.TH "uncsd2by1" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
uncsd2by1 \- {un,or}csd2by1: ??
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBcuncsd2by1\fP (jobu1, jobu2, jobv1t, m, p, q, x11, ldx11, x21, ldx21, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, work, lwork, rwork, lrwork, iwork, info)"
.br
.RI "\fBCUNCSD2BY1\fP "
.ti -1c
.RI "subroutine \fBdorcsd2by1\fP (jobu1, jobu2, jobv1t, m, p, q, x11, ldx11, x21, ldx21, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, work, lwork, iwork, info)"
.br
.RI "\fBDORCSD2BY1\fP "
.ti -1c
.RI "subroutine \fBsorcsd2by1\fP (jobu1, jobu2, jobv1t, m, p, q, x11, ldx11, x21, ldx21, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, work, lwork, iwork, info)"
.br
.RI "\fBSORCSD2BY1\fP "
.ti -1c
.RI "subroutine \fBzuncsd2by1\fP (jobu1, jobu2, jobv1t, m, p, q, x11, ldx11, x21, ldx21, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, work, lwork, rwork, lrwork, iwork, info)"
.br
.RI "\fBZUNCSD2BY1\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine cuncsd2by1 (character jobu1, character jobu2, character jobv1t, integer m, integer p, integer q, complex, dimension(ldx11,*) x11, integer ldx11, complex, dimension(ldx21,*) x21, integer ldx21, real, dimension(*) theta, complex, dimension(ldu1,*) u1, integer ldu1, complex, dimension(ldu2,*) u2, integer ldu2, complex, dimension(ldv1t,*) v1t, integer ldv1t, complex, dimension(*) work, integer lwork, real, dimension(*) rwork, integer lrwork, integer, dimension(*) iwork, integer info)"

.PP
\fBCUNCSD2BY1\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CUNCSD2BY1 computes the CS decomposition of an M-by-Q matrix X with
 orthonormal columns that has been partitioned into a 2-by-1 block
 structure:

                                [  I1 0  0 ]
                                [  0  C  0 ]
          [ X11 ]   [ U1 |    ] [  0  0  0 ]
      X = [-----] = [---------] [----------] V1**T \&.
          [ X21 ]   [    | U2 ] [  0  0  0 ]
                                [  0  S  0 ]
                                [  0  0  I2]

