unbdb1(3)

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version 3.12.1 · 9 versions
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💅 unbdb1(3) — {un,or}bdb1: step in uncsd2by1

okay so welcome to the unbdb1(3) page from the Library Functions Manual, bestie 💖 yes we're doing the whole thing, and yes it's giving linear algebra main character energy. This is the {un,or}bdb1 family — the step in uncsd2by1, no cap ✨

🌸 NAME

unbdb1 — {un,or}bdb1: step in uncsd2by1

✨ SYNOPSIS

🎀 Functions

  • cunbdb1 — CUNBDB1
       subroutine cunbdb1 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
           taup2, tauq1, work, lwork, info)
  • dorbdb1 — DORBDB1
       subroutine dorbdb1 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
           taup2, tauq1, work, lwork, info)
  • sorbdb1 — SORBDB1
       subroutine sorbdb1 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
           taup2, tauq1, work, lwork, info)
  • zunbdb1 — ZUNBDB1
       subroutine zunbdb1 (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1,
           taup2, tauq1, work, lwork, info)

🫶 DETAILED DESCRIPTION

the actual function lore is right below, bestie — stay with me 🦋

💖 FUNCTION DOCUMENTATION

💅 cunbdb1 — CUNBDB1

   subroutine cunbdb1 (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)

🎯 Purpose:

okay so CUNBDB1 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.

bestie, Q must be no larger than P, M-P, or M-Q — fr fr.

Routines CUNBDB2, CUNBDB3, and CUNBDB4 handle cases in which Q 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 Q-by-Q 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.
  • Q — Q is INTEGER. The number of columns in X11 and X21. 0 <= Q <= MIN(P,M-P,M-Q).
  • 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 cunbdb1.f.

💅 dorbdb1 — DORBDB1

   subroutine dorbdb1 (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)

🎯 Purpose:

okay so DORBDB1 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.

bestie, Q must be no larger than P, M-P, or M-Q — periodt.

Routines DORBDB2, DORBDB3, and DORBDB4 handle cases in which Q 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 Q-by-Q 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.
  • 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 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.

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 DORCSD for 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.

Okay so dorbdb1 is basically done — here’s the last little bibliography-and-where-she-lives moment before we move on, no cap:

  • References: [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.
  • Definition: at line 201 of file dorbdb1.f.

🌸 subroutine sorbdb1 — the single-precision icon 🌸

   subroutine sorbdb1 (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)

She’s SORBDB1, and her whole thing is simultaneously bidiagonalizing the blocks of a tall and skinny matrix X with orthonormal columns. The setup is giving:

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

X11 is P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P, M-P, or M-Q. Routines SORBDB2, SORBDB3, and SORBDB4 handle cases in which Q 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 Q-by-Q bidiagonal matrices represented implicitly by angles THETA, PHI.

🫶 Parameters 🫶

  • M — INTEGER. The number of rows X11 plus the number of rows in X21.
  • P — INTEGER. The number of rows in X11. 0 <= P <= M.
  • Q — INTEGER. The number of columns in X11 and X21. 0 <= Q <= MIN(P,M-P,M-Q).
  • 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. 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. 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).
  • 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 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 SORCSD for 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 201 of file sorbdb1.f.

💜 subroutine zunbdb1 — the complex*16 queen 💜

   subroutine zunbdb1 (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)

She’s ZUNBDB1, and she does the same simultaneous-bidiagonalization slay, but in complex*16. She 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. Q must be no larger than P, M-P, or M-Q. Routines ZUNBDB2, ZUNBDB3, and ZUNBDB4 handle cases in which Q 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 Q-by-Q bidiagonal matrices represented implicitly by angles THETA, PHI.

🫶 Parameters 🫶

  • M — INTEGER. The number of rows X11 plus the number of rows in X21.
  • P — INTEGER. The number of rows in X11. 0 <= P <= M.
  • Q — INTEGER. The number of columns in X11 and X21. 0 <= Q <= MIN(P,M-P,M-Q).
  • 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. 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. 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).
  • 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 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 ZUNCSD for 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 201 of file zunbdb1.f.

💖 Author 💖

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