Zgecov Zgecovs Zgecovy Zgels Zgelsd Zgelss Zgelsy Zgetsls

Zgelsy() Sub Zgelsy ( M As  Long, N As  Long, A() As  Complex, B() As  Complex, Jpvt() As  Long, RCond As  Double, Rank As  Long, Info As  Long, Optional Nrhs As  Long = 1  ) Solution to overdetermined or underdetermined linear equations Ax = b for complex matrices using a complete orthogonal factorization Purpose This routine computes the minimum-norm solution to a complex linear least squares problem: minimize || A * X - B || using a complete orthogonal factorization of A. A is an M x N matrix which may be rank-deficient. 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. The routine first computes a QR factorization with column pivoting: A * P = Q * [ R11 R12 ] [ 0 R22 ] with R11 defined as the largest leading submatrix whose estimated condition number is less than 1/rcond. The order of R11, rank, is the effective rank of A. Then, R22 is considered to be negligible, and R12 is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization: A * P = Q * [ T11 0 ] * Z [ 0 0 ] The minimum-norm solution is then X = P * Z^H [ T11^(-1)*Q1^H*B ] [ 0 ] where Q1 consists of the first rank columns of Q. Parameters [in] M Number of rows of the matrix A. (M >= 0) (If M = 0, returns with Rank = 0) [in] N Number of columns of the matrix A. (N >= 0) (If N = 0, returns with Rank = 0) [in,out] A() Array A(LA1 - 1, LA2 - 1) (LA1 >= M, LA2 >= N) [in] M x N matrix A. [out] A() has been overwritten by details of the complete orthogonal factorization. [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] M x Nrhs right hand side matrix B. [out] N x Nrhs solution matrix X. [in,out] Jpvt() Array Jpvt(LJpvt - 1) (LJpvt >= N) [in] If Jpvt(i-1) <> 0, the i-th column of A is permuted to the front of A*P, otherwise column i is a free column. [out] If Jpvt(i) = k, then the i-th column of A*P was the k-th column of A. [in] RCond RCond is used to determine the effective rank of A, which is defined as the order of the largest leading triangular submatrix R11 in the QR factorization with pivoting of A, whose estimated condition number < 1/RCond. [out] Rank The effective rank of A, i.e., the order of the submatrix R11. This is the same as the order of the submatrix T11 in the complete orthogonal factorization of A. [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 A() is invalid. = -4: The argument B() is invalid. = -5: The argument Jpvt() is invalid. = -9: The argument Nrhs had an illegal value. (Nrhs < 0) [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 with Rank = 0) (default = 1) Reference LAPACK Example Program Compute the least squares solution of the overdetermined linear equations Ax = b and its variance, where ( -0.82+0.83i 0.18-0.94i -0.18-0.12i ) A = ( -0.76-0.24i 0.57-0.16i -0.08-0.27i ) ( 1.90+0.26i -0.98+0.54i 0.21+0.28i ) ( 0.50-0.30i -0.31+0.37i 0.22+0.19i ) ( 1.7126-0.6648i ) B = ( 0.8697+0.7604i ) ( -2.1048-1.6171i ) ( -0.9297+0.1252i ) Sub Ex_Zgelsy() Const M = 4, N = 3 Dim A(M - 1, N - 1) As Complex, B(M - 1) As Complex, Ci(N - 1) As Complex Dim Jpvt(N - 1) As Long, RCond As Double, Rank As Long, Info As Long, I As Long A(0, 0) = Cmplx(-0.82, 0.83): A(0, 1) = Cmplx(0.18, -0.94): A(0, 2) = Cmplx(-0.18, -0.12) A(1, 0) = Cmplx(-0.76, -0.24): A(1, 1) = Cmplx(0.57, -0.16): A(1, 2) = Cmplx(-0.08, -0.27) A(2, 0) = Cmplx(1.9, 0.26): A(2, 1) = Cmplx(-0.98, 0.54): A(2, 2) = Cmplx(0.21, 0.28) A(3, 0) = Cmplx(0.5, -0.3): A(3, 1) = Cmplx(-0.31, 0.37): A(3, 2) = Cmplx(0.22, 0.19) B(0) = Cmplx(1.7126, -0.6648): B(1) = Cmplx(0.8697, 0.7604) B(2) = Cmplx(-2.1048, -1.6171): B(3) = Cmplx(-0.9297, 0.1252) For I = 0 To N - 1 Jpvt(I) = 0 Next RCond = 0.0001 Call Zgelsy(M, N, A(), B(), Jpvt(), RCond, Rank, Info) If Info <> 0 Then Debug.Print "Error in Zgelsy: Info =", Info Exit Sub End If Debug.Print "X =" Debug.Print Creal(B(0)), Cimag(B(0)), Creal(B(1)), Cimag(B(1)) Debug.Print Creal(B(2)), Cimag(B(2)) Call Zgecovy(0, N, A(), Jpvt(), Ci(), Info) Debug.Print "Var =" Debug.Print Creal(Ci(0)), Creal(Ci(1)), Creal(Ci(2)) Debug.Print "Rank =", Rank, "Info =", Info End Sub Example Results X = -0.819999999999999 -0.939999999999999 0.740000000000001 0.200000000000002 0.480000000000002 0.21 Var = 5.46169501938982 15.1880504061464 21.4241290120714 Rank = 3 Info = 0