# cuncsd.f

Section: LAPACK (3)
Updated: Tue Nov 14 2017
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cuncsd.f

## SYNOPSIS

### Functions/Subroutines

recursive subroutine cuncsd (JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, WORK, LWORK, RWORK, LRWORK, IWORK, INFO)
CUNCSD

## Function/Subroutine Documentation

### recursive subroutine cuncsd (character JOBU1, character JOBU2, character JOBV1T, character JOBV2T, character TRANS, character SIGNS, integer M, integer P, integer Q, complex, dimension( ldx11, * ) X11, integer LDX11, complex, dimension( ldx12, * ) X12, integer LDX12, complex, dimension( ldx21, * ) X21, integer LDX21, complex, dimension( ldx22, * ) X22, integer LDX22, real, dimension( * ) THETA, complex, dimension( ldu1, * ) U1, integer LDU1, complex, dimension( ldu2, * ) U2, integer LDU2, complex, dimension( ldv1t, * ) V1T, integer LDV1T, complex, dimension( ldv2t, * ) V2T, integer LDV2T, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer INFO)

CUNCSD

Purpose:

``` CUNCSD computes the CS decomposition of an M-by-M partitioned
unitary matrix X:

[  I  0  0 |  0  0  0 ]
[  0  C  0 |  0 -S  0 ]
[ X11 | X12 ]   [ U1 |    ] [  0  0  0 |  0  0 -I ] [ V1 |    ]**H
X = [-----------] = [---------] [---------------------] [---------]   .
[ X21 | X22 ]   [    | U2 ] [  0  0  0 |  I  0  0 ] [    | V2 ]
[  0  S  0 |  0  C  0 ]
[  0  0  I |  0  0  0 ]

X11 is P-by-Q. The unitary matrices U1, U2, V1, and V2 are P-by-P,
(M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
which R = MIN(P,M-P,Q,M-Q).
```

Parameters:

JOBU1

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

JOBU2

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

JOBV1T

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

JOBV2T

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

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.
```

SIGNS

```          SIGNS is CHARACTER
= 'O':      The lower-left block is made nonpositive (the
"other" convention);
otherwise:  The upper-right block is made nonpositive (the
"default" convention).
```

M

```          M is INTEGER
The number of rows and columns in X.
```

P

```          P is INTEGER
The number of rows in X11 and X12. 0 <= P <= M.
```

Q

```          Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.
```

X11

```          X11 is COMPLEX array, dimension (LDX11,Q)
On entry, part of the unitary matrix whose CSD is desired.
```

LDX11

```          LDX11 is INTEGER
The leading dimension of X11. LDX11 >= MAX(1,P).
```

X12

```          X12 is COMPLEX array, dimension (LDX12,M-Q)
On entry, part of the unitary matrix whose CSD is desired.
```

LDX12

```          LDX12 is INTEGER
The leading dimension of X12. LDX12 >= MAX(1,P).
```

X21

```          X21 is COMPLEX array, dimension (LDX21,Q)
On entry, part of the unitary matrix whose CSD is desired.
```

LDX21

```          LDX21 is INTEGER
The leading dimension of X11. LDX21 >= MAX(1,M-P).
```

X22

```          X22 is COMPLEX array, dimension (LDX22,M-Q)
On entry, part of the unitary matrix whose CSD is desired.
```

LDX22

```          LDX22 is INTEGER
The leading dimension of X11. LDX22 >= MAX(1,M-P).
```

THETA

```          THETA is REAL array, dimension (R), in which R =
MIN(P,M-P,Q,M-Q).
C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).
```

U1

```          U1 is COMPLEX array, dimension (LDU1,P)
If JOBU1 = 'Y', U1 contains the P-by-P unitary matrix U1.
```

LDU1

```          LDU1 is INTEGER
The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
MAX(1,P).
```

U2

```          U2 is COMPLEX array, dimension (LDU2,M-P)
If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) unitary
matrix U2.
```

LDU2

```          LDU2 is INTEGER
The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
MAX(1,M-P).
```

V1T

```          V1T is COMPLEX array, dimension (LDV1T,Q)
If JOBV1T = 'Y', V1T contains the Q-by-Q matrix unitary
matrix V1**H.
```

LDV1T

```          LDV1T is INTEGER
The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
MAX(1,Q).
```

V2T

```          V2T is COMPLEX array, dimension (LDV2T,M-Q)
If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) unitary
matrix V2**H.
```

LDV2T

```          LDV2T is INTEGER
The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
MAX(1,M-Q).
```

WORK

```          WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.

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.
```

RWORK

```          RWORK is REAL array, dimension MAX(1,LRWORK)
On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
If INFO > 0 on exit, RWORK(2:R) contains the values PHI(1),
..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
define the matrix in intermediate bidiagonal-block form
remaining after nonconvergence. INFO specifies the number
of nonzero PHI's.
```

LRWORK

```          LRWORK is INTEGER
The dimension of the array RWORK.

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.
```

IWORK

```          IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))
```

INFO

```          INFO is INTEGER
= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  CBBCSD did not converge. See the description of RWORK
above for details.
```

References:

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

Author:

Univ. of Tennessee

Univ. of California Berkeley