unbdb(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💅 unbdb(3) — your bestie's guide to bidiagonalizing that partitioned unitary matrix 💅

okay so welcome to unbdb(3), bestie — this is the Library Functions Manual page for {un,or}bdb: the bidiagonalize-partitioned-unitary-matrix step in uncsd. it's giving linear algebra main character energy, and we're gonna get through it together. ✨

✨ NAME ✨

unbdb — {un,or}bdb: bidiagonalize partitioned unitary matrix, step in uncsd

🦋 SYNOPSIS 🦋

Functions

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 dorbdb (trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21,
    ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work,
    lwork, info)
    DORBDB

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

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

🌸 Detailed Description 🌸

this is the detailed docs zone, bestie. it's mostly the function docs, and we're starting with cunbdb. buckle up. 🫶

🔥 Function Documentation 🔥

💖 subroutine cunbdb 💖

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

🎀 Purpose 🎀

okay so CUNBDB does the most: it simultaneously bidiagonalizes the blocks of an M-by-M partitioned unitary matrix X. the whole slay looks like this:

                                [ 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 that's 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.) no cap.

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. iconic.

B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented implicitly by angles THETA, PHI. periodt.

💜 Parameters 💜

  • TRANS — CHARACTER. if it's 'T', then X, U1, U2, V1T, and V2T are stored in row-major order; otherwise they're stored in column-major order. iykyk.
  • SIGNS — CHARACTER. 'O' makes the lower-left block nonpositive (the convention); otherwise the upper-right block is made nonpositive (the convention). yes, bestie, both branches say "the convention" — I'm not fixing the source, I'm just reporting it. 🫶
  • M — INTEGER. the number of rows and columns in X.
  • P — INTEGER. the number of rows in X11 and X12. 0 <= P <= M.
  • Q — INTEGER. the number of columns in X11 and X21. 0 <= Q <= MIN(P,M-P,M-Q).
  • X11 — 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, and the rows of triu(X11,1) specify reflectors for Q1; else TRANS = 'T', and the rows of triu(X11) specify reflectors for P1, and the columns of tril(X11,-1) specify reflectors for Q1.
  • LDX11 — INTEGER. the leading dimension of X11. if TRANS = 'N', then LDX11 >= P; else LDX11 >= Q.
  • X12 — 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 — INTEGER. the leading dimension of X12. if TRANS = 'N', then LDX12 >= P; else LDX11 >= M-Q. yes, the page says LDX11 right there — preserving it exactly. 🙈
  • X21 — 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 — INTEGER. the leading dimension of X21. if TRANS = 'N', then LDX21 >= M-P; else LDX21 >= Q.
  • X22 — 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 — INTEGER. the leading dimension of X22. if TRANS = 'N', then LDX22 >= M-P; else LDX22 >= M-Q.
  • THETA — 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 — REAL array, dimension (Q-1). same deal: the entries of the bidiagonal blocks B11, B12, B21, B22 can be computed from the angles THETA and PHI. see Further Details.
  • TAUP1 — COMPLEX array, dimension (P). the scalar factors of the elementary reflectors that define P1.
  • TAUP2 — COMPLEX array, dimension (M-P). the scalar factors of the elementary reflectors that define P2.
  • TAUQ1 — COMPLEX array, dimension (Q). the scalar factors of the elementary reflectors that define Q1.
  • TAUQ2 — COMPLEX array, dimension (M-Q). the scalar factors of the elementary reflectors that define Q2.
  • WORK — COMPLEX array, dimension (LWORK). that's it. that's the whole description. 💀
  • LWORK — 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 — 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. yes, B21 is listed in both — that's what the page says, bestie. 🙈 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 284 of file cunbdb.f. 💅

🦋 subroutine dorbdb 🦋

subroutine dorbdb (character trans, character signs, integer m, integer p,
    integer q, double precision, dimension( ldx11, * ) x11, integer ldx11,
    double precision, dimension( ldx12, * ) x12, integer ldx12, double
    precision, dimension( ldx21, * ) x21, integer ldx21, double precision,
    dimension( ldx22, * ) x22, integer ldx22, double precision, dimension( *
    ) theta, double precision, dimension( * ) phi, double precision,
    dimension( * ) taup1, double precision, dimension( * ) taup2, double
    precision, dimension( * ) tauq1, double precision, dimension( * ) tauq2,
    double precision, dimension( * ) work, integer lwork, integer info)

DORBDB

🎀 Purpose 🎀

DORBDB is out here doing the most: it simultaneously bidiagonalizes the blocks of an M-by-M partitioned orthogonal matrix X, and the glow-up looks like this:

                                 [ B11 | B12 0  0 ]
     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**T
 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 that's 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 DORCSD for details.)

The orthogonal 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 DOUBLE PRECISION array, dimension (LDX11,Q) On entry, the top-left block of the orthogonal 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 DOUBLE PRECISION array, dimension (LDX12,M-Q) On entry, the top-right block of the orthogonal 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 DOUBLE PRECISION array, dimension (LDX21,Q) On entry, the bottom-left block of the orthogonal 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 DOUBLE PRECISION array, dimension (LDX22,M-Q) On entry, the bottom-right block of the orthogonal 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 DOUBLE PRECISION array, dimension (P) The scalar factors of the elementary reflectors that define P1.
  • TAUP2 — TAUP2 is DOUBLE PRECISION array, dimension (M-P) The scalar factors of the elementary reflectors that define P2.
  • TAUQ1 — TAUQ1 is DOUBLE PRECISION array, dimension (Q) The scalar factors of the elementary reflectors that define Q1.
  • TAUQ2 — TAUQ2 is DOUBLE PRECISION array, dimension (M-Q) The scalar factors of the elementary reflectors that define Q2.
  • WORK — WORK is DOUBLE PRECISION 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:

So 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 DORCSD for details.

