---
name: "zunbdb.f"
section: "3"
platform: "linux"
package: "lapack-doc"
version: "3.12.1-2"
release: "Arch Linux 2026-11-01"
platforms: ["linux"]
url: "https://new.carta.tech/zunbdb.f"
---

# zunbdb.f(3)

## NAME

SRC/zunbdb.f

## SYNOPSIS

### Functions/Subroutines

subroutine **zunbdb** (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info)\
 **ZUNBDB**

## Function/Subroutine Documentation

### 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)

**ZUNBDB**

**Purpose:**

```
!>
!> 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.
!>
```

**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
!>                       convention);
!>          otherwise:  The upper-right block is made nonpositive (the
!>                       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*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.
!>
```

*LDX11*

```
!>          LDX11 is INTEGER
!>          The leading dimension of X11. If TRANS = 'N', then LDX11 >=
!>          P; else LDX11 >= Q.
!>
```

*X12*

```
!>          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.
!>
```

*LDX12*

```
!>          LDX12 is INTEGER
!>          The leading dimension of X12. If TRANS = 'N', then LDX12 >=
!>          P; else LDX11 >= M-Q.
!>
```

*X21*

```
!>          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.
!>
```

*LDX21*

```
!>          LDX21 is INTEGER
!>          The leading dimension of X21. If TRANS = 'N', then LDX21 >=
!>          M-P; else LDX21 >= Q.
!>
```

*X22*

```
!>          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.
!>
```

*LDX22*

```
!>          LDX22 is INTEGER
!>          The leading dimension of X22. If TRANS = 'N', then LDX22 >=
!>          M-P; else LDX22 >= M-Q.
!>
```

*THETA*

```
!>          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.
!>
```

*PHI*

```
!>          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.
!>
```

*TAUP1*

```
!>          TAUP1 is COMPLEX*16 array, dimension (P)
!>          The scalar factors of the elementary reflectors that define
!>          P1.
!>
```

*TAUP2*

```
!>          TAUP2 is COMPLEX*16 array, dimension (M-P)
!>          The scalar factors of the elementary reflectors that define
!>          P2.
!>
```

*TAUQ1*

```
!>          TAUQ1 is COMPLEX*16 array, dimension (Q)
!>          The scalar factors of the elementary reflectors that define
!>          Q1.
!>
```

*TAUQ2*

```
!>          TAUQ2 is COMPLEX*16 array, dimension (M-Q)
!>          The scalar factors of the elementary reflectors that define
!>          Q2.
!>
```

*WORK*

```
!>          WORK is COMPLEX*16 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.

**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 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.
!>
```

**References:**

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

Definition at line **284** of file **zunbdb.f**.

## Author

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