# cbbcsd.f

Section: LAPACK (3)
Updated: Tue Nov 14 2017
Page Index

cbbcsd.f

## SYNOPSIS

### Functions/Subroutines

subroutine 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, B22D, B22E, RWORK, LRWORK, INFO)
CBBCSD

## Function/Subroutine Documentation

### subroutine cbbcsd (character JOBU1, character JOBU2, character JOBV1T, character JOBV2T, character TRANS, integer M, integer P, integer Q, real, dimension( * ) THETA, real, dimension( * ) PHI, complex, dimension( ldu1, * ) U1, integer LDU1, complex, dimension( ldu2, * ) U2, integer LDU2, complex, dimension( ldv1t, * ) V1T, integer LDV1T, complex, dimension( ldv2t, * ) V2T, integer LDV2T, real, dimension( * ) B11D, real, dimension( * ) B11E, real, dimension( * ) B12D, real, dimension( * ) B12E, real, dimension( * ) B21D, real, dimension( * ) B21E, real, dimension( * ) B22D, real, dimension( * ) B22E, real, dimension( * ) RWORK, integer LRWORK, integer INFO)

CBBCSD

Purpose:

``` CBBCSD computes the CS decomposition of a unitary matrix in
bidiagonal-block form,

[ B11 | B12 0  0 ]
[  0  |  0 -I  0 ]
X = [----------------]
[ B21 | B22 0  0 ]
[  0  |  0  0  I ]

[  C | -S  0  0 ]
[ U1 |    ] [  0 |  0 -I  0 ] [ V1 |    ]**H
= [---------] [---------------] [---------]   .
[    | U2 ] [  S |  C  0  0 ] [    | V2 ]
[  0 |  0  0  I ]

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 CUNCSD 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.
```

Parameters:

JOBU1

```          JOBU1 is CHARACTER
= 'Y':      U1 is updated;
otherwise:  U1 is not updated.
```

JOBU2

```          JOBU2 is CHARACTER
= 'Y':      U2 is updated;
otherwise:  U2 is not updated.
```

JOBV1T

```          JOBV1T is CHARACTER
= 'Y':      V1T is updated;
otherwise:  V1T is not updated.
```

JOBV2T

```          JOBV2T is CHARACTER
= 'Y':      V2T is updated;
otherwise:  V2T is not updated.
```

TRANS

```          TRANS is CHARACTER
= 'T':      X, U1, U2, V1T, and V2T are stored in row-major
order;
otherwise:  X, U1, U2, V1T, and V2T are stored in column-
major order.
```

M

```          M is INTEGER
The number of rows and columns in X, the unitary matrix in
bidiagonal-block form.
```

P

```          P is INTEGER
The number of rows in the top-left block of X. 0 <= P <= M.
```

Q

```          Q is INTEGER
The number of columns in the top-left block of X.
0 <= Q <= MIN(P,M-P,M-Q).
```

THETA

```          THETA is REAL array, dimension (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.
```

PHI

```          PHI is REAL array, dimension (Q-1)
The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,
THETA(Q), define the matrix in bidiagonal-block form.
```

U1

```          U1 is COMPLEX array, dimension (LDU1,P)
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 ].
```

LDU1

```          LDU1 is INTEGER
The leading dimension of the array U1, LDU1 >= MAX(1,P).
```

U2

```          U2 is COMPLEX array, dimension (LDU2,M-P)
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 ].
```

LDU2

```          LDU2 is INTEGER
The leading dimension of the array U2, LDU2 >= MAX(1,M-P).
```

V1T

```          V1T is COMPLEX array, dimension (LDV1T,Q)
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 ].
```

LDV1T

```          LDV1T is INTEGER
The leading dimension of the array V1T, LDV1T >= MAX(1,Q).
```

V2T

```          V2T is COMPLEX array, dimension (LDV2T,M-Q)
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 ].
```

LDV2T

```          LDV2T is INTEGER
The leading dimension of the array V2T, LDV2T >= MAX(1,M-Q).
```

B11D

```          B11D is REAL array, dimension (Q)
When CBBCSD converges, B11D contains the cosines of THETA(1),
..., THETA(Q). If CBBCSD fails to converge, then B11D
contains the diagonal of the partially reduced top-left
block.
```

B11E

```          B11E is REAL array, dimension (Q-1)
When CBBCSD converges, B11E contains zeros. If CBBCSD fails
to converge, then B11E contains the superdiagonal of the
partially reduced top-left block.
```

B12D

```          B12D is REAL array, dimension (Q)
When CBBCSD converges, B12D contains the negative sines of
THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then
B12D contains the diagonal of the partially reduced top-right
block.
```

B12E

```          B12E is REAL array, dimension (Q-1)
When CBBCSD converges, B12E contains zeros. If CBBCSD fails
to converge, then B12E contains the subdiagonal of the
partially reduced top-right block.
```

B21D

```          B21D is REAL array, dimension (Q)
When CBBCSD converges, B21D contains the negative sines of
THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then
B21D contains the diagonal of the partially reduced bottom-left
block.
```

B21E

```          B21E is REAL array, dimension (Q-1)
When CBBCSD converges, B21E contains zeros. If CBBCSD fails
to converge, then B21E contains the subdiagonal of the
partially reduced bottom-left block.
```

B22D

```          B22D is REAL array, dimension (Q)
When CBBCSD converges, B22D contains the negative sines of
THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then
B22D contains the diagonal of the partially reduced bottom-right
block.
```

B22E

```          B22E is REAL array, dimension (Q-1)
When CBBCSD converges, B22E contains zeros. If CBBCSD fails
to converge, then B22E contains the subdiagonal of the
partially reduced bottom-right block.
```

RWORK

```          RWORK is REAL array, dimension (MAX(1,LRWORK))
On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
```

LRWORK

```          LRWORK is INTEGER
The dimension of the array RWORK. LRWORK >= MAX(1,8*Q).

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

INFO

```          INFO is INTEGER
= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  if CBBCSD did not converge, INFO specifies the number
of nonzero entries in PHI, and B11D, B11E, etc.,
contain the partially reduced matrix.
```

Internal Parameters:

```  TOLMUL  REAL, default = MAX(10,MIN(100,EPS**(-1/8)))
TOLMUL controls the convergence criterion of the QR loop.
Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they
are within TOLMUL*EPS of either bound.
```

References:

 Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Author:

Univ. of Tennessee

Univ. of California Berkeley