.TH "unbdb" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
unbdb \- {un,or}bdb: bidiagonalize partitioned unitary matrix, step in uncsd
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBcunbdb\fP (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info)"
.br
.RI "\fBCUNBDB\fP "
.ti -1c
.RI "subroutine \fBdorbdb\fP (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info)"
.br
.RI "\fBDORBDB\fP "
.ti -1c
.RI "subroutine \fBsorbdb\fP (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info)"
.br
.RI "\fBSORBDB\fP "
.ti -1c
.RI "subroutine \fBzunbdb\fP (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info)"
.br
.RI "\fBZUNBDB\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine cunbdb (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, real, dimension( * ) phi, complex, dimension( * ) taup1, complex, dimension( * ) taup2, complex, dimension( * ) tauq1, complex, dimension( * ) tauq2, complex, dimension( * ) work, integer lwork, integer info)"

.PP
\fBCUNBDB\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> CUNBDB simultaneously bidiagonalizes the blocks of an M-by-M
!> partitioned unitary matrix X:
!>
!>                                 [ B11 | B12 0  0 ]
!>     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**H
!> X = [-----------] = [---------] [----------------] [---------]   \&.
!>     [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
!>                                 [  0  |  0  0  I ]
!>
!> X11 is P-by-Q\&. Q must be no larger than P, M-P, or M-Q\&. (If this is
!> not the case, 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 unitary matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
!> (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively\&. They are
!> represented implicitly by Householder vectors\&.
!>
!> B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
!> implicitly by angles THETA, PHI\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fITRANS\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fISIGNS\fP 
.PP
.nf
!>          SIGNS is CHARACTER
!>          = 'O':      The lower-left block is made nonpositive (the
!>                       convention);
!>          otherwise:  The upper-right block is made nonpositive (the
!>                       convention)\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          M is INTEGER
!>          The number of rows and columns in X\&.
!> 
.fi
.PP
.br
\fIP\fP 
.PP
.nf
!>          P is INTEGER
!>          The number of rows in X11 and X12\&. 0 <= P <= M\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          Q is INTEGER
!>          The number of columns in X11 and X21\&. 0 <= Q <=
!>          MIN(P,M-P,M-Q)\&.
!> 
.fi
.PP
.br
\fIX11\fP 
.PP
.nf
!>          X11 is COMPLEX array, dimension (LDX11,Q)
!>          On entry, the top-left block of the unitary matrix to be
!>          reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the columns of tril(X11) specify reflectors for P1,
!>             the rows of triu(X11,1) specify reflectors for Q1;
!>          else TRANS = 'T', and
!>             the rows of triu(X11) specify reflectors for P1,
!>             the columns of tril(X11,-1) specify reflectors for Q1\&.
!> 
.fi
.PP
.br
\fILDX11\fP 
.PP
.nf
!>          LDX11 is INTEGER
!>          The leading dimension of X11\&. If TRANS = 'N', then LDX11 >=
!>          P; else LDX11 >= Q\&.
!> 
.fi
.PP
.br
\fIX12\fP 
.PP
.nf
!>          X12 is COMPLEX array, dimension (LDX12,M-Q)
!>          On entry, the top-right block of the unitary matrix to
!>          be reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the rows of triu(X12) specify the first P reflectors for
!>             Q2;
!>          else TRANS = 'T', and
!>             the columns of tril(X12) specify the first P reflectors
!>             for Q2\&.
!> 
.fi
.PP
.br
\fILDX12\fP 
.PP
.nf
!>          LDX12 is INTEGER
!>          The leading dimension of X12\&. If TRANS = 'N', then LDX12 >=
!>          P; else LDX11 >= M-Q\&.
!> 
.fi
.PP
.br
\fIX21\fP 
.PP
.nf
!>          X21 is COMPLEX array, dimension (LDX21,Q)
!>          On entry, the bottom-left block of the unitary matrix to
!>          be reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the columns of tril(X21) specify reflectors for P2;
!>          else TRANS = 'T', and
!>             the rows of triu(X21) specify reflectors for P2\&.
!> 
.fi
.PP
.br
\fILDX21\fP 
.PP
.nf
!>          LDX21 is INTEGER
!>          The leading dimension of X21\&. If TRANS = 'N', then LDX21 >=
!>          M-P; else LDX21 >= Q\&.
