unbdb3(3)

📦

version 3.12.1 · 9 versions

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3.12.1 · debian trixie

🌸 NAME 🌸

unbdb3 — {un,or}bdb3: step in uncsd2by1 💅

✨ SYNOPSIS ✨

🎀 Functions 🎀

   subroutine cunbdb3 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
       taup2, tauq1, work, lwork, info)
       CUNBDB3
   subroutine dorbdb3 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
       taup2, tauq1, work, lwork, info)
       DORBDB3
   subroutine sorbdb3 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
       taup2, tauq1, work, lwork, info)
       SORBDB3
   subroutine zunbdb3 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
       taup2, tauq1, work, lwork, info)
       ZUNBDB3

💖 Detailed Description 💖

okay so this is where the function-by-function details kick in, and they are very much giving linear algebra main character energy 💅

🔥 Function Documentation 🔥

🌈 subroutine cunbdb3 🌈

   subroutine cunbdb3 (integer m, integer p, integer q, complex,
       dimension(ldx11,*) x11, integer ldx11, complex, dimension(ldx21,*) x21,
       integer ldx21, real, dimension(*) theta, real, dimension(*) phi,
       complex, dimension(*) taup1, complex, dimension(*) taup2, complex,
       dimension(*) tauq1, complex, dimension(*) work, integer lwork, integer
       info)
       CUNBDB3

💫 Purpose

okay so CUNBDB3 is out here doing the absolute most in the best way: it simultaneously bidiagonalizes the blocks of a tall and skinny matrix X with orthonormal columns. The decomposition is giving:

                           [ B11 ]
     [ X11 ]   [ P1 |    ] [  0  ]
     [-----] = [---------] [-----] Q1**T .
     [ X21 ]   [    | P2 ] [ B21 ]
                           [  0  ]

X11 is P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P, Q, or M-Q. Routines CUNBDB1, CUNBDB2, and CUNBDB4 handle cases in which M-P is not the minimum dimension.

The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P), and (M-Q)-by-(M-Q), respectively. They are represented implicitly by Householder vectors.

B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented implicitly by angles THETA, PHI.

🫶 Parameters

M — M is INTEGER. The number of rows X11 plus the number of rows in X21.
P — P is INTEGER. The number of rows in X11. 0 <= P <= M. M-P <= min(P,Q,M-Q).
Q — Q is INTEGER. The number of columns in X11 and X21. 0 <= Q <= M.
X11 — X11 is COMPLEX array, dimension (LDX11,Q). On entry, the top block of the matrix X to be reduced. On exit, the columns of tril(X11) specify reflectors for P1 and the rows of triu(X11,1) specify reflectors for Q1.
LDX11 — LDX11 is INTEGER. The leading dimension of X11. LDX11 >= P.
X21 — X21 is COMPLEX array, dimension (LDX21,Q). On entry, the bottom block of the matrix X to be reduced. On exit, the columns of tril(X21) specify reflectors for P2.
LDX21 — LDX21 is INTEGER. The leading dimension of X21. LDX21 >= M-P.
THETA — THETA is REAL array, dimension (Q). The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.
PHI — PHI is REAL array, dimension (Q-1). The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.
TAUP1 — TAUP1 is COMPLEX array, dimension (P). The scalar factors of the elementary reflectors that define P1.
TAUP2 — TAUP2 is COMPLEX array, dimension (M-P). The scalar factors of the elementary reflectors that define P2.
TAUQ1 — TAUQ1 is COMPLEX array, dimension (Q). The scalar factors of the elementary reflectors that define Q1.
WORK — WORK is COMPLEX 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 upper-bidiagonal blocks B11, B21 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). 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, and Q1 are represented as products of elementary reflectors. See CUNCSD2BY1 for details on generating P1, P2, and Q1 using CUNGQR and CUNGLQ.

🍓 References

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

Definition at line 200 of file cunbdb3.f.

🌟 subroutine dorbdb3 🌟

   subroutine dorbdb3 (integer m, integer p, integer q, double precision,
       dimension(ldx11,*) x11, integer ldx11, double precision,
       dimension(ldx21,*) x21, integer ldx21, double precision, dimension(*)
       theta, double precision, dimension(*) phi, double precision,
       dimension(*) taup1, double precision, dimension(*) taup2, double
       precision, dimension(*) tauq1, double precision, dimension(*) work,
       integer lwork, integer info)
       DORBDB3

💫 Purpose

okay so DORBDB3 is the double-precision bestie doing the same simultaneous bidiagonalization glow-up on the blocks of a tall and skinny matrix X with orthonormal columns. The decomposition is giving:

                           [ B11 ]
     [ X11 ]   [ P1 |    ] [  0  ]
     [-----] = [---------] [-----] Q1**T .
     [ X21 ]   [    | P2 ] [ B21 ]
                           [  0  ]

X11 is P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P, Q, or M-Q. Routines DORBDB1, DORBDB2, and DORBDB4 handle cases in which M-P is not the minimum dimension.

