WDgesvd WDggsvd3

WDggsvd3() Function WDggsvd3 ( Jobu As  String, Jobv As  String, Jobq As  String, M As  Long, N As  Long, P As  Long, A As  Variant, B As  Variant  ) Generalized singular value decomposition (GSVD) Purpose WDggsvd3 computes the generalized singular value decomposition (GSVD) of an M x N real matrix A and P x N real matrix B: U^T*A*Q = D1*(0 R), V^T*B*Q = D2*(0 R) where, U, V and Q are orthogonal matrices. Let K+L = the effective numerical rank of the matrix (A^T, B^T)^T, then R is a K+L x K+L nonsingular upper triangular matrix, D1 and D2 are M x K+L and P x K+L "diagonal" matrices and of the following structures, respectively: If M-K-L >= 0, K L D1 = K (I 0) L (0 C) M-K-L (0 0) K L D2 = L (0 S) P-L (0 0) N-K-L K L (0 R) = K ( 0 R11 R12) L ( 0 0 R22) where C = diag(Alpha(K), ... , Alpha(K+L-1)), S = diag(Beta(K)), ... , Beta(K+L-1)), C^2 + S^2 = I. R is stored in A(0 to K+L-1, N-K-L to N-1) on exit. If M-K-L < 0, K M-K K+L-M D1 = K (I 0 0) M-K (0 C 0) K M-K K+L-M D2 = M-K (0 S 0) K+L-M (0 0 I) P-L (0 0 0) N-K-L K M-K K+L-M (0 R) = K ( 0 R11 R12 R13) M-K ( 0 0 R22 R23) K+L-M ( 0 0 0 R33) where C = diag(Alpha(K), ... , Alpha(M-1)), S = diag(Beta(K), ... , Beta(M-1)), C^2 + S^2 = I. (R11 R12 R13) is stored in A(0 to M-1, N-K-L to N-1), and R33 is stored ( 0 R22 R23) in B(M-K to L-1, N+M-K-L to N-1) on exit. The routine computes C, S, R, and optionally the orthogonal transformation matrices U, V and Q. Alpha() and Beta() contain the generalized singular value pairs of A and B.   Alpha(0 to K-1) = 1,   Beta(0 to K-1) = 0, and if M-K-L >= 0,   Alpha(K to K+L-1) = C,   Beta(K to K+L-1) = S, or if M-K-L < 0,   Alpha(K to M-1)= C, Alpha(M to K+L-1) = 0,   Beta(K to M-1) = S, Beta(M to K+L-1) = 1, and   Alpha(K+L to N-1) = 0,   Beta(K+L to N-1) = 0. In particular, if B is an N x N nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of A*inv(B). A*inv(B) = U*(D1*inv(D2))*V^T If (A^T, B^T)^T has orthonormal columns, then the GSVD of A and B is also equal to the CS decomposition of A and B. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem: A^T*A x = λ B^T*B x In some literature, the GSVD of A and B is presented in the form U^T*A*X = (0 D1), V^T*B*X = (0 D2) where U and V are orthogonal and X is nonsingular, D1 and D2 are "diagonal". The former GSVD form can be converted to the latter form by taking the nonsingular matrix X as X = Q*(I 0 ) (0 inv(R)) Returns N+1×K+L+2 (Jobu = "N", Jobv = "N", Jobq = "N") Column 1 Column 2 Columns 3 to K+L+2 Rows 1 to N Alpha(j) (j = 1 to N) Beta(j) (j = 1 to N) K+L x K+L upper triangular matrix R Row N+1 Return code Dimension of the subblocks K Dimension of the subblocks L (Column 3) max(M,N)+1×K+L+M+2 (Jobu = "U", Jobv = "N", Jobq = "N") Column 1 Column 2 Columns 3 to K+L+2 Columns K+L+3 to K+L+M+2 Rows 1 to max(M, N) Alpha(j) (j = 1 to N) Beta(j) (j = 1 to N) K+L x K+L upper triangular matrix R M x M orthogonal matrix U Row max(M, N) + 1 Return code Dimension of the subblocks K Dimension of the subblocks L (Column 3) 0 max(N,P)+1×K+L+P+2 (Jobu = "N", Jobv = "V", Jobq = "N") Column 1 Column 2 Columns 3 to K+L+2 Columns K+L+3 to K+L+P+2 Rows 1 to max(N, P) Alpha(j) (j = 1 to N) Beta(j) (j = 1 to N) K+L x K+L upper triangular matrix R P x P orthogonal matrix V Row max(N, P) + 1 Return code Dimension of the subblocks K Dimension of the subblocks L (Column 3) 0 N+1×K+L+N+2 (Jobu = "N", Jobv = "N", Jobq = "Q") Column 1 Column 2 Columns 3 to K+L+2 Columns K+L+3 to K+L+N+2 Rows 1 to N Alpha(j) (j = 1 to N) Beta(j) (j = 1 to N) K+L x K+L upper triangular matrix R N x N orthogonal matrix Q Row N+1 Return code Dimension of the subblocks K Dimension of the subblocks L (Column 3) 0 max(M,N,P)+1×K+L+M+P+2 (Jobu = "U", Jobv = "V", Jobq = "N") Column 1 Column 2 Columns 3 to K+L+2 Columns K+L+3 to K+L+M+2 Columns K+L+M+3 to K+L+M+P+2 Rows 1 to max(M, N, P) Alpha(j) (j = 1 to N) Beta(j) (j = 1 to N) K+L x K+L upper triangular matrix R M x M orthogonal matrix U P x P orthogonal matrix V Row max(M, N, P) + 1 Return code Dimension of the subblocks K Dimension of the subblocks L (Column 3) 0 0 max(M,N)+1×K+L+M+N+2 (Jobu = "U", Jobv = "N", Jobq = "Q") Column 1 Column 2 Columns 3 to K+L+2 Columns K+L+3 to K+L+M+2 Columns K+L+M+3 to K+L+M+N+2 Rows 1 to max(M, N) Alpha(j) (j = 1 to N) Beta(j) (j = 1 to N) K+L x K+L upper triangular matrix R M x M orthogonal matrix U N x N orthogonal matrix Q Row max(M, N) + 1 Return code Dimension of the subblocks K Dimension of the subblocks L (Column 3) 0 0 max(N,P)+1×K+L+N+P+2 (Jobu = "N", Jobv = "V", Jobq = "N") Column 1 Column 2 Columns 3 to K+L+2 Columns K+L+3 to K+L+P+2 Columns K+L+P+3 to K+L+N+P+2 Rows 1 to max(N, P) Alpha(j) (j = 1 to N) Beta(j) (j = 1 to N) K+L x K+L upper triangular matrix R P x P orthogonal matrix V N x N orthogonal matrix Q Row max(N, P) + 1 Return code Dimension of the subblocks K Dimension of the subblocks L (Column 3) 0 0 max(M,N,P)+1×K+L+M+N+P+2 (Jobu = "U", Jobv = "V", Jobq = "Q") Column 1 Column 2 Columns 3 to K+L+2 Columns K+L+3 to K+L+M+2 Columns K+L+M+3 to K+L+M+P+2 Columns K+L+M+P+3 to K+L+M+N+P+2 Rows 1 to max(M, N, P) Alpha(j) (j = 1 to N) Beta(j) (j = 1 to N) K+L x K+L upper triangular matrix R M x M orthogonal matrix U P x P orthogonal matrix V N x N orthogonal matrix Q Row max(M, N, P) + 1 Return code Dimension of the subblocks K Dimension of the subblocks L (Column 3) 0 0 0 Return code. = 0: Successful exit. = 1: The Jacobi-type procedure failed to converge. Parameters [in] Jobu = "U": Orthogonal matrix U is computed. = "N": U is not computed. [in] Jobv = "V": Orthogonal matrix V is computed. = "N": V is not computed. [in] Jobq = "Q": Orthogonal matrix Q is computed. = "N": Q is not computed. [in] M Number of rows of the matrix A. (M >= 1) [in] N Number of columns of the matrices A and B. (N >= 1) [in] P Number of rows of the matrix B. (P >= 1) [in] A (M x N) M x N matrix A. [in] B (P x N) P x N matrix B. Reference LAPACK Example Compute generalized singular value decomposition (GSVD) of matrix A and B, where ( 1 6 11 ) ( 1 5 9 ) ( 2 7 12 ) ( 2 6 10 ) A = ( 3 8 13 ), B = ( 3 7 11 ) ( 4 9 14 ) ( 4 8 12 ) ( 5 10 15 )