!> 
.fi
.PP
.br
\fIX22\fP 
.PP
.nf
!>          X22 is COMPLEX array, dimension (LDX22,M-Q)
!>          On entry, the bottom-right block of the unitary matrix to
!>          be reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
!>             M-P-Q reflectors for Q2,
!>          else TRANS = 'T', and
!>             the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
!>             M-P-Q reflectors for P2\&.
!> 
.fi
.PP
.br
\fILDX22\fP 
.PP
.nf
!>          LDX22 is INTEGER
!>          The leading dimension of X22\&. If TRANS = 'N', then LDX22 >=
!>          M-P; else LDX22 >= M-Q\&.
!> 
.fi
.PP
.br
\fITHETA\fP 
.PP
.nf
!>          THETA is REAL array, dimension (Q)
!>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
!>          be computed from the angles THETA and PHI\&. See Further
!>          Details\&.
!> 
.fi
.PP
.br
\fIPHI\fP 
.PP
.nf
!>          PHI is REAL array, dimension (Q-1)
!>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
!>          be computed from the angles THETA and PHI\&. See Further
!>          Details\&.
!> 
.fi
.PP
.br
\fITAUP1\fP 
.PP
.nf
!>          TAUP1 is COMPLEX array, dimension (P)
!>          The scalar factors of the elementary reflectors that define
!>          P1\&.
!> 
.fi
.PP
.br
\fITAUP2\fP 
.PP
.nf
!>          TAUP2 is COMPLEX array, dimension (M-P)
!>          The scalar factors of the elementary reflectors that define
!>          P2\&.
!> 
.fi
.PP
.br
\fITAUQ1\fP 
.PP
.nf
!>          TAUQ1 is COMPLEX array, dimension (Q)
!>          The scalar factors of the elementary reflectors that define
!>          Q1\&.
!> 
.fi
.PP
.br
\fITAUQ2\fP 
.PP
.nf
!>          TAUQ2 is COMPLEX array, dimension (M-Q)
!>          The scalar factors of the elementary reflectors that define
!>          Q2\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is COMPLEX array, dimension (LWORK)
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          LWORK is INTEGER
!>          The dimension of the array WORK\&. LWORK >= M-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\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit\&.
!>          < 0:  if INFO = -i, the i-th argument had an illegal value\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  The bidiagonal blocks B11, B12, B21, and B22 are represented
!>  implicitly by angles THETA(1), \&.\&.\&., THETA(Q) and PHI(1), \&.\&.\&.,
!>  PHI(Q-1)\&. B11 and B21 are upper bidiagonal, while B21 and B22 are
!>  lower bidiagonal\&. Every entry in each bidiagonal band is a product
!>  of a sine or cosine of a THETA with a sine or cosine of a PHI\&. See
!>  [1] or CUNCSD for details\&.
!>
!>  P1, P2, Q1, and Q2 are represented as products of elementary
!>  reflectors\&. See CUNCSD for details on generating P1, P2, Q1, and Q2
!>  using CUNGQR and CUNGLQ\&.
!> 
.fi
.PP
 
.RE
.PP
\fBReferences:\fP
.RS 4
[1] Brian D\&. Sutton\&. Computing the complete CS decomposition\&. Numer\&. Algorithms, 50(1):33-65, 2009\&. 
.RE
.PP

.PP
Definition at line \fB284\fP of file \fBcunbdb\&.f\fP\&.
.SS "subroutine dorbdb (character trans, character signs, integer m, integer p, integer q, double precision, dimension( ldx11, * ) x11, integer ldx11, double precision, dimension( ldx12, * ) x12, integer ldx12, double precision, dimension( ldx21, * ) x21, integer ldx21, double precision, dimension( ldx22, * ) x22, integer ldx22, double precision, dimension( * ) theta, double precision, dimension( * ) phi, double precision, dimension( * ) taup1, double precision, dimension( * ) taup2, double precision, dimension( * ) tauq1, double precision, dimension( * ) tauq2, double precision, dimension( * ) work, integer lwork, integer info)"

.PP
\fBDORBDB\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> DORBDB simultaneously bidiagonalizes the blocks of an M-by-M
!> partitioned orthogonal matrix X:
!>
!>                                 [ B11 | B12 0  0 ]
!>     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**T
!> X = [-----------] = [---------] [----------------] [---------]   \&.