The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P), and (M-Q)-by-(M-Q), respectively. They are represented implicitly by Householder vectors.

B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented implicitly by angles THETA, PHI.

🫶 Parameters

M — M is INTEGER. The number of rows X11 plus the number of rows in X21.
P — P is INTEGER. The number of rows in X11. 0 <= P <= M. M-P <= min(P,Q,M-Q).
Q — Q is INTEGER. The number of columns in X11 and X21. 0 <= Q <= M.
X11 — X11 is DOUBLE PRECISION array, dimension (LDX11,Q). On entry, the top block of the matrix X to be reduced. On exit, the columns of tril(X11) specify reflectors for P1 and the rows of triu(X11,1) specify reflectors for Q1.
LDX11 — LDX11 is INTEGER. The leading dimension of X11. LDX11 >= P.
X21 — X21 is DOUBLE PRECISION array, dimension (LDX21,Q). On entry, the bottom block of the matrix X to be reduced. On exit, the columns of tril(X21) specify reflectors for P2.
LDX21 — LDX21 is INTEGER. The leading dimension of X21. LDX21 >= M-P.
THETA — THETA is DOUBLE PRECISION array, dimension (Q). The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.
PHI — PHI is DOUBLE PRECISION array, dimension (Q-1). The entries of the bidiagonal blocks B11, B21 are defined by 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.
WORK — WORK is DOUBLE PRECISION array, dimension (LWORK).
LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= M-Q.

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

P1, P2, and Q1 are represented as products of elementary reflectors. See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR and DORGLQ.

References:

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

Definition at line 199 of file dorbdb3.f.

🦋 `sorbdb3` — the single-precision icon 🦋

   subroutine sorbdb3 (integer m, integer p, integer q, real,
       dimension(ldx11,*) x11, integer ldx11, real, dimension(ldx21,*) x21,
       integer ldx21, real, dimension(*) theta, real, dimension(*) phi, real,
       dimension(*) taup1, real, dimension(*) taup2, real, dimension(*) tauq1,
       real, dimension(*) work, integer lwork, integer info)

SORBDB3

✨ Purpose ✨

Okay so picture a tall, skinny matrix X with orthonormal columns, and SORBDB3 is out here simultaneously bidiagonalizing its blocks. It's giving main character energy:

                            [ B11 ]
      [ X11 ]   [ P1 |    ] [  0  ]
      [-----] = [---------] [-----] Q1**T .
      [ X21 ]   [    | P2 ] [ B21 ]
                            [  0  ]

X11 is P-by-Q, and X21 is (M-P)-by-Q. And listen: M-P must be no larger than P, Q, or M-Q. If M-P is not the minimum dimension, that's a different vibe — routines SORBDB1, SORBDB2, and SORBDB4 handle those cases.

The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P), and (M-Q)-by-(M-Q), respectively. They're represented implicitly by Householder vectors — iconic.

B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented implicitly by angles THETA, PHI. Yes bestie, B12 — that's what the page says, no cap.

🌸 Parameters 🌸

M (INTEGER) — the number of rows in X11 plus the number of rows in X21.
P (INTEGER) — the number of rows in X11. Constraint: 0 <= P <= M. M-P <= min(P,Q,M-Q).
Q (INTEGER) — the number of columns in X11 and X21. Constraint: 0 <= Q <= M.
X11 (REAL array, dimension (LDX11,Q)) — on entry, the top block of the matrix X to be reduced. On exit, the columns of tril(X11) specify reflectors for P1 and the rows of triu(X11,1) specify reflectors for Q1.
LDX11 (INTEGER) — the leading dimension of X11. Constraint: LDX11 >= P.
X21 (REAL array, dimension (LDX21,Q)) — on entry, the bottom block of the matrix X to be reduced. On exit, the columns of tril(X21) specify reflectors for P2.
LDX21 (INTEGER) — the leading dimension of X21. Constraint: LDX21 >= M-P.
THETA (REAL array, dimension (Q)) — the entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.
PHI (REAL array, dimension (Q-1)) — the entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.
TAUP1 (REAL array, dimension (P)) — the scalar factors of the elementary reflectors that define P1.
TAUP2 (REAL array, dimension (M-P)) — the scalar factors of the elementary reflectors that define P2.
TAUQ1 (REAL array, dimension (Q)) — the scalar factors of the elementary reflectors that define Q1.
WORK (REAL array, dimension (LWORK)) — the work array, bestie. That's it.
LWORK (INTEGER) — the dimension of the array WORK. Constraint: 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 🔍

P1, P2, and Q1 are represented as products of elementary reflectors. See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR and SORGLQ.