 X11 is P-by-Q\&. The unitary matrices U1, U2, and V1 are P-by-P,
 (M-P)-by-(M-P), and Q-by-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)\&. I1 is a K1-by-K1 identity matrix and I2 is a
 K2-by-K2 identity matrix, where K1 = MAX(Q+P-M,0), K2 = MAX(Q-P,0)\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU1\fP 
.PP
.nf
          JOBU1 is CHARACTER
          = 'Y':      U1 is computed;
          otherwise:  U1 is not computed\&.
.fi
.PP
.br
\fIJOBU2\fP 
.PP
.nf
          JOBU2 is CHARACTER
          = 'Y':      U2 is computed;
          otherwise:  U2 is not computed\&.
.fi
.PP
.br
\fIJOBV1T\fP 
.PP
.nf
          JOBV1T is CHARACTER
          = 'Y':      V1T is computed;
          otherwise:  V1T is not computed\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of rows in X\&.
.fi
.PP
.br
\fIP\fP 
.PP
.nf
          P is INTEGER
          The number of rows in X11\&. 0 <= P <= M\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is INTEGER
          The number of columns in X11 and X21\&. 0 <= Q <= M\&.
.fi
.PP
.br
\fIX11\fP 
.PP
.nf
          X11 is COMPLEX array, dimension (LDX11,Q)
          On entry, part of the unitary matrix whose CSD is desired\&.
.fi
.PP
.br
\fILDX11\fP 
.PP
.nf
          LDX11 is INTEGER
          The leading dimension of X11\&. LDX11 >= MAX(1,P)\&.
.fi
.PP
.br
\fIX21\fP 
.PP
.nf
          X21 is COMPLEX array, dimension (LDX21,Q)
          On entry, part of the unitary matrix whose CSD is desired\&.
.fi
.PP
.br
\fILDX21\fP 
.PP
.nf
          LDX21 is INTEGER
          The leading dimension of X21\&. LDX21 >= MAX(1,M-P)\&.
.fi
.PP
.br
\fITHETA\fP 
.PP
.nf
          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)) )\&.
.fi
.PP
.br
\fIU1\fP 
.PP
.nf
          U1 is COMPLEX array, dimension (P)
          If JOBU1 = 'Y', U1 contains the P-by-P unitary matrix U1\&.
.fi
.PP
.br
\fILDU1\fP 
.PP
.nf
          LDU1 is INTEGER
          The leading dimension of U1\&. If JOBU1 = 'Y', LDU1 >=
          MAX(1,P)\&.
.fi
.PP
.br
\fIU2\fP 
.PP
.nf
          U2 is COMPLEX array, dimension (M-P)
          If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) unitary
          matrix U2\&.
.fi
.PP
.br
\fILDU2\fP 
.PP
.nf
          LDU2 is INTEGER
          The leading dimension of U2\&. If JOBU2 = 'Y', LDU2 >=
          MAX(1,M-P)\&.
.fi
.PP
.br
\fIV1T\fP 
.PP
.nf
          V1T is COMPLEX array, dimension (Q)
          If JOBV1T = 'Y', V1T contains the Q-by-Q matrix unitary
          matrix V1**T\&.
.fi
.PP
.br
\fILDV1T\fP 
.PP
.nf
          LDV1T is INTEGER
          The leading dimension of V1T\&. If JOBV1T = 'Y', LDV1T >=
          MAX(1,Q)\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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 and RWORK
          arrays, returns this value as the first entry of the WORK
          and RWORK array, respectively, and no error message related
          to LWORK or LRWORK is issued by XERBLA\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILRWORK\fP 
.PP
.nf
          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 WORK and RWORK
          arrays, returns this value as the first entry of the WORK
          and RWORK array, respectively, and no error message related
          to LWORK or LRWORK is issued by XERBLA\&.
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))
.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\&.
          > 0:  CBBCSD did not converge\&. See the description of WORK
                above for details\&.
.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
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.SS "subroutine dorcsd2by1 (character jobu1, character jobu2, character jobv1t, integer m, integer p, integer q, double precision, dimension(ldx11,*) x11, integer ldx11, double precision, dimension(ldx21,*) x21, integer ldx21, double precision, dimension(*) theta, double precision, dimension(ldu1,*) u1, integer ldu1, double precision, dimension(ldu2,*) u2, integer ldu2, double precision, dimension(ldv1t,*) v1t, integer ldv1t, double precision, dimension(*) work, integer lwork, integer, dimension(*) iwork, integer info)"

.PP
\fBDORCSD2BY1\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DORCSD2BY1 computes the CS decomposition of an M-by-Q matrix X with
 orthonormal columns that has been partitioned into a 2-by-1 block
 structure:

                                [  I1 0  0 ]
                                [  0  C  0 ]
          [ X11 ]   [ U1 |    ] [  0  0  0 ]
      X = [-----] = [---------] [----------] V1**T \&.
          [ X21 ]   [    | U2 ] [  0  0  0 ]
                                [  0  S  0 ]
                                [  0  0  I2]