!>     [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
!>                                 [  0  |  0  0  I ]
!>
!> X11 is P-by-Q\&. Q must be no larger than P, M-P, or M-Q\&. (If this is
!> not the case, then X must be transposed and/or permuted\&. This can be
!> done in constant time using the TRANS and SIGNS options\&. See DORCSD
!> for details\&.)
!>
!> The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
!> (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively\&. They are
!> represented implicitly by Householder vectors\&.
!>
!> B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
!> implicitly by angles THETA, PHI\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fITRANS\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fISIGNS\fP 
.PP
.nf
!>          SIGNS is CHARACTER
!>          = 'O':      The lower-left block is made nonpositive (the
!>                       convention);
!>          otherwise:  The upper-right block is made nonpositive (the
!>                       convention)\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          M is INTEGER
!>          The number of rows and columns in X\&.
!> 
.fi
.PP
.br
\fIP\fP 
.PP
.nf
!>          P is INTEGER
!>          The number of rows in X11 and X12\&. 0 <= P <= M\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          Q is INTEGER
!>          The number of columns in X11 and X21\&. 0 <= Q <=
!>          MIN(P,M-P,M-Q)\&.
!> 
.fi
.PP
.br
\fIX11\fP 
.PP
.nf
!>          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
!>          On entry, the top-left block of the orthogonal matrix to be
!>          reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the columns of tril(X11) specify reflectors for P1,
!>             the rows of triu(X11,1) specify reflectors for Q1;
!>          else TRANS = 'T', and
!>             the rows of triu(X11) specify reflectors for P1,
!>             the columns of tril(X11,-1) specify reflectors for Q1\&.
!> 
.fi
.PP
.br
\fILDX11\fP 
.PP
.nf
!>          LDX11 is INTEGER
!>          The leading dimension of X11\&. If TRANS = 'N', then LDX11 >=
!>          P; else LDX11 >= Q\&.
!> 
.fi
.PP
.br
\fIX12\fP 
.PP
.nf
!>          X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
!>          On entry, the top-right block of the orthogonal matrix to
!>          be reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the rows of triu(X12) specify the first P reflectors for
!>             Q2;
!>          else TRANS = 'T', and
!>             the columns of tril(X12) specify the first P reflectors
!>             for Q2\&.
!> 
.fi
.PP
.br
\fILDX12\fP 
.PP
.nf
!>          LDX12 is INTEGER
!>          The leading dimension of X12\&. If TRANS = 'N', then LDX12 >=
!>          P; else LDX11 >= M-Q\&.
!> 
.fi
.PP
.br
\fIX21\fP 
.PP
.nf
!>          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
!>          On entry, the bottom-left block of the orthogonal matrix to
!>          be reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the columns of tril(X21) specify reflectors for P2;
!>          else TRANS = 'T', and
!>             the rows of triu(X21) specify reflectors for P2\&.
!> 
.fi
.PP
.br
\fILDX21\fP 
.PP
.nf
!>          LDX21 is INTEGER
!>          The leading dimension of X21\&. If TRANS = 'N', then LDX21 >=
!>          M-P; else LDX21 >= Q\&.
!> 
.fi
.PP
.br
\fIX22\fP 
.PP
.nf
!>          X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
!>          On entry, the bottom-right block of the orthogonal matrix to
!>          be reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
!>             M-P-Q reflectors for Q2,
!>          else TRANS = 'T', and
!>             the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
!>             M-P-Q reflectors for P2\&.
!> 
.fi
.PP
.br
\fILDX22\fP 
.PP
.nf
!>          LDX22 is INTEGER
!>          The leading dimension of X22\&. If TRANS = 'N', then LDX22 >=
!>          M-P; else LDX22 >= M-Q\&.
!> 
.fi
.PP
.br
\fITHETA\fP 
.PP
.nf
!>          THETA is DOUBLE PRECISION array, dimension (Q)
!>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
!>          be computed from the angles THETA and PHI\&. See Further
!>          Details\&.
!> 
.fi
.PP
.br
\fIPHI\fP 
.PP
.nf
!>          PHI is DOUBLE PRECISION array, dimension (Q-1)
!>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
!>          be computed from the angles THETA and PHI\&. See Further
!>          Details\&.
