Cosine-Sine (CS) decomposition

Functions
int64_t	lapack::bbcsd (lapack::Job jobu1, lapack::Job jobu2, lapack::Job jobv1t, lapack::Job jobv2t, lapack::Op trans, int64_t m, int64_t p, int64_t q, double *theta, double *phi, double *U1, int64_t ldu1, double *U2, int64_t ldu2, double *V1T, int64_t ldv1t, double *V2T, int64_t ldv2t, double *B11D, double *B11E, double *B12D, double *B12E, double *B21D, double *B21E, double *B22D, double *B22E)
int64_t	lapack::bbcsd (lapack::Job jobu1, lapack::Job jobu2, lapack::Job jobv1t, lapack::Job jobv2t, lapack::Op trans, int64_t m, int64_t p, int64_t q, double *theta, double *phi, std::complex< double > *U1, int64_t ldu1, std::complex< double > *U2, int64_t ldu2, std::complex< double > *V1T, int64_t ldv1t, std::complex< double > *V2T, int64_t ldv2t, double *B11D, double *B11E, double *B12D, double *B12E, double *B21D, double *B21E, double *B22D, double *B22E)
	Computes the CS decomposition of a unitary matrix in bidiagonal-block form,.
int64_t	lapack::bbcsd (lapack::Job jobu1, lapack::Job jobu2, lapack::Job jobv1t, lapack::Job jobv2t, lapack::Op trans, int64_t m, int64_t p, int64_t q, float *theta, float *phi, float *U1, int64_t ldu1, float *U2, int64_t ldu2, float *V1T, int64_t ldv1t, float *V2T, int64_t ldv2t, float *B11D, float *B11E, float *B12D, float *B12E, float *B21D, float *B21E, float *B22D, float *B22E)
int64_t	lapack::bbcsd (lapack::Job jobu1, lapack::Job jobu2, lapack::Job jobv1t, lapack::Job jobv2t, lapack::Op trans, int64_t m, int64_t p, int64_t q, float *theta, float *phi, std::complex< float > *U1, int64_t ldu1, std::complex< float > *U2, int64_t ldu2, std::complex< float > *V1T, int64_t ldv1t, std::complex< float > *V2T, int64_t ldv2t, float *B11D, float *B11E, float *B12D, float *B12E, float *B21D, float *B21E, float *B22D, float *B22E)

Detailed Description

Function Documentation

bbcsd()

int64_t lapack::bbcsd	(	lapack::Job	jobu1,
		lapack::Job	jobu2,
		lapack::Job	jobv1t,
		lapack::Job	jobv2t,
		lapack::Op	trans,
		int64_t	m,
		int64_t	p,
		int64_t	q,
		double *	theta,
		double *	phi,
		std::complex< double > *	U1,
		int64_t	ldu1,
		std::complex< double > *	U2,
		int64_t	ldu2,
		std::complex< double > *	V1T,
		int64_t	ldv1t,
		std::complex< double > *	V2T,
		int64_t	ldv2t,
		double *	B11D,
		double *	B11E,
		double *	B12D,
		double *	B12E,
		double *	B21D,
		double *	B21E,
		double *	B22D,
		double *	B22E
	)

Computes the CS decomposition of a unitary matrix in bidiagonal-block form,.

\[ X = \begin{bmatrix} B_{11} & B_{12} & 0 & 0 \\ 0 & 0 & -I & 0 \\ \hline B_{21} & B_{22} & 0 & 0 \\ 0 & 0 & 0 & I \end{bmatrix} = \begin{bmatrix} U_{1} & \\ \hline & U_{2} \end{bmatrix} \begin{bmatrix} C & -S & 0 & 0 \\ 0 & 0 & -I & 0 \\ \hline S & C & 0 & 0 \\ 0 & 0 & 0 & I \end{bmatrix} \begin{bmatrix} V_{1} & \\ \hline & V_{2} \end{bmatrix}^H \]

X is m-by-m, its top-left block is p-by-q, and q must be no larger than p, m-p, or m-q. (If q is not the smallest index, then X must be transposed and/or permuted. This can be done in constant time using the trans and signs options. See lapack::uncsd for details.)

The bidiagonal matrices B11, B12, B21, and B22 are represented implicitly by angles theta(1:q) and phi(1:q-1).

The unitary matrices U1, U2, V1T, and V2T are input/output. The input matrices are pre- or post-multiplied by the appropriate singular vector matrices.

Overloaded versions are available for float, double, std::complex<float>, and std::complex<double>.