 X11 is P-by-Q\&. The orthogonal matrices U1, U2, and V1 are P-by-P,
 (M-P)-by-(M-P), and Q-by-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)\&. I1 is a K1-by-K1 identity matrix and I2 is a
 K2-by-K2 identity matrix, where K1 = MAX(Q+P-M,0), K2 = MAX(Q-P,0)\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU1\fP 
.PP
.nf
          JOBU1 is CHARACTER
          = 'Y':      U1 is computed;
          otherwise:  U1 is not computed\&.
.fi
.PP
.br
\fIJOBU2\fP 
.PP
.nf
          JOBU2 is CHARACTER
          = 'Y':      U2 is computed;
          otherwise:  U2 is not computed\&.
.fi
.PP
.br
\fIJOBV1T\fP 
.PP
.nf
          JOBV1T is CHARACTER
          = 'Y':      V1T is computed;
          otherwise:  V1T is not computed\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of rows in X\&.
.fi
.PP
.br
\fIP\fP 
.PP
.nf
          P is INTEGER
          The number of rows in X11\&. 0 <= P <= M\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is INTEGER
          The number of columns in X11 and X21\&. 0 <= Q <= M\&.
.fi
.PP
.br
\fIX11\fP 
.PP
.nf
          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
          On entry, part of the orthogonal matrix whose CSD is desired\&.
.fi
.PP
.br
\fILDX11\fP 
.PP
.nf
          LDX11 is INTEGER
          The leading dimension of X11\&. LDX11 >= MAX(1,P)\&.
.fi
.PP
.br
\fIX21\fP 
.PP
.nf
          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
          On entry, part of the orthogonal matrix whose CSD is desired\&.
.fi
.PP
.br
\fILDX21\fP 
.PP
.nf
          LDX21 is INTEGER
          The leading dimension of X21\&. LDX21 >= MAX(1,M-P)\&.
.fi
.PP
.br
\fITHETA\fP 
.PP
.nf
          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)) )\&.
.fi
.PP
.br
\fIU1\fP 
.PP
.nf
          U1 is DOUBLE PRECISION array, dimension (P)
          If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1\&.
.fi
.PP
.br
\fILDU1\fP 
.PP
.nf
          LDU1 is INTEGER
          The leading dimension of U1\&. If JOBU1 = 'Y', LDU1 >=
          MAX(1,P)\&.
.fi
.PP
.br
\fIU2\fP 
.PP
.nf
          U2 is DOUBLE PRECISION array, dimension (M-P)
          If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
          matrix U2\&.
.fi
.PP
.br
\fILDU2\fP 
.PP
.nf
          LDU2 is INTEGER
          The leading dimension of U2\&. If JOBU2 = 'Y', LDU2 >=
          MAX(1,M-P)\&.
.fi
.PP
.br
\fIV1T\fP 
.PP
.nf
          V1T is DOUBLE PRECISION array, dimension (Q)
          If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
          matrix V1**T\&.
.fi
.PP
.br
\fILDV1T\fP 
.PP
.nf
          LDV1T is INTEGER
          The leading dimension of V1T\&. If JOBV1T = 'Y', LDV1T >=
          MAX(1,Q)\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))
.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\&.
          > 0:  DBBCSD did not converge\&. See the description of WORK
                above for details\&.
.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
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.SS "subroutine sorcsd2by1 (character jobu1, character jobu2, character jobv1t, integer m, integer p, integer q, real, dimension(ldx11,*) x11, integer ldx11, real, dimension(ldx21,*) x21, integer ldx21, real, dimension(*) theta, real, dimension(ldu1,*) u1, integer ldu1, real, dimension(ldu2,*) u2, integer ldu2, real, dimension(ldv1t,*) v1t, integer ldv1t, real, dimension(*) work, integer lwork, integer, dimension(*) iwork, integer info)"

.PP
\fBSORCSD2BY1\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SORCSD2BY1 computes the CS decomposition of an M-by-Q matrix X with
 orthonormal columns that has been partitioned into a 2-by-1 block
 structure:

                                [  I1 0  0 ]
                                [  0  C  0 ]
          [ X11 ]   [ U1 |    ] [  0  0  0 ]
      X = [-----] = [---------] [----------] V1**T \&.
          [ X21 ]   [    | U2 ] [  0  0  0 ]
                                [  0  S  0 ]
                                [  0  0  I2]