!> 
.fi
.PP
.br
\fITAUP1\fP 
.PP
.nf
!>          TAUP1 is DOUBLE PRECISION array, dimension (P)
!>          The scalar factors of the elementary reflectors that define
!>          P1\&.
!> 
.fi
.PP
.br
\fITAUP2\fP 
.PP
.nf
!>          TAUP2 is DOUBLE PRECISION array, dimension (M-P)
!>          The scalar factors of the elementary reflectors that define
!>          P2\&.
!> 
.fi
.PP
.br
\fITAUQ1\fP 
.PP
.nf
!>          TAUQ1 is DOUBLE PRECISION array, dimension (Q)
!>          The scalar factors of the elementary reflectors that define
!>          Q1\&.
!> 
.fi
.PP
.br
\fITAUQ2\fP 
.PP
.nf
!>          TAUQ2 is DOUBLE PRECISION array, dimension (M-Q)
!>          The scalar factors of the elementary reflectors that define
!>          Q2\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is DOUBLE PRECISION array, dimension (LWORK)
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          LWORK is INTEGER
!>          The dimension of the array WORK\&. LWORK >= M-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\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit\&.
!>          < 0:  if INFO = -i, the i-th argument had an illegal value\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  The bidiagonal blocks B11, B12, B21, and B22 are represented
!>  implicitly by angles THETA(1), \&.\&.\&., THETA(Q) and PHI(1), \&.\&.\&.,
!>  PHI(Q-1)\&. B11 and B21 are upper bidiagonal, while B21 and B22 are
!>  lower bidiagonal\&. Every entry in each bidiagonal band is a product
!>  of a sine or cosine of a THETA with a sine or cosine of a PHI\&. See
!>  [1] or DORCSD for details\&.
!>
!>  P1, P2, Q1, and Q2 are represented as products of elementary
!>  reflectors\&. See DORCSD for details on generating P1, P2, Q1, and Q2
!>  using DORGQR and DORGLQ\&.
!> 
.fi
.PP
 
.RE
.PP
\fBReferences:\fP
.RS 4
[1] Brian D\&. Sutton\&. Computing the complete CS decomposition\&. Numer\&. Algorithms, 50(1):33-65, 2009\&. 
.RE
.PP

.PP
Definition at line \fB284\fP of file \fBdorbdb\&.f\fP\&.
.SS "subroutine sorbdb (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( * ) phi, real, dimension( * ) taup1, real, dimension( * ) taup2, real, dimension( * ) tauq1, real, dimension( * ) tauq2, real, dimension( * ) work, integer lwork, integer info)"

.PP
\fBSORBDB\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> SORBDB simultaneously bidiagonalizes the blocks of an M-by-M
!> partitioned orthogonal matrix X:
!>
!>                                 [ B11 | B12 0  0 ]
!>     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**T
!> X = [-----------] = [---------] [----------------] [---------]   \&.
!>     [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
!>                                 [  0  |  0  0  I ]
!>
!> X11 is P-by-Q\&. Q must be no larger than P, M-P, or M-Q\&. (If this is
!> not the case, then X must be transposed and/or permuted\&. This can be
!> done in constant time using the TRANS and SIGNS options\&. See SORCSD
!> for details\&.)
!>
!> The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
!> (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively\&. They are
!> represented implicitly by Householder vectors\&.
!>
!> B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
!> implicitly by angles THETA, PHI\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fITRANS\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fISIGNS\fP 
.PP
.nf
!>          SIGNS is CHARACTER
!>          = 'O':      The lower-left block is made nonpositive (the
!>                       convention);
!>          otherwise:  The upper-right block is made nonpositive (the
!>                       convention)\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          M is INTEGER
!>          The number of rows and columns in X\&.
!> 
.fi
.PP
.br
\fIP\fP 
.PP
.nf
!>          P is INTEGER
!>          The number of rows in X11 and X12\&. 0 <= P <= M\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          Q is INTEGER
!>          The number of columns in X11 and X21\&. 0 <= Q <=
!>          MIN(P,M-P,M-Q)\&.
