cunbdb.f −

**Functions/Subroutines**

subroutine **cunbdb** (TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, PHI, TAUP1, TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO)

CUNBDB

**subroutine cunbdb (characterTRANS, characterSIGNS, integerM, integerP, integerQ, complex, dimension( ldx11, * )X11, integerLDX11, complex, dimension( ldx12, * )X12, integerLDX12, complex, dimension( ldx21, * )X21, integerLDX21, complex, dimension( ldx22, * )X22, integerLDX22, real, dimension( * )THETA, real, dimension( * )PHI, complex, dimension( * )TAUP1, complex, dimension( * )TAUP2, complex, dimension( * )TAUQ1, complex, dimension( * )TAUQ2, complex, dimension( * )WORK, integerLWORK, integerINFO)
CUNBDB**

**Purpose:**

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.

**Parameters:**

*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 <=

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

*X11*

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.

*LDX11*

LDX11 is INTEGER

The leading dimension of X11. If TRANS = ’N’, then LDX11 >=

P; else LDX11 >= Q.

*X12*

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.

*LDX12*

LDX12 is INTEGER

The leading dimension of X12. If TRANS = ’N’, then LDX12 >=

P; else LDX11 >= M-Q.

*X21*

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.

*LDX21*

LDX21 is INTEGER

The leading dimension of X21. If TRANS = ’N’, then LDX21 >=

M-P; else LDX21 >= Q.

*X22*

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.

*LDX22*

LDX22 is INTEGER

The leading dimension of X22. If TRANS = ’N’, then LDX22 >=

M-P; else LDX22 >= M-Q.

*THETA*

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.

*PHI*

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.

*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.

*TAUQ2*

TAUQ2 is COMPLEX array, dimension (M-Q)

The scalar factors of the elementary reflectors that define

Q2.

*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:**

November 2013

**Further Details:**

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.

**References:**

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

Definition at line 286 of file cunbdb.f.

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