unbdb#

Functions

void cunbdb(
    const char*          trans,
    const char*          signs,
    const INT            m,
    const INT            p,
    const INT            q,
          c64*  restrict X11,
    const INT            ldx11,
          c64*  restrict X12,
    const INT            ldx12,
          c64*  restrict X21,
    const INT            ldx21,
          c64*  restrict X22,
    const INT            ldx22,
          f32*  restrict theta,
          f32*  restrict phi,
          c64*  restrict taup1,
          c64*  restrict taup2,
          c64*  restrict tauq1,
          c64*  restrict tauq2,
          c64*  restrict work,
    const INT            lwork,
          INT*           info
);
void cunbdb(const char *trans, const char *signs, const INT m, const INT p, const INT q, c64 *restrict X11, const INT ldx11, c64 *restrict X12, const INT ldx12, c64 *restrict X21, const INT ldx21, c64 *restrict X22, const INT ldx22, f32 *restrict theta, f32 *restrict phi, c64 *restrict taup1, c64 *restrict taup2, c64 *restrict tauq1, c64 *restrict tauq2, c64 *restrict work, const INT lwork, INT *info)#

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

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 (the “other” convention); otherwise: The upper-right block is made nonpositive (the “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 <= MIN(P,M-P,M-Q).

inout
X11

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.

in
ldx11

The leading dimension of X11. If TRANS = ‘N’, then LDX11 >= P; else LDX11 >= Q.

inout
X12

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.

in
ldx12

The leading dimension of X12. If TRANS = ‘N’, then LDX12 >= P; else LDX12 >= M-Q.

inout
X21

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.

in
ldx21

The leading dimension of X21. If TRANS = ‘N’, then LDX21 >= M-P; else LDX21 >= Q.

inout
X22

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.

in
ldx22

The leading dimension of X22. If TRANS = ‘N’, then LDX22 >= M-P; else LDX22 >= M-Q.

out
theta

Single 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.

out
phi

Single 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.

out
taup1

Complex*16 array, dimension (P). The scalar factors of the elementary reflectors that define P1.

out
taup2

Complex*16 array, dimension (M-P). The scalar factors of the elementary reflectors that define P2.

out
tauq1

Complex*16 array, dimension (Q). The scalar factors of the elementary reflectors that define Q1.

out
tauq2

Complex*16 array, dimension (M-Q). The scalar factors of the elementary reflectors that define Q2.

out
work

Complex*16 array, dimension (LWORK).

in
lwork

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.

out
info

= 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

Functions

void zunbdb(
    const char*          trans,
    const char*          signs,
    const INT            m,
    const INT            p,
    const INT            q,
          c128* restrict X11,
    const INT            ldx11,
          c128* restrict X12,
    const INT            ldx12,
          c128* restrict X21,
    const INT            ldx21,
          c128* restrict X22,
    const INT            ldx22,
          f64*  restrict theta,
          f64*  restrict phi,
          c128* restrict taup1,
          c128* restrict taup2,
          c128* restrict tauq1,
          c128* restrict tauq2,
          c128* restrict work,
    const INT            lwork,
          INT*           info
);
void zunbdb(const char *trans, const char *signs, const INT m, const INT p, const INT q, c128 *restrict X11, const INT ldx11, c128 *restrict X12, const INT ldx12, c128 *restrict X21, const INT ldx21, c128 *restrict X22, const INT ldx22, f64 *restrict theta, f64 *restrict phi, c128 *restrict taup1, c128 *restrict taup2, c128 *restrict tauq1, c128 *restrict tauq2, c128 *restrict work, const INT lwork, INT *info)#

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

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 (the “other” convention); otherwise: The upper-right block is made nonpositive (the “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 <= MIN(P,M-P,M-Q).

inout
X11

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.

in
ldx11

The leading dimension of X11. If TRANS = ‘N’, then LDX11 >= P; else LDX11 >= Q.

inout
X12

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.

in
ldx12

The leading dimension of X12. If TRANS = ‘N’, then LDX12 >= P; else LDX12 >= M-Q.

inout
X21

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.

in
ldx21

The leading dimension of X21. If TRANS = ‘N’, then LDX21 >= M-P; else LDX21 >= Q.

inout
X22

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.

in
ldx22

The leading dimension of X22. If TRANS = ‘N’, then LDX22 >= M-P; else LDX22 >= M-Q.

out
theta

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.

out
phi

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.

out
taup1

Complex*16 array, dimension (P). The scalar factors of the elementary reflectors that define P1.

out
taup2

Complex*16 array, dimension (M-P). The scalar factors of the elementary reflectors that define P2.

out
tauq1

Complex*16 array, dimension (Q). The scalar factors of the elementary reflectors that define Q1.

out
tauq2

Complex*16 array, dimension (M-Q). The scalar factors of the elementary reflectors that define Q2.

out
work

Complex*16 array, dimension (LWORK).

in
lwork

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.

out
info

= 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

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