Zgejsv Zgesdd Zgesvd Zgesvdq Zgesvdx Zggsvd3

Zggsvd3() Sub Zggsvd3 ( Jobu As  String, Jobv As  String, Jobq As  String, M As  Long, N As  Long, P As  Long, K As  Long, L As  Long, A() As  Complex, B() As  Complex, Alpha() As  Double, Beta() As  Double, U() As  Complex, V() As  Complex, Q() As  Complex, Info As  Long  ) Generalized singular value decomposition (GSVD) of complex matrices Purpose This routine 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〜K+L-1, N-K-L〜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〜M-1, N-K-L〜N-1), and R33 is stored ( 0 R22 R23) in B(M-K〜L-1, N+M-K-L〜N-1) on exit. The routine computes C, S, R, and optionally the unitary transformation matrices U, V and Q. 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)) 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 >= 0) [in] N Number of columns of the matrices A and B. (N >= 0) [in] P Number of rows of the matrix B. (P >= 0) [out] K [out] L On exit, K and L specify the dimension of the subblocks described in Purpose. (K + L = effective numerical rank of (A^H, B^H)^H). [in,out] A() Array A(LA1 - 1, LA2 - 1) (LA1 >= M, LA2 >= N) [in] M x N matrix A. [out] A() contains the triangular matrix R, or part of R. See Purpose for details. [in,out] B() Array B(LB1 - 1, LB2 - 1) (LB1 >= P, LB2 >= N) [in] P x N matrix B. [out] B() contains the triangular matrix R if M-K-L < 0. See Purpose for details. [out] Alpha() Array Alpha(LAlpha - 1) (LAlpha >= N) [out] Beta() Array Beta(LBeta - 1) (LBeta >= N) 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. [out] U() Array U(LU1 - 1, LU2 - 1) (LU1 >= M, LU2 >= M) Jobu = 'U': U() contains the M x M unitary matrix U. Jobu = 'N': U() is not referenced. [out] V() Array V(LV1 - 1, LV2 - 1) (LV1 >= P, LV2 >= P) Jobv = 'V': V() contains the P x P unitary matrix V. Jobv = 'N': V() is not referenced. [out] Q() Array Q(LQ1 - 1, LQ2 - 1) (LQ1 >= N, LQ2 >= N) Jobq = 'Q': Q() contains the N x N unitary matrix Q. Jobq = 'N': Q() is not referenced. [out] Info = 0: Successful exit. = -1: The argument Jobu had an illegal value. (Jobu <> "U" nor "N") = -2: The argument Jobv had an illegal value. (Jobv <> "V" nor "N") = -3: The argument Jobq had an illegal value. (Jobq <> "Q" nor "N") = -4: The argument M had an illegal value. (M < 0) = -5: The argument N had an illegal value. (N < 0) = -6: The argument P had an illegal value. (P < 0) = -9: The argument A() is invalid. = -10: The argument B() is invalid. = -11: The argument Alpha() is invalid. = -12: The argument Beta() is invalid. = -13: The argument U() is invalid. = -14: The argument V() is invalid. = -15: The argument Q() is invalid. = 1: The Jacobi-type procedure failed to converge. Reference LAPACK Example Program 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 ) Sub Ex_Zggsvd3() Const M = 3, N = 3, P = 3 Dim A(M - 1, N - 1) As Complex, B(P - 1, N - 1) As Complex Dim Alpha(N - 1) As Double, Beta(N - 1) As Double Dim U(M - 1, M - 1) As Complex, V(P - 1, P - 1) As Complex, Q(N - 1, N - 1) As Complex Dim K As Long, L As Long, Info As Long A(0, 0) = cmplx(0.2, -0.11): A(0, 1) = cmplx(-0.93, -0.32): A(0, 2) = cmplx(0.81, 0.37) A(1, 0) = cmplx(-0.8, -0.92): A(1, 1) = cmplx(-0.29, 0.86): A(1, 2) = cmplx(0.64, 0.51) A(2, 0) = cmplx(0.71, 0.59): A(2, 1) = cmplx(-0.15, 0.19): A(2, 2) = cmplx(0.2, 0.94) B(0, 0) = cmplx(0.57, -0.91): B(0, 