 X11 is P-by-Q\&. The orthogonal matrices U1, U2, and V1 are P-by-P,
 (M-P)-by-(M-P), and Q-by-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)\&. I1 is a K1-by-K1 identity matrix and I2 is a
 K2-by-K2 identity matrix, where K1 = MAX(Q+P-M,0), K2 = MAX(Q-P,0)\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU1\fP 
.PP
.nf
          JOBU1 is CHARACTER
          = 'Y':      U1 is computed;
          otherwise:  U1 is not computed\&.
.fi
.PP
.br
\fIJOBU2\fP 
.PP
.nf
          JOBU2 is CHARACTER
          = 'Y':      U2 is computed;
          otherwise:  U2 is not computed\&.
.fi
.PP
.br
\fIJOBV1T\fP 
.PP
.nf
          JOBV1T is CHARACTER
          = 'Y':      V1T is computed;
          otherwise:  V1T is not computed\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of rows in X\&.
.fi
.PP
.br
\fIP\fP 
.PP
.nf
          P is INTEGER
          The number of rows in X11\&. 0 <= P <= M\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is INTEGER
          The number of columns in X11 and X21\&. 0 <= Q <= M\&.
.fi
.PP
.br
\fIX11\fP 
.PP
.nf
          X11 is REAL array, dimension (LDX11,Q)
          On entry, part of the orthogonal matrix whose CSD is desired\&.
.fi
.PP
.br
\fILDX11\fP 
.PP
.nf
          LDX11 is INTEGER
          The leading dimension of X11\&. LDX11 >= MAX(1,P)\&.
.fi
.PP
.br
\fIX21\fP 
.PP
.nf
          X21 is REAL array, dimension (LDX21,Q)
          On entry, part of the orthogonal matrix whose CSD is desired\&.
.fi
.PP
.br
\fILDX21\fP 
.PP
.nf
          LDX21 is INTEGER
           The leading dimension of X21\&. LDX21 >= MAX(1,M-P)\&.
.fi
.PP
.br
\fITHETA\fP 
.PP
.nf
          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)) )\&.
.fi
.PP
.br
\fIU1\fP 
.PP
.nf
          U1 is REAL array, dimension (P)
          If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1\&.
.fi
.PP
.br
\fILDU1\fP 
.PP
.nf
          LDU1 is INTEGER
          The leading dimension of U1\&. If JOBU1 = 'Y', LDU1 >=
          MAX(1,P)\&.
.fi
.PP
.br
\fIU2\fP 
.PP
.nf
          U2 is REAL array, dimension (M-P)
          If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
          matrix U2\&.
.fi
.PP
.br
\fILDU2\fP 
.PP
.nf
          LDU2 is INTEGER
          The leading dimension of U2\&. If JOBU2 = 'Y', LDU2 >=
          MAX(1,M-P)\&.
.fi
.PP
.br
\fIV1T\fP 
.PP
.nf
          V1T is REAL array, dimension (Q)
          If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
          matrix V1**T\&.
.fi
.PP
.br
\fILDV1T\fP 
.PP
.nf
          LDV1T is INTEGER
          The leading dimension of V1T\&. If JOBV1T = 'Y', LDV1T >=
          MAX(1,Q)\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))
.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\&.
          > 0:  SBBCSD did not converge\&. See the description of WORK
                above for details\&.
.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
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.SS "subroutine zuncsd2by1 (character jobu1, character jobu2, character jobv1t, integer m, integer p, integer q, complex*16, dimension(ldx11,*) x11, integer ldx11, complex*16, dimension(ldx21,*) x21, integer ldx21, 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(*) work, integer lwork, double precision, dimension(*) rwork, integer lrwork, integer, dimension(*) iwork, integer info)"

.PP
\fBZUNCSD2BY1\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZUNCSD2BY1 computes the CS decomposition of an M-by-Q matrix X with
 orthonormal columns that has been partitioned into a 2-by-1 block
 structure:

                                [  I1 0  0 ]
                                [  0  C  0 ]
          [ X11 ]   [ U1 |    ] [  0  0  0 ]
      X = [-----] = [---------] [----------] V1**T \&.
          [ X21 ]   [    | U2 ] [  0  0  0 ]
                                [  0  S  0 ]
                                [  0  0  I2]