Parameters

[in]	jobu1	lapack::Job::UpdateVec: U1 is updated; lapack::Job::NoVec: U1 is not updated.
[in]	jobu2	lapack::Job::UpdateVec: U2 is updated; lapack::Job::NoVec: U2 is not updated.
[in]	jobv1t	lapack::Job::UpdateVec: V1T is updated; lapack::Job::NoVec: V1T is not updated.
[in]	jobv2t	lapack::Job::UpdateVec: V2T is updated; lapack::Job::NoVec: V2T is not updated.
[in]	trans	lapack::Op::Trans: X, U1, U2, V1T, and V2T are stored in row-major order; lapack::Op::NoTrans: X, U1, U2, V1T, and V2T are stored in column-major order.
[in]	m	The number of rows and columns in X, the unitary matrix in bidiagonal-block form.
[in]	p	The number of rows in the top-left block of X. 0 <= p <= m.
[in]	q	The number of columns in the top-left block of X. 0 <= q <= min(p, m-p, m-q).
[in,out]	theta	The vector theta of length q. On entry, the angles theta(1), ..., theta(q) that, along with phi(1), ..., phi(q-1), define the matrix in bidiagonal-block form. On exit, the angles whose cosines and sines define the diagonal blocks in the CS decomposition.
[in,out]	phi	The vector phi of length q-1. The angles phi(1), ..., phi(q-1) that, along with theta(1), ..., theta(q), define the matrix in bidiagonal-block form.
[in,out]	U1	The p-by-p matrix U1, stored in an ldu1-by-p array. On entry, a p-by-p matrix. On exit, U1 is postmultiplied by the left singular vector matrix common to [ B11 ; 0 ] and [ B12 0 0 ; 0 -I 0 0 ].
[in]	ldu1	The leading dimension of the array U1, ldu1 >= max(1,p).
[in,out]	U2	The (m-p)-by-(m-p) matrix U2, stored in an ldu2-by-(m-p) array. On entry, an (m-p)-by-(m-p) matrix. On exit, U2 is postmultiplied by the left singular vector matrix common to [ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ].
[in]	ldu2	The leading dimension of the array U2, ldu2 >= max(1,m-p).
[in,out]	V1T	The q-by-q matrix V1T, stored in an ldv1t-by-q array. On entry, a q-by-q matrix. On exit, V1T is premultiplied by the conjugate transpose of the right singular vector matrix common to [ B11 ; 0 ] and [ B21 ; 0 ].
[in]	ldv1t	The leading dimension of the array V1T, ldv1t >= max(1,q).
[in,out]	V2T	The (m-q)-by-(m-q) matrix V2T, stored in an ldv2t-by-(m-q) array. On entry, an (m-q)-by-(m-q) matrix. On exit, V2T is premultiplied by the conjugate transpose of the right singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and [ B22 0 0 ; 0 0 I ].
[in]	ldv2t	The leading dimension of the array V2T, ldv2t >= max(1,m-q).
[out]	B11D	The vector B11D of length q. When bbcsd converges, B11D contains the cosines of theta(1), ..., theta(q). If bbcsd fails to converge, then B11D contains the diagonal of the partially reduced top-left block.
[out]	B11E	The vector B11E of length q-1. When bbcsd converges, B11E contains zeros. If bbcsd fails to converge, then B11E contains the superdiagonal of the partially reduced top-left block.
[out]	B12D	The vector B12D of length q. When bbcsd converges, B12D contains the negative sines of theta(1), ..., theta(q). If bbcsd fails to converge, then B12D contains the diagonal of the partially reduced top-right block.
[out]	B12E	The vector B12E of length q-1. When bbcsd converges, B12E contains zeros. If bbcsd fails to converge, then B12E contains the subdiagonal of the partially reduced top-right block.
[out]	B21D	The vector B21D of length q. When bbcsd converges, B21D contains the negative sines of theta(1), ..., theta(q). If bbcsd fails to converge, then B21D contains the diagonal of the partially reduced bottom-left block.
[out]	B21E	The vector B21E of length q-1. When bbcsd converges, B21E contains zeros. If bbcsd fails to converge, then B21E contains the subdiagonal of the partially reduced bottom-left block.
[out]	B22D	The vector B22D of length q. When bbcsd converges, B22D contains the negative sines of theta(1), ..., theta(q). If bbcsd fails to converge, then B22D contains the diagonal of the partially reduced bottom-right block.
[out]	B22E	The vector B22E of length q-1. When bbcsd converges, B22E contains zeros. If bbcsd fails to converge, then B22E contains the subdiagonal of the partially reduced bottom-right block.

Returns

= 0: successful exit.

> 0: did not converge; return value specifies the number of nonzero entries in phi, and B11D, B11E, etc., contain the partially reduced matrix.