetermined or underdetermined real linear systems involving an m x n matrix A, or its transpose, using a QR or LQ factorization of A. It is assumed that A has full rank. The following options are provided: If trans = 'N' and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - A*X ||. If trans = 'N' and m < n: find the minimum norm solution of an underdetermined system A * X = B. If trans = 'T' and m >= n: find the minimum norm solution of an underdetermined system A^T * X = B. If trans = 'T' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - A^T*X ||. Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the m x nrhs right hand side matrix B and the n x nrhs solution matrix X. Returns info (int) = 0: Successful exit = -1: The argument trans had an illegal value (trans != 'T' nor 'N') = -2: The argument m had an illegal value (m < 0) = -3: The argument n had an illegal value (n < 0) = -4: The argument a is invalid. = -5: The argument b is invalid. = -6: The argument nrhs had an illegal value (nrhs < 0) = i > 0: The i-th diagonal element of the triangular factor of A is zero, so that A does not have full rank. The least squares solution could not be computed. Parameters [in] trans = 'N': The linear system involves A. = 'T': The linear system involves A^T. [in] m Number of rows of the matrix A. (m >= 0) (If m = 0, returns zero vectors in b) [in] n Number of columns of the matrix A. (n >= 0) (If n = 0, returns zero vectors in b) [in,out] a Numpy ndarray (2-dimensional, float, m x n) [in] m x n matrix A. [out] m >= n: a is overwritten by details of its QR factorization as returned by dgeqrf.   m < n: a is overwritten by details of its LQ factorization as returned by dgelqf. [in,out] b[][] Numpy ndarray (1 or 2-dimensional, float, max(m, n) or max(m, n) x nrhs) [in] Matrix B of right hand side vectors, stored columnwise; B is m x nrhs if trans = 'N', or n x nrhs if trans = 'T'. [out] If info = 0, b is overwritten by the solution vectors, stored columnwise:   trans = 'N' and m >= n: Rows 0 to n-1 of b contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements n to m-1 in that column.   trans = 'N' and m < n: Rows 0 to n-1 of b contain the minimum norm solution vectors.   trans = 'T' and m >= n: Rows 0 to m-1 of b contain the minimum norm solution vectors.   trans = 'T' and m < n: Rows 0 to m-1 of b contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements m to n-1 in that column. [in] nrhs (Optional) Number of right hand sides, i.e., number of columns of the matrices B and X. (nrhs >= 0) (If nrhs = 0, returns without computation) (default = 1) Reference LAPACK Example Program Compute the least squares solution of the overdetermined linear equations Ax = b and its variance, where ( -1.06 0.48 -0.04 ) A = ( -1.19 0.73 -0.24 ) ( 1.97 -0.89 0.56 ) ( 0.68 -0.53 0.08 ) ( 0.3884 ) B = ( 0.1120 ) ( -0.3644 ) ( -0.0002 ) def TestDgels(): m = 4 n = 3 a = np.array([ [-1.06, -1.19, 1.97, 0.68], [0.48, 0.73, -0.89, -0.53], [-0.04, -0.24, 0.56, 0.08]]) b = np.array([0.3884, 0.1120, -0.3644, -0.0002]) info = dgels('N', m, n, a, b) print(b[0:n], info) if info == 0: ci = np.empty(n) info = dgecov(0, n, a, ci) print(ci, info) Example Results >>> TestDgels() [-0.82 -0.94 0.74] 0 [ 6.46959968 16.73504082 18.71775328] 0