orcsd#

Functions

void sorcsd(
    const char*          jobu1,
    const char*          jobu2,
    const char*          jobv1t,
    const char*          jobv2t,
    const char*          trans,
    const char*          signs,
    const INT            m,
    const INT            p,
    const INT            q,
          f32*  restrict X11,
    const INT            ldx11,
          f32*  restrict X12,
    const INT            ldx12,
          f32*  restrict X21,
    const INT            ldx21,
          f32*  restrict X22,
    const INT            ldx22,
          f32*  restrict theta,
          f32*  restrict U1,
    const INT            ldu1,
          f32*  restrict U2,
    const INT            ldu2,
          f32*  restrict V1T,
    const INT            ldv1t,
          f32*  restrict V2T,
    const INT            ldv2t,
          f32*  restrict work,
    const INT            lwork,
          INT*  restrict iwork,
          INT*           info
);
void sorcsd(const char *jobu1, const char *jobu2, const char *jobv1t, const char *jobv2t, const char *trans, const char *signs, const INT m, const INT p, const INT q, f32 *restrict X11, const INT ldx11, f32 *restrict X12, const INT ldx12, f32 *restrict X21, const INT ldx21, f32 *restrict X22, const INT ldx22, f32 *restrict theta, f32 *restrict U1, const INT ldu1, f32 *restrict U2, const INT ldu2, f32 *restrict V1T, const INT ldv1t, f32 *restrict V2T, const INT ldv2t, f32 *restrict work, const INT lwork, INT *restrict iwork, INT *info)#

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

in
jobu1

= ‘Y’: U1 is computed; otherwise: U1 is not computed.

in
jobu2

= ‘Y’: U2 is computed; otherwise: U2 is not computed.

in
jobv1t

= ‘Y’: V1T is computed; otherwise: V1T is not computed.

in
jobv2t

= ‘Y’: V2T is computed; otherwise: V2T is not computed.

in
trans

= ‘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.

in
signs

= ‘O’: The lower-left block is made nonpositive (“other” convention); otherwise: The upper-right block is made nonpositive (“default” convention).

in
m

The number of rows and columns in X.

in
p

The number of rows in X11 and X12. 0 <= p <= m.

in
q

The number of columns in X11 and X21. 0 <= q <= m.

inout
X11

Double precision array, dimension (ldx11, q). On entry, part of the orthogonal matrix whose CSD is desired.

in
ldx11

The leading dimension of X11. ldx11 >= max(1, p) (column-major) or ldx11 >= max(1, q) (row-major).

inout
X12

Double precision array, dimension (ldx12, m-q). On entry, part of the orthogonal matrix whose CSD is desired.

in
ldx12

The leading dimension of X12.

inout
X21

Double precision array, dimension (ldx21, q). On entry, part of the orthogonal matrix whose CSD is desired.

in
ldx21

The leading dimension of X21.

inout
X22

Double precision array, dimension (ldx22, m-q). On entry, part of the orthogonal matrix whose CSD is desired.

in
ldx22

The leading dimension of X22.

out
theta

Double precision array, dimension (r), where 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)) ).

out
U1

If jobu1 = ‘Y’, U1 contains the P-by-P orthogonal matrix U1.

in
ldu1

The leading dimension of U1.

out
U2

If jobu2 = ‘Y’, U2 contains the (M-P)-by-(M-P) orthogonal matrix U2.

in
ldu2

The leading dimension of U2.

out
V1T

If jobv1t = ‘Y’, V1T contains the Q-by-Q orthogonal matrix V1**T.

in
ldv1t

The leading dimension of V1T.

out
V2T

If jobv2t = ‘Y’, V2T contains the (M-Q)-by-(M-Q) orthogonal matrix V2**T.

in
ldv2t

The leading dimension of V2T.

out
work

Double precision array, dimension (max(1, lwork)). On exit, if info = 0, work[0] returns the optimal lwork.

in
lwork

The dimension of the array work. If lwork = -1, a workspace query is assumed.

out
iwork

Integer array, dimension (m - min(p, m-p, q, m-q)).

out
info

  • = 0: successful exit.

  • < 0: if info = -i, the i-th argument had an illegal value.

  • > 0: SBBCSD did not converge.

