There is one prototype of ggsvd
available, please see below.
ggsvd( const char jobu, const char jobv, const char jobq, int_t& k, int_t& l, MatrixA& a, MatrixB& b, VectorALPHA& alpha, VectorBETA& beta, MatrixU& u, MatrixV& v, MatrixQ& q );
ggsvd (short for $FRIENDLY_NAME)
provides a C++ interface to LAPACK routines SGGSVD, DGGSVD, CGGSVD, and
ZGGSVD. ggsvd computes
the generalized singular value decomposition (GSVD) of an M-by-N complex
matrix A and P-by-N complex matrix B:
U'A*Q = D1( 0 R ), V'B*Q = D2( 0 R )
where U, V and Q are unitary matrices, and Z' means the conjugate transpose of Z. Let K+L = the effective numerical rank of the matrix (A',B')', then R is a (K+L)-by-(K+L) nonsingular upper triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(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+1), ... , ALPHA(K+L) ), S = diag( BETA(K+1), ... , BETA(K+L) ), C*2 + S*2 = I.
R is stored in A(1:K+L,N-K-L+1:N) 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+1), ... , ALPHA(M) ), S = diag( BETA(K+1), ... , BETA(M) ), C*2 + S*2 = I.
(R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored ( 0 R22 R23 ) in B(M-K+1:L,N+M-K-L+1:N) 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-by-N nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of A*inv(B): Ainv(B) = U(D1*inv(D2))*V'. If ( A',B')' has orthnormal 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'A x = lambda B'*B x. In some literature, the GSVD of A and B is presented in the form U'*A*X = ( 0 D1 ), V'*B*X = ( 0 D2 ) where U and V are orthogonal and X is nonsingular, and 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) ) The selection of the LAPACK routine is done during compile-time, and is determined by the type of values contained in type `MatrixA`. The type of values is obtained through the `value_type` meta-function `typename value_type<MatrixA>::type`. The dispatching table below illustrates to which specific routine the code path will be generated. [table Dispatching of ggsvd [ [ Value type of MatrixA ] [LAPACK routine] ] [ [`float`][SGGSVD] ] [ [`double`][DGGSVD] ] [ [`complex<float>`][CGGSVD] ] [ [`complex<double>`][ZGGSVD] ] ] [heading Definition] Defined in header [headerref boost/numeric/bindings/lapack/driver/ggsvd.hpp]. [heading Parameters or Requirements on Types] [variablelist Parameters [[MatrixA] [The definition of term 1]] [[MatrixB] [The definition of term 2]] [[MatrixC] [ The definition of term 3. Definitions may contain paragraphs. ]] ] [heading Complexity] [heading Example]
#include <boost/numeric/bindings/lapack/driver/ggsvd.hpp> using namespace boost::numeric::bindings;
lapack::ggsvd( x, y, z );
this will output
[5] 0 1 2 3 4 5