WZgels WZgels2 WZgelss WZgelss2 WZgelsy WZgelsy2

WZgelss2() Function WZgelss2 ( M As  Long, N As  Long, A As  Variant, B As  Variant, Optional Nrhs As  Long = 1, Optional RCond As  Double = 0, Optional Cov As  String = "N", Optional Sing As  String = "N"  ) Solution to overdetermined or underdetermined linear equations Ax = b for complex matrices using the singular value decomposition (SVD) (complex numbers in pairs of cells) Purpose WZgelss2 computes the minimum norm solution to a complex linear least squares problem: minimize || A * X - B || using the singular value decomposition (SVD) 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 effective rank of A is determined by treating as zero those singular values which are less than RCond times the largest singular value. To represent complex numbers, a real part and an imaginary part are stored in a pair of adjacent cells (a real part in a left cell, and an imaginary part in a right cell). The computed results are stored in the same way. Returns If M >= N and Sing = "N" (N+1 x 2Nrhs (Cov = "N"), N+1 x 2Nrhs+1 (Cov = "D") or N+1 x 2Nrhs+2N (Cov = "C")) Columns 1 to 2Nrhs Column 2Nrhs+1 (if Cov = "D") Columns 2Nrhs+1 to 2Nrhs+2N (if Cov = "C") Rows 1 to N Least squares solution vector x Variance (diagonal elements of variance-covariance matrix) Variance-covariance matrix Row N+1 Effective rank (column 1), return code (column 2) 0 0 If M >= N and Sing = "S" (N+1 x 2Nrhs+1 (Cov = "N"), N+1 x 2Nrhs+2 (Cov = "D") or N+1 x 2Nrhs+2N+2 (Cov = "C")) Columns 1 to 2Nrhs Column 2Nrhs+1 Column 2Nrhs+2 (if Cov = "D") Columns 2Nrhs+2 to 2Nrhs+2N+1 (if Cov = "C") Rows 1 to N Least squares solution vector x Singular values of A in descending order Variance (diagonal elements of variance-covariance matrix) Variance-covariance matrix Row N+1 Effective rank (column 1), return code (column 2) 0 0 0 If M < N (N+1 x 2Nrhs (Sing = "N"), N+1 x 2Nrhs+1 (Sing = "S")) Columns 1 to 2Nrhs Column 2Nrhs+1 (if SIng = "S") Rows 1 to N Minimum norm solution vector x Singular values of A in descending order Row N+1 Effective rank (column 1), return code (column 2) 0 Return code = 0: Successful exit = i > 0: The algorithm for computing the SVD failed to converge; i off-diagonal elements of an intermediate bidiagonal form did not converge to zero. Parameters [in] M Number of rows of the matrix A. (M >= 1) [in] N Number of columns of the matrix A. (N >= 1) [in] A (M x 2N) M x N coefficient matrix A. (May be rank-deficient) [in] B (M x 2Nrhs) Right hand side matrix B. [in] Nrhs (Optional) Number of columns of right hand side matrix B. (Nrhs >= 1) (default = 1) [in] RCond (Optional) The parameter used to determine the effective rank of A. The effective rank is the number of singular values which are greater than RCond * largest singular value. (default = machine precision) [in] Cov (Optional) = "N": Do not compute variance-covariance matrix. = "D": Compute diagonal elements of variance-covariance matrix. (If M >= N) = "C": Compute variance-covariance matrix. (If M >= N) (default = "N") [in] Sing (Optional) = "N": Singular values are not returned = "S": Singular values are returned (default = "N") Reference LAPACK Example 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 )