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Manual

uncsd

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
recursive subroutine cuncsd (character jobu1, character jobu2, characterjobv1t, character jobv2t, character trans, character signs, integer m,integer p, integer q, complex, dimension( ldx11, * ) x11, integerldx11, 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, integerldu2, 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)
recursive subroutine dorcsd (character jobu1, character jobu2, characterjobv1t, character jobv2t, character trans, character signs, integer m,integer p, integer q, double precision, dimension( ldx11, * ) x11,integer ldx11, double precision, dimension( ldx12, * ) x12, integerldx12, double precision, dimension( ldx21, * ) x21, integer ldx21,double precision, dimension( ldx22, * ) x22,integer ldx22, double precision, dimension( * ) theta, doubleprecision, dimension( ldu1, * ) u1, integer ldu1, double precision,dimension( ldu2, * ) u2, integer ldu2, double precision, dimension(ldv1t, * ) v1t, integer ldv1t, double precision, dimension( ldv2t, * )v2t, integer ldv2t, double precision, dimension( * ) work, integerlwork, integer, dimension( * ) iwork, integer info)
recursive subroutine sorcsd (character jobu1, character jobu2, characterjobv1t, character jobv2t, character trans, character signs, integer m,integer p, integer q, real, dimension( ldx11, * ) x11, integer ldx11,real, dimension( ldx12, * ) x12, integer ldx12, real, dimension( ldx21,* ) x21, integer ldx21, real, dimension( ldx22,* ) x22, integer ldx22, real, dimension( * ) theta, real, dimension(ldu1, * ) u1, integer ldu1, real, dimension( ldu2, * ) u2, integerldu2, real, dimension( ldv1t, * ) v1t, integer ldv1t, real, dimension(ldv2t, * ) v2t, integer ldv2t, real, dimension( * ) work, integerlwork, integer, dimension( * ) iwork, integer info)
recursive subroutine zuncsd (character jobu1, character jobu2, characterjobv1t, character jobv2t, character trans, character signs, integer m,integer p, integer q, complex*16, dimension( ldx11, * ) x11, integerldx11, complex*16, dimension( ldx12, * ) x12, integer ldx12,complex*16, dimension( ldx21, * ) x21, integer ldx21, complex*16,dimension( ldx22, * ) x22, integer ldx22,double precision, dimension( * ) theta, complex*16, dimension( ldu1, *) u1, integer ldu1, complex*16, dimension( ldu2, * ) u2, integer ldu2,complex*16, dimension( ldv1t, * ) v1t, integer ldv1t, complex*16,dimension( ldv2t, * ) v2t, integer ldv2t, complex*16, dimension( * )work, integer lwork, double precision, dimension( * ) rwork, integerlrwork, integer, dimension( * ) iwork, integer info)
Author

NAME

uncsd - {un,or}csd: ??

SYNOPSIS

Functions

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
recursive subroutine dorcsd (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, iwork, info)
DORCSD
recursive subroutine sorcsd (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, iwork, info)
SORCSD
recursive subroutine zuncsd (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)
ZUNCSD

Detailed Description

Function Documentation

recursive subroutine cuncsd (character jobu1, character jobu2, characterjobv1t, character jobv2t, character trans, character signs, integer m,integer p, integer q, complex, dimension( ldx11, * ) x11, integerldx11, 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, integerldu2, 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 Cˆ2 + Sˆ2 = 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:

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

recursive subroutine dorcsd (character jobu1, character jobu2, characterjobv1t, character jobv2t, character trans, character signs, integer m,integer p, integer q, double precision, dimension( ldx11, * ) x11,integer ldx11, double precision, dimension( ldx12, * ) x12, integerldx12, double precision, dimension( ldx21, * ) x21, integer ldx21,double precision, dimension( ldx22, * ) x22,integer ldx22, double precision, dimension( * ) theta, doubleprecision, dimension( ldu1, * ) u1, integer ldu1, double precision,dimension( ldu2, * ) u2, integer ldu2, double precision, dimension(ldv1t, * ) v1t, integer ldv1t, double precision, dimension( ldv2t, * )v2t, integer ldv2t, double precision, dimension( * ) work, integerlwork, integer, dimension( * ) iwork, integer info)

DORCSD

Purpose:

DORCSD computes the CS decomposition of an M-by-M partitioned
orthogonal matrix X:

[ I 0 0 | 0 0 0 ]
[ 0 C 0 | 0 -S 0 ]
[ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T
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 orthogonal 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 Cˆ2 + Sˆ2 = 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 DOUBLE PRECISION array, dimension (LDX11,Q)
On entry, part of the orthogonal matrix whose CSD is desired.

