---
name: "sorcsd.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/sorcsd.f"
---

# sorcsd.f(3)

## NAME

SRC/sorcsd.f

## SYNOPSIS

### Functions/Subroutines

recursive subroutine **sorcsd** (jobu1, jobu2, jobv1t, jobv2t, trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t, ldv2t, work, lwork, iwork, info)\
 **SORCSD**

## Function/Subroutine Documentation

### recursive subroutine sorcsd (character jobu1, character jobu2, character jobv1t, character jobv2t, character trans, character signs, integer m, integer p, integer q, real, dimension( ldx11, \* ) x11, integer ldx11, real, dimension( ldx12, \* ) x12, integer ldx12, real, dimension( ldx21, \* ) x21, integer ldx21, real, dimension( ldx22, \* ) x22, integer ldx22, real, dimension( \* ) theta, real, dimension( ldu1, \* ) u1, integer ldu1, real, dimension( ldu2, \* ) u2, integer ldu2, real, dimension( ldv1t, \* ) v1t, integer ldv1t, real, dimension( ldv2t, \* ) v2t, integer ldv2t, real, dimension( \* ) work, integer lwork, integer, dimension( \* ) iwork, integer info)

**SORCSD**

**Purpose:**

```
!>
!> SORCSD computes the CS decomposition of an M-by-M partitioned
!> orthogonal matrix X:
!>
!>                                 [  I  0  0 |  0  0  0 ]
!>                                 [  0  C  0 |  0 -S  0 ]
!>     [ X11 | X12 ]   [ U1 |    ] [  0  0  0 |  0  0 -I ] [ V1 |    ]**T
!> X = [-----------] = [---------] [---------------------] [---------]   .
!>     [ X21 | X22 ]   [    | U2 ] [  0  0  0 |  I  0  0 ] [    | V2 ]
!>                                 [  0  S  0 |  0  C  0 ]
!>                                 [  0  0  I |  0  0  0 ]
!>
!> X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
!> (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-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).
!>
```

**Parameters**

*JOBU1*

```
!>          JOBU1 is CHARACTER
!>          = 'Y':      U1 is computed;
!>          otherwise:  U1 is not computed.
!>
```

*JOBU2*

```
!>          JOBU2 is CHARACTER
!>          = 'Y':      U2 is computed;
!>          otherwise:  U2 is not computed.
!>
```

*JOBV1T*

```
!>          JOBV1T is CHARACTER
!>          = 'Y':      V1T is computed;
!>          otherwise:  V1T is not computed.
!>
```

*JOBV2T*

```
!>          JOBV2T is CHARACTER
!>          = 'Y':      V2T is computed;
!>          otherwise:  V2T is not computed.
!>
```

*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 <= M.
!>
```

*X11*

```
!>          X11 is REAL array, dimension (LDX11,Q)
!>          On entry, part of the orthogonal matrix whose CSD is desired.
!>
```

*LDX11*

```
!>          LDX11 is INTEGER
!>          The leading dimension of X11. LDX11 >= MAX(1,P).
!>
```

*X12*

```
!>          X12 is REAL array, dimension (LDX12,M-Q)
!>          On entry, part of the orthogonal matrix whose CSD is desired.
!>
```

*LDX12*

```
!>          LDX12 is INTEGER
!>          The leading dimension of X12. LDX12 >= MAX(1,P).
!>
```

*X21*

```
!>          X21 is REAL array, dimension (LDX21,Q)
!>          On entry, part of the orthogonal matrix whose CSD is desired.
!>
```

*LDX21*

```
!>          LDX21 is INTEGER
!>          The leading dimension of X11. LDX21 >= MAX(1,M-P).
!>
```

*X22*

```
!>          X22 is REAL array, dimension (LDX22,M-Q)
!>          On entry, part of the orthogonal matrix whose CSD is desired.
!>
```

*LDX22*

```
!>          LDX22 is INTEGER
!>          The leading dimension of X11. LDX22 >= MAX(1,M-P).
!>
```

*THETA*

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

*U1*

```
!>          U1 is REAL array, dimension (LDU1,P)
!>          If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.
!>
```

*LDU1*

```
!>          LDU1 is INTEGER
!>          The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
!>          MAX(1,P).
!>
```

*U2*

```
!>          U2 is REAL array, dimension (LDU2,M-P)
!>          If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
!>          matrix U2.
!>
```

*LDU2*

```
!>          LDU2 is INTEGER
!>          The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
!>          MAX(1,M-P).
!>
```

*V1T*

```
!>          V1T is REAL array, dimension (LDV1T,Q)
!>          If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
!>          matrix V1**T.
!>
```

*LDV1T*

```
!>          LDV1T is INTEGER
!>          The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
!>          MAX(1,Q).
!>
```

*V2T*

```
!>          V2T is REAL array, dimension (LDV2T,M-Q)
!>          If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal
!>          matrix V2**T.
!>
```

*LDV2T*

```
!>          LDV2T is INTEGER
!>          The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
!>          MAX(1,M-Q).
!>
```

*WORK*

```
!>          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.
!>
```

*LWORK*

```
!>          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.
!>
```

*IWORK*

```
!>          IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))
!>
```

*INFO*

```
!>          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.
!>
```

**References:**

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

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **295** of file **sorcsd.f**.

## Author

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