SRC/zunbdb.f(3)

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🦋 SRC/zunbdb.f — Library Functions Manual 🦋

okay bestie, welcome to the SRC/zunbdb.f page from the Library Functions Manual 💖. this is the whole ZUNBDB moment, and yes, it's giving linear algebra main character energy.

🌸 NAME 🌸

SRC/zunbdb.f

✨ SYNOPSIS ✨

Functions/Subroutines

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

💅 Function/Subroutine Documentation 💅

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 🌈

okay so ZUNBDB simultaneously bidiagonalizes the blocks of an M-by-M partitioned unitary matrix X, and the diagram is literally iconic:

                                 [ 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. 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 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.

LDX11

LDX11 is INTEGER. The leading dimension of X11. If TRANS = 'N', then LDX11 >= P; else LDX11 >= Q.

X12

X12 is 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.

LDX12

LDX12 is INTEGER. The leading dimension of X12. If TRANS = 'N', then LDX12 >= P; else LDX11 >= M-Q.

X21

X21 is 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.

LDX21

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

X22

X22 is 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.

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 COMPLEX*16 array, dimension (P).

The scalar factors of the elementary reflectors that define P1.

TAUP2

TAUP2 is COMPLEX*16 array, dimension (M-P).

The scalar factors of the elementary reflectors that define P2.

TAUQ1

TAUQ1 is COMPLEX*16 array, dimension (Q).

The scalar factors of the elementary reflectors that define Q1.

TAUQ2

TAUQ2 is COMPLEX*16 array, dimension (M-Q).

The scalar factors of the elementary reflectors that define Q2.

WORK

WORK is COMPLEX*16 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 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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