!>
!> SORBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny
!> matrix X with orthonormal columns:
!>
!>                            [ B11 ]
!>      [ X11 ]   [ P1 |    ] [  0  ]
!>      [-----] = [---------] [-----] Q1**T .
!>      [ X21 ]   [    | P2 ] [ B21 ]
!>                            [  0  ]
!>
!> X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,
!> M-P, or Q. Routines SORBDB1, SORBDB2, and SORBDB3 handle cases in
!> which M-Q is not the minimum dimension.
!>
!> The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
!> and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
!> Householder vectors.
!>
!> B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
!> implicitly by angles THETA, PHI.
!>
!>

M

!>          M is INTEGER
!>           The number of rows X11 plus the number of rows in X21.
!> 

P

!>          P is INTEGER
!>           The number of rows in X11. 0 <= P <= M.
!> 

Q

!>          Q is INTEGER
!>           The number of columns in X11 and X21. 0 <= Q <= M and
!>           M-Q <= min(P,M-P,Q).
!> 

X11

!>          X11 is REAL array, dimension (LDX11,Q)
!>           On entry, the top block of the matrix X to be reduced. On
!>           exit, the columns of tril(X11) specify reflectors for P1 and
!>           the rows of triu(X11,1) specify reflectors for Q1.
!> 

LDX11

!>          LDX11 is INTEGER
!>           The leading dimension of X11. LDX11 >= P.
!> 

X21

!>          X21 is REAL array, dimension (LDX21,Q)
!>           On entry, the bottom block of the matrix X to be reduced. On
!>           exit, the columns of tril(X21) specify reflectors for P2.
!> 

LDX21

!>          LDX21 is INTEGER
!>           The leading dimension of X21. LDX21 >= M-P.
!> 

THETA

!>          THETA is REAL array, dimension (Q)
!>           The entries of the bidiagonal blocks B11, B21 are defined by
!>           THETA and PHI. See Further Details.
!> 

PHI

!>          PHI is REAL array, dimension (Q-1)
!>           The entries of the bidiagonal blocks B11, B21 are defined by
!>           THETA and PHI. See Further Details.
!> 

TAUP1

!>          TAUP1 is REAL array, dimension (M-Q)
!>           The scalar factors of the elementary reflectors that define
!>           P1.
!> 

TAUP2

!>          TAUP2 is REAL array, dimension (M-Q)
!>           The scalar factors of the elementary reflectors that define
!>           P2.
!> 

TAUQ1

!>          TAUQ1 is REAL array, dimension (Q)
!>           The scalar factors of the elementary reflectors that define
!>           Q1.
!> 

PHANTOM

!>          PHANTOM is REAL array, dimension (M)
!>           The routine computes an M-by-1 column vector Y that is
!>           orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
!>           PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
!>           Y(P+1:M), respectively.
!> 

WORK

!>          WORK is REAL array, dimension (LWORK)
!> 

LWORK

!>          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.
!> 

INFO

!>          INFO is INTEGER
!>           = 0:  successful exit.
!>           < 0:  if INFO = -i, the i-th argument had an illegal value.
!> 

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!>
!>  The upper-bidiagonal blocks B11, B21 are represented implicitly by
!>  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). 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, and Q1 are represented as products of elementary reflectors.
!>  See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
!>  and SORGLQ.
!> 

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