1) = cmplx(-0.28, -0.45): B(0, 2) = cmplx(0.25, 0.91) B(1, 0) = cmplx(0.83, 0.46): B(1, 1) = cmplx(0.63, -0.19): B(1, 2) = cmplx(-0.69, 0.09) B(2, 0) = cmplx(0.24, -1.33): B(2, 1) = cmplx(-0.56, -0.67): B(2, 2) = cmplx(0.9, 1.25) Call Zggsvd3("U", "V", "Q", M, N, P, K, L, A(), B(), Alpha(), Beta(), U(), V(), Q(), Info) Debug.Print "Alpha =", Alpha(0), Alpha(1), Alpha(2) Debug.Print "Beta =", Beta(0), Beta(1), Beta(2) Debug.Print "R =" Debug.Print Creal(A(0, 0)), Cimag(A(0, 0)), Creal(A(0, 1)), Cimag(A(0, 1)) Debug.Print Creal(A(1, 0)), Cimag(A(1, 0)), Creal(A(1, 1)), Cimag(A(1, 1)) Debug.Print Creal(A(2, 0)), Cimag(A(2, 0)), Creal(A(2, 1)), Cimag(A(2, 1)) Debug.Print Creal(A(0, 2)), Cimag(A(0, 2)) Debug.Print Creal(A(1, 2)), Cimag(A(1, 2)) Debug.Print Creal(A(2, 2)), Cimag(A(2, 2)) Debug.Print "U =" Debug.Print Creal(U(0, 0)), Cimag(U(0, 0)), Creal(U(0, 1)), Cimag(U(0, 1)) Debug.Print Creal(U(1, 0)), Cimag(U(1, 0)), Creal(U(1, 1)), Cimag(U(1, 1)) Debug.Print Creal(U(2, 0)), Cimag(U(2, 0)), Creal(U(2, 1)), Cimag(U(2, 1)) Debug.Print Creal(U(0, 2)), Cimag(U(0, 2)) Debug.Print Creal(U(1, 2)), Cimag(U(1, 2)) Debug.Print Creal(U(2, 2)), Cimag(U(2, 2)) Debug.Print "V =" Debug.Print Creal(V(0, 0)), Cimag(V(0, 0)), Creal(V(0, 1)), Cimag(V(0, 1)) Debug.Print Creal(V(1, 0)), Cimag(V(1, 0)), Creal(V(1, 1)), Cimag(V(1, 1)) Debug.Print Creal(V(2, 0)), Cimag(V(2, 0)), Creal(V(2, 1)), Cimag(V(2, 1)) Debug.Print Creal(V(0, 2)), Cimag(V(0, 2)) Debug.Print Creal(V(1, 2)), Cimag(V(1, 2)) Debug.Print Creal(V(2, 2)), Cimag(V(2, 2)) Debug.Print "Q =" Debug.Print Creal(Q(0, 0)), Cimag(Q(0, 0)), Creal(Q(0, 1)), Cimag(Q(0, 1)) Debug.Print Creal(Q(1, 0)), Cimag(Q(1, 0)), Creal(Q(1, 1)), Cimag(Q(1, 1)) Debug.Print Creal(Q(2, 0)), Cimag(Q(2, 0)), Creal(Q(2, 1)), Cimag(Q(2, 1)) Debug.Print Creal(Q(0, 2)), Cimag(Q(0, 2)) Debug.Print Creal(Q(1, 2)), Cimag(Q(1, 2)) Debug.Print Creal(Q(2, 2)), Cimag(Q(2, 2)) Debug.Print "K =", K, "L =", L, "Info =", Info End Sub Example Results Alpha = 0.430704325811232 0.976338298760024 0.991145496564202 Beta = 0.902493093451408 0.216248760399642 0.132780287093004 R = -2.49290056457435 0 0.457594814366194 0.194431879391052 0 0 -1.26470568603387 0 0 0 0 0 1.6180003034571 1.39174354257137 -7.00070473318143E-02 -0.174007882105091 -1.75471215104157 0 U = 0.568161065126274 0.474196630652255 -0.180338037878126 -6.24249232440974E-03 0.201921419044388 0.445329052169583 0.137381224968446 -0.594162404216661 -0.365049262631842 -0.282806243264565 -0.454409702123437 -0.623737510850184 0.622382357104722 0.180027732589302 -0.33819653446823 -0.524051826005947 0.431431460509124 -7.13435328533903E-02 V = -0.259639196635024 0.401093020738518 4.88190028247251E-02 3.25673943950224E-02 -0.400793045759536 -0.226433378198375 -0.714410691722334 0.432401521895175 -0.122502964271075 0.738104165913481 -0.437440072643577 -0.328403014673516 0.331492445838016 -0.811406620889563 -0.275482203370116 -0.121855253273566 2.47600244928659E-02 0.374672988073607 Q = 0.498168838100246 0.295619901645173 0.262656203742932 -0.420386203776387 0.659025789544342 -0.110123110678936 0.347354481147794 -5.02490827933195E-02 -0.285036186254502 -0.369795841286094 0.793403710637756 4.02194525204689E-02 -0.644893096482004 5.32610074710518E-02 0.613682926576038 -0.231885896388499 -6.10928315913905E-02 0.383622238414744 K = 0 L = 3 Info = 0