!> 
.fi
.PP
.br
\fIX11\fP 
.PP
.nf
!>          X11 is REAL array, dimension (LDX11,Q)
!>          On entry, the top-left block of the orthogonal matrix to be
!>          reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the columns of tril(X11) specify reflectors for P1,
!>             the rows of triu(X11,1) specify reflectors for Q1;
!>          else TRANS = 'T', and
!>             the rows of triu(X11) specify reflectors for P1,
!>             the columns of tril(X11,-1) specify reflectors for Q1\&.
!> 
.fi
.PP
.br
\fILDX11\fP 
.PP
.nf
!>          LDX11 is INTEGER
!>          The leading dimension of X11\&. If TRANS = 'N', then LDX11 >=
!>          P; else LDX11 >= Q\&.
!> 
.fi
.PP
.br
\fIX12\fP 
.PP
.nf
!>          X12 is REAL array, dimension (LDX12,M-Q)
!>          On entry, the top-right block of the orthogonal matrix to
!>          be reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the rows of triu(X12) specify the first P reflectors for
!>             Q2;
!>          else TRANS = 'T', and
!>             the columns of tril(X12) specify the first P reflectors
!>             for Q2\&.
!> 
.fi
.PP
.br
\fILDX12\fP 
.PP
.nf
!>          LDX12 is INTEGER
!>          The leading dimension of X12\&. If TRANS = 'N', then LDX12 >=
!>          P; else LDX11 >= M-Q\&.
!> 
.fi
.PP
.br
\fIX21\fP 
.PP
.nf
!>          X21 is REAL array, dimension (LDX21,Q)
!>          On entry, the bottom-left block of the orthogonal matrix to
!>          be reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the columns of tril(X21) specify reflectors for P2;
!>          else TRANS = 'T', and
!>             the rows of triu(X21) specify reflectors for P2\&.
!> 
.fi
.PP
.br
\fILDX21\fP 
.PP
.nf
!>          LDX21 is INTEGER
!>          The leading dimension of X21\&. If TRANS = 'N', then LDX21 >=
!>          M-P; else LDX21 >= Q\&.
!> 
.fi
.PP
.br
\fIX22\fP 
.PP
.nf
!>          X22 is REAL array, dimension (LDX22,M-Q)
!>          On entry, the bottom-right block of the orthogonal matrix to
!>          be reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
!>             M-P-Q reflectors for Q2,
!>          else TRANS = 'T', and
!>             the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
!>             M-P-Q reflectors for P2\&.
!> 
.fi
.PP
.br
\fILDX22\fP 
.PP
.nf
!>          LDX22 is INTEGER
!>          The leading dimension of X22\&. If TRANS = 'N', then LDX22 >=
!>          M-P; else LDX22 >= M-Q\&.
!> 
.fi
.PP
.br
\fITHETA\fP 
.PP
.nf
!>          THETA is REAL array, dimension (Q)
!>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
!>          be computed from the angles THETA and PHI\&. See Further
!>          Details\&.
!> 
.fi
.PP
.br
\fIPHI\fP 
.PP
.nf
!>          PHI is REAL array, dimension (Q-1)
!>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
!>          be computed from the angles THETA and PHI\&. See Further
!>          Details\&.
!> 
.fi
.PP
.br
\fITAUP1\fP 
.PP
.nf
!>          TAUP1 is REAL array, dimension (P)
!>          The scalar factors of the elementary reflectors that define
!>          P1\&.
!> 
.fi
.PP
.br
\fITAUP2\fP 
.PP
.nf
!>          TAUP2 is REAL array, dimension (M-P)
!>          The scalar factors of the elementary reflectors that define
!>          P2\&.
!> 
.fi
.PP
.br
\fITAUQ1\fP 
.PP
.nf
!>          TAUQ1 is REAL array, dimension (Q)
!>          The scalar factors of the elementary reflectors that define
!>          Q1\&.
!> 
.fi
.PP
.br
\fITAUQ2\fP 
.PP
.nf
!>          TAUQ2 is REAL array, dimension (M-Q)
!>          The scalar factors of the elementary reflectors that define
!>          Q2\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is REAL array, dimension (LWORK)
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          LWORK is INTEGER
!>          The dimension of the array WORK\&. LWORK >= M-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\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit\&.
