?bbcsd

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?bbcsd

Computes the CS decomposition of an orthogonal/unitary matrix in bidiagonal-block form.

Syntax

call sbbcsd( jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q, theta, phi, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, b11d, b11e, b12d, b12e, b21d, b21e, b21e, b22e, work, lwork, info )

call dbbcsd( jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q, theta, phi, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, b11d, b11e, b12d, b12e, b21d, b21e, b21e, b22e, work, lwork, info )

call cbbcsd( jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q, theta, phi, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, b11d, b11e, b12d, b12e, b21d, b21e, b21e, b22e, rwork, rlwork, info )

call zbbcsd( jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q, theta, phi, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, b11d, b11e, b12d, b12e, b21d, b21e, b21e, b22e, rwork, rlwork, info )

call bbcsd( theta,phi,u1,u2,v1t,v2t[,b11d][,b11e][,b12d][,b12e][,b21d][,b21e][,b22d][,b22e][,jobu1][,jobu2][,jobv1t][,jobv2t][,trans][,info] )

Include Files

mkl.fi, lapack.f90

Description

mkl_lapack.fiThe routine ?bbcsd computes the CS decomposition of an orthogonal or unitary matrix in bidiagonal-block form:

respectively.

x is m-by-m with the top-left block p-by-q. Note that q must not be larger than p, m-p, or m-q. If q is not the smallest index, x must be transposed and/or permuted in constant time using the trans option. See ?orcsd/?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 orthogonal/unitary matrices u₁, u₂, v₁^t, and v₂^t are input/output. The input matrices are pre- or post-multiplied by the appropriate singular vector matrices.

Input Parameters

jobu1

CHARACTER. If equals Y, then u₁ is updated. Otherwise, u₁ is not updated.

jobu2

CHARACTER. If equals Y, then u₂ is updated. Otherwise, u₂ is not updated.

jobv1t

CHARACTER. If equals Y, then v₁^t is updated. Otherwise, v₁^t is not updated.

jobv2t

CHARACTER. If equals Y, then v₂^t is updated. Otherwise, v₂^t is not updated.

trans

CHARACTER

= 'T':: x, u₁, u₂, v₁^t, v₂^t are stored in row-major order.
otherwise: x, u₁, u₂, v₁^t, v₂^t are stored in column-major order.

m

INTEGER. The number of rows and columns of the orthogonal/unitary matrix X in bidiagonal-block form.

p

INTEGER. The number of rows in the top-left block of x. 0 ≤p≤m.

≤

q

INTEGER. The number of columns in the top-left block of x. 0 q≤ min(p,m-p,m-q).

theta

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

Array, size (q).

On entry, the angles theta(1), ..., theta(q) that, along with phi(1), ..., phi(q-1), define the matrix in bidiagonal-block form as returned by orbdb/unbdb.

phi

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

Array, size (q-1).

The angles phi(1), ..., phi(q-1) that, along with theta(1), ..., theta(q), define the matrix in bidiagonal-block form as returned by orbdb/unbdb.

u1

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

Array, size (ldu1,p).

On entry, a p-by-p matrix.

ldu1

INTEGER. The leading dimension of the array u₁, ldu1≤ max(1, p).

u2

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

Array, size (ldu2,m-p).

On entry, an (m-p)-by-(m-p) matrix.

ldu2

INTEGER. The leading dimension of the array u₂, ldu2≤ max(1, m-p).

v1t

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

Array, size (ldv1t,q).

On entry, a q-by-q matrix.

ldv1t

INTEGER. The leading dimension of the array v1t, ldv1t≤ max(1, q).

v2t

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

Array, size (ldv2t,m-q).

On entry, an (m-q)-by-(m-q) matrix.

ldv2t

INTEGER. The leading dimension of the array v2t, ldv2t≤ max(1, m-q).

work

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

Workspace array, size (max(1,lwork)).

lwork

INTEGER. The size of the work array. lwork? max(1,8*q)

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla.

Output Parameters

theta

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

On exit, the angles whose cosines and sines define the diagonal blocks in the CS decomposition.

u1

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

On exit, u1 is postmultiplied by the left singular vector matrix common to [ b11 ; 0 ] and [ b12 0 0 ; 0 -I 0 ].

u2

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

On exit, u2 is postmultiplied by the left singular vector matrix common to [ b21 ; 0 ] and [ b22 0 0 ; 0 0 I ].

v1t

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

Array, size (q).

On exit, v1t is premultiplied by the transpose of the right singular vector matrix common to [ b11 ; 0 ] and [ b21 ; 0 ].

v2t

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

On exit, v2t is premultiplied by the transpose of the right singular vector matrix common to [ b12 0 0 ; 0 -I 0 ] and [ b22 0 0 ; 0 0 I ].

b11d

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

Array, size (q).

When ?bbcsd converges, b11d contains the cosines of theta(1), ..., theta(q). If ?bbcsd fails to converge, b11d contains the diagonal of the partially reduced top left block.

b11e

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

Array, size (q-1).

When ?bbcsd converges, b11e contains zeros. If ?bbcsd fails to converge, b11e contains the superdiagonal of the partially reduced top left block.

b12d

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

Array, size (q).

When ?bbcsd converges, b12d contains the negative sines of theta(1), ..., theta(q). If ?bbcsd fails to converge, b12d contains the diagonal of the partially reduced top right block.

b12e

REAL for sbbcsd

DOUBLE PRECISION for dbbcsd

COMPLEX for cbbcsd

DOUBLE COMPLEX for zbbcsd

Array, size (q-1).

When ?bbcsd converges, b12e contains zeros. If ?bbcsd fails to converge, b11e contains the superdiagonal of the partially reduced top right block.

info

INTEGER.

= 0: successful exit

< 0: if info = -i, the i-th argument has an illegal value

> 0: if ?bbcsd did not converge, info specifies the number of nonzero entries in phi, and b11d, b11e, etc. contain the partially reduced matrix.

Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see Fortran 95 Interface Conventions.

Specific details for the routine ?bbcsd interface are as follows:

theta: Holds the vector of length q.
phi: Holds the vector of length q-1.
u1: Holds the matrix of size (p,p).
u2: Holds the matrix of size (m-p,m-p).
v1t: Holds the matrix of size (q,q).
v2t: Holds the matrix of size (m-q,m-q).
b11d: Holds the vector of length q.
b11e: Holds the vector of length q-1.
b12d: Holds the vector of length q.
b12e: Holds the vector of length q-1.
b21d: Holds the vector of length q.
b21e: Holds the vector of length q-1.
b22d: Holds the vector of length q.
b22e: Holds the vector of length q-1.
jobsu1: Indicates whether u₁ is computed. Must be 'Y' or 'O'.
jobsu2: Indicates whether u₂ is computed. Must be 'Y' or 'O'.
jobv1t: Indicates whether v₁^t is computed. Must be 'Y' or 'O'.
jobv2t: Indicates whether v₂^t is computed. Must be 'Y' or 'O'.
trans: Must be 'N' or 'T'.

Parent topic: Cosine-Sine Decomposition: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?bbcsd

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