WZgesvd WZgesvd2 WZggsvd3 WZggsvd32

WZggsvd32() Function WZggsvd32 ( 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) of complex matrices (complex numbers in pairs of cells) Purpose WZggsvd3 computes the generalized singular value decomposition (GSVD) of an M x N complex matrix A and P x N complex matrix B: U^H*A*Q = D1*(0 R), V^H*B*Q = D2*(0 R) where, U, V and Q are unitary matrices. Let K+L = the effective numerical rank of the matrix (A^H, B^H)^H, 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 unitary 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^H If (A^H, B^H)^H 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^H*A x = λ B^H*B x In some literature, the GSVD of A and B is presented in the form U^H*A*X = (0 D1), V^H*B*X = (0 D2) where U and V are unitary 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)) 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 N+1×2(K+L)+2 (Jobu = "N", Jobv = "N", Jobq = "N") Column 1 Column 2 Columns 3 to 2(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)+1×2(K+L)+2M+2 (Jobu = "U", Jobv = "N", Jobq = "N") Column 1 Column 2 Columns 3 to 2(K+L)+2 Columns 2(K+L)+3 to 2(K+L)+2M+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 unitary 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)+1×2(K+L)+2P+2 (Jobu = "N", Jobv = "V", Jobq = "N") Column 1 Column 2 Columns 3 to 2(K+L)+2 Columns 2(K+L)+3 to 2(K+L)+2P+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 unitary matrix V Row max(N, P) + 1 Return code Dimension of the subblocks K Dimension of the subblocks L (Column 3) 0 N+1×2(K+L)+2N+2 (Jobu = "N", Jobv = "N", Jobq = "Q") Column 1 Column 2 Columns 3 to 2(K+L)+2 Columns 2(K+L)+3 to 2(K+L)+2N+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 unitary matrix Q Row N+1 Return code Dimension of the subblocks K Dimension of the subblocks L (Column 3) 0 max(M,N,P)+1×2(K+L)+2M+2P+2 (Jobu = "U", Jobv = "V", Jobq = "N") Column 1 Column 2 Columns 3 to 2(K+L)+2 Columns 2(K+L)+3 to 2(K+L)+2M+2 Columns 2(K+L)+2M+3 to 2(K+L)+2M+2P+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 unitary matrix U P x P unitary 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)+1×2(K+L)+2M+2N+2 (Jobu = "U", Jobv = "N", Jobq = "Q") Column 1 Column 2 Columns 3 to 2(K+L)+2 Columns 2(K+L)+3 to 2(K+L)+2M+2 Columns 2(K+L)+2M+3 to 2(K+L)+2M+2N+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 unitary matrix U N x N unitary 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)+1×2(K+L)+2N+2P+2 (Jobu = "N", Jobv = "V", Jobq = "N") Column 1 Column 2 Columns 3 to 2(K+L)+2 Columns 2(K+L)+3 to 2(K+L)+2P+2 Columns 2(K+L)+2P+3 to 2(K+L)+2N+2P+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 unitary matrix V N x N unitary 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)+1×2(K+L)+2M+2N+2P+2 (Jobu = "U", Jobv = "V", Jobq = "Q") Column 1 Column 2 Columns 3 to 2(K+L)+2 Columns 2(K+L)+3 to 2(K+L)+2M+2 Columns 2(K+L)+2M+3 to 2(K+L)+2M+2P+2 Columns 2(K+L)+2M+2P+3 to 2(K+L)+2M+2N+2P+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 unitary matrix U P x P unitary matrix V N x N unitary 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': Unitary matrix U is computed. = 'N': U is not computed. [in] Jobv = 'V': Unitary matrix V is computed. = 'N': V is not computed. [in] Jobq = "Q": Unitary 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 2N) M x N complex matrix A. [in] B (P x 2N) P x N complex matrix B. Reference LAPACK Example Compute generalized singular value decomposition (GSVD) of matrix A and B, where ( 0.20-0.11i -0.93-0.32i 0.81+0.37i ) A = ( -0.80-0.92i -0.29+0.86i 0.64+0.51i ) ( 0.71+0.59i -0.15+0.19i 0.20+0.94i ) ( 0.57-0.91i -0.28-0.45i 0.25+0.91i ) B = ( 0.83+0.46i 0.63-0.19i -0.69+0.09i ) ( 0.24-1.33i -0.56-0.67i 0.90+1.25i )