!>          < 0:  if INFO = -i, the i-th argument had an illegal value\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  The bidiagonal blocks B11, B12, B21, and B22 are represented
!>  implicitly by angles THETA(1), \&.\&.\&., THETA(Q) and PHI(1), \&.\&.\&.,
!>  PHI(Q-1)\&. B11 and B21 are upper bidiagonal, while B21 and B22 are
!>  lower bidiagonal\&. Every entry in each bidiagonal band is a product
!>  of a sine or cosine of a THETA with a sine or cosine of a PHI\&. See
!>  [1] or SORCSD for details\&.
!>
!>  P1, P2, Q1, and Q2 are represented as products of elementary
!>  reflectors\&. See SORCSD for details on generating P1, P2, Q1, and Q2
!>  using SORGQR and SORGLQ\&.
!> 
.fi
.PP
 
.RE
.PP
\fBReferences:\fP
.RS 4
[1] Brian D\&. Sutton\&. Computing the complete CS decomposition\&. Numer\&. Algorithms, 50(1):33-65, 2009\&. 
.RE
.PP

.PP
Definition at line \fB284\fP of file \fBsorbdb\&.f\fP\&.
.SS "subroutine zunbdb (character trans, character signs, integer m, integer p, integer q, complex*16, dimension( ldx11, * ) x11, integer ldx11, 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, double precision, dimension( * ) phi, complex*16, dimension( * ) taup1, complex*16, dimension( * ) taup2, complex*16, dimension( * ) tauq1, complex*16, dimension( * ) tauq2, complex*16, dimension( * ) work, integer lwork, integer info)"

.PP
\fBZUNBDB\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> ZUNBDB simultaneously bidiagonalizes the blocks of an M-by-M
!> partitioned unitary matrix X:
!>
!>                                 [ B11 | B12 0  0 ]
!>     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**H
!> X = [-----------] = [---------] [----------------] [---------]   \&.
!>     [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
!>                                 [  0  |  0  0  I ]
!>
!> X11 is P-by-Q\&. Q must be no larger than P, M-P, or M-Q\&. (If this is
!> not the case, then X must be transposed and/or permuted\&. This can be
!> done in constant time using the TRANS and SIGNS options\&. See ZUNCSD
!> for details\&.)
!>
!> The unitary matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
!> (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively\&. They are
!> represented implicitly by Householder vectors\&.
!>
!> B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
!> implicitly by angles THETA, PHI\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fITRANS\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fISIGNS\fP 
.PP
.nf
!>          SIGNS is CHARACTER
!>          = 'O':      The lower-left block is made nonpositive (the
!>                       convention);
!>          otherwise:  The upper-right block is made nonpositive (the
!>                       convention)\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          M is INTEGER
!>          The number of rows and columns in X\&.
!> 
.fi
.PP
.br
\fIP\fP 
.PP
.nf
!>          P is INTEGER
!>          The number of rows in X11 and X12\&. 0 <= P <= M\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          Q is INTEGER
!>          The number of columns in X11 and X21\&. 0 <= Q <=
!>          MIN(P,M-P,M-Q)\&.
!> 
.fi
.PP
.br
\fIX11\fP 
.PP
.nf
!>          X11 is COMPLEX*16 array, dimension (LDX11,Q)
!>          On entry, the top-left block of the unitary matrix to be
!>          reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the columns of tril(X11) specify reflectors for P1,
!>             the rows of triu(X11,1) specify reflectors for Q1;
!>          else TRANS = 'T', and
!>             the rows of triu(X11) specify reflectors for P1,
!>             the columns of tril(X11,-1) specify reflectors for Q1\&.
!> 
.fi
.PP
.br
\fILDX11\fP 
.PP
.nf
!>          LDX11 is INTEGER
!>          The leading dimension of X11\&. If TRANS = 'N', then LDX11 >=
!>          P; else LDX11 >= Q\&.
!> 
.fi
.PP
.br
\fIX12\fP 
.PP
.nf
!>          X12 is COMPLEX*16 array, dimension (LDX12,M-Q)
!>          On entry, the top-right block of the unitary matrix to
!>          be reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the rows of triu(X12) specify the first P reflectors for
!>             Q2;
!>          else TRANS = 'T', and
!>             the columns of tril(X12) specify the first P reflectors
!>             for Q2\&.
!> 
.fi
.PP
.br
\fILDX12\fP 
.PP
.nf
!>          LDX12 is INTEGER
!>          The leading dimension of X12\&. If TRANS = 'N', then LDX12 >=
!>          P; else LDX11 >= M-Q\&.
