cunbdb1.f −

**Functions/Subroutines**

subroutine **cunbdb1** (M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO)

CUNBDB1

**subroutine cunbdb1 (integerM, integerP, integerQ, complex, dimension(ldx11,*)X11, integerLDX11, complex, dimension(ldx21,*)X21, integerLDX21, real, dimension(*)THETA, real, dimension(*)PHI, complex, dimension(*)TAUP1, complex, dimension(*)TAUP2, complex, dimension(*)TAUQ1, complex, dimension(*)WORK, integerLWORK, integerINFO)
CUNBDB1**

CUNBDB1 simultaneously bidiagonalizes the blocks of a tall and skinny

matrix X with orthonomal columns:

[ B11 ]

[ X11 ] [ P1 | ] [ 0 ]

[-----] = [---------] [-----] Q1**T .

[ X21 ] [ | P2 ] [ B21 ]

[ 0 ]

X11 is P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P,

M-P, or M-Q. Routines CUNBDB2, CUNBDB3, and CUNBDB4 handle cases in

which Q is not the minimum dimension.

The unitary 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 Q-by-Q bidiagonal matrices represented implicitly by

angles THETA, PHI..fi

**Parameters:**

*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 <=

MIN(P,M-P,M-Q).

*X11*

X11 is COMPLEX 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 COMPLEX 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 COMPLEX array, dimension (P)

The scalar factors of the elementary reflectors that define

P1.

*TAUP2*

TAUP2 is COMPLEX array, dimension (M-P)

The scalar factors of the elementary reflectors that define

P2.

*TAUQ1*

TAUQ1 is COMPLEX array, dimension (Q)

The scalar factors of the elementary reflectors that define

Q1.

*WORK*

WORK is COMPLEX 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.

**Author:**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

July 2012

**Further Details:**

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 CUNCSD for details.

P1, P2, and Q1 are represented as products of elementary reflectors. See CUNCSD2BY1 for details on generating P1, P2, and Q1 using CUNGQR and CUNGLQ.

**References:**

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

Definition at line 202 of file cunbdb1.f.

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