P1, P2, Q1, and Q2 are represented as products of elementary reflectors. See DORCSD for details on generating P1, P2, Q1, and Q2 using DORGQR and DORGLQ.

References:

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

Definition at line 284 of file dorbdb.f.

   subroutine sorbdb (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( * ) phi, real, dimension( *
       ) taup1, real, dimension( * ) taup2, real, dimension( * ) tauq1, real,
       dimension( * ) tauq2, real, dimension( * ) work, integer lwork, integer
       info)

🌸 SORBDB 🌸

Purpose:

SORBDB simultaneously bidiagonalizes the blocks of an M-by-M partitioned orthogonal matrix X:

                                 [ B11 | B12 0  0 ]
     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**T
 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 SORCSD for details.)

The orthogonal 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 — a REAL array, dimension (LDX11,Q). On entry, this is the top-left block of the orthogonal matrix to be reduced. On exit, the form depends on TRANS:
    • If TRANS = 'N', then the columns of tril(X11) specify reflectors for P1, and the rows of triu(X11,1) specify reflectors for Q1.
    • else TRANS = 'T', and the rows of triu(X11) specify reflectors for P1, and the columns of tril(X11,-1) specify reflectors for Q1.
  • LDX11 — LDX11 is INTEGER. It's the leading dimension of X11. If TRANS = 'N', then LDX11 >= P; else LDX11 >= Q.
  • X12 — a REAL array, dimension (LDX12,M-Q). On entry, this is the top-right block of the orthogonal 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. It's the leading dimension of X12. If TRANS = 'N', then LDX12 >= P; else LDX11 >= M-Q. (yes, that says LDX11, bestie — copying it exactly as the page does 💅)
  • X21 — a REAL array, dimension (LDX21,Q). On entry, this is the bottom-left block of the orthogonal 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. It's the leading dimension of X21. If TRANS = 'N', then LDX21 >= M-P; else LDX21 >= Q.
  • X22 — a REAL array, dimension (LDX22,M-Q). On entry, this is the bottom-right block of the orthogonal 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. It's the leading dimension of X22. If TRANS = 'N', then LDX22 >= M-P; else LDX22 >= M-Q.
  • THETA — a 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 — a 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 — a REAL array, dimension (P). These are the scalar factors of the elementary reflectors that define P1.
  • TAUP2 — a REAL array, dimension (M-P). These are the scalar factors of the elementary reflectors that define P2.
  • TAUQ1 — a REAL array, dimension (Q). These are the scalar factors of the elementary reflectors that define Q1.
  • TAUQ2 — a REAL array, dimension (M-Q). These are the scalar factors of the elementary reflectors that define Q2.
  • WORK — a REAL array, dimension (LWORK).
  • LWORK — LWORK is INTEGER. It's the dimension of the array WORK, and 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 SORCSD for details.

P1, P2, Q1, and Q2 are represented as products of elementary reflectors. See SORCSD for details on generating P1, P2, Q1, and Q2 using SORGQR and SORGLQ.

📚 References:

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

Definition at line 284 of file sorbdb.f.

🦋 ZUNBDB

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. It's the number of rows and columns in X.
  • P — P is INTEGER. It's the number of rows in X11 and X12. 0 <= P <= M.
  • Q — Q is INTEGER. It's the number of columns in X11 and X21. 0 <= Q <= MIN(P,M-P,M-Q).
  • X11 — a COMPLEX*16 array, dimension (LDX11,Q). On entry, this is 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, and the rows of triu(X11,1) specify reflectors for Q1.
    • else TRANS = 'T', and the rows of triu(X11) specify reflectors for P1, and the columns of tril(X11,-1) specify reflectors for Q1.
  • LDX11 — LDX11 is INTEGER. It's the leading dimension of X11. If TRANS = 'N', then LDX11 >= P; else LDX11 >= Q.
  • X12 — a COMPLEX*16 array, dimension (LDX12,M-Q). On entry, this is 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. It's the leading dimension of X12. If TRANS = 'N', then LDX12 >= P; else LDX11 >= M-Q. (again, exactly what the page says 🫶)
  • X21 — a COMPLEX*16 array, dimension (LDX21,Q). On entry, this is 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. It's the leading dimension of X21. If TRANS = 'N', then LDX21 >= M-P; else LDX21 >= Q.
  • X22 —
  • X22 — okay so this one is a COMPLEX*16 array, dimension (LDX22,M-Q). On entry, it’s 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 — INTEGER. The leading dimension of X22. If TRANS = 'N', then LDX22 >= M-P; else LDX22 >= M-Q.
  • 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.
  • 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.
  • TAUP1 — COMPLEX*16 array, dimension (P). The scalar factors of the elementary reflectors that define P1.
  • TAUP2 — COMPLEX*16 array, dimension (M-P). The scalar factors of the elementary reflectors that define P2.
  • TAUQ1 — COMPLEX*16 array, dimension (Q). The scalar factors of the elementary reflectors that define Q1.
  • TAUQ2 — COMPLEX*16 array, dimension (M-Q). The scalar factors of the elementary reflectors that define Q2.
  • WORK — COMPLEX*16 array, dimension (LWORK).
  • LWORK — 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 — 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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