!> 
.fi
.PP
.br
\fIX21\fP 
.PP
.nf
!>          X21 is COMPLEX*16 array, dimension (LDX21,Q)
!>          On entry, the bottom-left block of the unitary matrix to
!>          be reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the columns of tril(X21) specify reflectors for P2;
!>          else TRANS = 'T', and
!>             the rows of triu(X21) specify reflectors for P2\&.
!> 
.fi
.PP
.br
\fILDX21\fP 
.PP
.nf
!>          LDX21 is INTEGER
!>          The leading dimension of X21\&. If TRANS = 'N', then LDX21 >=
!>          M-P; else LDX21 >= Q\&.
!> 
.fi
.PP
.br
\fIX22\fP 
.PP
.nf
!>          X22 is COMPLEX*16 array, dimension (LDX22,M-Q)
!>          On entry, the bottom-right block of the unitary matrix to
!>          be reduced\&. On exit, the form depends on TRANS:
!>          If TRANS = 'N', then
!>             the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
!>             M-P-Q reflectors for Q2,
!>          else TRANS = 'T', and
!>             the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
!>             M-P-Q reflectors for P2\&.
!> 
.fi
.PP
.br
\fILDX22\fP 
.PP
.nf
!>          LDX22 is INTEGER
!>          The leading dimension of X22\&. If TRANS = 'N', then LDX22 >=
!>          M-P; else LDX22 >= M-Q\&.
!> 
.fi
.PP
.br
\fITHETA\fP 
.PP
.nf
!>          THETA is DOUBLE PRECISION array, dimension (Q)
!>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
!>          be computed from the angles THETA and PHI\&. See Further
!>          Details\&.
!> 
.fi
.PP
.br
\fIPHI\fP 
.PP
.nf
!>          PHI is DOUBLE PRECISION array, dimension (Q-1)
!>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
!>          be computed from the angles THETA and PHI\&. See Further
!>          Details\&.
!> 
.fi
.PP
.br
\fITAUP1\fP 
.PP
.nf
!>          TAUP1 is COMPLEX*16 array, dimension (P)
!>          The scalar factors of the elementary reflectors that define
!>          P1\&.
!> 
.fi
.PP
.br
\fITAUP2\fP 
.PP
.nf
!>          TAUP2 is COMPLEX*16 array, dimension (M-P)
!>          The scalar factors of the elementary reflectors that define
!>          P2\&.
!> 
.fi
.PP
.br
\fITAUQ1\fP 
.PP
.nf
!>          TAUQ1 is COMPLEX*16 array, dimension (Q)
!>          The scalar factors of the elementary reflectors that define
!>          Q1\&.
!> 
.fi
.PP
.br
\fITAUQ2\fP 
.PP
.nf
!>          TAUQ2 is COMPLEX*16 array, dimension (M-Q)
!>          The scalar factors of the elementary reflectors that define
!>          Q2\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is COMPLEX*16 array, dimension (LWORK)
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          LWORK is INTEGER
!>          The dimension of the array WORK\&. LWORK >= M-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\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit\&.
!>          < 0:  if INFO = -i, the i-th argument had an illegal value\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  The bidiagonal blocks B11, B12, B21, and B22 are represented
!>  implicitly by angles THETA(1), \&.\&.\&., THETA(Q) and PHI(1), \&.\&.\&.,
!>  PHI(Q-1)\&. B11 and B21 are upper bidiagonal, while B21 and B22 are
!>  lower bidiagonal\&. Every entry in each bidiagonal band is a product
!>  of a sine or cosine of a THETA with a sine or cosine of a PHI\&. See
!>  [1] or ZUNCSD for details\&.
!>
!>  P1, P2, Q1, and Q2 are represented as products of elementary
!>  reflectors\&. See ZUNCSD for details on generating P1, P2, Q1, and Q2
!>  using ZUNGQR and ZUNGLQ\&.
!> 
.fi
.PP
 
.RE
.PP
\fBReferences:\fP
.RS 4
[1] Brian D\&. Sutton\&. Computing the complete CS decomposition\&. Numer\&. Algorithms, 50(1):33-65, 2009\&. 
.RE
.PP

.PP
Definition at line \fB284\fP of file \fBzunbdb\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.