Dgecov Dgecovs Dgecovy Dgels Dgelsd Dgelss Dgelsy Dgetsls

Dgetsls() Sub Dgetsls ( Trans As  String, M As  Long, N As  Long, A() As  Double, B() As  Double, Info As  Long, Optional Nrhs As  Long = 1  ) Solution to overdetermined or underdetermined linear equations Ax = b (Full rank) (Tall skinny QR or short wide LQ factorization) Purpose This routine solves overdetermined or underdetermined real linear systems involving an M x N matrix A, using a tall skinny QR or short wide 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. 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 without computation) [in] N Number of columns of the matrix A. (N >= 0) (If N = 0, returns without computation) [in,out] A() Array A(LA1 - 1, LA2 - 1) (LA1 >= M, LA2 >= N) [in] M x N matrix A. [out] A() is overwritten by details of its QR or LQ factorization as returned by Dgeqr or Dgelq. [in,out] B() Array B(LB1 - 1, LB2 - 1) (LB1 >= max(M, N), LB2 >= Nrhs) (2D array) or B(LB - 1) (LB >= max(M, N), Nrhs = 1) (1D array) [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.   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. [out] Info = 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. = -7: 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. [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 ) Sub Ex_Dgetsls() Const M = 4, N = 3 Dim A(M - 1, N - 1) As Double, B(M - 1) As Double, Ci(N - 1) As Double Dim Info As Long A(0, 0) = -1.06: A(0, 1) = 0.48: A(0, 2) = -0.04 A(1, 0) = -1.19: A(1, 1) = 0.73: A(1, 2) = -0.24 A(2, 0) = 1.97: A(2, 1) = -0.89: A(2, 2) = 0.56 A(3, 0) = 0.68: A(3, 1) = -0.53: A(3, 2) = 0.08 B(0) = 0.3884: B(1) = 0.112: B(2) = -0.3644: B(3) = -0.0002 Call Dgetsls("N", M, N, A(), B(), Info) If Info <> 0 Then Debug.Print "Error in Dgels: Info =", Info Exit Sub End If Debug.Print "X =", B(0), B(1), B(2) Call Dgecov(0, N, A(), Ci(), Info) Debug.Print "Var =", Ci(0), Ci(1), Ci(2) Debug.Print "Info =", Info End Sub Example Results X = -0.820000000000001 -0.94 0.740000000000001 Var = 6.46959967542666 16.7350408218919 18.7177532773229 Info = 0