Dggglm Dgglse

Dgglse() Sub Dgglse ( M As  Long, N As  Long, P As  Long, A() As  Double, B() As  Double, C() As  Double, D() As  Double, X() As  Double, Info As  Long  ) Linear equality-constrained least squares (LSE) problem Purpose This routine solves the linear equality-constrained least squares (LSE) problem: minimize || c - Ax ||_2 subject to B*x = d where A is an m x n matrix, B is a p x n matrix, c is a given m-vector, and d is a given p-vector. It is assumed that p <= n <= m + p, and rank(B) = p and rank( (A) ) = n ( (B) ) These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized RQ factorization of the matrices (B, A) given by B = (0 R)*Q, A = Z*T*Q Parameters [in] M Number of rows of the matrix A. (M >= 0) [in] N Number of columns of the matrices A and B. (N >= 0) (If N = 0, returns without computation) [in] P Number of rows of the matrix B. (0 <= P <= N <= M + P) [in,out] A() Array A(LA1 - 1, LA2 - 1) (LA1 >= M, LA2 >= N) [in] M x N matrix A [out] The elements on and above the diagonal of the array contain the min(M, N) x N upper trapezoidal matrix T. [in,out] B() Array B(LB1 - 1, LB2 - 1) (LB1 >= P, LB2 >= N) [in] P x N matrix B [out] The upper triangle of the subarray B(0 to P-1, N-P to N-1) contains the P x P upper triangular matrix R. [in,out] C() Array C(LC - 1) (LC >= M) [in] The right hand side vector for the least squares part of the LSE problem. [out] The residual sum of squares for the solution is given by the sum of squares of elements N-P to M-1 of vector C. [in,out] D() Array D(LD - 1) (LD >= P) [in] The right hand side vector for the constrained equation. [out] D() is destroyed. [out] X() Array X(LX - 1) (LX >= N) The solution of the LSE problem. [out] Info = 0: Successful exit. = -1: The argument M had an illegal value. (M < 0) = -2: The argument N had an illegal value. (N < 0) = -3: The argument P had an illegal value. (P < 0 or P > N or P < N - M) = -4: The argument A() is invalid. = -5: The argument B() is invalid. = -6: The argument C() is invalid. = -7: The argument D() is invalid. = -8: The argument X() is invalid. = 1: The upper triangular factor R associated with B in the generalized RQ factorization of the pair (B, A) is singular, so that rank(B) < P. The least squares solution could not be computed. = 2: The N-P x N-P part of the upper trapezoidal factor T associated with A in the generalized RQ factorization of the pair (B, A) is singular, so that rank((A^T B^T)^T) < N. The least squares solution could not be computed. Reference LAPACK Example Program Solve the linear equality-constrained least squares (LSE) problem, i.e. minimize || c - Ax ||_2 subject to B*x = d, 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 ) ( -0.58 -0.79 0.82 ) B = ( 0.77 0.71 -0.55 ) ( -1.36 -1.22 1.66 ) ( 0.3884 ) c = ( 0.1120 ) ( -0.3644 ) ( -0.0002 ) ( 1.8250 ) d = ( -1.7058 ) ( 3.4904 ) Sub Ex_Dgglse() Const M = 4, N = 3, P = 3 Dim A(M - 1, N - 1) As Double, B(P - 1, N - 1) As Double Dim C(M - 1) As Double, D(P - 1) As Double, X(N - 1) As Double Dim S As Double, 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) = -0.58: B(0, 1) = -0.79: B(0, 2) = 0.82 B(1, 0) = 0.77: B(1, 1) = 0.71: B(1, 2) = -0.55 B(2, 0) = -1.36: B(2, 1) = -1.22: B(2, 2) = 1.66 C(0) = 0.3884: C(1) = 0.112: C(2) = -0.3644: C(3) = -0.0002 D(0) = 1.825: D(1) = -1.7058: D(2) = 3.4904 Call Dgglse(M, N, P, A(), B(), C(), D(), X(), Info) S = Dnrm2(M - N + P, C(N - P)) Debug.Print "X =", X(0), X(1), X(2) Debug.Print "SumSq =", S, "Info =", Info End Sub Example Results X = -0.820000000000002 -0.939999999999998 0.74 SumSq = 7.1082457242971E-15 Info = 0