 X11 is P-by-Q\&. The unitary matrices U1, U2, and V1 are P-by-P,
 (M-P)-by-(M-P), and Q-by-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)\&. I1 is a K1-by-K1 identity matrix and I2 is a
 K2-by-K2 identity matrix, where K1 = MAX(Q+P-M,0), K2 = MAX(Q-P,0)\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOBU1\fP 
.PP
.nf
          JOBU1 is CHARACTER
          = 'Y':      U1 is computed;
          otherwise:  U1 is not computed\&.
.fi
.PP
.br
\fIJOBU2\fP 
.PP
.nf
          JOBU2 is CHARACTER
          = 'Y':      U2 is computed;
          otherwise:  U2 is not computed\&.
.fi
.PP
.br
\fIJOBV1T\fP 
.PP
.nf
          JOBV1T is CHARACTER
          = 'Y':      V1T is computed;
          otherwise:  V1T is not computed\&.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          M is INTEGER
          The number of rows in X\&.
.fi
.PP
.br
\fIP\fP 
.PP
.nf
          P is INTEGER
          The number of rows in X11\&. 0 <= P <= M\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is INTEGER
          The number of columns in X11 and X21\&. 0 <= Q <= M\&.
.fi
.PP
.br
\fIX11\fP 
.PP
.nf
          X11 is COMPLEX*16 array, dimension (LDX11,Q)
          On entry, part of the unitary matrix whose CSD is desired\&.
.fi
.PP
.br
\fILDX11\fP 
.PP
.nf
          LDX11 is INTEGER
          The leading dimension of X11\&. LDX11 >= MAX(1,P)\&.
.fi
.PP
.br
\fIX21\fP 
.PP
.nf
          X21 is COMPLEX*16 array, dimension (LDX21,Q)
          On entry, part of the unitary matrix whose CSD is desired\&.
.fi
.PP
.br
\fILDX21\fP 
.PP
.nf
          LDX21 is INTEGER
          The leading dimension of X21\&. LDX21 >= MAX(1,M-P)\&.
.fi
.PP
.br
\fITHETA\fP 
.PP
.nf
          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)) )\&.
.fi
.PP
.br
\fIU1\fP 
.PP
.nf
          U1 is COMPLEX*16 array, dimension (P)
          If JOBU1 = 'Y', U1 contains the P-by-P unitary matrix U1\&.
.fi
.PP
.br
\fILDU1\fP 
.PP
.nf
          LDU1 is INTEGER
          The leading dimension of U1\&. If JOBU1 = 'Y', LDU1 >=
          MAX(1,P)\&.
.fi
.PP
.br
\fIU2\fP 
.PP
.nf
          U2 is COMPLEX*16 array, dimension (M-P)
          If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) unitary
          matrix U2\&.
.fi
.PP
.br
\fILDU2\fP 
.PP
.nf
          LDU2 is INTEGER
          The leading dimension of U2\&. If JOBU2 = 'Y', LDU2 >=
          MAX(1,M-P)\&.
.fi
.PP
.br
\fIV1T\fP 
.PP
.nf
          V1T is COMPLEX*16 array, dimension (Q)
          If JOBV1T = 'Y', V1T contains the Q-by-Q matrix unitary
          matrix V1**T\&.
.fi
.PP
.br
\fILDV1T\fP 
.PP
.nf
          LDV1T is INTEGER
          The leading dimension of V1T\&. If JOBV1T = 'Y', LDV1T >=
          MAX(1,Q)\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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 and RWORK
          arrays, returns this value as the first entry of the WORK
          and RWORK array, respectively, and no error message related
          to LWORK or LRWORK is issued by XERBLA\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILRWORK\fP 
.PP
.nf
          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 WORK and RWORK
          arrays, returns this value as the first entry of the WORK
          and RWORK array, respectively, and no error message related
          to LWORK or LRWORK is issued by XERBLA\&.
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))
.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\&.
          > 0:  ZBBCSD did not converge\&. See the description of WORK
                above for details\&.
.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
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.SH "Author"
.PP 
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