📚 References 📚

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

Definition at line 200 of file sorbdb3.f.

🦋 `zunbdb3` — the complex*16 edition 🦋

   subroutine zunbdb3 (integer m, integer p, integer q, complex*16,
       dimension(ldx11,*) x11, integer ldx11, complex*16, dimension(ldx21,*)
       x21, integer ldx21, double precision, dimension(*) theta, double
       precision, dimension(*) phi, complex*16, dimension(*) taup1, complex*16,
       dimension(*) taup2, complex*16, dimension(*) tauq1, complex*16,
       dimension(*) work, integer lwork, integer info)

ZUNBDB3

✨ Purpose ✨

Same energy, but now with COMPLEX*16 and unitary matrices. ZUNBDB3 simultaneously bidiagonalizes the blocks of a tall and skinny matrix X with orthonormal columns:

                            [ B11 ]
      [ X11 ]   [ P1 |    ] [  0  ]
      [-----] = [---------] [-----] Q1**T .
      [ X21 ]   [    | P2 ] [ B21 ]
                            [  0  ]

X11 is P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P, Q, or M-Q. Routines ZUNBDB1, ZUNBDB2, and ZUNBDB4 handle cases in which M-P is not the minimum dimension.

The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P), and (M-Q)-by-(M-Q), respectively. They are represented implicitly by Householder vectors.

B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented implicitly by angles THETA, PHI. Again, the page says B12 — we don't make the rules.

🌸 Parameters 🌸

M (INTEGER) — the number of rows in X11 plus the number of rows in X21.
P (INTEGER) — the number of rows in X11. Constraint: 0 <= P <= M. M-P <= min(P,Q,M-Q).
Q (INTEGER) — the number of columns in X11 and X21. Constraint: 0 <= Q <= M.
X11 (COMPLEX*16 array, dimension (LDX11,Q)) — on entry, the top block of the matrix X to be reduced. On exit, the columns of tril(X11) specify reflectors for P1 and the rows of triu(X11,1) specify reflectors for Q1.
LDX11 (INTEGER) — the leading dimension of X11. Constraint: LDX11 >= P.
X21 (COMPLEX*16 array, dimension (LDX21,Q)) — on entry, the bottom block of the matrix X to be reduced. On exit, the columns of tril(X21) specify reflectors for P2.
LDX21 (INTEGER) — the leading dimension of X21. Constraint: LDX21 >= M-P.
THETA (DOUBLE PRECISION array, dimension (Q)) — the entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.
PHI (DOUBLE PRECISION array, dimension (Q-1)) — the entries of the bidiagonal blocks B11, B21 are defined by 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.
WORK (COMPLEX*16 array, dimension (LWORK)) — the work array, bestie.
LWORK (INTEGER) — the dimension of the array WORK. Constraint: 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 🔍

P1, P2, and Q1 are represented as products of elementary reflectors. See ZUNCSD2BY1 for details on generating P1, P2, and Q1 using ZUNGQR and ZUNGLQ.

📚 References 📚

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

Definition at line 199 of file zunbdb3.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

💖 And that's a wrap, bestie! This whole glow-up is from LAPACK, version 3.12.0, and it's the unbdb3(3) manual page. Slay. 💖

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

unbdb3(3)

🌸 NAME 🌸

✨ SYNOPSIS ✨

🎀 Functions 🎀

💖 Detailed Description 💖

🔥 Function Documentation 🔥

🌈 subroutine cunbdb3 🌈

💫 Purpose

🫶 Parameters

👑 Author

🦋 Further Details

🍓 References

🌟 subroutine dorbdb3 🌟

💫 Purpose

🫶 Parameters

👑 Author

🦋 Further Details

🦋 sorbdb3 — the single-precision icon 🦋

✨ Purpose ✨

🌸 Parameters 🌸

👑 Author 👑

🔍 Further Details 🔍

📚 References 📚

🦋 zunbdb3 — the complex*16 edition 🦋

✨ Purpose ✨

🌸 Parameters 🌸

👑 Author 👑

🔍 Further Details 🔍

📚 References 📚

👑 Author 👑

🦋 `sorbdb3` — the single-precision icon 🦋

🦋 `zunbdb3` — the complex*16 edition 🦋