Functions

void dorcsd(
    const char*          jobu1,
    const char*          jobu2,
    const char*          jobv1t,
    const char*          jobv2t,
    const char*          trans,
    const char*          signs,
    const INT            m,
    const INT            p,
    const INT            q,
          f64*  restrict X11,
    const INT            ldx11,
          f64*  restrict X12,
    const INT            ldx12,
          f64*  restrict X21,
    const INT            ldx21,
          f64*  restrict X22,
    const INT            ldx22,
          f64*  restrict theta,
          f64*  restrict U1,
    const INT            ldu1,
          f64*  restrict U2,
    const INT            ldu2,
          f64*  restrict V1T,
    const INT            ldv1t,
          f64*  restrict V2T,
    const INT            ldv2t,
          f64*  restrict work,
    const INT            lwork,
          INT*  restrict iwork,
          INT*           info
);
void dorcsd(const char *jobu1, const char *jobu2, const char *jobv1t, const char *jobv2t, const char *trans, const char *signs, const INT m, const INT p, const INT q, f64 *restrict X11, const INT ldx11, f64 *restrict X12, const INT ldx12, f64 *restrict X21, const INT ldx21, f64 *restrict X22, const INT ldx22, f64 *restrict theta, f64 *restrict U1, const INT ldu1, f64 *restrict U2, const INT ldu2, f64 *restrict V1T, const INT ldv1t, f64 *restrict V2T, const INT ldv2t, f64 *restrict work, const INT lwork, INT *restrict iwork, INT *info)#

DORCSD 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 = [——–&#8212;] = [——&#8212;] [——————&#8212;] [——&#8212;] . [ 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

in
jobu1

= ‘Y’: U1 is computed; otherwise: U1 is not computed.

in
jobu2

= ‘Y’: U2 is computed; otherwise: U2 is not computed.

in
jobv1t

= ‘Y’: V1T is computed; otherwise: V1T is not computed.

in
jobv2t

= ‘Y’: V2T is computed; otherwise: V2T is not computed.

in
trans

= ‘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.

in
signs

= ‘O’: The lower-left block is made nonpositive (“other” convention); otherwise: The upper-right block is made nonpositive (“default” convention).

in
m

The number of rows and columns in X.

in
p

The number of rows in X11 and X12. 0 <= p <= m.

in
q

The number of columns in X11 and X21. 0 <= q <= m.

inout
X11

Double precision array, dimension (ldx11, q). On entry, part of the orthogonal matrix whose CSD is desired.

in
ldx11

The leading dimension of X11. ldx11 >= max(1, p) (column-major) or ldx11 >= max(1, q) (row-major).

inout
X12

Double precision array, dimension (ldx12, m-q). On entry, part of the orthogonal matrix whose CSD is desired.

in
ldx12

The leading dimension of X12.

inout
X21

Double precision array, dimension (ldx21, q). On entry, part of the orthogonal matrix whose CSD is desired.

in
ldx21

The leading dimension of X21.

inout
X22

Double precision array, dimension (ldx22, m-q). On entry, part of the orthogonal matrix whose CSD is desired.

in
ldx22

The leading dimension of X22.

out
theta

Double precision array, dimension (r), where 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)) ).

out
U1

If jobu1 = ‘Y’, U1 contains the P-by-P orthogonal matrix U1.

in
ldu1

The leading dimension of U1.

out
U2

If jobu2 = ‘Y’, U2 contains the (M-P)-by-(M-P) orthogonal matrix U2.

in
ldu2

The leading dimension of U2.

out
V1T

If jobv1t = ‘Y’, V1T contains the Q-by-Q orthogonal matrix V1**T.

in
ldv1t

The leading dimension of V1T.

out
V2T

If jobv2t = ‘Y’, V2T contains the (M-Q)-by-(M-Q) orthogonal matrix V2**T.

in
ldv2t

The leading dimension of V2T.

out
work

Double precision array, dimension (max(1, lwork)). On exit, if info = 0, work[0] returns the optimal lwork.

in
lwork

The dimension of the array work. If lwork = -1, a workspace query is assumed.

out
iwork

Integer array, dimension (m - min(p, m-p, q, m-q)).

out
info

  • = 0: successful exit.

  • < 0: if info = -i, the i-th argument had an illegal value.

  • > 0: DBBCSD did not converge.

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