LDX11

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

X12

X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
On entry, part of the orthogonal matrix whose CSD is desired.

LDX12

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

X21

X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
On entry, part of the orthogonal matrix whose CSD is desired.

LDX21

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

X22

X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
On entry, part of the orthogonal matrix whose CSD is desired.

LDX22

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

THETA

THETA is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDU1,P)
If JOBU1 = ’Y’, U1 contains the P-by-P orthogonal matrix U1.

LDU1

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

U2

U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
If JOBU2 = ’Y’, U2 contains the (M-P)-by-(M-P) orthogonal
matrix U2.

LDU2

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

V1T

V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
If JOBV1T = ’Y’, V1T contains the Q-by-Q matrix orthogonal
matrix V1**T.

LDV1T

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

V2T

V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q)
If JOBV2T = ’Y’, V2T contains the (M-Q)-by-(M-Q) orthogonal
matrix V2**T.

LDV2T

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

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
If INFO > 0 on exit, WORK(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.

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.

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: DBBCSD did not converge. See the description of WORK
above for details.

References:

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

recursive subroutine sorcsd (character jobu1, character jobu2, characterjobv1t, character jobv2t, character trans, character signs, integer m,integer p, integer q, real, dimension( ldx11, * ) x11, integer ldx11,real, dimension( ldx12, * ) x12, integer ldx12, real, dimension( ldx21,* ) x21, integer ldx21, real, dimension( ldx22,* ) x22, integer ldx22, real, dimension( * ) theta, real, dimension(ldu1, * ) u1, integer ldu1, real, dimension( ldu2, * ) u2, integerldu2, real, dimension( ldv1t, * ) v1t, integer ldv1t, real, dimension(ldv2t, * ) v2t, integer ldv2t, real, dimension( * ) work, integerlwork, integer, dimension( * ) iwork, integer info)

SORCSD

Purpose:

SORCSD computes the CS decomposition of an M-by-M partitioned
orthogonal matrix X:

[ I 0 0 | 0 0 0 ]
[ 0 C 0 | 0 -S 0 ]
[ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T
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 orthogonal 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 Cˆ2 + Sˆ2 = 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 REAL array, dimension (LDX11,Q)
On entry, part of the orthogonal matrix whose CSD is desired.

LDX11

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

X12

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

LDX12

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

X21

X21 is REAL array, dimension (LDX21,Q)
On entry, part of the orthogonal matrix whose CSD is desired.

LDX21

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

X22

X22 is REAL array, dimension (LDX22,M-Q)
On entry, part of the orthogonal 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 REAL array, dimension (LDU1,P)
If JOBU1 = ’Y’, U1 contains the P-by-P orthogonal matrix U1.

LDU1

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

U2

U2 is REAL array, dimension (LDU2,M-P)
If JOBU2 = ’Y’, U2 contains the (M-P)-by-(M-P) orthogonal
matrix U2.

LDU2

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

V1T

V1T is REAL array, dimension (LDV1T,Q)
If JOBV1T = ’Y’, V1T contains the Q-by-Q matrix orthogonal
matrix V1**T.

LDV1T

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

V2T

V2T is REAL array, dimension (LDV2T,M-Q)
If JOBV2T = ’Y’, V2T contains the (M-Q)-by-(M-Q) orthogonal
matrix V2**T.

LDV2T

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

WORK

WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
If INFO > 0 on exit, WORK(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.

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.

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: SBBCSD did not converge. See the description of WORK
above for details.

References:

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

recursive subroutine zuncsd (character jobu1, character jobu2, characterjobv1t, character jobv2t, character trans, character signs, integer m,integer p, integer q, complex*16, dimension( ldx11, * ) x11, integerldx11, complex*16, dimension( ldx12, * ) x12, integer ldx12,complex*16, dimension( ldx21, * ) x21, integer ldx21, complex*16,dimension( ldx22, * ) x22, integer ldx22,double precision, dimension( * ) theta, complex*16, dimension( ldu1, *) u1, integer ldu1, complex*16, dimension( ldu2, * ) u2, integer ldu2,complex*16, dimension( ldv1t, * ) v1t, integer ldv1t, complex*16,dimension( ldv2t, * ) v2t, integer ldv2t, complex*16, dimension( * )work, integer lwork, double precision, dimension( * ) rwork, integerlrwork, integer, dimension( * ) iwork, integer info)

ZUNCSD

Purpose:

ZUNCSD 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 Cˆ2 + Sˆ2 = 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*16 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*16 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*16 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*16 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 DOUBLE PRECISION 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*16 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*16 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*16 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*16 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*16 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 DOUBLE PRECISION 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: ZBBCSD did not converge. See the description of RWORK
above for details.

References:

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Author

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